Abstract:
In the method for constrained optimization for compander design for OFDM PAPR, wherein the improvement comprises the step of converting a Rayleigh amplitude distribution from which superior performing companders are derived, whereby a constrained optimization problem may be solved using Lagrange multipliers.

Description:
RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 62/036,388, filed Aug. 12, 2014. This application is herein incorporated by reference in its entirety for all purposes. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention relates to signal processing and, more particularly, to companders for reducing peak to average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) signals. 
       BACKGROUND OF THE INVENTION 
       [0003]    Orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL Internet access, wireless networks, powerline networks, and 4G mobile communications. 
         [0004]    For a sufficiently large number of tones, OFDM signal values are Gaussian, or normally, distributed, by the central limit theorem, which states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. 
         [0005]    The Gaussian distributed signal components generate Rayleigh distributed signal amplitude values, Rayleigh distributions being continuous probability distributions for positive-valued random variables, specifically the distribution of the magnitude of a two-dimensional random vector whose coordinates are independent, identically distributed, mean 0 normal variables. In signals typical of the current state of the art, the Rayleigh distribution is typically heavy-tailed, i.e. a larger proportion of its population rests within its tail than would under a normal distribution. The long tail of the Rayleigh distribution, i.e. the portion of the distribution having a large number of occurrences far from the “head” or central part of the distribution, indicates the presence of large amplitude values, resulting from constructive interference. These large amplitude values can saturate the power amplifier, used in the processing chain, thus creating signal distortion which reduces demodulation performance. 
         [0006]    This causes existing techniques for Orthogonal Frequency-Division Multiplexing (OFDM) to suffer from large peak-to-average power ratio (PAPR). The PAPR is the peak amplitude squared (giving the peak power) divided by the mean square value squared (giving the average power). 
         [0007]    Many techniques have been developed to mitigate the PAPR problem, including signal companding. Companders are attractive because of their simplicity and effectiveness. Companders apply an amplitude weight to the signal, generally downweighting large amplitude values, while upweighting smaller values to maintain constant power. The large amplitude downweighting, while simultaneously maintaining the same power level, reduces the PAPR. The amplitude weighting introduces signal distortion, so it is important to design companders that introduce minimal distortion to limit demodulation errors, while also providing out-of-band power rejection. 
         [0008]    Early solutions adapted companders from speech and audio coding, such as μ-law and A-law companders. Subsequent approaches decomposed the amplitude interval into disjoint regions over which different amplitude weighting functions were used. Modern companders are derived by transforming the Rayleigh amplitude distribution into a more favorable distribution to reduce PAPR. Among these transformations, piecewise transformations, i.e. a function composed of straight-line sections, have been used, either uniform, linear, nonlinear, or piecewise linear. 
         [0009]    A need still exists, however, for solutions which minimally alter the original Rayleigh amplitude distribution. 
       SUMMARY OF THE INVENTION 
       [0010]    One embodiment of the present invention provides a signal transmitting apparatus for an orthogonal frequency division multiplexing (OFDM) system, comprising: a compander configured to compress and expand a transmitted signal wherein the compander comprises a Rayleigh amplitude distribution to a constrained optimization problem which may be solved using Lagrange multipliers. 
         [0011]    One embodiment of the present invention provides a signal transmitting apparatus for orthogonal frequency division multiplexing (OFDM) system, comprising: a companding feedback module configured to compress and expand a transmitted signal; wherein the companding feedback module comprises a Rayleigh amplitude distribution to a constrained optimization problem which may be solved using Lagrange multipliers. 
         [0012]    One embodiment of the present invention provides a method of deriving a compander for orthogonal frequency division multiplexing comprising: converting a Rayleigh amplitude distribution to a constrained optimization problem which may be solved using Lagrange multipliers. 
         [0013]    Another embodiment of the present invention provides such a method wherein the Rayleigh amplitude distribution is decomposed into disjoint regions over which a plurality of amplitude weighting functions are used. 
         [0014]    A further embodiment of the present invention provides such a method wherein the constrained optimization approach comprises optimizing an objective function with respect to changeable elements in the presence of constraints on those elements. 
         [0015]    Yet another embodiment of the present invention provides such a method wherein the Rayleigh amplitude distribution is a probability distribution function and a first the constraint is that the Rayleigh amplitude distribution must integrate to unity. 
         [0016]    A yet further embodiment of the present invention provides such a method wherein a second the constraint is that power must be constant across the distribution. 
         [0017]    Still another embodiment of the present invention provides such a method wherein an optimal solution is minimally perturbed to generate only non-negative solutions. 
         [0018]    A still further embodiment of the present invention provides such a method of claim  5  wherein the Rayleigh amplitude distribution is defined as: 
         [0000]    
       
         
           
             
               
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         [0019]    Even another embodiment of the present invention provides such a method wherein further comprising determining a function g(x) which minimizes: 
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         [0020]    An even further embodiment of the present invention provides such a method wherein g(x) is chosen to be of piecewise linear form. 
         [0021]    A still even another embodiment of the present invention provides such a method wherein g(x) is defined as 
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         [0022]    A still even further embodiment of the present invention provides such a method wherein the dynamics of the system are defined as: Λ(β 1 , . . . β N+1 , λ 1 , λ 2 )=F(β 1 , . . . β N+1 )+λ 1 [g 1 (β 1 , . . . β N+1 )−1]+λ 2 [g 2 (β 1 , . . . β N+1 )−σ 2 ]. 
         [0023]    Still yet another embodiment of the present invention provides such a method wherein a first constraint, that the cumulative distribution function must integrate to unity, is expressed as: g 1 (β 1 , . . . β N+1 )=1. 
         [0024]    A still yet further embodiment of the present invention provides such a method wherein a second constraint, that power must be constant, is expressed as: g 2 (β 1 , . . . β N+1 )=σ 2 . 
         [0000]    Even yet another embodiment of the present invention provides such a method wherein the constrained optimization problem can be defined as: Minimize: 
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         [0000]    for β i , subject to 
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         [0000]    using Lagrange multipliers. 
         [0025]    An even yet further embodiment of the present invention provides such a method wherein the compander is defined as: 
         [0000]    
       
         
           
             
               
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         [0000]    for a i ≦x&lt;a i+1 , wherein 
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         [0026]    Still even yet another embodiment of the present invention provides such a method wherein a decompander for use with the compander is defined as: 
         [0000]    
       
         
           
             
               x 
               = 
               
                 
                   sgn 
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         [0000]    where the sgn function refers to retaining either the positive or negative square-root, depending on the sign of x; and wherein the range of the compander, when a i ≦x&lt;a i+1 , then y MIN ≦y&lt;y MAX  where: y MIN =min{y(a i ),y(a i+1 )}, y MAX =max{y(a i ),y(a i+1 )}, can be defined as 
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         [0027]    A still even yet further embodiment of the present invention provides such a method wherein a Hessian Matrix is used to determine critical point types. 
         [0028]    The features and advantages described herein are not all-inclusive and, in particular, many additional features and advantages will be apparent to one of ordinary skill in the art in view of the drawings, specification, and claims. Moreover, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and not to limit the scope of the inventive subject matter. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0029]      FIG. 1  is a graph showing piecewise linear approximation to a preferred embodiment of the invention; 
           [0030]      FIG. 2  is a graph showing optimal solution, A=1.1; 
           [0031]      FIG. 3  is a graph showing optimal solution, A=1.2; 
           [0032]      FIG. 4  is a graph showing optimal solution, A=1.3; 
           [0033]      FIG. 5  is a graph showing optimal solution, A=1.4; 
           [0034]      FIG. 6  is a graph showing optimal solution, A=1.5; 
           [0035]      FIG. 7  is a graph showing optimal solution, A=1.6; 
           [0036]      FIG. 8  is a graph showing optimal solution, A=1.7; 
           [0037]      FIG. 9  is a graph showing optimal solution, A=1.8; 
           [0038]      FIG. 10  is a graph showing positive perturbed solution; A=1.4; 
           [0039]      FIG. 11  is a graph showing positive perturbed function; 
           [0040]      FIG. 12  is a graph showing perturbation solution; A=1.4, k=5; 
           [0041]      FIG. 13  is a graph showing perturbation solution; A=1.4, k=6; 
           [0042]      FIG. 14  is a graph showing perturbation solution; A=1.4, k=7; 
           [0043]      FIG. 15  is a graph showing perturbation solution; A=1.4, k=8; 
           [0044]      FIG. 16  is a graph showing symbol error rate performance; 
           [0045]      FIG. 17  is a graph showing PAPR reduction performance; 
           [0046]      FIG. 18  is a graph showing out-of-band power rejection; and 
           [0047]      FIG. 19  is a graph showing sample Lagrange Companders. 
       
    
    
     DETAILED DESCRIPTION 
       [0048]    The present disclosure describes new companders that use a constrained optimization approach for reducing peak-to-average power ratio (PAPR) in orthogonal frequency-division multiplexing (OFDM) signals. A constrained optimization approach, in this context, involves optimizing an objective function, specifically an energy function, with respect to changeable elements, i.e. variables, in the presence of constraints on those variables. 
         [0049]    Companders, devices which use both compression and expansion to improve dynamic range and signal-to-noise ratio, in accordance with the present disclosure provide symbol error rate performance improvements over current state-of-the art companders. In digital communications, symbol rate (also known as baud or modulation rate) is the number of symbol changes (waveform changes or signalling events) made to the transmission medium per second using a digitally modulated signal. The symbol rate is measured in baud (Bd) or symbols per second. Each symbol can represent or convey one or several bits of data. The symbol error rate is simply the rate at which signalling events fail to convey the intended data. 
         [0050]    The newly-designed companders also provide design flexibility, thereby expanding the space of tradeoffs between demodulation performance, PAPR reduction, and out-of-band power rejection. Furthermore, the new companders provide solutions in operating regions where certain current companders fail to exit; solutions may be derived for cutoff amplitude values that are unobtainable using other companders. The new compander solutions enable tuning the cutoff amplitude value based on power amplifier bandwidth. 
         [0051]    Below, we formulate the constrained optimization problem and derive the compander and decompander forms. Through numerical simulation, we generate performance results demonstrating the capability of the new companders. 
       1. COMPANDER DESIGN USING CONSTRAINED OPTIMIZATION 
       [0052]    This section describes the constrained optimization problem and solution, and derives the compander and decompander. 
       1.1. Constrained Optimization Problem Definition 
       [0053]    Denote the Rayleigh amplitude distribution by ƒ R (x) for: 
         [0000]    
       
         
           
             
               
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         [0054]    In the above equation, ƒ R (x) represents the probability density function of the amplitude values of the OFDM signal. This is a statistical description of the spread of amplitude values of the OFDM signal, which follow a Rayleigh distribution. We then look for a function g(x)which minimizes: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0055]    This function is used because we are making the assumption, or operating under the hypothesis, that minimal distortion of the natural signal distribution (i.e., the Rayleigh distribution) will incur minimal signal distortion of the companded function and hence reduce the demodulation errors (i.e., reduce the symbol error rate). 
         [0056]    We then choose g(x) to be of piecewise linear form: 
         [0000]    
       
         
           
             
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         [0057]    In the above equation, the piecewise linear form was chosen since it extends previous work on using single linear components, with the view that the increased flexibility of the piecewise approach will lead to better performing companders. Another reason it was used is that it allows a closed form expression for the compander weights to be found, by solving a linear system of equations. 
         [0058]    Still referring to the above equation, each U i (x) is a compactly-supported linear segment, in this context, meaning a line segment of finite extent, i.e., of finite length. The U i (x) is defined by equation (2) and for each index i, represents the equation for one of the linear segments shown in  FIG. 1 . 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0059]    and (a 1 , . . . a N+1 ) is a partition of the interval [0, A] and the β i  are the ordinates at x=a i  ( FIG. 1 ). This interval is a standard interval used in PAPR reduction techniques, and represents an interval of amplitude values, covering amplitude values between 0 and A. Any amplitude value in the original OFDM signal that is larger than A gets mapped to something less than or equal to A, so A is the maximum value the companded value can have. 
         [0060]    With the assumed form of g(x), we want to minimize: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0061]    as a function of the β i . The function that we are minimizing represents a measure of the deviation of the new, companded function amplitude value distribution from the original OFDM Rayleigh amplitude distribution. The beta values are the values of the piecewise linear component functions at the endpoints, as shown in  FIG. 1 . (i.e., the ordinates), and described by the functions U i (x). 
         [0062]    There are two constraints on g(x); g(x) is a probability density function and so must integrate to unity value over xε[0, A]. This is because, by definition, a probability density function gives an amount of probability per unit of something, in this case, per unit of signal amplitude value. Since the total probability of all of the possibilities adds up (integrates up) to one, then the integral of the probability density function over all of its values must equal the total probability value of one. 
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         [0063]    This is the unity cumulative distribution function (c.d.f.) constraint. The c.d.f. itself describes the probability that a real-valued random variable x with a given probability distribution will be found to have a value less than or equal to x. In the case of a continuous distribution, it gives the area under the probability density function (p.d.f.) from minus infinity to x. 
         [0064]    The second constraint is the constant power constraint: 
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                   5 
                   ) 
                 
               
             
           
         
       
     
         [0065]    In the above equation, equation 5, the integral on the right represents the total power across all possible amplitude values, and its value is equal to the constant value sigma times sigma, hence the total power is the constant value sigma times sigma. 
         [0066]    There is actually a third constraint on g(x); because g(x) is a p.d.f., we have a non-negativity constraint: 
         [0000]        g ( x )≧0  (6)
 
         [0067]    However, we forgo the non-negativity constraint, because, as is discussed later, we can slightly perturb the optimal solution to generate non-negative solutions. The constrained optimization problem then becomes: minimize (3) for the β i  subject to (4) and (5). We solve this using Lagrange multipliers, a strategy for finding the local maxima and minima of a function subject to equality constraints. 
       1.2. Calculation of Partial Derivatives 
       [0068]    To solve the constrained optimization problem, we need to calculate the partial derivatives of F. First consider: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                      
                     
                       ( 
                       
                         
                           β 
                           1 
                         
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           
                             β 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       A 
                     
                      
                     
                       
                         
                           ( 
                           
                             
                               
