Abstract:
The invention relates to apparatus for the receipt of digital data which is transmitted as a data signal from a remote location and a method by which the receiving apparatus can be designed with a configuration to improve the ability to receive the data signal. The design of the configuration is generated with reference to at least one decision metric which is adapted to suit the known characteristics of the data signal which is being received.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to United Kingdom Patent Application No. 1004946.8 filed Mar. 24, 2010 which is incorporated herein by reference. 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT 
     Not Applicable 
     INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC 
     Not Applicable 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention to which this application relates is a broadcast data receiver apparatus which is provided in a form so as to have an improved response to long and short term fading of the data channels. 
     2. Prior Art 
     The demand for increased capacity in data receiving apparatus, and/or reduced information errors continues to grow and there is also an increasing demand for generally efficient operation of wireless communication systems. However, a fundamental constraint to the capacity of the data transmission and receiving system is one or both of the digital receiver and the propagation medium (i.e the channel on which the data is transmitted). 
     Broadcast signals which are transmitted are typically distorted by atmospheric and environmental impairments, thereby degrading the quality of the signal which is subsequently received. In general, one form of deterioration is fading which is typically caused by physical phenomena such as reflection, diffraction and scattering. According to the number of the scattering phenomena which may be present in a wireless channel, the effect of these phenomena is either rich or poor. This is the case in the occurrence of multipath and shadowing fading, respectively. 
     It is known from the theory of telecommunications that multipath fading and shadowing of propagated signals, (also known as short-term fading and long-term fading respectively), can be effectively described and represented by statistical distributions. 
     Conventionally used distributions such as those known as Rayleigh, Weibull and Nakagami -m, -n, -q distributions, are adequate when describing multipath fading. However, they are found not to be optimal and/or robust because they do not additionally consider the simultaneous occurrence of the shadowing effect. 
     Another known distribution known as the Suzuki statistical distribution has the advantage that it is a composite Rayleigh/Lognormal distribution. As a consequence, it is capable of describing at once both the multipath and the shadowing of communication signals which occur simultaneously during wireless propagation. However, the probability density function (PDF) of the Suzuki distribution is described by the integral of the synthesis of Lognormal and Rayleigh, distribution. An explicit solution to this integral is not known in the prior art and thus it has not been possible to implement a receiver for a digital data system which is operable over such a fading channel. 
     An existing method of compensating for shadowing is by providing the receiver with empirical data, i.e. actual data, relating to the geographical area in which the receiver is to be used. However, it, is time-consuming to produce such data, the data is not always specific enough to the receiver location, and if the receiver is moved to a new location, new empirical data has to be obtained. 
     An aim of the present invention is to provide a receiver which can characterise accurately the signal degradation caused by the presence of multipath fading and shadowing. Another aim is to provide a method whereby the characterisation of the signal degradation can be accurately determined. 
     BRIEF SUMMARY OF THE INVENTION 
     In one aspect of the invention, there is provided a broadcast receiver wherein the configuration of the apparatus and/or processing means used to receive the data signal is assessed with respect to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form: 
                 p     a   l       ⁡     (     a   l     )       =         (     5   2     )       1     8   ⁢     σ   2           ⁢         a   l     ⁢     ⅇ     -       μ   2       2   ⁢     σ   2                 σ   ⁢       2   ⁢   π           ⁢     Γ   ⁡     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )       ⁢       (       -     1     8   ⁢     σ   2           -       a   l   2     2       )         -   1     +     1     8   ⁢     σ   2         +     μ     2   ⁢     σ   2                     
where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.
 
     Thus the receiver can be optimised in a number of conditions such that the requirement for empirically modelling the channel before deploying a new network is removed. As such the receiver has improved performance over fading channels and is adaptive to the location of the receiver. 
     In one embodiment, where amplitudes are independent with first order probability density functions {pα l (α l )} l=1   Lp , and the phase and delay values are known the decision metric to be used which is derived from the Suzuki distribution is:
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{θ l } l=1   Lp ,{τ l } l=1   Lp )]
 
     The decision metric below allows implementation of a receiver design in the case of known phases but unknown delays and amplitudes.
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{θ l } l=1   Lp )]=
 
     From the theory of digital communications, the decision metric for the case of known delays and amplitudes but unknown phases is: 
     
       
         
           
             
               
                 
                   
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     In one embodiment the decision metric derived from the case of known delays but unknown amplitudes and phases is:
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{τ l } l=1   Lp )]
 
     This decision metric allows implementation of a receiver configuration design in the case of known delays but unknown phases and amplitudes. 
     In one embodiment the decision metric derived from the case of unknown delays, amplitudes and phases is:
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t )]=
 
     This decision metric allows implementation of a receiver configuration design for the case of unknown delays, phases and amplitudes. 
     In one embodiment the design or configuration of the receiving apparatus include details of coherent detection as well as non-coherent detection. 
     In one embodiment the designs include details of the possible modulation schemes, typically M-PSK, M-QAM, ASK/PAM, FSK, MSK and OFDM. 
     In one embodiment the designs include a plurality of shaping pulses, typically square, sinusoidal and cosine. 
     In one embodiment the broadcast receiver characterises signal impairment optimally by utilising any, or any combination of, the decision metrics described herein. 
     Typically, according to the level of channel knowledge, the decision metric allows optimisation of the signal reception parameters or design. 
     In a further aspect of the invention there is provided a method of configuring receiving and/or processing apparatus for at least one data signal received from a remote transmitting location, said method including the steps of: 
     identifying the geographical location at which the said apparatus is to be used; 
     identifying the data signal which is to be received and wherein the said apparatus is further configured with reference to a decision metric derived upon utilising the probability density function of a Suzuki distribution in a closed form: 
                 p     a   l       ⁡     (     a   l     )       =         (     5   2     )       1     8   ⁢     σ   2           ⁢         a   l     ⁢     ⅇ     -       μ   2       2   ⁢     σ   2                 σ   ⁢       2   ⁢   π           ⁢     Γ   ⁡     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )       ⁢       (       -     1     8   ⁢     σ   2           -       a   i   2     2       )         -   1     +     1     8   ⁢     σ   2         +     μ     2   ⁢     σ   2                     
where σ and μ denote the variance and mean value respectively of the Lognormal distribution, in order to take into account potential impairment of the said data signal.
 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S) 
       Further details of specific embodiments are now described wherein:— 
         FIG. 1  illustrates a receiver apparatus configuration design for the case of known delay and phase values but unknown amplitude values for a data signal to be received. 
         FIG. 2  illustrates a receiver apparatus configuration design for the case of known phase value but unknown delay and amplitude values for a received data signal. 
         FIG. 3  illustrates a receiver apparatus configuration design for the case of a known delay value but unknown phase and amplitude values of a received data signal. 
         FIG. 4  illustrates a receiver apparatus configuration design for receipt of a data signal for which the delay, phase and amplitude values are unknown. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the transmission of a data signal to one or more receiver apparatus then during a symbol period of ‘Ts’ seconds of the data signal the transmitter transmits a real bandpass signal which can be represented as:
 
 S   k ( t )= Re{{tilde over (s)}   k ( t )}= Re{{tilde over (S)}   k ( t ) e   j2nf     c     t }  (1)
 
where {tilde over (s)} k  is the k th  complex bandpass signal and {tilde over (S)} k (t) is the corresponding k th , complex baseband signal chosen from the set of M equi-probable message waveforms which represents the transmitted information.
 
     The above signal (1) is transmitted over the fading channel which is characterized by L p , independent diversity paths. Each of these paths is a time varying channel that attenuates, delays, phase-shifts the signal while it also adds an AWGN source. Thus, the received data signal at the receiving apparatus is in fact a set of “noisy” replicas of the transmitted signal. 
     It should be noted here that a so-called “one-shot” approach for the transmission of the data signal is assumed to be used herein i.e. a single transmission wherein ISI that would be produced by the presence of the path delays on continuous transmission is ignored. Hence, the received signal will be expressed as follows:
 
 r   l ( t )= Re{α   l   {tilde over (s)}   k ( t−τ   l ) e   jθ     l     +n   l ( t )}= Re{α   l   {tilde over (S)}   k ( t−τ   l ) e   j(2nf     c     t+θ     l     )   +N   l ( t ) e   j2nf     c     t }  (2)
 
             r   l ( t )= Re{{tilde over (r)}   l ( t )}= Re{{tilde over (R)}   l ( t ) e   j2nf     c     t   }l= 1,2,3 , . . . ,L   p   (3)
 
where {N l (t)}hd l=1 L     p    is a set of statistically independent complex Additive White Gaussian Noise (AWGN) processes, each with PSD 2N l  Watts/Hertz.

