Abstract:
A method for suppressing repetition jammer signals and/or clutter signals by using a radar system wherein said radar system comprises, at least one wideband arbitrary waveform generator, for creating arbitrary waveforms, and at least one antenna operating with wide instantaneous bandwidth, enabling direction dependent waveform radiation, wherein said method comprises the steps of transmitting different waveforms in different directions; adapting transmitted waveforms in different directions depending on received waveforms such that a normalised signal to noise-jammer ratio, NSNJR, is maximized; and thereby providing received signals from repetition jammers and/or clutter that are substantially orthogonal to a received signal from a target; thereby achieving suppression of the repetition jammers and/or clutter.

Description:
TECHNICAL FIELD 
       [0001]    The present invention relates to the field of radar systems, and more particularly, to systems and techniques for minimizing system performance degradation caused by jamming or clutter. 
         [0002]    The present invention relates to a method and a radar system used to suppress repetition jammer signals and/or clutter signals. 
       BACKGROUND ART 
       [0003]    There exist various examples of methods and systems for repetition jammer suppression and clutter suppression. Today, the known solutions for repetition jammer suppression and clutter suppression are usually based on adaption of the transmitter and/or receiver to form nulls in the antenna pattern in one or more directions corresponding to jammers and/or clutter. A weakness of using these present solutions, however, is that they require advanced control of the transmit and/or receive antenna pattern. 
         [0004]    Further, a common way to minimize system performance degradation caused by jamming is to create deterministic and/or adaptive cancellation directions in all jammed directions. Deterministic cancellation directions are mainly used during transmit and adaptive cancellation directions mainly during receive. 
         [0005]    These present solutions are restricted to using both expensive and heavy equipments. Further, present solutions are normally not effective enough in suppressing repetition jammer signals. 
         [0006]    There is thus a need for an improved solution for providing an effective suppression of repetition jammer signals and/or clutter signals in order to cancel jammer and/or clutter interference. 
       SUMMARY 
       [0007]    The present invention is defined by the appended independent claims. Various examples of the invention are set forth by the appended dependent claims as well as by the following description and the accompanying drawings. 
         [0008]    With the above description in mind, then, an aspect of the present invention is to provide an improved solution of repetition jammer suppression and/or clutter suppression which seeks to mitigate, alleviate, or eliminate one or more of the above-identified deficiencies in the art and disadvantages singly or in any combination. 
         [0009]    The object of the present invention is to provide an inventive method and system where previously mentioned problems are partly avoided. This object is achieved by the features of claim  1  wherein a radar system for suppressing repetition jammer signals and/or clutter signals characterized in that said radar system comprises:
       at least one wideband arbitrary waveform generator, for creating arbitrary waveforms; and   at least one antenna operating with wide instantaneous bandwidth, enabling direction dependent waveform radiation;
 
wherein said radar system is adapted to transmit different waveforms in different directions depending on received waveforms, such that a normalised signal to noise-jammer ratio, NSNJR, is maximized, which provides that received signals from repetition jammers and/or clutter are substantially orthogonal to a received signal from a target and thereby achieving suppression of the repetition jammers and/or clutter. The present invention provides a solution which minimizes radar system performance degradation due to repetition jammer and clutter by adaptively modifying the transmitted waveforms from the radar system. The antenna may be a large antenna adapted for wideband operation. The arbitrary waveform generator used in the present invention enables the creation of arbitrary waveforms. The radar system described in the present invention is thus a radar system which is capable of suppressing repetition jammer signals and/or clutter signals.
       
 
         [0012]    Said object is further achieved by the features of claim  9 , wherein a method for suppressing repetition jammer signals and/or clutter signals by using a radar system characterized in that said radar system comprises: at least one wideband arbitrary waveform generator, for creating arbitrary waveforms; and at least one antenna operating with wide instantaneous bandwidth, enabling direction dependent waveform radiation;
       wherein said method comprises the steps of:
           transmitting different waveforms in different directions;   adapting transmitted waveforms in different directions depending on received waveforms such that a normalised signal to noise-jammer ratio, NSNJR, is maximized; and thereby   providing received signals from repetition jammers and/or clutter that are substantially orthogonal to a received signal from a target; thereby   achieving suppression of the repetition jammers and/or clutter.   
               