                                 f 
                                 R 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   U 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0069]    This equation results from eq. (1), where we use the piecewise linear form for function g, and call the whole thing F, and in F we explicitly identify the fact that the expression is now a function of the beta parameters, as shown from eq 2. 
         [0070]    Because the integrand in (7) is continuously differentiable and integrable for x≠β i , then differentiation is permitted under the integral sign and: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             F 
                           
                           
                             ∂ 
                             
                               β 
                               j 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             
                               ∂ 
                               
                                 ∂ 
                                 
                                   β 
                                   j 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       f 
                                       R 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                      
                                     
                                       
                                         U 
                                         i 
                                       
                                        
                                       
                                         ( 
                                         x 
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                              
                             
                                 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     f 
                                     R 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       U 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   - 
                                   
                                       
                                   
                                    
                                   
                                     ∂ 
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       U 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     f 
                                     R 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       U 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 - 
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       ∂ 
                                       
                                         
                                           U 
                                           i 
                                         
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               
                                 ∫ 
                                 0 
                                 A 
                               
                                
                               
                                 
                                   - 
                                   
                                     
                                       f 
                                       R 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       ∂ 
                                       
                                         
                                           U 
                                           i 
                                         
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 
                                   U 
                                   k 
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                  
                                 
                                   
                                     
                                       ∂ 
                                       
                                         
                                           U 
                                           i 
                                         
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                    
                                   
                                      
                                     x 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 A 
                               
                                
                               
                                 
                                   
                                     f 
                                     R 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       
                                         ∂ 
                                         
                                           
                                             U 
                                             i 
                                           
                                            
                                           
                                             ( 
                                             x 
                                             ) 
                                           
                                         
                                       
                                       
                                         ∂ 
                                         
                                           β 
                                           j 
                                         
                                       
                                     
                                      
                                     
                                        
                                       x 
                                     
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           2 
                            
                           
                             
                               ∫ 
                               0 
                               A 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   
                                     U 
                                     k 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     N 
                                   
                                    
                                   
                                     
                                       
                                         ∂ 
                                         
                                           
                                             U 
                                             i 
                                           
                                            
                                           
                                             ( 
                                             x 
                                             ) 
                                           
                                         
                                       
                                       
                                         ∂ 
                                         
                                           β 
                                           j 
                                         
                                       
                                     
                                      
                                     
                                        
                                       x 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   ∫ 
                                   
                                     a 
                                     i 
                                   
                                   
                                     a 
                                     
                                       i 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   
                                     
                                       f 
                                       R 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       ∂ 
                                       
                                         
                                           U 
                                           i 
                                         
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                    
                                   
                                      
                                     x 
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           2 
                            
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ∫ 
                                 0 
                                 A 
                               
                                
                               
                                 
                                   
                                     U 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∂ 
                                     
                                       
                                         U 
                                         i 
                                       
                                        
                                       
                                         ( 
                                         x 
                                         ) 
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       β 
                                       j 
                                     
                                   
                                 
                                  
                                 
                                    
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               2 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   ∫ 
                                   
                                     a 
                                     i 
                                   
                                   
                                     a 
                                     
                                       i 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   
                                     
                                       f 
                                       R 
                                     
                                      
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       ∂ 
                                       
                                         
                                           U 
                                           i 
                                         
                                          
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         β 
                                         j 
                                       
                                     
                                   
                                    
                                   
                                      
                                     x 
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           2 
                            
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   a 
                                   i 
                                 
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     U 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∂ 
                                     
                                       
                                         U 
                                         i 
                                       
                                        
                                       
                                         ( 
                                         x 
                                         ) 
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       β 
                                       j 
                                     
                                   
                                 
                                  
                                 
                                    
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0071]    Now from (2), U i (x) can be rewritten as: 
         [0000]    
       
         
           
             
               
                 U 
                 i 
               
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       x 
                       - 
                       
                         a 
                         i 
                       
                     
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                   
                   ) 
                 
                  
                 
                   β 
                   
                     i 
                     + 
                     1 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         x 
                         - 
                         
                           a 
                           i 
                         
                       
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           a 
                           i 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   β 
                   i 
                 
               
             
           
         
       
     
         [0072]    so that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         U 
                         i 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             x 
                             - 
                             
                               a 
                               i 
                             
                           
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ∂ 
                           
                             β 
                             
                               i 
                               + 
                               1 
                             
                           
                         
                         
                           ∂ 
                           
                             β 
                             j 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               x 
                               - 
                               
                                 a 
                                 i 
                               
                             
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           ∂ 
                           
                             β 
                             i 
                           
                         
                         
                           ∂ 
                           
                             β 
                             j 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0073]    In the above equation, j represents an index on the beta parameters. For j=1, equation (9), once simplified, becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         U 
                         i 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                             - 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   1 
                                 
                               
                               
                                 
                                   a 
                                   2 
                                 
                                 - 
                                 
                                   a 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               if 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0074]    For 1&lt;j≦N, (9) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         U 
                         i 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                             - 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   j 
                                 
                               
                               
                                 
                                   a 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   a 
                                   j 
                                 
                               
                             
                           
                         
                         
                           
                             
                               if 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               x 
                               - 
                               
                                 a 
                                 
                                   j 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 a 
                                 j 
                               
                               - 
                               
                                 a 
                                 
                                   j 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               if 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             
                               j 
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0075]    For j=N+1, (9) becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         U 
                         i 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                               
                                 
                                   a 
                                   
                                     N 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                             
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               = 
                               N 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             otherwise 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0076]    So, from (8), and (10), for j=1 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       F 
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         2 
                       
                        
                       
                         
                           ∫ 
                           
                             a 
                             1 
                           
                           
                             a 
                             2 
                           
                         
                          
                         
                           
                             
                               f 
                               R 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     1 
                                   
                                 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         
                           ∫ 
                           
                             a 
                             1 
                           
                           
                             a 
                             2 
                           
                         
                          
                         
                           
                             
                               U 
                               1 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     1 
                                   
                                 
                                 
                                   
                                     a 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0077]    From (8), and (11), for 1&lt;j≦N 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       F 
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         2 
                       
                        
                       
                         
                           ∫ 
                           
                             a 
                             
                               j 
                               - 
                               1 
                             
                           
                           
                             a 
                             j 
                           
                         
                          
                         
                           
                             
                               f 
                               R 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 
                                   a 
                                   j 
                                 
                                 - 
                                 
                                   a 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     - 
                     
                       2 
                        
                       
                         
                           ∫ 
                           
                             a 
                             j 
                           
                           
                             a 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                          
                         
                           
                             
                               f 
                               R 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         
                           ∫ 
                           
                             a 
                             
                               j 
                               - 
                               1 
                             
                           
                           
                             a 
                             j 
                           
                         
                          
                         
                           
                             
                               U 
                               
                                 j 
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 
                                   a 
                                   j 
                                 
                                 - 
                                 
                                   a 
                                   
                                     j 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         
                           ∫ 
                           
                             a 
                             j 
                           
                           
                             a 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                          
                         
                           
                             
                               U 
                               j 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0078]    From (8), and (12), for j=N+1 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       F 
                     
                     
                       ∂ 
                       
                         β 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         2 
                       
                        
                       
                         
                           ∫ 
                           
                             a 
                             N 
                           
                           
                             a 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                          
                         
                           
                             
                               f 
                               R 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                               
                                 
                                   a 
                                   
                                     N 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         
                           ∫ 
                           
                             a 
                             N 
                           
                           
                             a 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                          
                         
                           
                             
                               U 
                               N 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                               
                                 
                                   a 
                                   
                                     N 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   a 
                                   N 
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0079]    Inspection of (13)-(15) show that the following integral forms are needed: 
         [0000]    
       
         
           
             
               
                 K 
                 R 
                 1 
               
                
               
                 ( 
                 
                   c 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 c 
                 d 
               
                
               
                 
                   
                     f 
                     R 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
           
         
       
       
         
           
             
               
                 K 
                 R 
                 2 
               
                
               
                 ( 
                 
                   c 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 c 
                 d 
               
                
               
                 
                   
                     xf 
                     R 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
           
         
       
       
         
           
             
               
                 K 
                 U 
                 1 
               
                
               
                 ( 
                 
                   c 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 c 
                 d 
               
                
               
                 
                   
                     U 
                     j 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
           
         
       
       
         
           
             
               
                 K 
                 U 
                 2 
               
                
               
                 ( 
                 
                   c 
                   , 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 c 
                 d 
               
                
               
                 
                   
                     xU 
                     j 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
           
         
       
     
         [0080]    In the above equations, K is used as a compact way to represent the integral on the right side of the equation. After much simplification, straightforward integration produces: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         ∫ 
                         c 
                         d 
                       
                        
                       
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                     = 
                     
                       
                          
                         
                           - 
                           
                             
                               c 
                               2 
                             
                             
                               σ 
                               2 
                             
                           
                         
                       
                       - 
                       
                          
                         
                           - 
                           
                             
                               d 
                               2 
                             
                             
                               σ 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         ∫ 
                         c 
                         d 
                       
                        
                       
                         
                           
                             xf 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                     = 
                     
                       
                         c 
                          
                         
                             
                         
                          
                         
                            
                           
                             - 
                             
                               
                                 c 
                                 2 
                               
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       - 
                       
                         d 
                          
                         
                             
                         
                          
                         
                            
                           
                             - 
                             
                               
                                 d 
                                 2 
                               
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             σ 
                              
                             
                               π 
                             
                           
                           2 
                         
                          
                         
                           [ 
                           
                             
                               Φ 
                                
                               
                                 ( 
                                 
                                   d 
                                   σ 
                                 
                                 ) 
                               
                             
                             - 
                             
                               Φ 
                                
                               
                                 ( 
                                 
                                   c 
                                   σ 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∫ 
                       c 
                       d 
                     
                      
                     
                       
                         
                           U 
                           j 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 a 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   d 
                                   - 
                                   c 
                                 
                                 ) 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           - 
                           
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   c 
                                   2 
                                 
                                 2 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           + 
                           
                             ( 
                             
                               d 
                               - 
                               c 
                             
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         β 
                         j 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   c 
                                   2 
                                 
                                 2 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           - 
                           
                             
                               
                                 a 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   d 
                                   - 
                                   c 
                                 
                                 ) 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         β 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∫ 
                       c 
                       d 
                     
                      
                     
                       
                         
                           xU 
                           j 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 a 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       2 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       c 
                                       2 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           - 
                           
                             
                               
                                 
                                   d 
                                   3 
                                 
                                 3 
                               
                               - 
                               
                                 
                                   c 
                                   3 
                                 
                                 3 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   c 
                                   2 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                        
                       
                         β 
                         j 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   d 
                                   3 
                                 
                                 3 
                               
                               - 
                               
                                 
                                   c 
                                   3 
                                 
                                 3 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                           - 
                           
                             
                               
                                 a 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       2 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       c 
                                       2 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 a 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 j 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         β 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0081]    where Φ(x) denotes the Gaussian error function, a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and other disciplines: 
         [0000]    
       
         
           
             
               Φ 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 0 
                 x 
               
                
               
                 
                   2 
                   
                     π 
                   
                 
                  
                 
                    
                   
                     - 
                     
                       ξ 
                       2 
                     
                   
                 
                  
                 
                    
                   ξ 
                 
               
             
           
         
       
     
         [0082]    From (18) and (19) we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           K 
                           U 
                           1 
                         
                         = 
                           
                          
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       a 
                                       j 
                                     
                                      
                                     
                                       ( 
                                       
                                         d 
                                         - 
                                         c 
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       j 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       
                                         d 
                                         2 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         c 
                                         2 
                                       
                                       2 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       j 
                                     
                                   
                                 
                                 + 
                                 
                                   ( 
                                   
                                     d 
                                     - 
                                     c 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                              
                             
                               β 
                               j 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       d 
                                       2 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       c 
                                       2 
                                     
                                     2 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     a 
                                     j 
                                   
                                    
                                   
                                     ( 
                                     
                                       d 
                                       - 
                                       c 
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             β 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 M 
                                 j 
                                 
                                   U 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     a 
                                     j 
                                   
                                   , 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               β 
                               j 
                             
                           
                           + 
                           
                             
                               
                                 N 
                                 j 
                                 
                                   U 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     a 
                                     j 
                                   
                                   , 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               β 
                               
                                 j 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           K 
                           U 
                           2 
                         
                         = 
                           
                          
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       a 
                                       j 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             d 
                                             2 
                                           
                                           2 
                                         
                                         - 
                                         
                                           
                                             c 
                                             2 
                                           
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       j 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       
                                         d 
                                         3 
                                       
                                       3 
                                     
                                     - 
                                     
                                       
                                         c 
                                         3 
                                       
                                       3 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       j 
                                     
                                   
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       
                                         d 
                                         2 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         c 
                                         2 
                                       
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                              
                             
                               β 
                               j 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       d 
                                       3 
                                     
                                     3 
                                   
                                   - 
                                   
                                     
                                       c 
                                       3 
                                     
                                     3 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     a 
                                     j 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           d 
                                           2 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           c 
                                           2 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     j 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             β 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 M 
                                 j 
                                 
                                   U 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     a 
                                     j 
                                   
                                   , 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               β 
                               j 
                             
                           
                           + 
                           
                             
                               
                                 N 
                                 j 
                                 
                                   U 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     a 
                                     j 
                                   
                                   , 
                                   
                                     a 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               β 
                               
                                 j 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0083]    Using (16), (17), (20), and (21) in (13) through (15) after much simplification produces for j=1: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       F 
                     
                     
                       ∂ 
                       
                         β 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         P 
                         1 
                       
                        
                       
                         β 
                         1 
                       
                     
                     + 
                     
                       
                         P 
                         2 
                       
                        
                       
                         β 
                         2 
                       
                     
                     + 
                     
                       P 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0084]    Where: 
         [0000]    
       
         
           
             
               P 
               1 
             
             = 
             
               
                 2 
                  
                 
                   
                     M 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ( 
                   
                     2 
                     
                       
                         a 
                         2 
                       
                       - 
                       
                         a 
                         1 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     M 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         1 
                       
                     
                     
                       
                         a 
                         2 
                       
                       - 
                       
                         a 
                         1 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     M 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 2 
                  
                 
                   