     The sets {a l } l=1   L     p   , {θ l } l=1   L     p    and {τ l } l=1   L     p    are the random channel amplitudes, phases and delays respectively and they are assumed to be constant over one or two symbol periods i.e. T g  and 2T g  for non-coherent and coherent detection respectively. 
     Another important assumption is that τ 1 &lt;τ 2 &lt; . . . &lt;τ Lp . The optimum receiver computes the set of a posteriori probabilities p(s k (t)|{r l (t)} l=1   Lp ), k=1, 2, . . . , M and chooses as its decision that message whose signal s k (t) corresponds to the largest of these probabilities. In addition, the receiver is assumed to be time—synchronised to the transmitted signal. 
     Since the messages (signals) are assumed to be equi-probable, then, by Bayes&#39; rule, the equivalent decision rule is to choose s k (t) corresponding to the largest of the conditional probabilities (likelihoods)
 
 p ({ r   l ( t )} l=1   Lp   |s   k ( t )), k= 1,2 ; . . . ,M  
 
which is the Maximum-Likelihood (ML) decision rule. By making use of the law of conditional probability, each of these conditional probabilities can be expressed by equation (4) as follows:
 
∫∫∫ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{α l } l=1   Lp ,{θ l } l=1   Lp ,{τ l } l=1   Lp ) p ({α l } l=1   Lp ,{θ l } l=1   Lp ,{τ l } l=1   Lp ) d{α   l } l=1   Lp   d{θ   l } l=1   Lp   d{τ   l } l=1   Lp  
 
which is dependent upon the degree of knowledge i.e. amount of Channel State Information (CSI) available on the parameter sets) {α l } l=1   Lp ,{θ l } l=1   Lp ,{τ l } l=1   Lp .
 
     In case that any of the three parameter sets are assumed to be known, then the statistical averages on that set of parameters need not be performed. Of course, in the case where all parameters are assumed to be known to the receiver, none of the statistical averages in (4) need be performed and as a consequence the ML decision rule simplifies to choosing the largest of:
 
 p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{α l } l=1   Lp ,{θ l } l=1   Lp ,{τ l } l=1   Lp ), k= 1,2 , . . . ,M  
 
     Receivers that make use of CSI have been termed self-adaptive in that the estimates of the system parameters are utilized to adjust the decision structure, thereby, improving the system performance by adaptation to variation of the channel changes. 
     Example 1 
     Known Amplitudes, Phases and Delays; Non-Coherent Detection 
     Conditioned on perfect knowledge of the amplitudes, phases and delays, the conditional probability p({r l (t)} l=1   Lp |s k (t),{α l } l=1   Lp ,{θ l } l=1   Lp ,{τ l } l=1   Lp ) is a joint Gaussian Probability Density Function (PDF) that because of the independence assumption on the additive noise components can be written as: 
                     p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     ❘       s   k     ⁡     (   t   )         ,       {     α   l     }       l   =   1     Lp     ,       {     θ   l     }       l   =   1     Lp     ,       {     τ   l     }       l   =   1     Lp       )       =         ∏     l   =   1       L   p       ⁢       K   1     ⁢     ⅇ     -     1     2   ⁢     N   i             ⁢       ∫     τ   l       Ts   +     τ   l         ⁢                  r   l     ⁡     (   t   )       -       α   l     ⁢         s   _     k     ⁡     (     t   -     τ   l       )       ⁢     ⅇ     jθ   l                2     ⁢     ⅆ   t             =       ∏     l   =   1       L   p       ⁢       K   1     ⁢     ⅇ       -     1     2   ⁢     N   i           ⁢       ∫     τ   l       Ts   +     τ   l         ⁢                    R   ~     l     ⁡     (   t   )       -       α   l     ⁢         S   ~     k     ⁡     (     t   -     τ   l       )       ⁢     ⅇ     jθ   l                2     ⁢     ⅆ   t                           (   5   )               
where K is an integration constant. After simplification of equation (5) we obtain:
 
                       p   ⁡     (           [       r   l     ⁡     (   t   )       ]       l   =   1     Lp     ❘       s   k     ⁡     (   t   )         ,         {     α   l     }       l   =   1     Lp     ⁢       {     θ   l     }       l   =   1     Lp       ,       {     τ   l     }       l   =   1     Lp       )       =       K   ⁢       ∏     l   =   1       L   p       ⁢     ⅇ       Re   ⁡     [         α   l       N   l       ⁢     ⅇ     -     jθ   l         ⁢       y   kl     ⁡     (   τ   )         ]       -         α   l   2     ⁢     E   k         N   i               =     K   ⁢           ⁢     ⅇ         ∑     l   =   1       L   p       ⁢     Re   ⁡     [         α   l       N   l       ⁢     ⅇ     -     jθ   l         ⁢       y   kl     ⁡     (   τ   )         ]         -         α   l   2     ⁢     E   k         N   i                 ⁢     
     ⁢           ⁢     where   ,             (   6   )                       ⁢         y   kl     ⁡     (   τ   )       =         ∫     τ   l         T   s     +     τ   l         ⁢           R   ~     l     ⁡     (   t   )       ⁢         S   ~     k   *     ⁡     (     t   -     τ   l       )       ⁢     ⅆ   t         =       ∫   0     T   s       ⁢           R   ~     l     ⁡     (     t   +     τ   l       )       ⁢         S   ~     k   *     ⁡     (   t   )       ⁢     ⅆ   t                     (   7   )               
is the complex cross correlation of the l th  received signal and the k th , signal waveform and
 
                     E   k     =         1   2     ⁢       ∫   0     T   s       ⁢                  S   ~     k     ⁡     (   t   )            2     ⁢     ⅆ   t           =       1   2     ⁢       ∫     τ   l         T   s     +     τ   l         ⁢                  S   ~     k     ⁡     (     t   -     τ   l       )            2     ⁢     ⅆ   t                     (   8   )               
is the energy of the k th  signal {tilde over (s)}s k (t). Furthermore, the constant K absorbs all the K l .s as well as the factor
 
               ⅇ     ∑     l   =   1       L   p         ⁢     1     2   ⁢     N   i         ⁢       ∫   0     T   S       ⁢                R   l     ⁡     (   t   )            2     ⁢       ⅆ   t     .               
which is independent of K and thus has no bearing on the decision. Since, the natural logarithm is a monotonic function of its argument, we can equivalently maximize with respect to k and obtain (9) as follows:
 
                           Λ   k     =       ⁢     ln   ⁡     [     p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     α   l     }       l   =   1     Lp     ,       {     θ   l     }       l   =   1     Lp     ,       {     τ   l     }       l   =   1     Lp       )       ]                   =       ⁢         ∑     l   =   1       L   p       ⁢     Re   ⁡     [         α   l       N   l       ⁢     ⅇ     -     jθ   l         ⁢     ykl   ⁡     (   τ   )         ]         -         α   l   2     ⁢     E   k         N   i                             (   9   )                                     (   10   )                           
where K is ignored due to its independence with k.
 
     Example 2 
     Known Phases and Delays, Unknown Amplitudes; Non-Coherent Detection 
     When the amplitudes of the received data signal are unknown, then the conditional probability of equation (6), must be averaged over their Joint Probability Density Function (JPDF) to arrive at the decision metric. Assuming independent amplitudes with first-order PDFs {pα l (α l )} l=1   Lp  we obtain: equation (11) as follows: 
     
       
         
           
             
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     According to the invention, the PDF of Suzuki distribution is expressed mathematically as: 
                       p     a   l       ⁡     (     a   l     )       =         (     5   2     )       1     8   ⁢     σ   2           ⁢         a   l     ⁢     ⅇ     -       μ   2       2   ⁢     σ   2                 σ   ⁢       2   ⁢   n           ⁢     Γ   ⁡     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )       ⁢       (       -     1     8   ⁢     σ   2           -       a   l   2     2       )         -   1     +     1     8   ⁢     σ   2         +     μ     2   ⁢     σ   2                       (   12   )               
where,
 
σ=the variance of Lognormal distribution
 
μ=mean value of lognormal distribution
 
     Therefore, by inserting equation (12) into equation (11) we obtain analytically: 
     
       
         
           
             
               
                 
                   
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                                 - 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   8 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 μ 
                                 
                                   2 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               a 
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The integral in (13) can not be solved in closed-form as currently expressed, mainly due to the summation term under the power term. By assuming that 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             
                               8 
                               ⁢ 
                               
                                 σ 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               a 
                               l 
                               2 
                             
                             2 
                           
                         
                         ) 
                       
                       ≈ 
                       
                         a 
                         l 
                         2 
                       
                     
                     , 
                     
                       0 
                       ≤ 
                       
                         α 
                         l 
                       
                       &lt; 
                       1 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           1 
                           
                             8 
                             ⁢ 
                             
                               σ 
                               2 
                             
                           
                         
                         + 
                         
                           
                             a 
                             l 
                             2 
                           
                           2 
                         
                       
                       ) 
                     
                     ≈ 
                     
                       
                         a 
                         l 
                         2 
                       
                       2 
                     
                   
                   , 
                   
                     
                       a 
                       l 
                     
                     ≥ 
                     1 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Equation (13) can become respectively, 
     
       
         
           
             
               
                 
                   
                     p 
                     ( 
                     
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         | 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       , 
                       
                         
                           { 
                           
                             θ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                       , 
                       
                         
                           { 
                           
                             τ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                     
                     ) 
                   
                   = 
                   
                     K 
                     ⁢ 
                     
                       
                         ∏ 
                         
                           l 
                           = 
                           1 
                         
                         
                           L 
                           p 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   
                                     Re 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           
                                             a 
                                             l 
                                           