 
         [0018]    According to a further advantageous aspect of the invention, said wideband arbitrary waveform generator comprises a noise generator, NG, and FIR-filters comprising FIR-filter coefficients. 
         [0019]    According to a further advantageous aspect of the invention, said antenna is a large antenna adapted for wideband operation. Said large antenna is adapted for operation with wide instantaneous bandwidth. 
         [0020]    According to a further advantageous aspect of the invention, said values for said FIR-filter coefficients are adapted to maximize the normalised signal to noise-jammer ratio, NSNJR. 
         [0021]    According to a further advantageous aspect of the invention, said maximized normalised signal to noise-jammer ratio, NSNJR, is obtained after convergence of an iterative process performed for the transmit FIR-filter coefficients values, where the FIR-filter coefficients values are optimized to obtain the maximized NSNJR. Maximum NSNJR is obtained by iteratively converging and optimizing the FIR-filter coefficient values. 
         [0022]    According to a further advantageous aspect of the invention, said maximized normalised signal to noise-jammer ratio, NSNJR, is obtained after solution of a FIR-filter coefficient equation, where the FIR-filter coefficients values are optimized to obtain the maximized NSNJR. NSNJR is obtained after solution of an equation system created by setting all partial derivatives of NSNJR as a function of all FIR-filter coefficients equal to zero. 
         [0023]    According to a further advantageous aspect of the invention, said wideband arbitrary waveform generator comprises one noise generator connected to several FIR-filters. 
         [0024]    According to a further advantageous aspect of the invention, said wideband arbitrary waveform generators are uncorrelated noise sources with one noise source connected to each FIR-filter. 
         [0025]    Any of the advantageous features of the present invention above may be combined in any suitable way. 
         [0026]    A number of advantages are provided by means of the present invention, for example:
       effective suppression of repetition jammers and/or clutter is allowed;   no additional radar hardware development is needed to provide the improved suppression of repetition jammers and/or clutter in accordance with the present invention;   a solution for suppression of repetition jammers and/or clutter which does not require expensive and advanced control of neither the transmit nor the receive antenna pattern is allowed;   adaptive modification of the transmitted waveform is allowed;       
 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0031]    The present invention will now be described in detail with reference to the figures, wherein: 
           [0032]      FIG. 1  schematically shows a pictorial representation of a scenario for suppression of jammers and/or clutter according to the present invention. 
           [0033]      FIG. 2  schematically show a layout of a calculation model according to the present invention. 
           [0034]      FIG. 3  schematically show a layout of a calculation model according to the present invention. 
           [0035]      FIG. 4  schematically show a layout of a calculation model according to the present invention. 
           [0036]      FIG. 5  schematically show initial FIR-filter coefficients for examples. 
           [0037]      FIG. 6  schematically show NSNJR convergence as a function of iteration with correlated sources. 
           [0038]      FIG. 7  schematically show wideband antenna pattern with correlated sources. 
           [0039]      FIG. 8  schematically show main beam transmitted power spectrum with correlated sources. 
           [0040]      FIG. 9  schematically show main beam transmitted waveform autocorrelation function with correlated sources. 
           [0041]      FIG. 10  schematically show NSNJR convergence as a function of iteration with uncorrelated sources. 
           [0042]      FIG. 11  schematically show wideband antenna pattern with uncorrelated sources. 
           [0043]      FIG. 12  schematically show main beam transmitted power spectrum with uncorrelated sources. 
           [0044]      FIG. 13  schematically main beam transmitted waveform autocorrelation function with uncorrelated sources. 
       
    
    