                     N 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ( 
                   
                     2 
                     
                       
                         a 
                         2 
                       
                       - 
                       
                         a 
                         1 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     N 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         1 
                       
                     
                     
                       
                         a 
                         2 
                       
                       - 
                       
                         a 
                         1 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     N 
                     1 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         1 
                       
                       , 
                       
                         a 
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 P 
                 0 
               
               = 
               
                 
                   
                     - 
                     2 
                   
                    
                   
                     
                       K 
                       R 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           1 
                         
                         , 
                         
                           a 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   
                     ( 
                     
                       2 
                       
                         
                           a 
                           2 
                         
                         - 
                         
                           a 
                           1 
                         
                       
                     
                     ) 
                   
                    
                   
                     
                       K 
                       R 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           1 
                         
                         , 
                         
                           a 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   
                     ( 
                     
                       
                         2 
                          
                         
                           a 
                           1 
                         
                       
                       
                         
                           a 
                           2 
                         
                         - 
                         
                           a 
                           1 
                         
                       
                     
                     ) 
                   
                    
                   
                     
                       
                         K 
                         R 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             a 
                             1 
                           
                           , 
                           
                             a 
                             2 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0085]    In equation 22, P is used as a convenient and compact way to represent the expressions shown above. 
         [0086]    For 1&lt;j≦N: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 F 
               
               
                 ∂ 
                 
                   β 
                   j 
                 
               
             
             = 
             
               
                 
                   Q 
                   0 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
               + 
               
                 
                   
                     Q 
                     1 
                   
                    
                   
                     ( 
                     j 
                     ) 
                   
                 
                  
                 
                   β 
                   
                     j 
                     - 
                     1 
                   
                 
               
               + 
               
                 
                   
                     Q 
                     2 
                   
                    
                   
                     ( 
                     j 
                     ) 
                   
                 
                  
                 
                   β 
                   j 
                 
               
               + 
               
                 
                   
                     Q 
                     3 
                   
                    
                   
                     ( 
                     j 
                     ) 
                   
                 
                  
                 
                   β 
                   
                     j 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
         [0087]    Where: 
         [0000]    
       
         
           
             
               
                 Q 
                 0 
               
                
               
                 ( 
                 j 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       - 
                       2 
                     
                     
                       
                         a 
                         j 
                       
                       - 
                       
                         a 
                         
                           j 
                           - 
                           1 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     K 
                     R 
                     2 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           j 
                           - 
                           1 
                         
                       
                       , 
                       
                         a 
                         j 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         
                           j 
                           - 
                           1 
                         
                       
                     
                     
                       
                         a 
                         j 
                       
                       - 
                       
                         a 
                         
                           j 
                           - 
                           1 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     K 
                     R 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           j 
                           - 
                           1 
                         
                       
                       , 
                       
                         a 
                         j 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 2 
                  
                 
                   
                     K 
                     R 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         j 
                       
                       , 
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     2 
                     
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         j 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     K 
                     R 
                     2 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         j 
                       
                       , 
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         j 
                       
                     
                     
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         j 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     K 
                     R 
                     1 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         j 
                       
                       , 
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   Q 
                   1 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
               = 
               
                 
                   
                     2 
                      
                     
                       
                         M 
                         
                           j 
                           - 
                           1 
                         
                         
                           U 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             a 
                             
                               j 
                               - 
                               1 
                             
                           
                           , 
                           
                             a 
                             j 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       a 
                       j 
                     
                     - 
                     
                       a 
                       
                         j 
                         - 
                         1 
                       
                     
                   
                 
                 - 
                 
                   
                     2 
                      
                     
                       a 
                       
                         j 
                         - 
                         1 
                       
                     
                      
                     
                       
                         M 
                         
                           j 
                           - 
                           1 
                         
                         
                           U 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             a 
                             
                               j 
                               - 
                               1 
                             
                           
                           , 
                           
                             a 
                             j 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       a 
                       j 
                     
                     - 
                     
                       a 
                       
                         j 
                         - 
                         1 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Q 
                 2 
               
                
               
                 ( 
                 j 
                 ) 
               
             
             = 
             
               
                 
                   2 
                    
                   
                     
                       N 
                       
                         j 
                         - 
                         1 
                       
                       
                         U 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           a 
                           
                             j 
                             - 
                             1 
                           
                         
                         , 
                         
                           a 
                           j 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     j 
                   
                   - 
                   
                     a 
                     
                       j 
                       - 
                       1 
                     
                   
                 
               
               - 
               
                 
                   2 
                    
                   
                     a 
                     
                       j 
                       - 
                       1 
                     
                   
                    
                   
                     
                       N 
                       
                         j 
                         - 
                         1 
                       
                       
                         U 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           a 
                           
                             j 
                             - 
                             1 
                           
                         
                         , 
                         
                           a 
                           j 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     j 
                   
                   - 
                   
                     a 
                     
                       j 
                       - 
                       1 
                     
                   
                 
               
               + 
               
                 2 
                  
                 
                   
                     M 
                     j 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         j 
                       
                       , 
                       
                         a 
                         
                           j 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   2 
                    
                   
                     
                       M 
                       j 
                       
                         U 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           a 
                           j 
                         
                         , 
                         
                           a 
                           
                             j 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     
                       j 
                       + 
                       1 
                     
                   
                   - 
                   
                     a 
                     j 
                   
                 
               
               + 
               
                 
                   2 
                    
                   
                     a 
                     j 
                   
                    
                   
                     
                       M 
                       j 
                       
                         U 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           a 
                           j 
                         
                         , 
                         
                           a 
                           
                             j 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     
                       j 
                       + 
                       1 
                     
                   
                   - 
                   
                     a 
                     j 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   Q 
                   3 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
               = 
               
                 
                   2 
                    
                   
                     
                       N 
                       j 
                       
                         U 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           a 
                           j 
                         
                         , 
                         
                           a 
                           
                             j 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   
                     2 
                      
                     
                       
                         N 
                         j 
                         
                           U 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             a 
                             j 
                           
                           , 
                           
                             a 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       a 
                       
                         j 
                         + 
                         1 
                       
                     
                     - 
                     
                       a 
                       j 
                     
                   
                 
                 + 
                 
                   
                     2 
                      
                     
                       a 
                       j 
                     
                      
                     
                       
                         N 
                         j 
                         
                           U 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             a 
                             j 
                           
                           , 
                           
                             a 
                             
                               j 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       a 
                       
                         j 
                         + 
                         1 
                       
                     
                     - 
                     
                       a 
                       j 
                     
                   
                 
               
             
           
         
       
     
         [0088]    In the equation following paragraph [0071], Q is used as a convenient and compact way to represent the expressions shown above. 
         [0089]    M, as used above, is similarly a convenient, compact expression, seen in eq. 20. It is the term inside the first set of brackets in eq. 20. 
         [0090]    N, as used above, is also a convenient, compact expression, seen in eq. 20. It is the term inside the second set of brackets in eq. 20. 
         [0091]    For j=N+1: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 F 
               
               
                 ∂ 
                 
                   β 
                   
                     N 
                     + 
                     1 
                   
                 
               
             
             = 
             
               
                 
                   R 
                   1 
                 
                  
                 
                   β 
                   N 
                 
               
               + 
               
                 
                   R 
                   2 
                 
                  
                 
                   β 
                   
                     N 
                     + 
                     1 
                   
                 
               
               + 
               
                 R 
                 0 
               
             
           
         
       
     
         [0092]    Where: 
         [0000]    
       
         
           
             
               R 
               0 
             
             = 
             
               
                 
                   
                     - 
                     2 
                   
                    
                   
                     
                       K 
                       R 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           N 
                         
                         , 
                         
                           a 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     
                       N 
                       + 
                       1 
                     
                   
                   - 
                   
                     a 
                     N 
                   
                 
               
               + 
               
                 
                   2 
                    
                   
                     a 
                     N 
                   
                    
                   
                     
                       K 
                       R 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           N 
                         
                         , 
                         
                           a 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     a 
                     
                       N 
                       + 
                       1 
                     
                   
                   - 
                   
                     a 
                     N 
                   
                 
               
             
           
         
       
       
         
           
             
               R 
               1 
             
             = 
             
               
                 
                   ( 
                   
                     2 
                     
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         N 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     M 
                     N 
                     
                       U 
                        
                       
                           
                       
                        
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         N 
                       
                       , 
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         N 
                       
                     
                     
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         N 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     M 
                     N 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         N 
                       
                       , 
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               R 
               2 
             
             = 
             
               
                 
                   ( 
                   
                     2 
                     
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         N 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     N 
                     N 
                     
                       U 
                        
                       
                           
                       
                        
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         N 
                       
                       , 
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                         a 
                         N 
                       
                     
                     
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         N 
                       
                     
                   
                   ) 
                 
                  
                 
                   
                     N 
                     N 
                     
                       U 
                        
                       
                           
                       
                        
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         N 
                       
                       , 
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0093]    The R variables used in the preceding paragraphs are just convenient, compact expressions for the quantities shown directly above this paragraph. 
         [0094]    To summarize: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       F 
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 P 
                                 1 
                               
                                
                               
                                 β 
                                 1 
                               
                             
                             + 
                             
                               
                                 P 
                                 2 
                               
                                
                               
                                 β 
                                 2 
                               
                             
                             + 
                             
                               P 
                               0 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 Q 
                                 0 
                               
                                
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   Q 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                                
                               
                                 β 
                                 
                                   j 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   Q 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                                
                               
                                 β 
                                 j 
                               
                             
                             + 
                             
                               
                                 
                                   Q 
                                   3 
                                 
                                  
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                                
                               
                                 β 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               1 
                             
                             &lt; 
                             j 
                             ≤ 
                             N 
                           
                         
                       
                       
                         
                           
                             
                               
                                 R 
                                 1 
                               
                                
                               
                                 β 
                                 N 
                               
                             
                             + 
                             
                               
                                 R 
                                 2 
                               
                                
                               
                                 β 
                                 
                                   N 
                                   + 
                                   1 
                                 
                               
                             
                             + 
                             
                               R 
                               0 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
       1.3. Unity C.D.F. Constraint 
       [0095]    From (4), we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             
                               g 
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                                
                               x 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 
                                   U 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                                
                               
                                  
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               ∫ 
                               0 
                               A 
                             
                              
                             
                               
                                 
                                   U 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                                
                               
                                  
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   a 
                                   i 
                                 
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         β 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         β 
                                         i 
                                       
                                     
                                     
                                       
                                         a 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         a 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       a 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               β 
                               i 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   a 
                                   i 
                                 
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         β 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         β 
                                         i 
                                       
                                     
                                     
                                       
                                         a 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         a 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 x 
                               
                             
                           
                           + 
                           
                             
                               [ 
                               
                                 
                                   β 
                                   i 
                                 
                                 - 
                                 
                                   
                                     
                                       a 
                                       i 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           β 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                         - 
                                         
                                           β 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                               
                               ] 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0096]    Let: 
         [0000]    
       
         
           
             
               
                 m 
                 i 
               
               = 
               
                 
                   
                     β 
                     
                       i 
                       + 
                       1 
                     
                   
                   - 
                   
                     β 
                     i 
                   
                 
                 
                   
                     a 
                     
                       i 
                       + 
                       1 
                     
                   
                   - 
                   
                     a 
                     i 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 b 
                 i 
               
               = 
               
                 
                   β 
                   i 
                 
                 - 
                 
                   
                     
                       
                         a 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             β 
                             
                               i 
                               + 
                               1 
                             
                           
                           - 
                           
                             β 
                             i 
                           
                         
                         ) 
                       
                     
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         
                           i 
                            
                           
                               
                           
                         
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0097]    Then (24) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∫ 
                             0 
                             A 
                           
                            
                           
                             
                               g 
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                              
                             
                                
                               x 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ∫ 
                                 
                                   a 
                                   i 
                                 
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   m 
                                   i 
                                 
                                  
                                 x 
                               
                             
                           
                           + 
                           
                             
                               b 
                               i 
                             
                              
                             
                                
                               x 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   m 
                                   i 
                                 
                                  
                                 
                                   
                                     x 
                                     2 
                                   
                                   2 
                                 
                               
                             
                             + 
                             
                               
                                 b 
                                 i 
                               
                                
                               x 
                             
                           
                            
                           
                              
                             
                               a 
                               i 
                             
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 m 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                       2 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       a 
                                       i 
                                       2 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               b 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   a 
                                   
                                     i 
                                      
                                     
                                         
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       β 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       β 
                                       i 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       a 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                       2 
                                     
                                     2 
                                   
                                   - 
                                   
                                     
                                       a 
                                       i 
                                       2 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 β 
                                 i 
                               
                               - 
                               
                                 
                                   
                                     a 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         β 
                                         
                                           i 
                                           + 
                                           1 
                                         
                                       
                                       - 
                                       
                                         β 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     a 
                                     
                                       i 
                                       + 
                                       1 
                                     
                                   
                                   - 
                                   
                                     a 
                                     i 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             ( 
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0098]    After further simplification, (25) becomes: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 A 
               
                
               
                 
                   g 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   N 
                 
                  
                 
                   
                     T 
                     i 
                     1 
                   
                    
                   
                     β 
                     i 
                   
                 
               
               + 
               
                 
                   T 
                   i 
                   2 
                 
                  
                 
                   β 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
         [0099]    Where: 
         [0000]    
       
         
           
             
               
                 
                   
                     T 
                     i 
                     1 
                   
                   = 
                   
                     
                       a 
                       
                         i 
                         + 
                         1 
                       
                     
                     - 
                     
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                         - 
                         
                           a 
                           i 
                           2 
                         
                       
                       
                         2 
                          
                         
                           ( 
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     T 
                     i 
                     2 
                   
                   = 
                   
                     
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                         - 
                         
                           a 
                           i 
                           2 
                         
                       
                       
                         2 
                          
                         
                           ( 
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       a 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0100]    Define: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       g 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           β 
                           1 
                         
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           
                             B 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           T 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         T 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0101]    Then the unity c.d.f. constraint, (4), is expressed as 
         [0000]        g   1 (β 1 , . . . β N+1 )=1  (29)
 
       1.4. Constant Power Constraint 
       [0102]    Consider (5): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         g 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       ∫ 
                       0 
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                           
                             
                               U 
                               
                                 i 
                                  
                                 
                                     