                                           
                                             N 
                                             l 
                                           
                                         
                                         ⁢ 
                                         
                                           ⅇ 
                                           
                                             - 
                                             
                                               jθ 
                                               l 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             y 
                                             kl 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         α 
                                         l 
                                         2 
                                       
                                       ⁢ 
                                       
                                         E 
                                         k 
                                       
                                     
                                     
                                       N 
                                       i 
                                     
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   5 
                                   2 
                                 
                                 ) 
                               
                             
                             
                               1 
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 a 
                                 l 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   - 
                                   
                                     
                                       μ 
                                       2 
                                     
                                     
                                       2 
                                       ⁢ 
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             
                               σ 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             Γ 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   1 
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   μ 
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 - 
                                 
                                   a 
                                   l 
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   8 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 μ 
                                 
                                   2 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               a 
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     p 
                     ( 
                     
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         | 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       , 
                       
                         
                           { 
                           
                             θ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                       , 
                       
                         
                           { 
                           
                             τ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                     
                     ) 
                   
                   = 
                   
                     K 
                     ⁢ 
                     
                       
                         ∏ 
                         
                           l 
                           = 
                           1 
                         
                         
                           L 
                           p 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   
                                     Re 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           
                                             a 
                                             l 
                                           
                                           
                                             N 
                                             l 
                                           
                                         
                                         ⁢ 
                                         
                                           ⅇ 
                                           
                                             - 
                                             
                                               jθ 
                                               l 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             y 
                                             kl 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         α 
                                         l 
                                         2 
                                       
                                       ⁢ 
                                       
                                         E 
                                         k 
                                       
                                     
                                     
                                       N 
                                       i 
                                     
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   5 
                                   2 
                                 
                                 ) 
                               
                             
                             
                               1 
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 a 
                                 l 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   - 
                                   
                                     
                                       μ 
                                       2 
                                     
                                     
                                       2 
                                       ⁢ 
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             
                               σ 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             Γ 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   1 
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   μ 
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     a 
                                     l 
                                     2 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   8 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 μ 
                                 
                                   2 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               a 
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The closed-form solution of each of the above two equations are given below as follows 
     
       
         
           
             
               
                 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               { 
                               
                                 
                                   r 
                                   l 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               } 
                             
                             
                               l 
                               = 
                               1 
                             
                             Lp 
                           
                           | 
                           
                             
                               s 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         , 
                         
                           
                             { 
                             
                               θ 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         , 
                         
                           
                             { 
                             
                               τ 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 5 
                                 2 
                               
                               ) 
                             
                             
                               1 
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     μ 
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               σ 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 Γ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       1 
                                       
                                         8 
                                         ⁢ 
                                         
                                           σ 
                                           2 
                                         
                                       
                                     
                                     - 
                                     
                                       μ 
                                       
                                         2 
                                         ⁢ 
                                         
                                           σ 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     N 
                                     l 
                                   
                                   
                                     E 
                                     k 
                                   
                                 
                                 ] 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                               + 
                               
                                 
                                   1 
                                   + 
                                   
                                     4 
                                     ⁢ 
                                     μ 
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             Γ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     1 
                                     + 
                                     
                                       4 
                                       ⁢ 
                                       μ 
                                     
                                   
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             * 
                             1 
                           
                           ⁢ 
                           
                             
                               F 
                               1 
                             
                             ( 
                             
                               
                                 
                                   
                                     1 
                                     + 
                                     
                                       4 
                                       ⁢ 
                                       μ 
                                     
                                   
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 
                                   
                                     ⅇ 
                                     
                                       - 
                                       
                                         j2θ 
                                         l 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       y 
                                       kl 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                   ⁢ 
                                   
                                     N 
                                     l 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   jθ 
                                   l 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 y 
                                 kl 
                               
                               ⁡ 
                               
                                 ( 
                                 T 
                                 ) 
                               
                             
                           
                           
                             N 
                             l 
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         
                           
                             N 
                             l 
                           
                           
                             E 
                             k 
                           
                         
                       
                       ⁢ 
                       
                         
                           Γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 1 
                                 + 
                                 
                                   4 
                                   ⁢ 
                                   μ 
                                 
                               
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         1 
                       
                       ⁢ 
                       
                         
                           F 
                           1 
                         
                         ( 
                         
                           
                             
                               1 
                               + 
                               
                                 4 
                                 ⁢ 
                                 μ 
                               
                             
                             
                               8 
                               ⁢ 
                               
                                 σ 
                                 2 
                               
                             
                           
                           , 
                           
                             3 
                             2 
                           
                           , 
                           
                             
                               
                                 e 
                                 
                                   - 
                                   
                                     j2θ 
                                     l 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   y 
                                   ki 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   T 
                                   ) 
                                 
                               
                             
                             
                               4 
                               ⁢ 
                               
                                 E 
                                 k 
                               
                               ⁢ 
                               
                                 N 
                                 l 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     p 
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               { 
                               
                                 
                                   r 
                                   l 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               } 
                             
                             
                               l 
                               = 
                               1 
                             
                             Lp 
                           
                           | 
                           
                             
                               s 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         , 
                         
                           
                             { 
                             
                               θ 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         , 
                         
                           
                             { 
                             
                               τ 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 5 
                                 2 
                               
                               ) 
                             
                             
                               1 
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     μ 
                                     2 
                                   
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               σ 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 Γ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       1 
                                       
                                         8 
                                         ⁢ 
                                         
                                           σ 
                                           2 
                                         
                                       
                                     
                                     - 
                                     
                                       μ 
                                       
                                         2 
                                         ⁢ 
                                         
                                           σ 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     N 
                                     l 
                                   
                                   
                                     E 
                                     k 
                                   
                                 
                                 ] 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                               + 
                               
                                 
                                   1 
                                   + 
                                   
                                     4 
                                     ⁢ 
                                     μ 
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             Γ 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     1 
                                     + 
                                     
                                       4 
                                       ⁢ 
                                       μ 
                                     
                                   
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                               ) 
                             
                           
                           * 
                           
                             
                               2 
                               
                                 1 
                                 - 
                                 
                                   1 
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   μ 
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                             
                             1 
                           
                           ⁢ 
                           
                             
                               F 
                               1 
                             
                             ( 
                             
                               
                                 
                                   
                                     1 
                                     + 
                                     
                                       4 
                                       ⁢ 
                                       μ 
                                     
                                   
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   1 
                                   2 
                                 
                               
                               , 
                               
                                 1 
                                 2 
                               
                               , 
                               
                                 
                                   
                                     ⅇ 
                                     
                                       - 
                                       
                                         j2θ 
                                         l 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       y 
                                       kl 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 
                                   4 
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                   ⁢ 
                                   
                                     N 
                                     l 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   jθ 
                                   l 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 y 
                                 kl 
                               
                               ⁡ 
                               
                                 ( 
                                 T 
                                 ) 
                               
                             
                           
                           
                             N 
                             l 
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         
                           
                             N 
                             l 
                           
                           
                             E 
                             k 
                           
                         
                       
                       ⁢ 
                       
                         
                           Γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 1 
                                 + 
                                 
                                   4 
                                   ⁢ 
                                   μ 
                                 
                               
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         1 
                       
                       ⁢ 
                       
                         
                           F 
                           1 
                         
                         ( 
                         
                           
                             
                               1 
                               + 
                               
                                 4 
                                 ⁢ 
                                 μ 
                               
                             
                             
                               8 
                               ⁢ 
                               
                                 σ 
                                 2 
                               
                             
                           
                           , 
                           
                             3 
                             2 
                           
                           , 
                           
                             
                               
                                 e 
                                 
                                   - 
                                   
                                     j2θ 
                                     l 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   y 
                                   kl 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   T 
                                   ) 
                                 
                               
                             
                             
                               4 
                               ⁢ 
                               
                                 E 
                                 k 
                               
                               ⁢ 
                               
                                 N 
                                 l 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Having managed to obtain (13) in a closed-form, the natural logarithm of it is taken in order to derive the decision metric of a receiver that has knowledge of the delays and phases of a Suzuki fading channel but not amplitudes. Therefore, one obtains:
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{θ l } l=1   Lp ,{τ l } l=1   Lp )]  (20)
 
     Based on that decision metric the receiver apparatus configuration design illustrated in  FIG. 1  can be implemented for the case of known delay and phase values but unknown amplitude values for the data signal to be received. This design contains a block that theoretically includes the modulation schemes that can be possibly used. It also contains three different shaping pulses. Furthermore, non-coherent detection as well as uniform PDF are assumed. 
     Example 3 
     Known Data Signal Phase Value, Unknown Data Signal Delay and Amplitude Values; Non-Coherent Detection 
     After deriving successfully the decision metric for the case of known phases and delays but unknown amplitudes, we can derive next the decision metric for the case of known phases but unknown delays and amplitudes. However, it should be mentioned here that in typical practical applications, the fact that the phases are known while the delays are unknown is considered unorthodox. However, in quite narrowband systems (like OFDM) recent developments can make the realisation of such schemes possible. This can actually happen by determining the delays by the phase difference. Therefore, this design will be implemented since such case may attract more attention in future applications. 
     It is obvious that the difference between the current and the previous scheme is the lack of knowledge of the path delays. Hence, one has to average over the statistics of both amplitude and time, i.e, p α     l   (α 1 ) and p τ     l   (τ 1 ). 
     The former is expressed by equation (12) while the latter is expressed by Alouini—initially Charash—as: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             p 
                             