     DETAILED DESCRIPTION 
       [0045]    Examples of the present invention relate, in general, to the field radar systems, in particularly, to methods and systems for deceptive transmission and suppression of jammers and/or clutter when detecting the presence and determining the direction of targets. 
         [0046]    Examples of the present invention will be described more fully hereinafter with reference to the accompanying drawings, in which examples of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the examples set forth herein. Rather, these examples are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like reference signs refer to like elements throughout. 
         [0047]    All the  FIGS. 1 to 13  are schematically illustrated. 
         [0048]    A scenario for suppression of jammers and/or clutter according to the present invention is schematically shown in  FIG. 1 . The radar system  12  in the present invention comprises an arbitrary waveform generator, such as for example a High Frequency Generator (HFG) which is able to create arbitrary waveforms and large antennas operating with wide instantaneous bandwidth which results in direction dependent waveform radiation. An arbitrary waveform generator, AWG, may generate any arbitrarily defined waveform as its output. 
         [0049]    In communications, wideband is a relative term used to describe a wide range of frequencies in a spectrum. A system is typically described as wideband if the message bandwidth significantly exceeds the channel&#39;s coherence bandwidth. Wideband conditions are assumed to be present if the size-instantaneous bandwidth product of the antenna are approaching or exceeding the velocity of light. The antenna, used in the present invention may be a large antenna. A large antenna can be defined as an antenna not being small, where a small antenna can be defined as an antenna with a maximal physical extension considerably smaller than the ratio between the velocity of light and the instantaneous bandwidth. 
         [0050]    The large wideband antennas operated with an instantaneous wideband waveform radiate different waveforms in different directions, as shown in  FIG. 1 . The present invention use this feature to ensure that the waveform radiated in the direction of localized side lobe clutter and active repetition jammers will after reception be orthogonal or close to orthogonal to the waveform received in the main beam direction from a point target, which result in suppression of jammers and/or clutter. Relating to  FIG. 1 , aircraft  11  comprises a radar system  12  according to the present invention. Aircraft  14  comprises a repetition jammer  13  which can actively disturb signals, contaminate and interfere in the performance of the radar system  12 . The repetition jammer  13  may reduce the performance of the radar system. The radar system  12  transmit a first transmitted sidelobe waveform  17   a  which may be contaminated and then reflected due to a repetition jammer  13  and a first received sidelobe waveform  17   b  is received by the radar system  12 . The transmitted main beam waveform  18   a  is transmitted and reflected due to the target  15  and a received main beam waveform  18   b  is received in the radar system  12 . 
         [0051]    Ground clutter is a factor which affects radar performance. Ground clutter is disturbance on a ground or airborne radar due to reflection of signals from the ground  16  or objects on the ground  16 . Ground clutter is an unavoidable form of radar contamination. It occurs when fixed objects, such as buildings, trees or terrain obstruct the radar beam and produce non-meteorological echoes. Echoes resulting from ground clutter may cause radar systems to overestimate precipitation intensity near the radar.  FIG. 1  further show, the radar system  12  transmitting a second transmitted sidelobe waveform  19   a  which is reflected due to ground clutter and a second received sidelobe waveform  19   b  which is received by the radar system  2 . 
         [0052]    The solution in the present invention calculates suitable power spectra and auto correlation functions corresponding to waveforms suitable for suppression of localized side lobe clutter and active repetition jammers. For a given signal, the power spectrum gives a plot of the portion of a signal&#39;s power (energy per unit frequency) falling within given frequency bins. 
         [0053]      FIG. 2  schematically show a calculation model relating to how the calculations of the present invention are organized. The calculation model comprises at least one noise generator, NG,  21 , at least one transmit finite impulse response, FIR, filter  22 , at least one transmitting and receiving antenna  23  with sub arrays and at least one matching filter, MF,  20 . The wideband arbitrary waveform generator  30  comprises for example a noise generator, NG,  21  and FIR-filters  22  comprising FIR-filter coefficients as indicated in  FIG. 2 . The transmitting antenna and the receiving antenna may be two separate antennas. The FIR-filter  22  shapes the waveform spectrum and comprises FIR-filter coefficients which are adaptable and programmable. Matched filters are commonly used in radar systems, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal. A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. The transfer function of the matched filter, MF,  20  in the receiver is normally equal to the complex conjugate of the spectrum of the waveform received in the main beam direction from a point target which provides high attenuation of localized side lobe clutter and repetition jammers. The transfer function of each block in  FIG. 2  is calculated and the problem is formulated as a cascade connection of filters with corresponding transfer functions. In the receiver antenna  23  is the inevitable receiver noise added. Further is the receiver noise onwards assumed to be white and Gaussian. This assumption simplifies the calculations without limiting the generality of the results, as other spectral distribution can be used if so desired. The radar system  12  comprises at least one wideband arbitrary waveform generator, and at least one antenna  23  operating with wide instantaneous bandwidth, wherein said wideband arbitrary waveform generator comprises at least one noise generator  21 , NG, and at least one FIR-filter  22 , but observe other constructions of the wideband arbitrary waveform generator/generators are conceivable. 
         [0054]    The block for illustrating ground clutter  16  is dotted to indicate that it is excluded in the calculations to follow. The reason for excluding the ground clutter  16  is that the repetition jammer path is considered to be more severe than the ground clutter path as the output power from the repetition jammer  13 , in contrast to the ground clutter  16 , is assumed to be independent of input power. This also ensures that no suppression is achieved by forming cancellation directions in the transmit pattern. Further, there are no receiving FIR-filters in the following calculations and the reason for excluding the receive FIR-filter is to ensure that no cancellation directions are formed in the receive pattern. We have thus, by these two exclusions, refined the problem to ensure that only the waveform influences the final suppression. Normalized signal to noise-jammer ration, NSNJR, is a ratio which describes the signal noise ratio, SNR, reduction due to the jamming and is always less than unity (0 dB). NSNJR equals the following equation 2.7.7: 
         [0000]    
       