                                 
                               
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         0 
                         A 
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             U 
                             
                               i 
                                
                               
                                   
                               
                             
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           i 
                         
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             U 
                             
                               i 
                                
                               
                                   
                               
                             
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           i 
                         
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                        
                       
                         
                           
                             x 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   m 
                                   i 
                                 
                                  
                                 x 
                               
                               + 
                               
                                 b 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                           
                             m 
                             i 
                           
                            
                           
                             
                               x 
                               4 
                             
                             4 
                           
                         
                       
                       + 
                       
                         
                           b 
                           i 
                         
                          
                         
                           
                             x 
                             3 
                           
                           3 
                         
                       
                     
                      
                     
                       | 
                       
                         a 
                         i 
                       
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           m 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                                 4 
                               
                               4 
                             
                             - 
                             
                               
                                 a 
                                 i 
                                 4 
                               
                               4 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         b 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                               3 
                             
                             3 
                           
                           - 
                           
                             
                               a 
                               i 
                               3 
                             
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0103]    This equation comes directly from equation 5, where we replaced the g(x) with the assumed form shown earlier. After much simplification, this equation becomes: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 A 
               
                
               
                 
                   x 
                   2 
                 
                  
                 
                   g 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   N 
                 
                  
                 
                   
                     W 
                     i 
                     1 
                   
                    
                   
                     β 
                     i 
                   
                 
               
               + 
               
                 
                   W 
                   i 
                   2 
                 
                  
                 
                   β 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
         [0104]    Where: 
         [0000]    
       
         
           
             
               W 
               i 
               1 
             
             = 
             
               
                 
                   
                     a 
                     
                       i 
                       + 
                       1 
                     
                     3 
                   
                   - 
                   
                     a 
                     i 
                     3 
                   
                 
                 3 
               
               - 
               
                 
                   
                     a 
                     
                       i 
                       + 
                       1 
                     
                     4 
                   
                   - 
                   
                     a 
                     i 
                     4 
                   
                 
                 
                   4 
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     a 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                         3 
                       
                       - 
                       
                         a 
                         i 
                         3 
                       
                     
                     ) 
                   
                 
                 
                   3 
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               W 
               i 
               2 
             
             = 
             
               
                 
                   
                     a 
                     
                       i 
                       + 
                       1 
                     
                     4 
                   
                   - 
                   
                     a 
                     i 
                     4 
                   
                 
                 
                   4 
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   
                     a 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                         3 
                       
                       - 
                       
                         a 
                         i 
                         3 
                       
                     
                     ) 
                   
                 
                 
                   3 
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0105]    Define: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       g 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           B 
                           1 
                         
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           
                             β 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           W 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         W 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0106]    Then, the constant power constraint (5) becomes: 
         [0000]        g   2 (β 1 , . . . β N+1 )=σ 2 .
 
       1.5. Lagrangian Definition 
       [0107]    When we take the partial derivatives of the Lagrangian and set them to zero to find the stationary points, we get a system of equations which includes the original system of equations (i.e., for F), plus the two constraint equations. Hence, the Lagrangian contains or includes or encompasses the constraints along with the original system, allowing the Lagrangian, i.e. the mathematical function that summarizes the dynamics of the system, to be defined as: 
         [0000]      λ(β 1 , . . . β N+1 ,β 1 ,β 2 )= F (β 1 , . . . β N+1 )+λ 1   [g   1 (β 1 , . . . β N+1 )−1]+λ 2   [g   2 (β 1 , . . . β N+1 )−σ 2 ]
 
         [0108]    Solve the set of equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           β 
                           j 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           λ 
                           1 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           λ 
                           2 
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0109]    We have: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           β 
                           j 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           F 
                         
                         
                           ∂ 
                           
                             β 
                             j 
                           
                         
                       
                       + 
                       
                         
                           λ 
                           1 
                         
                          
                         
                           
                             ∂ 
                             
                               g 
                               1 
                             
                           
                           
                             ∂ 
                             
                               β 
                               j 
                             
                           
                         
                       
                       + 
                       
                         
                           λ 
                           2 
                         
                          
                         
                           
                             ∂ 
                             
                               g 
                               2 
                             
                           
                           
                             ∂ 
                             
                               β 
                               j 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           λ 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           g 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               β 
                               1 
                             
                             , 
                             
                               … 
                                
                               
                                   
                               
                                
                               
                                 B 
                                 
                                   N 
                                   + 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       1 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         Λ 
                       
                       
                         ∂ 
                         
                           λ 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           g 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               β 
                               1 
                             
                             , 
                             
                               … 
                                
                               
                                   
                               
                                
                               
                                 B 
                                 
                                   N 
                                   + 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           σ 
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0110]    From (32), we see that we need the partial derivatives of g 1 ,g 2 . We set the partial derivatives to zero, then solve for the betas to give values that under certain conditions represent the maximum or minimum of the function. We do this for the Lagrangian, because we want to include the constraints also, so we solve for beta parameter values which maximize or minimize the system and which also satisfy the constraints. From (28), we have, for j=1: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         g 
                         1 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             T 
                             1 
                             1 
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               T 
                               
                                 j 
                                 - 
                                 1 
                               
                               2 
                             
                             + 
                             
                               T 
                               j 
                               1 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               1 
                             
                             &lt; 
                             j 
                             ≤ 
                             N 
                           
                         
                       
                       
                         
                           
                             T 
                             N 
                             2 
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0111]    From (30), we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         g 
                         2 
                       
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             W 
                             1 
                             1 
                           
                         
                         
                           
                             
                               
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 j 
                               
                               = 
                               1 
                             
                              
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             
                               W 
                               
                                 j 
                                 - 
                                 1 
                               
                               2 
                             
                             + 
                             
                               W 
                               j 
                               1 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               1 
                             
                             &lt; 
                             j 
                             ≤ 
                             N 
                           
                         
                       
                       
                         
                           
                             W 
                             N 
                             2 
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0112]    Combining all the partial derivative expressions from (23), (33) and (34) gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       Λ 
                     
                     
                       ∂ 
                       
                         β 
                         j 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   P 
                                   0 
                                 
                                 + 
                                 
                                   
                                     P 
                                     1 
                                   
                                    
                                   
                                     β 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     P 
                                     2 
                                   
                                    
                                   
                                     β 
                                     2 
                                   
                                 
                               
                               ) 
                             
                             + 
                             
                               
                                 λ 
                                 1 
                               
                                
                               
                                 T 
                                 1 
                                 1 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                                
                               
                                 W 
                                 1 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           Q 
                                           0 
                                         
                                          
                                         
                                           ( 
                                           j 
                                           ) 
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             Q 
                                             1 
                                           
                                            
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                          
                                         
                                           β 
                                           
                                             j 
                                             - 
                                             1 
                                           
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             Q 
                                             2 
                                           
                                            
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                          
                                         
                                           β 
                                           j 
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             Q 
                                             3 
                                           
                                            
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                          
                                         
                                           β 
                                           
                                             j 
                                             + 
                                             1 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       λ 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           T 
                                           
                                             j 
                                             - 
                                             1 
                                           
                                           2 
                                         
                                         + 
                                         
                                           T 
                                           j 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       λ 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           W 
                                           
                                             j 
                                             - 
                                             1 
                                           
                                           2 
                                         
                                         + 
                                         
                                           W 
                                           j 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               1 
                             
                             &lt; 
                             j 
                             ≤ 
                             N 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   
                                     0 
                                      
                                     
                                         
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     R 
                                     1 
                                   
                                    
                                   
                                     β 
                                     N 
                                   
                                 
                                 + 
                                 
                                   
                                     R 
                                     2 
                                   
                                    
                                   
                                     β 
                                     
                                       N 
                                       + 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                             + 
                             
                               
                                 λ 
                                 1 
                               
                                
                               
                                 T 
                                 N 
                                 2 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                                
                               
                                 W 
                                 N 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               j 
                             
                             = 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0113]    From (35) and the system (31), we have the system of equations: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     ( 
                     
                       T 
                       1 
                       1 
                     
                     ) 
                   
                    
                   λ 
                 
                 + 
                 
                   
                     ( 
                     
                       W 
                       1 
                       1 
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       P 
                       1 
                     
                     ) 
                   
                    
                   
                     β 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       P 
                       2 
                     
                     ) 
                   
                    
                   
                     β 
                     2 
                   
                 
               
               = 
               
                 - 
                 
                   P 
                   0 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     ⋮ 
                      
                     
                       
 
                     
                     ( 
                     
                       
                         T 
                         
                           j 
                           - 
                           1 
                         
                         2 
                       
                       + 
                       
                         T 
                         j 
                         1 
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         W 
                         
                           j 
                           - 
                           1 
                         
                         2 
                       
                       + 
                       
                         W 
                         j 
                         1 
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         Q 
                         1 
                       
                        
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ) 
                   
                    
                   
                     β 
                     
                       j 
                       - 
                       1 
                     
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         Q 
                         2 
                       
                        
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ) 
                   
                    
                   
                     β 
                     j 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         Q 
                         3 
                       
                        
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ) 
                   
                    
                   
                     β 
                     
                       j 
                       + 
                       1 
                     
                   
                 
               
               = 
               
                 - 
                 
                   
                     Q 
                     0 
                   
                    
                   
                     ( 
                     j 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     ⋮ 
                      
                     
                       
 
                     
                      
                     
                         
                     
                     ( 
                     
                       T 
                       N 
                       2 
                     
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       W 
                       N 
                       2 
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       R 
                       1 
                     
                     ) 
                   
                    
                   
                     β 
                     N 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       R 
                       2 
                     
                     ) 
                   
                    
                   
                     β 
                     
                       N 
                       + 
                       1 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       - 
                       
                         
                           R 
                           0 
                         
                          
                         
                           
 
                         
                         ( 
                         0 
                         ) 
                       
                     
                      
                     
                       λ 
                       1 
                     
                   
                   + 
                   
                     
                       ( 
                       0 
                       ) 
                     
                      
                     
                       λ 
                       2 
                     
                   
                   + 
                   
                     
                       T 
                       1 
                       1 
                     
                      
                     
                       β 
                       1 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         
                           T 
                           1 
                           2 
                         
                         + 
                         
                           T 
                           2 
                           1 
                         
                       
                       ) 
                     
                      
                     
                       β 
                       2 
                     
                   
                   + 
                   … 
                   + 
                   
                     
                       ( 
                       
                         
                           T 
                           
                             N 
                             - 
                             1 
                           
                           2 
                         
                         + 
                         
                           T 
                           N 
                           1 
                         
                       
                       ) 
                     
                      
                     
                       β 
                       N 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         T 
                         N 
                         2 
                       
                       ) 
                     
                      
                     
                       β 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                 
                 = 
                 1 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   0 
                   ) 
                 
                  
                 
                   λ 
                   1 
                 
               
               + 
               
                 
                   ( 
                   0 
                   ) 
                 
                  
                 
                   λ 
                   2 
                 
               
               + 
               
                 
                   W 
                   1 
                   1 
                 
                  
                 
                   β 
                   1 
                 
               
               + 
               
                 
                   ( 
                   
                     
                       W 
                       1 
                       2 
                     
                     + 
                     
                       W 
                       2 
                       1 
                     
                   
                   ) 
                 
                  
                 
                   β 
                   2 
                 
               
               + 
               … 
               + 
               
                 
                   ( 
                   
                     
                       W 
                       
                         N 
                         - 
                         1 
                       
                       2 
                     
                     + 
                     
                       W 
                       N 
                       1 
                     
                   
                   ) 
                 
                  
                 
                   β 
                   N 
                 
               
               + 
               
                 
                   ( 
                   
                     W 
                     N 
                     2 
                   
                   ) 
                 
                  
                 
                   β 
                   
                     N 
                     + 
                     1 
                   
                 
               
             
             = 
             
               
                 σ 
                 2 
               
               . 
             
           
         
       
     
         [0114]    The vertical three dots, as used above, signify that you continue on with the same pattern for the next value of the index j. So, the first equation gives the expression for j=1, then you continue with the same pattern for the remaining j values until the next expression. The reason why the three dots are used is two-fold: first it is compact notation, and second you don&#39;t have an exact number for the maximum value of j (it is a variable) so you can&#39;t write out all of the equations, in general. 
         [0115]    In matrix form, the system of equations becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               T 
                               1 
                               1 
                             
                           
                           
                             
                               W 
                               1 
                               1 
                             
                           
                           
                             
                               P 
                               1 
                             
                           
                           
                             
                               P 
                               2 
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 T 
                                 1 
                                 2 
                               
                               + 
                               
                                 T 
                                 2 
                                 1 
                               
                             
                           
                           
                             
                               
                                 W 
                                 1 
                                 2 
                               
                               + 
                               
                                 W 
                                 2 
                                 1 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 1 
                               
                                
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 2 
                               
                                
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 3 
                               
                                
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 
                                   j 
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 T 
                                 j 
                                 1 
                               
                             
                           
                           
                             
                               
                                 W 
                                 
                                   j 
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 W 
                                 j 
                                 1 
                               
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                           
                             
                               
                                 Q 
                                 1 
                               
                                
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 2 
                               
                                
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 3 
                               
                                
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               T 
                               N 
                               2 
                             
                           
                           
                             
                               W 
                               N 
                               2 
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                           
                             
                               R 
                               1 
                             
                           
                           
                             
                               R 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               T 
                               1 
                               1 
                             
                           
                           
                             
                               
                                 T 
                                 1 
                                 2 
                               
                               + 
                               
                                 T 
                                 2 
                                 1 
                               
                             
                           
                           
                             
                               
                                 T 
                                 2 
                                 2 
                               
                               + 
                               
                                 T 
                                 3 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               
                                 T 
                                 
                                   N 
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 T 
                                 N 
                                 1 
                               
                             
                           
                           
                             
                               T 
                               N 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               W 
                               1 
                               1 
                             
                           
                           
                             
                               
                                 W 
                                 1 
                                 2 
                               
                               + 
                               
                                 W 
                                 2 
                                 1 
                               
                             
                           
                           
                             
                               
                                 W 
                                 2 
                                 2 
                               
                               + 
                               
                                 W 
                                 3 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               
                                 W 
                                 
                                   N 
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 W 
                                 N 
                                 1 
                               