                               τ 
                               l 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               τ 
                               l 
                             
                             ) 
                           
                         
                         = 
                         
                           1 
                           
                             B 
                             - 
                             A 
                           
                         
                       
                       , 
                       
                         A 
                         &lt; 
                         
                           τ 
                           l 
                         
                         &lt; 
                         B 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         p 
                         ( 
                         
                           
                             
                               
                                 { 
                                 
                                   
                                     r 
                                     l 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 } 
                               
                               
                                 l 
                                 = 
                                 1 
                               
                               Lp 
                             
                             | 
                             
                               
                                 s 
                                 k 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           , 
                           
                             
                               { 
                               
                                 θ 
                                 l 
                               
                               } 
                             
                             
                               l 
                               = 
                               1 
                             
                             Lp 
                           
                         
                         ) 
                       
                       = 
                       
                         
                           K 
                           ⁢ 
                           
                             
                               ∏ 
                               
                                 l 
                                 = 
                                 1 
                               
                               
                                 L 
                                 p 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 A 
                                 B 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   ∞ 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅇ 
                                     
                                       
                                         Re 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               
                                                 a 
                                                 l 
                                               
                                               
                                                 N 
                                                 l 
                                               
                                             
                                             ⁢ 
                                             
                                               ⅇ 
                                               
                                                 - 
                                                 
                                                   jθ 
                                                   l 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               
                                                 y 
                                                 kl 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 τ 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             α 
                                             l 
                                             2 
                                           
                                           ⁢ 
                                           
                                             E 
                                             k 
                                           
                                         
                                         
                                           N 
                                           i 
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       p 
                                       
                                         α 
                                         l 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         α 
                                         l 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       p 
                                       
                                         τ 
                                         l 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         τ 
                                         l 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     
                                       α 
                                       l 
                                     
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     
                                       τ 
                                       l 
                                     
                                   
                                 
                               
                             
                           
                         
                         = 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Thus, the decision metric is expressed mathematically as follows: 
                       p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     θ   l     }       l   =   1     Lp       )       =     K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   A   B     ⁢       p   (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     θ   l     }       l   =   1     Lp     ,       {     τ   l     }       l   =   1     Lp       )     ⁢     ⅆ     τ   l                 ⁢     
     ⁢     Λ   k     =       ⁢       ln   ⁡     [     p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     θ   l     }       l   =   1     Lp       )       ]       =     ln   ⁢           [     K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   A   B     ⁢       p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     θ   l     }       l   =   1     Lp     ,       {     τ   l     }       l   =   1     Lp       )       ⁢     ⅆ     τ   l               ]               (   22   )               
where the term
 
             1     B   -   A           
from p τ     l   (τ l ) is absorbed within the constant ‘K’.
 
     Therefore, it is straightforward to obtain the closed form expression for the decision metric: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Λ 
                             k 
                           
                           = 
                             
                           ⁢ 
                           
                             ln 
                             ⁡ 
                             
                               [ 
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           { 
                                           
                                             
                                               r 
                                               l 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               t 
                                               ) 
                                             
                                           
                                           } 
                                         
                                         
                                           l 
                                           = 
                                           1 
                                         
                                         Lp 
                                       
                                       | 
                                       
                                         
                                           s 
                                           k 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     , 
                                     
                                       
                                         { 
                                         
                                           θ 
                                           l 
                                         
                                         } 
                                       
                                       
                                         l 
                                         = 
                                         1 
                                       
                                       Lp 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             ln 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         K 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             B 
                                             - 
                                             A 
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           ∏ 
                                           
                                             l 
                                             = 
                                             1 
                                           
                                           
                                             L 
                                             p 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               ( 
                                               
                                                 5 
                                                 2 
                                               
                                               ) 
                                             
                                             
                                               1 
                                               
                                                 8 
                                                 ⁢ 
                                                 
                                                   σ 
                                                   2 
                                                 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               ⅇ 
                                               
                                                 - 
                                                 
                                                   
                                                     μ 
                                                     2 
                                                   
                                                   
                                                     2 
                                                     ⁢ 
                                                     
                                                       σ 
                                                       2 
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               2 
                                               ⁢ 
                                               σ 
                                               ⁢ 
                                               
                                                 
                                                   2 
                                                   ⁢ 
                                                   n 
                                                 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               
                                                 Γ 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       1 
                                                       
                                                         8 
                                                         ⁢ 
                                                         
                                                           σ 
                                                           2 
                                                         
                                                       
                                                     
                                                     - 
                                                     
                                                       μ 
                                                       
                                                         2 
                                                         ⁢ 
                                                         
                                                           σ 
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁡ 
                                               
                                                 [ 
                                                 
                                                   
                                                     N 
                                                     l 
                                                   
                                                   
                                                     E 
                                                     k 
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             
                                               
                                                 - 
                                                 1 
                                               
                                               + 
                                               
                                                 
                                                   1 
                                                   + 
                                                   
                                                     4 
                                                     ⁢ 
                                                     μ 
                                                   
                                                 
                                                 
                                                   4 
                                                   ⁢ 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             Γ 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       1 
                                                       + 
                                                       
                                                         4 
                                                         ⁢ 
                                                         μ 
                                                       
                                                     
                                                     
                                                       8 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                   -- 
                                                 
                                                 ⁢ 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 F 
                                                 1 
                                                   
                                                   
                                                 1 
                                                   
                                               
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       1 
                                                       + 
                                                       
                                                         4 
                                                         ⁢ 
                                                         μ 
                                                       
                                                     
                                                     
                                                       8 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 , 
                                                 
                                                   1 
                                                   2 
                                                 
                                                 , 
                                                 
                                                   
                                                     
                                                       ⅇ 
                                                       
                                                         - 
                                                         
                                                           j2θ 
                                                           l 
                                                         
                                                       
                                                     
                                                     ⁢ 
                                                     
                                                       
                                                         y 
                                                         kl 
                                                         2 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         T 
                                                         ) 
                                                       
                                                     
                                                   
                                                   
                                                     4 
                                                     ⁢ 
                                                     
                                                       E 
                                                       k 
                                                     
                                                     ⁢ 
                                                     
                                                       N 
                                                       l 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     ⅇ 
                                                     
                                                       - 
                                                       
                                                         jθ 
                                                         l 
                                                       
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     
                                                       y 
                                                       kl 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       T 
                                                       ) 
                                                     
                                                   
                                                 
                                                 
                                                   N 
                                                   l 
                                                 
                                               
                                               ] 
                                             
                                             ⁢ 
                                             
                                               
                                                 
                                                   N 
                                                   l 
                                                 
                                                 
                                                   E 
                                                   k 
                                                 
                                               
                                             
                                             ⁢ 
                                             
                                               
                                                 Γ 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       1 
                                                       + 
                                                       
                                                         4 
                                                         ⁢ 
                                                         μ 
                                                       
                                                     
                                                     
                                                       8 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               1 
                                             
                                             ⁢ 
                                             
                                               
                                                 F 
                                                 1 
                                               
                                               ( 
                                               
                                                 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       4 
                                                       ⁢ 
                                                       μ 
                                                     
                                                   
                                                   
                                                     8 
                                                     ⁢ 
                                                     
                                                       σ 
                                                       2 
                                                     
                                                   
                                                 
                                                 , 
                                                 
                                                   3 
                                                   2 
                                                 
                                                 , 
                                                 
                                                   
                                                     
                                                       e 
                                                       
                                                         - 
                                                         
                                                           j2θ 
                                                           l 
                                                         
                                                       
                                                     
                                                     ⁢ 
                                                     
                                                       
                                                         y 
                                                         ki 
                                                         2 
                                                       
                                                       ⁡ 
                                                       
                                                         ( 
                                                         T 
                                                         ) 
                                                       
                                                     
                                                   
                                                   
                                                     4 
                                                     ⁢ 
                                                     
                                                       E 
                                                       k 
                                                     
                                                     ⁢ 
                                                     
                                                       N 
                                                       l 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Λ 
                           k 
                         
                         = 
                           
                         ⁢ 
                         
                           ln 
                           ⁡ 
                           
                             [ 
                             
                               p 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         { 
                                         
                                           
                                             r 
                                             l 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             t 
                                             ) 
                                           
                                         
                                         } 
                                       
                                       
                                         l 
                                         = 
                                         1 
                                       
                                       Lp 
                                     
                                     | 
                                     
                                       
                                         s 
                                         k 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       { 
                                       
                                         θ 
                                         l 
                                       
                                       } 
                                     
                                     
                                       l 
                                       = 
                                       1 
                                     
                                     Lp 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       K 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           B 
                                           - 
                                           A 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ∏ 
                                         
                                           l 
                                           = 
                                           1 
                                         