         
           
             
               
                 
                   NSNJR 
                   = 
                   
                     
                       
                         S 
                         
                           N 
                           + 
                           J 
                         
                       
                       
                         
                           S 
                           
                             m 
                              
                             
                                 
                             
                              
                             ax 
                           
                         
                         N 
                       
                     
                     = 
                     
                       
                         S 
                         
                           S 
                           
                             m 
                              
                             
                                 
                             
                              
                             ax 
                           
                         
                       
                       · 
                       
                         1 
                         
                           1 
                           + 
                           
                             J 
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.7 
                      
                     .7 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein S is defined as signal power, with both jamming and clutter present and S max  is defined as the maximum signal power, with neither jamming nor clutter present, J is defined as the combined jammer and clutter power and N is defined as the receiver noise power. There is no performance degradation caused by repetition jamming or clutter if the normalised signal to noise-jammer ratio, NSNJR, equals 0 dB. 
         [0055]    The actual waveform is of no importance, the important functions are the Fourier transform pair formed by the waveform autocorrelation function and power spectrum. It is thus sufficient to shape the power spectrum from a “white” noise generator  21  in a transmit FIR-filter  22  as indicated in  FIG. 2 . The theory of the present invention is valid both for correlated and uncorrelated noise generators feeding the FIR-filter behind each antenna sub array. 
         [0056]    In the transmit antenna there is either one noise generator (NG) connected to all FIR-filters (η=0) or independent noise generators (NG) are connected to each FIR-filter (η=1), where η is equal to unity if V {right arrow over (n)}     s   (t) are uncorrelated wideband white noise waveforms and zero if V {right arrow over (n)}     s   (t) are correlated wideband white noise waveforms and where V {right arrow over (n)}     s   (t) is the waveform connected to transmit sub array {right arrow over (n)} s . The receive antenna could also be equipped with FIR-filters to further increase the suppression, but this option has not been used to ensure that no cancellation directions are formed in the receive pattern and thus will only the waveform/waveforms influence the final suppression. 
         [0057]      FIG. 3  schematically show a calculation model according to the present invention comprising one noise generator  21 . The calculation model of the example in  FIG. 3  comprises one noise generator  21  connected to several FIR-filters  22  which are connected to sub array antenna elements in the antenna  23 . 
         [0058]      FIG. 4  schematically show a calculation model according to the present invention comprising several noise generators. The calculation model of the example in  FIG. 4  comprise several different uncorrelated noise generators  21  each connected to several different FIR-filters  22  which are connected to sub array antenna elements in the antenna  23 . The noise to each FIR-filter is thus uncorrelated to the noise to any other FIR-filter. 
         [0059]    The theoretical analysis describing in detail the theory of the present invention contains the assumptions that:
       there are no errors in the array;   mutual coupling could be ignored;   the observation time is long enough to contain the entire waveform, range straddling losses is not included in this preliminary analysis;   a unique waveform can be radiated from each sub array;   element and sub array positions can be chosen arbitrarily.       
 
       Transmitted Waveform 
       [0065]    Start with the assumption that the waveform V {right arrow over (n)}     s   (t) is connected to transmit sub array {right arrow over (n)} s . The right hand pointing arrow (→) indicates transmit direction. The resulting waveform, V Tx , including FIR filtering with {right arrow over (M)} taps, after summation over all, {right arrow over (N)} s , transmit sub arrays as a function of direction ({circumflex over (r)}) main beam direction ({right arrow over (r)} MB ), and time (t) can be calculated according to the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       Tx 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         
                           
                             n 
                             → 
                           
                           s 
                         
                         = 
                         0 
                       
                       
                         
                           
                             N 
                             → 
                           
                           s 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           
                             h 
                             → 
                           
                           
                             
                               n 
                               → 
                             
                             s 
                           
                         
                          
                         
                           ( 
                           
                             
                               r 
                               ^ 
                             
                             , 
                             
                               
                                 r 
                                 ^ 
                               
                               MB 
                             
                             , 
                             
                               t 
                               + 
                               
                                 
                                   
                                     τ 
                                     → 
                                   
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     , 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       MB 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                       * 
                       
                         
                           ∑ 
                           
                             
                               m 
                               → 
                             
                             = 
                             0 
                           
                           
                             