                             
                           
                           
                             
                               W 
                               N 
                               2 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               λ 
                               1 
                             
                           
                         
                         
                           
                             
                               λ 
                               2 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               β 
                               j 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               β 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               β 
                               N 
                             
                           
                         
                         
                           
                             
                               β 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             - 
                             
                               P 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 Q 
                                 0 
                               
                                
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 Q 
                                 0 
                               
                                
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             - 
                             
                               R 
                               0 
                             
                           
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           
                             σ 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0116]    The matrix form can then be solved numerically. 
       1.6. Hessian Matrix 
       [0117]    A Hessian Matrix, sometimes referred to as simply a Hessian, is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. This was also previously known by the term “functional determinants”. 
         [0118]    In this case, a Hessian Matrix is useful in determination of the critical point type, which, after solving the system (36), can be obtained from the sign test on the sequence of principal minors of the bordered Hessian [23]. To calculate the Hessian, we need the second partial derivatives. For the Lagrangian system described above, the Hessian becomes the bordered Hessian, calculated as: 
         [0000]    
       
         
           
             H 
             = 
             
               ( 
               
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           1 
                         
                       
                       
                         ∂ 
                         
                           β 
                           1 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           1 
                         
                       
                       
                         ∂ 
                         
                           β 
                           j 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           1 
                         
                       
                       
                         ∂ 
                         
                           β 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           2 
                         
                       
                       
                         ∂ 
                         
                           β 
                           1 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           2 
                         
                       
                       
                         ∂ 
                         
                           β 
                           j 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∂ 
                         
                           g 
                           2 
                         
                       
                       
                         ∂ 
                         
                           β 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋱ 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋱ 
                   
                   
                     
                         
                     
                   
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           F 
                         
                         
                           
                             ∂ 
                             
                               β 
                               j 
                             
                           
                            
                           
                             ∂ 
                             
                               β 
                               i 
                             
                           
                         
                       
                       + 
                       
                         
                           λ 
                           1 
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               g 
                               1 
                             
                           
                           
                             
                               ∂ 
                               
                                 β 
                                 j 
                               
                             
                              
                             
                               ∂ 
                               
                                 β 
                                 i 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           λ 
                           2 
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               g 
                               2 
                             
                           
                           
                             
                               ∂ 
                               
                                 β 
                                 j 
                               
                             
                              
                             
                               ∂ 
                               
                                 β 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                   
                     ⋱ 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0119]    From (33) and (34), we see that: 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                  
                 
                   g 
                   1 
                 
               
               
                 
                   ∂ 
                   
                     β 
                     j 
                   
                 
                  
                 
                   ∂ 
                   
                     β 
                     i 
                   
                 
               
             
             = 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     g 
                     2 
                   
                 
                 
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       i 
                     
                   
                 
               
               = 
               0 
             
           
         
       
     
         [0120]    From (23) we see that: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   ∂ 
                   
                     β 
                     1 
                     2 
                   
                 
               
               = 
               
                 P 
                 1 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       2 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       1 
                     
                   
                 
               
               = 
               
                 P 
                 2 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       1 
                     
                   
                 
               
               = 
               
                 
                   0 
                    
                   
                       
                   
                    
                   for 
                    
                   
                       
                   
                    
                   j 
                 
                 &gt; 
                 2 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   ∂ 
                   
                     β 
                     j 
                     2 
                   
                 
               
               = 
               
                 
                   Q 
                   2 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       
                         j 
                         - 
                         1 
                       
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                 
               
               = 
               
                 
                   Q 
                   1 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       
                         j 
                         + 
                         1 
                       
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                 
               
               = 
               
                 
                   Q 
                   3 
                 
                  
                 
                   ( 
                   j 
                   ) 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       k 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                 
               
               = 
               
                 
                   0 
                    
                   
                       
                   
                    
                   for 
                    
                   
                       
                   
                    
                   k 
                 
                 ≠ 
                 
                   { 
                   
                     
                       j 
                       - 
                       1 
                     
                     , 
                     j 
                     , 
                     
                       j 
                       + 
                       1 
                     
                   
                   } 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   ∂ 
                   
                     β 
                     
                       N 
                       + 
                       1 
                     
                     2 
                   
                 
               
               = 
               
                 R 
                 2 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       N 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                 
               
               = 
               
                 R 
                 1 
               
             
             , 
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   F 
                 
                 
                   
                     ∂ 
                     
                       β 
                       j 
                     
                   
                    
                   
                     ∂ 
                     
                       β 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                 
               
               = 
               
                 
                   0 
                    
                   
                       
                   
                    
                   for 
                    
                   
                       
                   
                    
                   j 
                 
                 &lt; 
                 N 
               
             
           
         
       
     
         [0121]    Performing the sign test numerically shows that the solution to ( 36 ) is a minimum. 
       1.7 Numerical Solutions and Analysis 
       [0122]    We now can solve system (36) numerically. Throughout, we take σ=√{square root over (2)}/2 to generate a normalized complex signal with unit power.  FIGS. 2 through 9  contain plots of the optimal solution compared against the baseline Rayleigh distribution, as a function of cutoff amplitude value A. Values of N range from 8 to 11, depending on A. Above these N values, numerical problems occur causing the determinant of the system to be close to zero and thus precluding numerical solution. 
         [0123]    Two observations are clear from the plots in  FIGS. 2 through 9 . The first observation is that as the value of A increases the solution approaches the baseline Rayleigh distribution. This is most apparent when comparing the solution prior to the tail of the Rayleigh distribution, since most of the solutions appear to match the Rayleigh distribution reasonably well before the tail. On the tail, the solution tends to curve upward to satisfy the unity c.d.f. and constant power constraints. This behavior suggests that closely matching the Rayleigh distribution when possible, is a desirable solution property. 
         [0124]    The second observation is that for almost all of the solutions, the first ordinate, i.e. the y-coordinate, representing the distance from a point to the horizontal or x-axis measured parallel to the vertical or y axis, in this case β 1 , is negative and thus a portion of the solution has negative values. This condition violates the non-negativity constraint on probability density functions. 
         [0125]    There are two methods for overcoming the partial negativity of the optimal solution. The first method increases the first abscissa, the perpendicular distance of a point from the vertical axis, a 1 =0 value to a small but positive value, specifically until β 1  in the solution becomes positive. In this case, the solution consists of the piecewise solution for x≧a 1 , and the up-front portion of the Rayleigh distribution for 0≦x&lt;a 1 . The perturbation of a 1  has generally been observed to be small. 
         [0126]    As an example,  FIG. 10  contains the solution for A=1.4. In that instance, a value of a 1 =0.0082 is sufficient to make β 1 &gt;0. The solution shown in  FIG. 10  is negligibly different from the optimal solution shown in  FIG. 5 . 
         [0127]    The second method for overcoming the negativity of the solution is to use the Rayleigh distribution as the front segment of the distribution, and then use a perturbed version of the back-end of the optimal solution. In this approach, the tail of the optimal solution must be perturbed to meet the new unity c.d.f. and constant power constraints, which are generated by replacing the front-end of the optimal solution with the Rayleigh distribution. This perturbation approach is discussed in the next section. 
         [0000]    1.8. Perturbations from the Optimal Solution 
         [0128]    Solutions of the system (36) tend to give a negative value for the first ordinate value β 1 . Therefore, to determine a solution for which the non-negativity condition (6) holds, we perturb the optimal solution. We choose a k  starting at the distribution tail to perturb the solution. For xε[0,a 1 ] we use the original Rayleigh distribution ƒ R (x), and we perturb the ordinate values for a j  with j&gt;k for some k&lt;N as depicted in  FIG. 11 . 
         [0129]    We now define a perturbed function as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       g 
                       ɛ 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         U 
                         i 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           
                             ɛ 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0130]    We define the perturbed function this way, because we intend to change the beta values by small amounts, i.e. by the epsilon values. 
         [0131]    For some 1&lt;k&lt;N, where the U i (x,ε i ) are perturbed, compactly-supported linear segments: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       U 
                       i 
                     
                      
                     
                       ( 
                       
                         x 
                         , 
                         
                           ɛ 
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           β 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                         + 
                                         
                                           ɛ 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                     - 
                                     
                                       ( 
                                       
                                         
                                           β 
                                           i 
                                         
                                         + 
                                         
                                           ɛ 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       a 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   - 
                                   
                                     a 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               ( 
                               
                                 
                                   β 
                                   i 
                                 
                                 + 
                                 
                                   ɛ 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               x 
                             
                             ∈ 
                             
                               [ 
                               
                                 
                                   a 
                                   i 
                                 
                                 , 
                                 
                                   a 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0132]    Here, k represents the index value of the a value immediately before the a values corresponding to the beta values that we are going to change, see  FIG. 11 . We pick an index k, past which we adjust the beta parameters. 
         [0133]    We then take ε k =0 so that (a k , β k +ε k )=(a k ,β k ) thus leaving the point at index k fixed, maintaining a continuous function. Unity c.d.f and constant power constraints must still be satisfied for the perturbed solution; these are examined next. 
       1.8.1 Unity C.D.F. Constraint 
       [0134]    From the unity c.d.f. condition on g ε (x), we have: 
         [0000]    
       
         
           
             
               
                 
                   1 
                   = 
                     
                    
                   
                     
                       ∫ 
                       0 
                       A 
                     
                      
                     
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           a 
                           k 
                         
                         A 
                       
                        
                       
                         
                           
                             g 
                             ɛ 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0135]    which gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0136]    From integration of the Rayleigh distribution in (16), we obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                     
                    
                   
                     1 
                     - 
                     
                       [ 
                       
                         1 
                         - 
                         
                            
                           
                             - 
                             
                               
                                 a 
                                 k 
                                 2 
                               
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                      
                     
                       - 
                       
                         
                           a 
                           k 
                           2 
                         
                         
                           σ 
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0137]    Now from (37): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           
                             U 
                             i 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 ɛ 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           k 
                         
                         A 
                       
                        
                       
                         
                           
                             U 
                             i 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 ɛ 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           i 
                         
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                        
                       
                         
                           
                             U 
                             i 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 ɛ 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           ∫ 
                           
                             a 
                             i 
                           
                           
                             a 
                             
                               i 
                               + 
                               1 
                             
                           
                         
                          
                         
                           
                             m 
                             i 
                           
                            
                           x 
                         
                       
                     
                     + 
                     
                       
                         b 
                         i 
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
               
             
           
         
       
     
         [0138]    Where: 
         [0000]    
       
         
           
             
               m 
               i 
             
             = 
             
               
                 
                   ( 
                   
                     
                       β 
                       
                         i 
                         + 
                         1 
                       
                     
                     - 
                     
                       β 
                       i 
                     
                   
                   ) 
                 
                 + 
                 
                   ( 
                   
                     
                       ɛ 
                       
                         i 
                         + 
                         1 
                       
                     
                     - 
                     
                       ɛ 
                       i 
                     
                   
                   ) 
                 
               
               
                 
                   a 
                   
                     i 
                     + 
                     1 
                   
                 
                 - 
                 
                   a 
                   i 
                 
               
             
           
         
       
       
         
           
             
               b 
               i 
             
             = 
             
               
                 β 
                 i 
               
               + 
               
                 ɛ 
                 i 
               
               - 
               
                 
                   
                     
                       a 
                       i 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               β 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               β 
                               i 
                             
                           
                           ) 
                         
                         + 
                         
                           ( 
                           
                             
                               ɛ 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               ɛ 
                               i 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   
                     
                       a 
                       
                         i 
                         + 
                         1 
                       
                     
                     - 
                     
                       a 
                       i 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0139]    Thus: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               k 
                             
                             N 
                           
                            
                           
                             
                               ∫ 
                               
                                 a 
                                 i 
                               
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               
                                 m 
                                 i 
                               
                                
                               x 
                             
                           
                         
                         + 
                         
                           
                             b 
                             i 
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                       = 
                         
                        
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               k 
                             
                             N 
                           
                            
                           
                             
                               m 
                               i 
                             
                              
                             
                               
                                 x 
                                 2 
                               
                               2 
                             
                           
                         
                         + 
                         
                           
                             b 
                             i 
                           
                            
                           x 
                         
                       
                     
                      
                   
                   
                     a 
                     i 
                   
                   
                     a 
                     
                       i 
                       + 
                       1 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           m 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                                 2 
                               
                               2 
                             
                             - 
                             
                               
                                 a 
                                 i 
                                 2 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         b 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             a 
                             
                               i 
                               + 
                               1 
                             
                           
                           - 
                           
                             a 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0140]    Which, after much simplification, gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           T 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         T 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     + 
                     
                       
                         ɛ 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                           2 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0141]    for T i   1  and T i   2  defined in (26) and (27). 
         [0142]    Hence from (39) and (40) we get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           ɛ 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                             2 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               a 
                               i 
                             
                           
                           2 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           T 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         T 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
       1.8.2 Constant Power Constraint 
       [0143]    In general, we want to keep the average power level the same as for the original, uncompanded OFDM signal, while simultaneously reducing the peak power, so as to lower the PAPR value of the companded OFDM signal relative to the nominal OFDM signal, so a constant power constraint is used. 
         [0144]    From the constant power constraint on g ε (x), we have: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     2 
                   
                   = 
                     
                    
                   
                     
                       ∫ 
                       0 
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           a 
                           k 
                         
                         A 
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             g 
                             ɛ 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0145]    which gives: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     
                       σ 
                       2 
                     
                     - 
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
         [0146]    Using integration by parts, we get: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 c 
                 d 
               
                
               
                 
                   x 
                   2 
                 
                  
                 
                   
                     f 
                     R 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   x 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       c 
                       2 
                     
                     + 
                     
                       σ 
                       2 
                     
                   
                   ) 
                 
                  
                 
                    
                   
                     - 
                     
                       
                         c 
                         2 
                       
                       
                         σ 
                         2 
                       
                     
                   
                 
               
               - 
               
                 
                   ( 
                   
                     
                       d 
                       2 
                     
                     + 
                     
                       σ 
                       2 
                     
                   
                   ) 
                 
                  
                 
                    
                   
                     
                       d 
                       2 
                     
                     
                       σ 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0147]    and so: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 
                   a 
                   k 
                 
               
                
               
                 
                   x 
                   2 
                 
                  
                 