                                         
                                           L 
                                           p 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             ( 
                                             
                                               5 
                                               2 
                                             
                                             ) 
                                           
                                           
                                             1 
                                             
                                               8 
                                               ⁢ 
                                               
                                                 σ 
                                                 2 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             ⅇ 
                                             
                                               - 
                                               
                                                 
                                                   μ 
                                                   2 
                                                 
                                                 
                                                   2 
                                                   ⁢ 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             2 
                                             ⁢ 
                                             σ 
                                             ⁢ 
                                             
                                               
                                                 2 
                                                 ⁢ 
                                                 n 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               Γ 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   1 
                                                   - 
                                                   
                                                     1 
                                                     
                                                       8 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     μ 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁡ 
                                             
                                               [ 
                                               
                                                 
                                                   N 
                                                   l 
                                                 
                                                 
                                                   E 
                                                   k 
                                                 
                                               
                                               ] 
                                             
                                           
                                           
                                             
                                               - 
                                               1 
                                             
                                             + 
                                             
                                               
                                                 1 
                                                 + 
                                                 
                                                   4 
                                                   ⁢ 
                                                   μ 
                                                 
                                               
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   σ 
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Γ 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       4 
                                                       ⁢ 
                                                       μ 
                                                     
                                                   
                                                   
                                                     8 
                                                     ⁢ 
                                                     
                                                       σ 
                                                       2 
                                                     
                                                   
                                                 
                                                 -- 
                                               
                                               ⁢ 
                                               
                                                 1 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                         * 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 2 
                                                 
                                                   1 
                                                   - 
                                                   
                                                     1 
                                                     
                                                       8 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     μ 
                                                     
                                                       2 
                                                       ⁢ 
                                                       
                                                         σ 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                         
                                                     
                                                     1 
                                                   
                                                   ⁢ 
                                                   
                                                     
                                                       F 
                                                       1 
                                                     
                                                     ( 
                                                     
                                                       
                                                         
                                                           
                                                             1 
                                                             + 
                                                             
                                                             4 
                                                             ⁢ 
                                                             μ 
                                                             
                                                           
                                                           
                                                             8 
                                                             ⁢ 
                                                             
                                                             σ 
                                                             2 
                                                             
                                                           
                                                         
                                                         - 
                                                         
                                                           1 
                                                           2 
                                                         
                                                       
                                                       , 
                                                       
                                                         1 
                                                         2 
                                                       
                                                       , 
                                                       
                                                         
                                                           
                                                             ⅇ 
                                                             
                                                             - 
                                                             
                                                             j2θ 
                                                             l 
                                                             
                                                             
                                                           
                                                           ⁢ 
                                                           
                                                             
                                                             y 
                                                             kl 
                                                             2 
                                                             
                                                             ⁡ 
                                                             
                                                             ( 
                                                             T 
                                                             ) 
                                                             
                                                           
                                                         
                                                         
                                                           4 
                                                           ⁢ 
                                                           
                                                             E 
                                                             k 
                                                           
                                                           ⁢ 
                                                           
                                                             N 
                                                             l 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             [ 
                                             
                                               
                                                 
                                                   ⅇ 
                                                   
                                                     - 
                                                     
                                                       jθ 
                                                       l 
                                                     
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   
                                                     y 
                                                     kl 
                                                   
                                                   ⁡ 
                                                   
                                                     ( 
                                                     T 
                                                     ) 
                                                   
                                                 
                                               
                                               
                                                 N 
                                                 l 
                                               
                                             
                                             ] 
                                           
                                           ⁢ 
                                           
                                             
                                               
                                                 N 
                                                 l 
                                               
                                               
                                                 E 
                                                 k 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               Γ 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     1 
                                                     + 
                                                     
                                                       4 
                                                       ⁢ 
                                                       μ 
                                                     
                                                   
                                                   
                                                     8 
                                                     ⁢ 
                                                     
                                                       σ 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             1 
                                           
                                           ⁢ 
                                           
                                             
                                               F 
                                               1 
                                             
                                             ( 
                                             
                                               
                                                 
                                                   1 
                                                   + 
                                                   
                                                     4 
                                                     ⁢ 
                                                     μ 
                                                   
                                                 
                                                 
                                                   8 
                                                   ⁢ 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                 
                                               
                                               , 
                                               
                                                 3 
                                                 2 
                                               
                                               , 
                                               
                                                 
                                                   
                                                     e 
                                                     
                                                       - 
                                                       
                                                         j2θ 
                                                         l 
                                                       
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     
                                                       y 
                                                       ki 
                                                       2 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       T 
                                                       ) 
                                                     
                                                   
                                                 
                                                 
                                                   4 
                                                   ⁢ 
                                                   
                                                     E 
                                                     k 
                                                   
                                                   ⁢ 
                                                   
                                                     N 
                                                     l 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Based on the decision metric expressed by equations (23) and (24), the receiver design illustrated in  FIG. 2  can be implemented for the case of known phases but unknown delays and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with the shaping pulses. Furthermore, non-coherent detection as well as uniform power decay profile (PDP) have been assumed. 
     Example 4 
     Known Delay and Amplitude Values, Unknown Phase Value; Non-Coherent Detection 
     In the case of known delay and amplitude values but unknown phase value the scheme is independent of the novel expression for the PDF of Suzuki distribution. However, the analysis of this scheme is vital since the derived results are used in the derivation process of any scheme that the phases are considered unknown. 
     When the phases are unknown, then the conditional probability p({r l (t)} l=1   Lp |s k (t),{α l } l=1   Lp ,{τ l } l=1   Lp ) has to be averaged over their joint probability density function (JPDF) in order to arrive at the decision metric. Assuming independent phases with PDFs over the interval {0, 2π}, one obtains: 
     
       
         
           
             
               
                 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               { 
                               
                                 
                                   r 
                                   l 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               } 
                             
                             
                               l 
                               = 
                               1 
                             
                             Lp 
                           
                           | 
                           
                             
                               s 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         , 
                         
                           
                             { 
                             
                               α 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         , 
                         
                           
                             { 
                             
                               τ 
                               l 
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   Re 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         
                                           a 
                                           l 
                                         
                                         
                                           N 
                                           l 
                                         
                                       
                                       ⁢ 
                                       
                                         ⅇ 
                                         
                                           - 
                                           
                                             jθ 
                                             l 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           y 
                                           kl 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           τ 
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       α 
                                       l 
                                       2 
                                     
                                     ⁢ 
                                     
                                       E 
                                       k 
                                     
                                   
                                   
                                     N 
                                     i 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               
                                 p 
                                 
                                   θ 
                                   l 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   θ 
                                   l 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               
                                 θ 
                                 l 
                               
                             
                           
                         
                       
                     
                     = 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     α 
                                     l 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                 
                                 
                                   N 
                                   i 
                                 
                               
                             
                           
                           ⁢ 
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ⅇ 
                                 
                                   Re 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         
                                           a 
                                           l 
                                         
                                         
                                           N 
                                           l 
                                         
                                       
                                       ⁢ 
                                       
                                         ⅇ 
                                         
                                           - 
                                           
                                             jθ 
                                             l 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           y 
                                           kl 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           τ 
                                           ) 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 
                                   θ 
                                   l 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     α 
                                     l 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                 
                                 
                                   N 
                                   i 
                                 
                               
                             
                           
                           ⁢ 
                           
                             1 
                             
                               2 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 ⅇ 
                                 
                                   Re 
                                   [ 
                                   
                                     
                                       
                                         a 
                                         l 
                                       
                                       
                                         N 
                                         l 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         y 
                                         kl 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         T 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                        
                                       
                                         cos 
                                         ( 
                                         
                                           
                                             θ 
                                             l 
                                           
                                           - 
                                           
                                             arg 
                                             ( 
                                             
                                               
                                                 y 
                                                 kl 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 T 
                                                 ) 
                                               
                                             
                                             ] 
                                           
                                         
                                          
                                       
                                       ] 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 
                                   θ 
                                   l 
                                 
                               
                             
                           
                         
                       
                     
                     ⇒ 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     ( 
                     
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         | 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       , 
                       
                         
                           { 
                           
                             α 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                       , 
                       
                         
                           { 
                           
                             τ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                     
                     ) 
                   
                   = 
                   
                     
                       K 
                       ⁢ 
                       
                         
                           ∏ 
                           
                             l 
                             = 
                             1 
                           
                           
                             L 
                             p 
                           
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     α 
                                     l 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                 
                                 
                                   N 
                                   i 
                                 
                               
                             
                           
                           ⁢ 
                           
                             I 
                             o 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   a 
                                   l 
                                 
                                 
                                   N 
                                   l 
                                 
                               
                               | 
                               
                                 
                                   y 
                                   kl 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     τ 
                                     l 
                                   
                                   ) 
                                 
                               
                               | 
                             
                             ) 
                           
                         
                       
                     
                     ⇒ 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Λ 
                       
                         k 
                         = 
                       
                     
                     ⁢ 
                     
                       ln 
                       ⁡ 
                       
                         [ 
                         
                           p 
                           ( 
                           
                             
                               
                                 
                                   { 
                                   
                                     
                                       r 
                                       l 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   } 
                                 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 Lp 
                               
                               | 
                               
                                 
                                   s 
                                   k 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             , 
                             
                               
                                 { 
                                 
                                   α 
                                   l 
                                 
                                 } 
                               
                               
                                 l 
                                 = 
                                 1 
                               
                               Lp 
                             
                             , 
                             
                               
                                 { 
                                 
                                   τ 
                                   l 
                                 
                                 } 
                               
                               
                                 l 
                                 = 
                                 1 
                               
                               Lp 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                   = 
                   
                     K 
                     ⁢ 
                     
                       
                         ∏ 
                         
                           l 
                           = 
                           1 
                         
                         
                           L 
                           p 
                         
                       
                       ⁢ 
                       
                         ln 
                         [ 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     α 
                                     l 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                 
                                 
                                   N 
                                   i 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               I 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     l 
                                   
                                   
                                     N 
                                     l 
                                   
                                 
                                 | 
                                 
                                   
                                     y 
                                     kl 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       τ 
                                       l 
                                     
                                     ) 
                                   
                                 
                                 | 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     Example 5 
     Known Delay Values Unknown Phase and Amplitude Values for a Received Data Signal; Non-Coherent Detection 
     The case of known delay values but unknown phase and amplitude values is perhaps the most critical design to be considered for wireless communication engineering applications and the appropriate configuration of the receiving apparatus for the data signals, and can be derived from the closed form solution for the decision metric Of the case of known delays and amplitude values but unknown phase values. 
     The equation (28) is averaged over the PDF of Suzuki distribution p α   l (α l ). In other words, the mathematical expression of the decision metric of this scheme is the following: 
                     p   ⁢     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     τ   l     }       l   =   1     Lp       )       =       K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   0   ∞     ⁢       ∫   0     2   ⁢   π       ⁢       ⅇ       Re   ⁡     [         a   l       N   l       ⁢     ⅇ     -     jθ   l         ⁢       y   kl     ⁡     (   τ   )         ]       -         α   l   2     ⁢     E   k         N   i           ⁢       p     θ   l       ⁡     (     θ   l     )       ⁢       p     α   l       ⁡     (     α   l     )       ⁢     ⅆ     θ   l       ⁢     ⅆ     α   l                 =             (   30   )                 p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     τ   l     }       l   =   1     Lp       )       =       K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   0   ∞     ⁢       p   ⁡     (                 {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁢     (   t   )         ,                   {     α   l     }       l   =   1     Lp     ,       {     τ   l     }       l   =   1     Lp             )       ⁢       p     α   l       ⁡     (     α   l     )       ⁢     ⅆ     α   l               =             (   31   )                 p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     |       s   k     ⁡     (   t   )         ,       {     τ   l     }       l   =   1     Lp       )       =       K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   0   ∞     ⁢       ⅇ     -         α   l   2     ⁢     E   k         N   i           ⁢       I   o     ⁡     (         a   l       N   l       |       y   kl     ⁡     (     τ   l     )       |     )       ⁢       p     α   l       ⁡     (     α   l     )       ⁢     ⅆ     α   l               =             (   32   )               
and by inserting the equation (12) into equation (32) we obtain (33) as follows:
 
     
       
         
           
             
               
                 
                   
                     p 
                     ( 
                     
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         | 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       , 
                       
                         
                           { 
                           
                             τ 
                             l 
                           
                           } 
                         
                         
                           l 
                           = 
                           1 
                         
                         Lp 
                       
                     
                     ) 
                   
                   = 
                   
                     K 
                     ⁢ 
                     
                       
                         ∏ 
                         
                           l 
                           = 
                           1 
                         
                         
                           L 
                           p 
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     α 
                                     l 
                                     2 
                                   
                                   ⁢ 
                                   
                                     E 
                                     k 
                                   
                                 
                                 
                                   N 
                                   i 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               I 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     a 
                                     l 
                                   
                                   
                                     N 
                                     l 
                                   
                                 
                                 | 
                                 
                                   
                                     y 
                                     kl 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       τ 
                                       l 
                                     
                                     ) 
                                   
                                 
                                 | 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 5 
                                 2 
                               
                               ) 
                             
                             
                               1 
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 a 
                                 l 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   - 
                                   
                                     
                                       μ 
                                       2 
                                     
                                     
                                       2 
                                       ⁢ 
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             
                               σ 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             Γ 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   1 
                                   
                                     8 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                                 - 
                                 
                                   μ 
                                   
                                     2 
                                     ⁢ 
                                     
                                       σ 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       8 
                                       ⁢ 
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     · 
                                     
                                       a 
                                       l 
                                       2 
                                     
                                   
                                   2 
                                 
                               
                               ) 
                             
                             
                               
                                 - 
                                 1 
                               
                               + 
                               
                                 1 
                                 
                                   8 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 μ 
                                 
                                   2 
                                   ⁢ 
                                   
                                     σ 
                                     2 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             
                               a 
                               l 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Equation (33) cannot be solved in closed-form as currently expressed. The additional factor of complexity in this case is the modified Bessel function. 
     In order to simplify the summation term under the power term of Suzuki PDF, two assumptions were made based on the value of α 1 . The range of values of each of these two assumptions quite coincide with the range of values of the asymptotes of the modified Bessel function. In more detail, the modified Bessel function can be also expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       b 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                 
                                   Γ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       β 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     x 
                                     2 
                                   
                                   ) 
                                 
                                 b 
                               
                             
                             , 
                           
                         
                         
                           
                             0 
                             &lt; 
                             x 
                             ⪡ 
                             
                               
                                 b 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     x 
                                   
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 x 
                               
                             
                             , 
                           
                         
                         
                           
                             x 
                             ⪢ 
                             
                                
                               
                                 
                                   b 
                                   2 
                                 
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Therefore, by taking into consideration equation (34) as well as the assumptions in (14) and (15), we obtain respectively, 
                       p   ⁡     (           {     r   l     (   t   )       }       l   =   1     Lp     ❘       s   k     ⁡     (   t   )         ,       {     τ   l     }       l   =   1     Lp       )       =       K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   0   ∞     ⁢           ⅇ     -         a   l   2     ⁢     E   k         N   l           ⁡     (     5   2     )         1     8   ⁢     σ   2           ⁢         a   l     ⁢     ⅇ     -       μ   2       2   ⁢     σ   2                 σ   ⁢       2   ⁢   π           ⁢     Γ   ⁡     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )       ⁢       (     a   l   2     )         -   1     +     1     8   ⁢     σ   2         +     μ     2   ⁢     σ   2             ⁢     ⅆ     α   l               =     
     ⁢           ⁢   and       ⁢     
     ⁢     K   ⁢       ∏     l   =   1       L   p       ⁢       ∫   0   ∞     ⁢           ⅇ     -         a   l   2     ⁢     E   k         N   l           ⁡     (     5   2     )         1     8   ⁢     σ   2           ⁢         a   l     ⁢     ⅇ     -       μ   2       2   ⁢     σ   2                 σ   ⁢       2   ⁢   π           ⁢       ⅇ         a   l       N   l       ⁢            y   kl     ⁡     (     τ   l     )                    2   ⁢   π   ⁢       a   l       N   l       ⁢            y   kl     ⁡     (     τ   l     )                  ⁢   Γ   ⁢     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )     ⁢       (       a   l   2     2     )         -   1     +     1     8   ⁢     σ   2         +     μ     2   ⁢     σ   2             ⁢     ⅆ     α   l                       (   35   )               
which has the following closed-form solutions, i.e.
 
                       p   ⁡     (           {       r   l     ⁡     (   t   )       }       l   =   1     Lp     ❘       s   k     ⁡     (   t   )         ,       {     τ   l     }       l   =   1     Lp       )       =     K   ⁢       ∏     l   =   1       L   p       ⁢         (     5   2     )       1     8   ⁢     σ   2           ⁢       ⅇ     -       μ   2       2   ⁢     σ   2               σ   ⁢       2   ⁢   π           ⁢     Γ   ⁡     (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )       ⁢         1   2     ⁡     [       E   k       N   l       ]         -       1   +     4   ⁢   μ         8   ⁢     σ   2             ⁢     Γ   ⁡     (       1   +     4   ⁢   μ         8   ⁢     σ   2         )               ⁢     
     ⁢           ⁢   and   ⁢     
     ⁢     K   ⁢       ∏     l   =   1       L   p       ⁢         (     5   2     )       1     8   ⁢     σ   2           ⁢       ⅇ     -       μ   2       2   ⁢     σ   2               4   ⁢   σπ   ⁢     2         ⁢   Γ   ⁢         (     1   -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2           )     ⁡     [       E   k       N   l       ]           1   4     -       1   +     4   ⁢   μ         8   ⁢     σ   2             ⁢     2       1   2     -     1     8   ⁢     σ   2         -     μ     2   ⁢     σ   2             ⁢                y   kl     ⁡     (     τ   l     )              N   l         ⁢     (         Γ   ⁡     (         1   +     4   ⁢   μ         8   ⁢     σ   2         -     1   4       )       ⁢       F   1           1         ⁡     (           1   +     4   ⁢   μ         8   ⁢     σ   2         -     1   4       ,     1   2     ,         y   kl   2     ⁡     (   τ   )         4   ⁢     E   k     ⁢     N   l           )         -       [         y   kl     ⁡     (   τ   )             E   k     ⁢     N   l           ]     ⁢     Γ   ⁡     (         1   +     4   ⁢   μ         8   ⁢     σ   2         +     1   4       )       ⁢       F   1           1         ⁡     (           1   +     4   ⁢   μ         8   ⁢     σ   2         +     1   4       ,     3   2     ,         y   kl   2     ⁡     (   τ   )         4   ⁢     E   k     ⁢     N   l           )           )                   (   36   )               
and by taking the natural logarithm of (36) we obtain the decision metric for the case of known delays but unknown amplitudes and phases as follows:
 