                   
                     f 
                     R 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   x 
                 
               
             
             = 
             
               
                 σ 
                 2 
               
               - 
               
                 
                   ( 
                   
                     
                       a 
                       k 
                       2 
                     
                     + 
                     
                       σ 
                       2 
                     
                   
                   ) 
                 
                  
                 
                    
                   
                     - 
                     
                       
                         a 
                         k 
                         2 
                       
                       
                         σ 
                         2 
                       
                     
                   
                 
               
             
           
         
       
     
         [0148]    from which follows, using (42): 
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   a 
                   k 
                 
                 A 
               
                
               
                 
                   x 
                   2 
                 
                  
                 
                   
                     g 
                     ɛ 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   x 
                 
               
             
             = 
             
               
                 ( 
                 
                   
                     a 
                     k 
                     2 
                   
                   + 
                   
                     σ 
                     2 
                   
                 
                 ) 
               
                
               
                  
                 
                   - 
                   
                     
                       a 
                       k 
                       2 
                     
                     
                       σ 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0149]    Now: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             k 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               U 
                               i 
                             
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 
                                   ɛ 
                                   i 
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           k 
                         
                         A 
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                             
                         
                          
                         
                           
                             U 
                             i 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 ɛ 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           i 
                         
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                             
                         
                          
                         
                           
                             U 
                             i 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               
                                 ɛ 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         k 
                       
                       N 
                     
                      
                     
                       
                         ∫ 
                         
                           a 
                           i 
                         
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                        
                       
                         
                           
                             x 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   m 
                                   i 
                                 
                                  
                                 x 
                               
                               + 
                               
                                 b 
                                 i 
                               
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0150]    Which, after much manipulation, becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         a 
                         k 
                       
                       A 
                     
                      
                     
                       
                         x 
                         2 
                       
                        
                       
                         
                           g 
                           ɛ 
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           W 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         W 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     + 
                     
                       
                         ɛ 
                         i 
                       
                        
                       
                         η 
                         i 
                         1 
                       
                     
                     + 
                     
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         η 
                         i 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
         [0151]    Where: 
         [0000]    
       
         
           
             
               η 
               i 
               1 
             
             = 
             
               
                 
                   
                     
                       a 
                       
                         i 
                         + 
                         1 
                       
                       3 
                     
                     - 
                     
                       a 
                       i 
                       3 
                     
                   
                   3 
                 
                 - 
                 
                   
                     
                       a 
                       
                         i 
                         + 
                         1 
                       
                       4 
                     
                     - 
                     
                       a 
                       i 
                       4 
                     
                   
                   
                     4 
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           a 
                           i 
                         
                       
                       ) 
                     
                   
                 
                 + 
                 
                   
                     
                       a 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                           3 
                         
                         - 
                         
                           a 
                           i 
                           3 
                         
                       
                       ) 
                     
                   
                   
                     3 
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           a 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 W 
                 i 
                 1 
               
             
           
         
       
       
         
           
             
               η 
               i 
               2 
             
             = 
             
               
                 
                   
                     
                       a 
                       
                         i 
                         + 
                         1 
                       
                       4 
                     
                     - 
                     
                       a 
                       i 
                       4 
                     
                   
                   
                     4 
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           a 
                           i 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   
                     
                       a 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                           3 
                         
                         - 
                         
                           a 
                           i 
                           3 
                         
                       
                       ) 
                     
                   
                   
                     3 
                      
                     
                       ( 
                       
                         
                           a 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           a 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   W 
                   i 
                   2 
                 
                 . 
               
             
           
         
       
     
         [0152]    From (43) using (42), we get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           ɛ 
                           i 
                         
                          
                         
                           η 
                           i 
                           1 
                         
                       
                     
                     + 
                     
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         η 
                         i 
                         2 
                       
                     
                   
                   = 
                   
                     
                       σ 
                       2 
                     
                     - 
                     
                       
                         ∫ 
                         0 
                         
                           a 
                           k 
                         
                       
                        
                       
                         
                           x 
                           2 
                         
                          
                         
                           
                             f 
                             R 
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           x 
                         
                       
                     
                     - 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           W 
                           i 
                           1 
                         
                          
                         
                           β 
                           i 
                         
                       
                     
                     + 
                     
                       
                         W 
                         i 
                         2 
                       
                        
                       
                         β 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0153]    Combining (41) and (44), we get the under-constrained system of equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           ξ 
                           i 
                         
                          
                         
                           ɛ 
                           i 
                         
                       
                     
                     + 
                     
                       
                         ξ 
                         i 
                       
                        
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                   = 
                   
                     C 
                     1 
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           k 
                         
                         N 
                       
                        
                       
                         
                           η 
                           i 
                           1 
                         
                          
                         
                           ɛ 
                           i 
                         
                       
                     
                     + 
                     
                       
                         η 
                         i 
                         2 
                       
                        
                       
                         ɛ 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                   = 
                   
                     C 
                     2 
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
         [0154]    Where: 
         [0000]    
       
         
           
             
               ξ 
               i 
             
             = 
             
               
                 
                   a 
                   
                     i 
                     + 
                     1 
                   
                 
                 - 
                 
                   a 
                   i 
                 
               
               2 
             
           
         
       
       
         
           
             
               C 
               1 
             
             = 
             
               1 
               - 
               
                 
                   ∫ 
                   0 
                   
                     a 
                     k 
                   
                 
                  
                 
                   
                     f 
                     R 
                   
                    
                   
                       
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                    
                   
                       
                   
                    
                   
                      
                     x 
                   
                 
               
               - 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     k 
                   
                   N 
                 
                  
                 
                   
                     T 
                     i 
                     1 
                   
                    
                   
                     β 
                     i 
                   
                 
               
               + 
               
                 
                   T 
                   i 
                   2 
                 
                  
                 
                   β 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
               2 
             
             = 
             
               
                 σ 
                 2 
               
               - 
               
                 
                   ∫ 
                   0 
                   
                     a 
                     k 
                   
                 
                  
                 
                   
                     x 
                     2 
                   
                    
                   
                     f 
                     R 
                   
                    
                   
                       
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                    
                   
                       
                   
                    
                   
                      
                     x 
                   
                 
               
               - 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     k 
                   
                   N 
                 
                  
                 
                   
                     W 
                     i 
                     1 
                   
                    
                   
                     β 
                     i 
                   
                 
               
               + 
               
                 
                   W 
                   i 
                   2 
                 
                  
                 
                   
                     β 
                     
                       i 
                       + 
                       1 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0155]    Next, the optimal values of ε i  are found. 
       1.8.3 Optimal Perturbation Parameters 
       [0156]    Our goal here is to find the minimal perturbation g ε (x) away from g(x). To do so, we&#39;ll choose parameters (ε k+1 , . . . ε N ) to minimize: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       ɛ 
                     
                      
                     
                       ( 
                       
                         
                           
                             ɛ 
                             
                               
                                 k 
                                 + 
                                 1 
                               
                               , 
                             
                           
                            
                           
                               
                           
                            
                           … 
                         
                          
                         
                             
                         
                         , 
                         
                           ɛ 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         
                           k 
                           + 
                           1 
                         
                       
                       
                         N 
                         + 
                         1 
                       
                     
                      
                     
                       ɛ 
                       i 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0157]    First we rewrite (45) and (46) as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           C 
                           1 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 k 
                               
                               N 
                             
                              
                             
                               
                                 ξ 
                                 i 
                               
                                
                               
                                 ɛ 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               ξ 
                               i 
                             
                              
                             
                               ɛ 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ξ 
                               k 
                             
                              
                             
                               ɛ 
                               k 
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ξ 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     ɛ 
                                     i 
                                   
                                 
                               
                               + 
                               
                                 
                                   ξ 
                                   i 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             ( 
                             
                               
                                 
                                   ξ 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   ɛ 
                                   N 
                                 
                               
                               + 
                               
                                 
                                   ξ 
                                   N 
                                 
                                  
                                 
                                   ɛ 
                                   N 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               ξ 
                               N 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ξ 
                               k 
                             
                              
                             
                               ɛ 
                               k 
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       ξ 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       ξ 
                                       i 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 + 
                                 
                                   ξ 
                                   N 
                                 
                               
                               ) 
                             
                              
                             
                               ɛ 
                               N 
                             
                           
                           + 
                           
                             
                               ξ 
                               N 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       ξ 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       ξ 
                                       i 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 + 
                                 
                                   ξ 
                                   N 
                                 
                               
                               ) 
                             
                              
                             
                               ɛ 
                               N 
                             
                           
                           + 
                           
                             
                               ξ 
                               N 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0158]    because, by definition, we took ε k =0. Similarly from (46): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           C 
                           2 
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 k 
                               
                               N 
                             
                              
                             
                               
                                 η 
                                 i 
                                 1 
                               
                                
                               
                                 ɛ 
                                 i 
                               
                             
                           
                           + 
                           
                             
                               η 
                               i 
                               2 
                             
                              
                             
                               ɛ 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               η 
                               k 
                               1 
                             
                              
                             
                               ɛ 
                               k 
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     η 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     2 
                                   
                                    
                                   
                                     ɛ 
                                     i 
                                   
                                 
                               
                               + 
                               
                                 
                                   η 
                                   i 
                                   1 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             ( 
                             
                               
                                 
                                   η 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   2 
                                 
                                  
                                 
                                   ɛ 
                                   N 
                                 
                               
                               + 
                               
                                 
                                   η 
                                   N 
                                   1 
                                 
                                  
                                 
                                   ɛ 
                                   N 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               η 
                               N 
                               2 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               η 
                               k 
                               1 
                             
                              
                             
                               ɛ 
                               k 
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       η 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       2 
                                     
                                     + 
                                     
                                       η 
                                       i 
                                       1 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   η 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   N 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ɛ 
                               N 
                             
                           
                           + 
                           
                             
                               η 
                               N 
                               2 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       η 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       2 
                                     
                                     + 
                                     
                                       η 
                                       i 
                                       1 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   ɛ 
                                   i 
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   η 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   N 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ɛ 
                               N 
                             
                           
                           + 
                           
                             
                               η 
                               N 
                               2 
                             
                              
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0159]    Now we define functions: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       h 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           ɛ 
                           
                             k 
                             + 
                             1 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           ɛ 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             
                               k 
                               + 
                               1 
                             
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ξ 
                                 
                                   i 
                                   - 
                                   1 
                                 
                               
                               + 
                               
                                 ξ 
                                 i 
                               
                             
                             ) 
                           
                            
                           
                             ɛ 
                             i 
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             ξ 
                             
                               N 
                               - 
                               1 
                             
                           
                           + 
                           
                             ξ 
                             N 
                           
                         
                         ) 
                       
                        
                       
                         ɛ 
                         N 
                       
                     
                     + 
                     
                       
                         ξ 
                         N 
                       
                        
                       
                         ɛ 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       h 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           ɛ 
                           
                             k 
                             + 
                             1 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           ɛ 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             
                               k 
                               + 
                               1 
                             
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 η 
                                 
                                   i 
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 η 
                                 i 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             ɛ 
                             i 
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             η 
                             
                               N 
                               - 
                               1 
                             
                             2 
                           
                           + 
                           
                             η 
                             N 
                             1 
                           
                         
                         ) 
                       
                        
                       
                         ɛ 
                         N 
                       
                     
                     + 
                     
                       
                         η 
                         N 
                         2 
                       
                        
                       
                         ɛ 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0160]    Then from (48) and (49) we get two constraint equations: 
         [0000]        h   1 (ε k+1 , . . . ε N+1 )= C   1  
 
         [0000]        h   2 (ε k+1 , . . . ε N+1 )= C   2  
 
         [0161]    Next, we define the Lagrangian: 
         [0000]      Λ h (ε k+1 , . . . ,ε N+1 ,λ 1 ,λ 2 )= F   ε (ε k+1 , . . . ,ε N+1 )+λ 1   [h   1 (ε k+1 , . . . ,ε N+1 ) C   1 ]+λ 2   [h   2 (ε k+1 , . . . ε N+1 )− C   2 ]
 
         [0162]    Now, we solve the set of equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           Λ 
                           h 
                         
                       
                       
                         ∂ 
                         
                           ɛ 
                           j 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         
                           Λ 
                           h 
                         
                       
                       
                         ∂ 
                         
                           λ 
                           1 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         
                           Λ 
                           h 
                         
                       
                       
                         ∂ 
                         
                           λ 
                           2 
                         
                       
                     
                     = 
                     0. 
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
         [0163]    We have: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   Λ 
                   h 
                 
               
               
                 ∂ 
                 
                   ɛ 
                   j 
                 
               
             
             = 
             
               
                 
                   ∂ 
                   
                     F 
                     ɛ 
                   
                 
                 
                   ∂ 
                   
                     ɛ 
                     j 
                   
                 
               
               + 
               
                 
                   λ 
                   1 
                 
                  
                 
                   
                     ∂ 
                     
                       h 
                       1 
                     
                   
                   
                     ∂ 
                     
                       ɛ 
                       j 
                     
                   
                 
               
               + 
               
                 
                   λ 
                   2 
                 
                  
                 
                   
                     ∂ 
                     
                       h 
                       2 
                     
                   
                   
                     ∂ 
                     
                       ɛ 
                       j 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ∂ 
                 
                   Λ 
                   h 
                 
               
               
                 ∂ 
                 
                   λ 
                   1 
                 
               
             
             = 
             
               
                 
                   h 
                   1 
                 
                  
                 
                   ( 
                   
                     
                       ɛ 
                       
                         k 
                         + 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       ɛ 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 C 
                 1 
               
             
           
         
       
       
         
           
             
               
                 ∂ 
                 
                   Λ 
                   h 
                 
               
               
                 ∂ 
                 
                   λ 
                   2 
                 
               
             
             = 
             
               
                 
                   h 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       ɛ 
                       
                         k 
                         + 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       ɛ 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 C 
                 2 
               
             
           
         
       
     
         [0164]    From (47): 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   F 
                   ɛ 
                 
               
               
                 ∂ 
                 
                   ɛ 
                   j 
                 
               
             
             = 
             
               2 
                
               
                 ɛ 
                 j 
               
             
           
         
       
     
         [0165]    From (50) and (51): 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   h 
                   1 
                 
               
               
                 ∂ 
                 
                   ɛ 
                   i 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           ξ 
                           
                             i 
                             - 
                             1 
                           
                         
                         + 
                         
                           ξ 
                           i 
                         
                       
                     
                     
                       
                         
                           
                             if 
                              
                             
                                 
                             
                              
                             k 
                           
                           + 
                           1 
                         
                         ≤ 
                         i 
                         ≤ 
                         N 
                       
                     
                   
                   
                     
                       
                         ξ 
                         N 
                       
                     
                     
                       
                         
                           if 
                            
                           
                               
                           
                            
                           i 
                         
                         = 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                   
                 
                 , 
                 
                   
 
                 
                  
                 
                   
                     
                       ∂ 
                       
                         h 
                         2 
                       
                     
                     
                       ∂ 
                       
                         ɛ 
                         i 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               η 
                               
                                 i 
                                 - 
                                 1 
                               
                               2 
                             
                             + 
                             
                               η 
                               i 
                               1 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 k 
                               
                               + 
                               1 
                             
                             ≤ 
                             i 
                             ≤ 
                             N 
                           
                         
                       
                       
                         
                           
                             η 
                             N 
                             2 
                           
                         
                         
                           
                             
                               if 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             
                               N 
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0166]    Thus: 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   Λ 
                   j 
                 
               
               
                 ∂ 
                 
                   ɛ 
                   j 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         2 
                          
                         
                           ɛ 
                           i 
                         
                       
                       + 
                       
                         
                           λ 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               ξ 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               ξ 
                               
                                 i 
                                  
                                 
                                     
                                 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           λ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               η 
                               
                                 i 
                                 - 
                                 1 
                               
                               2 
                             
                             + 
                             
                               η 
                               i 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           if 
                            
                           
                               
                           
                            
                           k 
                         
                         + 
                         1 
                       
                       ≤ 
                       i 
                       ≤ 
                       N 
                     
                   
                 
                 
                   
                     
                       
                         2 
                          
                         
                           ɛ 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       + 
                       
                         
                           λ 
                           1 
                         
                          
                         
                           ( 
                           
                             ξ 
                             N 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           λ 
                           2 
                         
                          
                         
                           ( 
                           
                             η 
                             N 
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         if 
                          
                         
                             
                         
                          
                         i 
                       
                       = 
                       
                         N 
                         + 
                         1 
                       
                     
                   
                 
               
             
           
         
       
     
         [0167]    Hence, we need to solve the following system of equations for ε i , λ 1 , λ 2 : 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     ( 
                     
                       
                         ξ 
                         k 
                       
                       + 
                       
                         ξ 
                         
                           k 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         η 
                         k 
                         2 
                       
                       + 
                       
                         η 
                         
                           k 
                           + 
                           1 
                         
                         1 
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     2 
                     ) 
                   
                    
                   
                     ɛ 
                     
                       k 
                       + 
                       1 
                     
                   
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     ⋮ 
                      
                     
                       
 
                     
                      
                     
                         
                     
                     ( 
                     
                       
                         ξ 
                         
                           i 
                           - 
                           1 
                         
                       
                       + 
                       
                         ξ 
                         i 
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         η 
                         
                           i 
                           - 
                           1 
                         
                         2 
                       
                       + 
                       
                         η 
                         i 
                         1 
                       
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     2 
                     ) 
                   
                    
                   
                     ɛ 
                     i 
                   
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     ⋮ 
                      
                     
                       
 
                     
                      
                     
                         
                     
                     ( 
                     
                       ξ 
                       N 
                     
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       η 
                       N 
                       2 
                     
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   
                     ( 
                     2 
                     ) 
                   
                    
                   
                     ɛ 
                     
                       N 
                       + 
                       1 
                     
                   
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   0 
                   ) 
                 
                  
                 
                   λ 
                   1 
                 
               
               + 
               
                 
                   ( 
                   0 
                   ) 
                 
                  
                 
                   λ 
                   2 
                 
               
               + 
               
                 ( 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       
                         k 
                         + 
                         1 
                       
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       ( 
                       
                         
                           ξ 
                           
                             i 
                             - 
                             1 
                           
                         
                         + 
                         
                           ξ 
                           i 
                         
                       
                       ) 
                     
                      
                     
                       ɛ 
                       
                         i 
                          
                         
                             
                         
                       
                     
                   
                 
                 ) 
               
               + 
               
                 
                   ( 
                   
                     
                       ξ 
                       
                         N 
                         - 
                         1 
                       
                     
                     + 
                     
                       ξ 
                       N 
                     
                   
                   ) 
                 
                  
                 
                   ɛ 
                   N 
                 
               
               + 
               
                 
                   ξ 
                   N 
                 
                  
                 
                   ɛ 
                   
                     N 
                     + 
                     1 
                   
                 
               
             
             = 
             
               
                 
                   
                     
                       C 
                       1 
                     
                      
                     
                       
 
                     
                     ( 
                     0 
                     ) 
                   
                    
                   
                     λ 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     0 
                     ) 
                   
                    
                   
                     λ 
                     2 
                   
                 
                 + 
                 
                   ( 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         
                           k 
                           + 
                           1 
                         
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             η 
                             
                               i 
                               - 
                               1 
                             
                             2 
                           
                           + 
                           
                             η 
                             i 
                             1 
                           
                         
                         ) 
                       
                        
                       
                         ɛ 
                         
                           i 
                            
                           
                               
                           
                         
                       
                     
                   
                   ) 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         η 
                         
                           N 
                           - 
                           1 
                         
                         2 
                       
                       + 
                       
                         η 
                         N 
                         1 
                       
                     
                     ) 
                   
                    
                   
                     ɛ 
                     N 
                   
                 
                 + 
                 
                   
                     η 
                     N 
                     2 
                   
                    
                   
                     ɛ 
                     
                       N 
                       + 
                       1 
                     
                   
                 
               
               = 
               
                 C 
                 2 
               
             
           
         
       
     
         [0168]    Or: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   k 
                                 
                                 + 
                                 
                                   ξ 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   η 
                                   k 
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             2 
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                                 + 
                                 
                                   ξ 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   η 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             0 
                           
                           
                             2 
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             ⋱ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             ⋱ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 + 
                                 
                                   ξ 
                                   i 
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   η 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   i 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             2 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             ⋱ 
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               ξ 
                               N 
                             
                           
                           
                             
                               η 
                               N 
                               2 
                             
                           
                           
                             0 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                           
                             2 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               ( 
                               
                                 
                                   ξ 
                                   k 
                                 
                                 + 
                                 
                                   ξ 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             … 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               ξ 
                               N 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               ( 
                               
                                 
                                   η 
                                   k 
                                   2 
                                 
                                 + 
                                 
                                   η 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             … 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               η 
                               N 
                               2 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               λ 
                               1 
                             
                           
                         
                         
                           
                             
                               λ 
                               2 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               
                                   
                               
                                
                               
                                 ɛ 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                               ɛ 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               ɛ 
                               N 
                             
                           
                         
                         
                           
                             
                               ɛ 
                               
                                 N 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             C 
                             1 
                           
                         
                       
                       
                         
                           
                             C 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
         [0169]    This system of equations (53) may then be solved numerically. 
       1.9. Perturbed Solution Numerical Results 
       [0170]    Perturbation solutions for the system in (53) are shown for A=1.4 and kε{5,6,7,8} in  FIGS. 12 through 15 . In these Figures, the perturbation solution with the Rayleigh front-end is plotted against the optimal solution. 
       1.10. Compander Derivation 
       [0171]    Next, we denote the companding function by C, which operates on the input sequence x(n) by modifying the amplitude to produce the companded output sequence y(n)=C{x(n)}. The compander is monotonic, i.e. it maintains order of the function (i.e., either increasing or decreasing), over its domain xε[0,A], so the compander may be derived as: 
         [0000]        C{x ( n )}= sgn{x ( n )} F   |y|   −1   F   |x|   {x ( n )}  (54)
 
         [0172]    Where F |x|  denotes the c.d.f of the input and F |y|  denotes the c.d.f of the output companded signal. To derive the compander, expressions for each c.d.f in (54) are needed. The p.d.f. of the companded signal is given by: 
         [0000]    
       
         
           
             
               
                 f 
                 
                    
                   y 
                    
                 
               
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           2 
                            
                           x 
                         
                         
                           σ 
                           2 
                         
                       
                        
                       
                          
                         
                           - 
                           
                             
                               x 
                               2 
                             
                             
                               σ 
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         0 
                       
                       ≤ 
                       x 
                       &lt; 
                       
                         a 
                         1 
                       
                     
                   
                 
                 
                   
                     
                       
                         f 
                         
                           a 
                           1 
                         
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         
                           a 
                           1 
                         
                       
                       ≤ 
                       x 
                       &lt; 
                       
                         a 
                         2 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         f 
                         
                           a 
                           i 
                         
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         
                           a 
                           i 
                         
                       
                       ≤ 
                       x 
                       &lt; 
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         f 
                         
                           a 
                           N 
                         
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         
                           a 
                           N 
                         
                       
                       ≤ 
                       x 
                       &lt; 
                       
                         a 
                         
                           N 
                           + 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         
                           a 
                           
                             N 
                             + 
                             1 
                           
                         
                       
                       ≤ 
                       x 
                     
                   
                 
               
             
           
         
       
     
         [0173]    For a i ≦x&lt;a i+1 , the c.d.f. is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             F 
                             
                                
                               y 
                                
                             
                           
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             x 
                           
                            
                           
                             
                               
                                 f 
                                 
                                    
                                   y 
                                    
                                 
                               
                                
                               
                                 ( 
                                 ξ 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             
                                
                               ξ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   ∫ 
                                   0 
                                   
                                     a 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     
                                       f 
                                       R 
                                     
                                      
                                     
                                       ( 
                                       ξ 
                                       ) 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     ξ 
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     1 
                                   
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∫ 
                                     
                                       a 
                                       j 
                                     
                                     
                                       a 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                   
                                    
                                   
                                     
                                       
                                         f 
                                         
                                           a 
                                           j 
                                         
                                       
                                        
                                       
                                           
                                       
                                       ( 
                                       ξ 
                                       ) 
                                     
                                      
                                     
                                        
                                       ξ 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               ∫ 
                               
                                 a 
                                 i 
                               
                               x 
                             
                              
                             
                               
                                 
                                   f 
                                   
                                     a 
                                     i 
                                   
                                 
                                  
                                 
                                   ( 
                                   ξ 
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 ξ 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             Z 
                             
                               i 
                               - 
                               1 
                             
                           
                           + 
                           
                             
                               ∫ 
                               
                                 a 
                                 i 
                               
                               x 
                             
                              
                             
                               
                                 
                                   f 
                                   
                                     a 
                                     i 
                                   
                                 
                                  
                                 
                                   ( 
                                   ξ 
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 ξ 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         ( 
                         55 
                         ) 
                       
                     
                   
                   
                     
                       
                         ( 
                         56 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0174]    Where Z i−1  denotes the term in brackets. Thus: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               F 
                               
                                  
                                 y 
                                  
                               
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                           - 
                           
                             Z 
                             
                               i 
                               - 
                               1 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             0 
                             x 
                           
                            
                           
                             
                               
                                 f 
                                 
                                   a 
                                   i 
                                 
                               
                                
                               
                                 ( 
                                 ξ 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             
                                
                               ξ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∫ 
                               
                                 a 
                                 i 
                               
                               x 
                             
                              
                             
                               
                                 m 
                                 i 
                               
                                
                               ξ 
                             
                           
                           + 
                           
                             
                               b 
                               i 
                             
                              
                             
                                 
                             
                              
                             
                                
                               ξ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               m 
                               i 
                             
                              
                             
                               
                                 x 
                                 2 
                               
                               2 
                             
                           
                           + 
                           
                             
                               b 
                               i 
                             
                              
                             x 
                           
                           - 
                           
                             
                               m 
                               i 
                             
                              
                             
                               
                                 a 
                                 i 
                                 2 
                               
                               2 
                             
                           
                           - 
                           
                             
                               b 
                               i 
                             
                              
                             
                               a 
                               i 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   m 
                                   i 
                                 
                                 2 
                               
                               ) 
                             
                              
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 b 
                                 i 
                               
                               ) 
                             
                              
                             x 
                           
                           + 
                           
                             [ 
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       i 
                                     
                                      
                                     
                                       
                                         a 
                                         i 
                                         2 
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       b 
                                       i 
                                     
                                      
                                     
                                       a 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               γ 
                               ) 
                             
                              
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               δ 
                               ) 
                             
                              
                             x 
                           
                           + 
                           e 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
         [0175]    with the obvious definitions for γ, δ, e, i.e. they correspond to the coefficients of the x times x term, the x term, and the constant term. Now, γ might be negative, so divide both sides of (57) by γ to get: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           F 
                           
                              
                             y 
                              
                           
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       - 
                       
                         Z 
                         
                           i 
                           - 
                           1 
                         
                       
                     
                     γ 
                   
                   = 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           δ 
                           γ 
                         
                         ) 
                       
                        
                       x 
                     
                     + 
                     
                       ( 
                       
                         e 
                         γ 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
         [0176]    Complete the square in (58): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         x 
                         + 
                         
                           δ 
                           
                             2 
                              
                             γ 
                           
                         
                       
                       ) 
                     
                     2 
                   
                   = 
                     
                    
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       
                         
                           2 
                            
                           δ 
                         
                         
                           2 
                            
                           γ 
                         
                       
                        
                       x 
                     
                     + 
                     
                       
                         δ 
                         2 
                       
                       
                         4 
                          
                         
                           γ 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           δ 
                           γ 
                         
                         ) 
                       
                        
                       x 
                     
                     + 
                     
                       ( 
                       
                         
                           δ 
                           2 
                         
                         
                           4 
                            
                           
                             γ 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0177]    Which implies: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 x 
                                 + 
                                 
                                   δ 
                                   
                                     2 
                                      
                                     γ 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             ( 
                             
                               
                                 e 
                                 γ 
                               
                               - 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 δ 
                                 γ 
                               
                               ) 
                             
                              
                             x 
                           
                           + 
                           
                             
                               δ 
                               2 
                             
                             
                               4 
                                
                               
                                 γ 
                                 2 
                               
                             
                           