Λ k =ln [ p ({ r   l ( t )} l=1   Lp   |s   k ( t ),{τ l } l=1   Lp )]  (37)
 
     Based on that decision metric the receiver design illustrated in  FIG. 3  can be implemented for the case of known delays but unknown phases and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with shaping pulses. Furthermore, coherent detection as well as uniform PDP have been selected in this design. 
     Example 6 
     Unknown Phase, Delay and Amplitude Values for a Received Data Signal; Non-Coherent Detection 
     When all channel parameters are unknown, the conditional probability has to be averaged over the statistics of the amplitudes, phases and delays which all are assumed statistically independent. However, since the decision metric has been derived for the case of known delays but unknown amplitudes and phases, we only need to average this metric over the PDF of time. The same methodology was followed above in the case of known phases but unknown amplitudes and delays. Therefore, the following expression must be solved in closed-form: 
     
       
         
           
             
               
                 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         ❘ 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         A 
                         B 
                       
                       ⁢ 
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     { 
                                     
                                       
                                         r 
                                         l 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     } 
                                   
                                   
                                     l 
                                     = 
                                     1 
                                   
                                   Lp 
                                 
                                 ❘ 
                                 
                                   
                                     s 
                                     k 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 
                                   { 
                                   
                                     τ 
                                     l 
                                   
                                   } 
                                 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 Lp 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             p 
                             
                               τ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               l 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               τ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               l 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           τ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         l 
                       
                     
                     ⇒ 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             { 
                             
                               
                                 r 
                                 l 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             } 
                           
                           
                             l 
                             = 
                             1 
                           
                           Lp 
                         
                         ❘ 
                         
                           
                             s 
                             k 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   { 
                                   
                                     
                                       r 
                                       l 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   } 
                                 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 Lp 
                               
                               ❘ 
                               
                                 
                                   s 
                                   k 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             , 
                             
                               
                                 { 
                                 
                                   τ 
                                   l 
                                 
                                 } 
                               
                               
                                 l 
                                 = 
                                 1 
                               
                               Lp 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           A 
                           B 
                         
                         ⁢ 
                         
                           
                             
                               p 
                               
                                 τ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 l 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 τ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 l 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             τ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           l 
                         
                       
                     
                     ⇒ 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Λ 
                       k 
                     
                     = 
                     
                       
                         ln 
                         ⁡ 
                         
                           [ 
                           
                             p 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     { 
                                     
                                       
                                         r 
                                         l 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     } 
                                   
                                   
                                     l 
                                     = 
                                     1 
                                   
                                   Lp 
                                 
                                 ❘ 
                                 
                                   
                                     s 
                                     k 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           ln 
                           [ 
                           
                             
                               K 
                               ⁡ 
                               
                                 ( 
                                 
                                   B 
                                   - 
                                   A 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∏ 
                                 
                                   l 
                                   = 
                                   1 
                                 
                                 
                                   L 
                                   p 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     ( 
                                     
                                       5 
                                       2 
                                     
                                     ) 
                                   
                                   
                                     1 
                                     
                                       8 
                                       ⁢ 
                                       
                                         σ 
                                         2 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ⅇ 
                                     
                                       - 
                                       
                                         
                                           μ 
                                           2 
                                         
                                         
                                           2 
                                           ⁢ 
                                           
                                             σ 
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                                     ⁢ 
                                     
                                       
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                                         ⁢ 
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                                 ⁢ 
                                 
                                   Γ 
                                   ⁡ 
                                   
                                     ( 
                                     
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                                           ⁢ 
                                           
                                             σ 
                                             2 
                                           
                                         
                                       
                                       - 
                                       
                                         μ 
                                         
                                           2 
                                           ⁢ 
                                           
                                             σ 
                                             2 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             
                               1 
                               2 
                             
                             [ 
                             
                               
                                 E 
                                 k 
                               
                               
                                 N 
                                 l 
                               
                             
                             ] 
                           
                           
                             - 
                             
                               
                                 1 
                                 + 
                                 
                                   4 
                                   ⁢ 
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                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           Γ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 1 
                                 + 
                                 
                                   4 
                                   ⁢ 
                                   μ 
                                 
                               
                               
                                 8 
                                 ⁢ 
                                 
                                   σ 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ln 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   K 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       B 
                                       - 
                                       A 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ∏ 
                                     
                                       l 
                                       = 
                                       1 
                                     
                                     
                                       L 
                                       p 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         ( 
                                         
                                           5 
                                           2 
                                         
                                         ) 
                                       
                                       
                                         1 
                                         
                                           8 
                                           ⁢ 
                                           
                                             σ 
                                             2 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ⅇ 
                                         
                                           - 
                                           
                                             
                                               μ 
                                               2 
                                             
                                             
                                               2 
                                               ⁢ 
                                               
                                                 σ 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                       
                                         4 
                                         ⁢ 
                                         σπ 
                                         ⁢ 
                                         
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           Γ 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               1 
                                               - 
                                               
                                                 1 
                                                 
                                                   8 
                                                   ⁢ 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                 
                                               
                                               - 
                                               
                                                 μ 
                                                 
                                                   2 
                                                   ⁢ 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               E 
                                               k 
                                             
                                             
                                               N 
                                               l 
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         
                                           1 
                                           4 
                                         
                                         - 
                                         
                                           
                                             1 
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                                               4 
                                               ⁢ 
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                                             8 
                                             ⁢ 
                                             
                                               σ 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
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                                           1 
                                           2 
                                         
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                                             ⁢ 
                                             
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                                         - 
                                         
                                           
                                             μ 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 σ 
                                                 2 
                                               
                                             
                                           
                                           ⁢ 
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     y 
                                                     kl 
                                                   
                                                   ⁡ 
                                                   
                                                     ( 
                                                     
                                                       τ 
                                                       l 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                               
                                               
                                                 N 
                                                 l 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
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                                             1 
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                                               ⁢ 
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                                             ⁢ 
                                             
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                                         1 
                                           
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 1 
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                                                 ⁢ 
                                                 
                                                   σ 
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                                             - 
                                             
                                               1 
                                               4 
                                             
                                           
                                           , 
                                           
                                             1 
                                             2 
                                           
                                           , 
                                           
                                             
                                               
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                                                 2 
                                               
                                               ⁡ 
                                               
                                                 ( 
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                                                 ) 
                                               
                                             
                                             
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                                               ⁢ 
                                               
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                                                 l 
                                               
                                             
                                           
                                         
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                                   - 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             y 
                                             kl 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                         
                                           
                                             
                                               E 
                                               k 
                                             
                                             ⁢ 
                                             
                                               N 
                                               l 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     ⁢ 
                                     
                                       Γ 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               1 
                                               + 
                                               
                                                 4 
                                                 ⁢ 
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                                               ⁢ 
                                               
                                                 σ 
                                                 2 
                                               
                                             
                                           
                                           + 
                                           
                                             1 
                                             4 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         F 
                                         1 
                                           
                                           
                                         1 
                                           
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 1 
                                                 + 
                                                 
                                                   4 
                                                   ⁢ 
                                                   μ 
                                                 
                                               
                                               
                                                 8 
                                                 ⁢ 
                                                 
                                                   σ 
                                                   2 
                                                 
                                               
                                             
                                             + 
                                             
                                               1 
                                               4 
                                             
                                           
                                           , 
                                           
                                             3 
                                             2 
                                           
                                           , 
                                           
                                             
                                               
                                                 y 
                                                 kl 
                                                 2 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 τ 
                                                 ) 
                                               
                                             
                                             
                                               4 
                                               ⁢ 
                                               
                                                 E 
                                                 k 
                                               
                                               ⁢ 
                                               
                                                 N 
                                                 l 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     Based on this decision metric we can implement the receiver design as illustrated in  FIG. 4  for the case of unknown delays, phases and amplitudes. As in the above case, this design contains a block that theoretically includes the modulation schemes that can be possibly used together with the shaping pulses. Furthermore, coherent detection as well as uniform PDP have been selected in this design. 
     Power Decay Profile 
     The Power Decay Profile (PDP) critically affects the performance of wireless communications systems and it is always considered as a factor of significant importance during the system design process. In the majority of systems, the PDP is typically considered to be flat i.e., Ω l =Ω; l=1, 2, . . . , L p . The PDP is considered to be of the same nature in all diversity paths between transmitter and receiver. 
     Uniform PDP indicates the uniform rate of variation of the power of the signal while Exponential PDP indicates its exponential variation. The PDP is described mathematically as follows:
 
γ l =γ 1   e   −δ(l−1)   ,l= 1,2 , . . . ,L   p   (41)
 
where γ l  is the signal-to-noise ratio in the lth diversity path—
 
               (     SNR   l     )     =         E   k       N   l       ⁢     -             
and δ is the fading power decay profile factor which typically takes values between zero and two.
 
     In the present case we will consider the values δ=0, 0.1, 0.5, 1.0, 2.0, while it is obvious that for δ=0, the exponential PDP becomes uniform PDP. 
     In addition, the nature of the PDP determines the degree of optimisation of a receiver design. For uniform PDP, the receiver design is considered optimum, while, for non-uniform PDP (exponential in our case), the receiver design is considered suboptimum. 
     Coherent Detection 
     Hereinabove it is assumed that the channel is constant for one symbol period i.e. the detection is considered non-coherent. However, it is well known that coherent detection is always desired ideally. Coherent detection assumes that the channel remains constant for at least two symbol periods. This can be realised mathematically by changing the upper limit of the integral in equation (7) from ‘T s +τ l ’ to ‘2T s +τ l ’. Therefore; only the term ‘y kl ’ changes and varies proportionally the value decision metric. Hence, this small change of ‘T s ’ to ‘2T s ’ is the difference between the optimum/suboptimum receiver designs with non-coherent and coherent detection respectively. 
     Shaping Pulses 
     Using shaping pulses in the transmitter, the data signal which is transmitted can be shaped in a desired waveform which will Make it relatively easier for the receiver to filter the received signals effectively and extract the information included in it. The typical shaping pulses that are used are the square, sinusoidal and cosine. These three pulses are expressed mathematically, respectively, as follows: 
                       g   ⁡     (   t   )       ⁢     ❘   r       =         2   ⁢     E   b         T   b                 (   42   )                   g   ⁡     (   t   )       ⁢     ❘   s       =     {               sin   ⁡     (     π   ⁢           ⁢   τ     )         2   ⁢   T       ,           0   ≤   t   ≤     2   ⁢   T                 0   ,         otherwise         }             (   43   )                   g   ⁡     (   t   )       ⁢     ❘   c       =     {               1     2   ⁢   L   ⁢           ⁢   T       ⁡     [     1   -       cos   ⁡     (     2   ⁢   πτ     )         L   ⁢           ⁢   T         ]       ,           0   ≤   t   ≤     2   ⁢   T                 0   ,         otherwise         }             (   44   )               
where L=1, for full response and L&gt;1 for partial response, result in smaller bandwidth efficiency, hence, greater bandwidth efficiency than the use of rectangular pulses.
 
Digital Modulation Schemes
 
     Digital modulation techniques can be an important factor in the transmission of digital data in order to achieve the highest possible data transmission rates. The higher the order of the modulation, the higher the spectral efficiency i.e. the higher the number of bits per transmitted symbol. However, in high order modulation schemes the complexity, and hence, the cost increase dramatically. In addition, the information symbols are located very close to each other which results in inter symbol interference (ISO. This increases the error rate and raises the need for more sophisticated, complex and costly receiver designs. Therefore, the choice of the appropriate modulation scheme for each system is relative to the performance requirements of it. Six types of digital modulations, namely, M-PSK, M-QAM, ASK/PAM, FSK, MSK and OFDM can be expressed mathematically as follows: 
                     M   ⁢     -     ⁢   PSK     ⁢                                           S   k     ⁡     (   t   )       ⁢     ❘     M   ⁢     -     ⁢   PSK         =         g   ⁡     (   t   )       ⁢     cos   ⁡     [       2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     +         2   ⁢   π     M     ⁢     (     m   -   1     )         ]       ⁢           ⁢   m     =   1       ,   2   ,   …   ⁢           ,   M           (   45   )               
where g(t) stands for the shaping pulse, f c  is the carrier frequency, M is the number of symbol states
 
                   M   ⁢     -     ⁢   QAM                                 S   k     ⁡     (   t   )       ⁢     ❘     M   ⁢     -     ⁢   QAM         =         Re   ⁡     [       A   mc     +     j   ⁢           ⁢     A   ms         ]       ⁢     g   ⁡     (   t   )       ⁢     ⅇ     2   ⁢   π   ⁢           ⁢   f       ⁢     c   t       =             (   46   )               =         A   mc     ⁢     g   ⁡     (   t   )       ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )         -       A   ms     ⁢     g   ⁡     (   t   )       ⁢     sin   ⁡     (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )                   (   47   )               
where A mc  and A ms  stand for the mapping amplitudes in the real and imaginary axis, respectively.
 
                   ASK   ⁢     /     ⁢   PAM                                 S   k     ⁡     (   t   )       ⁢     ❘     ASK   /   PAM         =       Re   ⁡     [       A   m     ⁢     g   ⁡     (   t   )       ⁢     ⅇ     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t         ]       =             (   48   )                 =         A   m     ⁢     g   ⁡     (   t   )       ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )       ⁢           ⁢   m     =   1       ,   2   ,   …   ⁢           ,   M           (   49   )             MSK                               S   k     ⁡     (   t   )       ⁢     ❘   MSK       =         A   mc     ⁢     cos   ⁡     (       π   ⁢           ⁢   t     2     )       ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )         -       A   ms     ⁢     sin   ⁡     (       π   ⁢           ⁢   t     2     )       ⁢     sin   ⁡     (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )                   (   50   )             FSK                                 S   k     ⁡     (   t   )       ⁢     ❘   FSK       =         g   ⁡     (   t   )       ⁢     cos   ⁡     [       (     2   ⁢   π   ⁢           ⁢     f   c     ⁢   t     )     +     2   ⁢   π   ⁢           ⁢   m   ⁢           ⁢   Δ   ⁢           ⁢   f   ⁢           ⁢   t       ]       ⁢           ⁢   m     =   1       ,   2   ,   …   ⁢           ,   M           (   51   )             OFDM                               S   k     ⁡     (   t   )       ⁢     |   OFDM       =       1   2     ⁢       ∑     k   =   0       N   -   1       ⁢       X   k     ⁢     ⅇ       j2π   ⁢           ⁢   k   ⁢           ⁢   t     T       ⁢     ⅇ     j2π   ⁢           ⁢     f   c     ⁢   t                     (   52   )               
where X k  is the k th  transmitted symbol and T is the period of each OFDM symbol. 1/T stands for the subcarrier spacing and it is actually the factor that provides the orthogonality between the subcarriers consisted within each OFDM.
 
     Thus in accordance with the invention the configuration design of the receiving apparatus can be achieved by referring to the appropriate decision metric. As indicated above and in  FIGS. 1-4 , four basic receiver designs are produced directly from the four different derived decision metrics in equations (20), (24), (37), (40). 
     However, where non-coherent detection is provided in these designs, four modified designs can be produced for the cases of coherent detection. 
     In addition, the PDP was considered uniform in the four initial designs. Hence receiver designs with coherent and non-coherent detection under exponential PDP can be produced in a straightforward manner. In addition, each of the designs can be changed to take into account the six modulation schemes, and different shaping pulses can be used in all schemes except MSK and OFDM. Therefore, the number of receiver configuration designs which can be achieved are: 
     
       
         
           
             
               
                 
                   4 
                   ⁢ 
                   
                     ( 
                     Initial 
                     ) 
                   
                   * 
                   4 
                   ⁢ 
                   
                     ( 
                     modulations 
                     ) 
                   
                   * 
                   2 
                   ⁢ 
                   
                     ( 
                     PDP 
                     ) 
                   
                   * 
                   2 
                   ⁢ 
                   
                     ( 
                     detections 
                     ) 
                   
                   * 
                   3 
                   ⁢ 
                   
                     
                       ( 
                       pulses 
                       ) 
                     
                     ⁡ 
                     
                       [ 
                       
                         = 
                         192 
                       
                       ] 
                     
                   
                 
                 + 
                 
                   4 
                   ⁢ 
                   
                     ( 
                     Initial 
                     ) 
                   
                   * 
                   2 
                   ⁢ 
                   
                     ( 
                     modulations 
                     ) 
                   
                   * 
                   2 
                   ⁢ 
                   
                     ( 
                     PDP 
                     ) 
                   
                   * 
                   2 
                   ⁢ 
                   
                     
                       ( 
                       detections 
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     [ 
                     
                       = 
                       32 
                     
                     ] 
                   
                 
               
               ⁢ 
               
                   
               
               = 
               
                 
                   
                     112 
                     ⁢ 
                     
                       ( 
                       optimum 
                       ) 
                     
                   
                   + 
                   
                     112 
                     ⁢ 
                     
                       ( 
                       suboptimum 
                       ) 
                     
                   
                 
                 = 
                 
                   224 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   receiver 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   designs 
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     It will be appreciated by persons skilled in the art that the present invention may also include further additional modifications made to the device which does not affect the overall functioning of the device. 
     SEQUENCE LISTING 
     Not Applicable