                           + 
                           
                             ( 
                             
                               
                                 e 
                                 γ 
                               
                               - 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 δ 
                                 γ 
                               
                               ) 
                             
                              
                             x 
                           
                           + 
                           
                             ( 
                             
                               e 
                               γ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
         [0178]    Using (59), (58) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           F 
                           
                              
                             y 
                              
                           
                         
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       - 
                       
                         Z 
                         
                           i 
                           - 
                           1 
                         
                       
                     
                     γ 
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               x 
                               + 
                               
                                 δ 
                                 
                                   2 
                                    
                                   γ 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           ( 
                           
                             
                               e 
                               γ 
                             
                             - 
                             
                               
                                 δ 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   γ 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         
 
                       
                       ⇒ 
                       
                         
 
                       
                        
                       
                         
                           ( 
                           
                             x 
                             + 
                             
                               δ 
                               
                                 2 
                                  
                                 γ 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   F 
                                   
                                      
                                     y 
                                      
                                   
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               - 
                               
                                 Z 
                                 
                                   i 
                                   - 
                                   1 
                                 
                               
                             
                             γ 
                           
                           - 
                           
                             ( 
                             
                               
                                 e 
                                 γ 
                               
                               - 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           
 
                         
                         ⇒ 
                         
                           
 
                         
                          
                         
                           x 
                           + 
                           
                             δ 
                             
                               γ 
                               2 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               
                                 
                                   F 
                                   
                                      
                                     y 
                                      
                                   
                                 
                                  
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               - 
                               
                                 Z 
                                 
                                   i 
                                   - 
                                   1 
                                 
                               
                             
                             γ 
                           
                           - 
                           
                             ( 
                             
                               
                                 e 
                                 γ 
                               
                               - 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ± 
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    (The above arrows are mathematical shorthand for the phrase: “which implies”) 
         [0179]    And thus: 
         [0000]    
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 F 
                                 
                                    
                                   y 
                                    
                                 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             - 
                             
                               Z 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                           
                           γ 
                         
                         - 
                         
                           ( 
                           
                             
                               e 
                               γ 
                             
                             - 
                             
                               
                                 δ 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   γ 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ± 
                     
                     - 
                     
                       δ 
                       
                         2 
                          
                         γ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
         [0180]    In (61) we need to decide whether to take the positive or negative square-root. The decision is made from (60) by examining the sign of 
         [0000]    
       
         
           
             x 
             + 
             
               
                 δ 
                 
                   2 
                    
                   γ 
                 
               
               . 
             
           
         
       
     
         [0000]    If positive, the positive square-root is chosen, if negative, the negative square-root should be chosen. Of course, if x≧A, then we consider the sign on 
         [0000]    
       
         
           
             A 
             + 
             
               
                 δ 
                 
                   2 
                    
                   γ 
                 
               
               . 
             
           
         
       
     
         [0000]    Finally, the compander is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       
                          
                         y 
                          
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         F 
                         R 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             1 
                             - 
                             
                                
                               
                                 - 
                                 
                                   
                                     x 
                                     2 
                                   
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                             - 
                             
                               Z 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                           
                           γ 
                         
                         - 
                         
                           ( 
                           
                             
                               e 
                               γ 
                             
                             - 
                             
                               
                                 δ 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   γ 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ± 
                     
                     - 
                     
                       δ 
                       
                         2 
                          
                         γ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
         [0181]    for a i ≦x&lt;a a+1 . To complete the compander derivation, we need to calculate Z i  From (16) and (55): 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     0 
                   
                   = 
                     
                    
                   
                     
                       ∫ 
                       0 
                       
                         a 
                         1 
                       
                     
                      
                     
                       
                         
                           f 
                           R 
                         
                          
                         
                           ( 
                           ξ 
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         ξ 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     - 
                     
                        
                       
                         - 
                         
                           
                             a 
                             1 
                             2 
                           
                           
                             σ 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0182]    For i&gt;0, Z i  contains integral terms of the form: 
         [0000]    
       
         
           
             
               ∫ 
               
                 a 
                 j 
               
               
                 a 
                 
                   j 
                   + 
                   1 
                 
               
             
              
             
               
                 
                   f 
                   
                     a 
                     j 
                   
                 
                  
                 
                   ( 
                   ξ 
                   ) 
                 
               
                
               
                   
               
                
               
                  
                 ξ 
               
             
           
         
       
     
         [0183]    which represents a trapezoidal area and can be shown to be: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 
                   a 
                   j 
                 
                 
                   a 
                   
                     j 
                     + 
                     1 
                   
                 
               
                
               
                 
                   
                     f 
                     
                       a 
                       j 
                     
                   
                    
                   
                     ( 
                     ξ 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                    
                   ξ 
                 
               
             
             = 
             
               
                 
                   
                     β 
                     
                       i 
                       + 
                       1 
                     
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     β 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         a 
                         
                           i 
                           + 
                           1 
                         
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
               2 
             
           
         
       
     
         [0184]    Hence: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     i 
                   
                   = 
                   
                     
                       Z 
                       
                         i 
                         - 
                         1 
                       
                     
                     + 
                     
                       
                         
                           
                             β 
                             
                               i 
                               + 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             β 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 a 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 a 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
       1.11 Decompander Derivation 
       [0185]    To find the decompander, we need to invert (62): 
         [0000]    
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     - 
                                     
                                       
                                         x 
                                         2 
                                       
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   Z 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                               
                               γ 
                             
                             - 
                             
                               ( 
                               
                                 
                                   e 
                                   γ 
                                 
                                 - 
                                 
                                   
                                     δ 
                                     2 
                                   
                                   
                                     4 
                                      
                                     
                                       γ 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ± 
                         
                         - 
                         
                           δ 
                           
                             2 
                              
                             γ 
                           
                         
                       
                        
                       
                         
 
                       
                       ⇒ 
                       
                         
 
                       
                        
                       
                         
                           ( 
                           
                             y 
                             + 
                             
                               δ 
                               
                                 2 
                                  
                                 γ 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               1 
                               - 
                               
                                  
                                 
                                   - 
                                   
                                     
                                       x 
                                       2 
                                     
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                               - 
                               
                                 Z 
                                 
                                   i 
                                   - 
                                   1 
                                 
                               
                             
                             γ 
                           
                           - 
                           
                             ( 
                             
                               
                                 e 
                                 γ 
                               
                               - 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           
 
                         
                         ⇒ 
                         
                           
 
                         
                          
                         
                           
                             γ 
                              
                             
                               ( 
                               
                                 y 
                                 + 
                                 
                                   δ 
                                   
                                     2 
                                      
                                     γ 
                                   
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                       = 
                       
                         
                           1 
                           - 
                           
                              
                             
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           - 
                           
                             Z 
                             
                               i 
                               - 
                               1 
                             
                           
                           - 
                           e 
                           + 
                           
                             
                               δ 
                               2 
                             
                             
                               4 
                                
                               γ 
                             
                           
                         
                          
                         
                           
 
                         
                         ⇒ 
                         
                           
 
                         
                          
                         
                           
                              
                             
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           - 
                           
                             
                               γ 
                                
                               
                                 ( 
                                 
                                   y 
                                   + 
                                   
                                     δ 
                                     
                                       2 
                                        
                                       γ 
                                     
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                           - 
                           e 
                           + 
                           
                             
                               δ 
                               2 
                             
                             
                               4 
                                
                               γ 
                             
                           
                           + 
                           1 
                           - 
                           
                             Z 
                             
                               i 
                               - 
                               1 
                             
                           
                         
                          
                         
                           
 
                         
                         ⇒ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   x 
                   = 
                   
                     
                       sgn 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                      
                     
                       
                         
                           - 
                           
                             σ 
                             2 
                           
                         
                          
                         
                           ln 
                            
                           
                             ( 
                             
                               
                                 - 
                                 
                                   
                                     γ 
                                      
                                     
                                       ( 
                                       
                                         y 
                                         + 
                                         
                                           δ 
                                           
                                             2 
                                              
                                             γ 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               - 
                               e 
                               + 
                               
                                 
                                   δ 
                                   2 
                                 
                                 
                                   4 
                                    
                                   γ 
                                 
                               
                               + 
                               1 
                               - 
                               
                                 Z 
                                 
                                   i 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
         [0186]    Note that, in (64), the sgn function refers to retaining either the positive or negative square-root, depending on the sign of X. 
         [0187]    Finally, we need to find the range of the compander for use in the decompander. When a i ≦x&lt;a i+1 , then y MIN ≦y&lt;y MAX  where: 
         [0000]        y   MIN =min{ y ( a   i ), y ( a   i+1 )} 
         [0000]        y   MAX =max{ y ( a   i ), y ( a   i+1 )} 
         [0000]    and from (63): 
         [0000]    
       
         
           
             
               y 
                
               
                 ( 
                 
                   a 
                   i 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       1 
                       - 
                       
                          
                         
                           - 
                           
                             
                               a 
                               i 
                               2 
                             
                             
                               σ 
                               2 
                             
                           
                         
                       
                       - 
                       
                         Z 
                         
                           i 
                           - 
                           1 
                         
                       
                     
                     γ 
                   
                   - 
                   
                     ( 
                     
                       
                         e 
                         γ 
                       
                       - 
                       
                         
                           δ 
                           2 
                         
                         
                           4 
                            
                           
                             γ 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
                 ± 
               
               - 
               
                 δ 
                 
                   2 
                    
                   γ 
                 
               
             
           
         
       
       
         
           
             
               y 
                
               
                 ( 
                 
                   a 
                   
                     i 
                     + 
                     1 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       1 
                       - 
                       
                          
                         
                           - 
                           
                             
                               a 
                               
                                 i 
                                 + 
                                 1 
                               
                               2 
                             
                             
                               σ 
                               2 
                             
                           
                         
                       
                       - 
                       
                         Z 
                         
                           i 
                           - 
                           1 
                         
                       
                     
                     γ 
                   
                   - 
                   
                     ( 
                     
                       
                         e 
                         γ 
                       
                       - 
                       
                         
                           δ 
                           2 
                         
                         
                           4 
                            
                           
                             γ 
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
                 ± 
               
               - 
               
                 δ 
                 
                   2 
                    
                   γ 
                 
               
             
           
         
       
     
       2. SIMULATION 
       [0188]    Simulation results were generated for a Quadrature Phase Shift Keying (QPSK) (a type of phase shift keying) modulated OFDM signal with sixty four subcarriers over an additive white Gaussian noise channel. Ten thousand realizations of signal-plus-noise were generated for each noise power. 
         [0189]    For the Rayleigh p.d.f., σ=√{square root over (2)}/2 was chosen. An oversampling factor of four was used so that the discrete PAPR reasonably approximates the PAPR from a continuous system as is described Y. Wang, L.-H. Wang, J.-H. Ge, and B. Ai., “An Efficient Nonlinear Companding Transform for Reducing PAPR of OFDM Signals”,  IEEE Trans. Broadcast ., vol. 58, no. 4, pp.: 677-684, December 2012. PAPR reduction was measured using the complimentary cumulative distribution function (CCDF), which gives the probability that the PAPR is above a given threshold value. Symbol error rates, to measure demodulation performance, and power spectrums, to measure out-of-band power reduction, were also generated. 
         [0190]    Lagrange compander results were then generated and compared against two state-of-the-art companders; the linear compander (LIN) in Y. Wang, L.-H. Wang, J.-H. Ge, and B. Ai., “An Efficient Nonlinear Companding Transform for Reducing PAPR of OFDM Signals”,  IEEE Trans. Broadcast ., vol. 58, no. 4, pp.: 677-684, December 2012, and the two-component, pieceweise linear compander (LIN2) in Y. Wang, J.-H. Ge, L.-H. Wang, J. Li, and B. Ai., “Nonlinear Companding Transform for Reduction of Peak-to-Average Power ratio in OFDM Systems”,  IEEE Trans. Broadcast ., vol. 59, no. 2, pp.: 369-375, June 2013. The LIN compander was state-of-the-art as of December 2012. The LIN compander was improved to create the LIN2 compander. A cutoff value of A=1.6281 was chosen, as this appears to be the maximum cutoff value obtainable with the LIN2 compander. The LIN compander can attain a maximum cutoff value of about A=1.44, and this value was used for the LIN compander results. Therefore the Lagrange compander solution can provide higher cutoff values than can be obtained with the LIN and LIN2 solutions, thereby providing additional compander design flexibility. Two Lagrange companders were used; the optimal solution (LG) with a perturbed a 1 &gt;0 to generate a non-negative solution, and the epsilon-perturbed compander (EP), which takes the optimal Lagrange compander and perturbs the tail using optimal perturbation values. For each plot, the curve labels have the following meaning: LG=optimal Lagrange, EP=epsilon-perturbed Lagrange, LW=LIN2, WG=LIN. 
         [0191]      FIG. 16  contains symbol error plots for the four companders.  FIG. 16  shows that the LG and EP companders provide an improvement in symbol error rate performance over the LIN and LIN2 companders. 
         [0192]    This improvement in symbol error rate comes at the cost of a small reduction in PAPR performance relative to the LIN2 approach, shown in  FIG. 17  which contains the CCDF results. The LIN approach, under the present scenario, gives the best performance due to the significantly reduced cutoff value, A=1.44 versus A=1.6281. 
         [0193]    The Lagrange companders, however, provide a small performance improvement in out-of-band power rejection over the LIN2 and a significant performance advantage over the LIN approach ( FIG. 18 ). The Lagrange approach shows an improvement of about 0.4 dB over the LIN2 approach and over 1.0 dB in out-of-band power rejection over the LIN approach. 
         [0194]    Sample Lagrange companders, for different cutoff values, are shown in  FIG. 19 . 
       3. CONCLUSION 
       [0195]    Those skilled in the art will appreciate that we have developed a family of companders for PAPR reduction in OFDM signals using a constrained optimization approach. We derived companders and decompanders and demonstrated that the new companders can provide performance improvements over current state-of-the art solutions. The new companders can also provide solutions over ranges of the cutoff values where the current state-of-the-art companders fail to exist. The set of companders developed in this paper increase the solution set of companders to tradeoff between demodulation performance, PAPR reduction, and out-of-band power rejection. 
         [0196]    The foregoing description of the embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of this disclosure. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.