                               M 
                               → 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               w 
                               → 
                             
                             
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                               , 
                               
                                 m 
                                 → 
                               
                             
                           
                           · 
                           
                             
                               V 
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     · 
                                     Δ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                                 + 
                                 
                                   
                                     
                                       τ 
                                       → 
                                     
                                     
                                       
                                         n 
                                         → 
                                       
                                       s 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       , 
                                       
                                         
                                           r 
                                           ^ 
                                         
                                         MB 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.2 
                      
                     .1 
                   
                   ) 
                 
               
             
           
         
       
       
         Where: 
         {right arrow over (h)} p ({circumflex over (r)},{right arrow over (r)} MB ,t) is the transmit sub array impulse response for transmit sub array p 
         {right arrow over (τ)} p ({circumflex over (r)},{circumflex over (r)} MB ) is equal to: 
       
     
         [0000]    
       
         
           
             
               
                 
                   r 
                   → 
                 
                 p 
                 sT 
               
               · 
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   - 
                   
                     
                       r 
                       ^ 
                     
                     MB 
                   
                 
                 ) 
               
             
             
               c 
               0 
             
           
         
       
       
         • is the scalar product operator 
         · is the multiplication operator 
         * is the convolution operator 
         {right arrow over (w)} p,q  Is the transmit FIR-filter coefficient for transmit sub array p and transmit FIR-filter tap q 
                     p   s  is the vector from the antenna origin to the phase centre of transmit sub array p 
         {circumflex over (r)} MB  is the unit vector in the main beam direction (θ MB ,φ MB ) 
         {circumflex over (r)} is the unit vector in the observation direction (θ,φ) 
         c o  is the speed of light 
       
     
       Received Waveform 
       [0077]    The transmitted waveform V Tx ({circumflex over (r)}, {circumflex over (r)} MB ,t) reaches sub array            s  with impulse response            {right arrow over (n)}     s   ({circumflex over (r)},{circumflex over (r)} MB ,t). The left hand pointing arrow (←) indicating receive direction. The received waveform, V Rx , including FIR-filtering, after summation over all,            s , receive sub arrays as a function of direction ({circumflex over (r)}), main beam direction ({circumflex over (r)} MB ), and time (t) can be calculated according to the following equation 2.3.1: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       Rx 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       D 
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               ← 
                             
                             s 
                           
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               ← 
                             
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               m 
                               ← 
                             
                             = 
                             0 
                           
                           
                             
                               M 
                               ← 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               
                                 w 
                                 ← 
                               
                               
                                 
                                   
                                     n 
                                     ← 
                                   
                                   s 
                                 
                                  
                                 
                                   m 
                                   ← 
                                 
                               
                             
                             · 
                             
                               
                                 
                                   h 
                                   ← 
                                 
                                 
                                   
                                     n 
                                     ← 
                                   
                                   s 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   , 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     MB 
                                   
                                   , 
                                   
                                     t 
                                     - 
                                     
                                       
                                         
                                           m 
                                           ← 
                                         
                                         · 
                                         Δ 
                                       
                                        
                                       
                                           
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       
                                         
                                           τ 
                                           ← 
                                         
                                         
                                           
                                             n 
                                             ← 
                                           
                                           s 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             ^ 
                                           
                                           , 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             MB 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           * 
                           
                             
                               V 
                               Tx 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                                 , 
                                 
                                   t 
                                   - 
                                   
                                     
                                       
                                         m 
                                         ← 
                                       
                                       · 
                                       Δ 
                                     
                                      
                                     
                                         
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       
                                         τ 
                                         ← 
                                       
                                       
                                         
                                           n 
                                           ← 
                                         
                                         s 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           ^ 
                                         
                                         , 
                                         
                                           
                                             r 
                                             ^ 
                                           
                                           MB 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.3 
                      
                     .1 
                   
                   ) 
                 
               
             
           
         
       
       
         Where: 
                     p ({circumflex over (r)},{circumflex over (r)} MB ,t) is the receive sub array impulse response for receive sub array p 
                     p ({circumflex over (r)},{circumflex over (r)} MB ) is equal to: 
       
     
         [0000]    
       
         
           
             
               
                 
                   r 
                   ← 
                 
                 p 
                 sT 
               
               · 
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   - 
                   
                     
                       r 
                       ^ 
                     
                     MB 
                   
                 
                 ) 
               
             
             
               c 
               0 
             
           
         
       
       
                     p,q  is the receive FIR-filter coefficient for receive sub array p and receive FIR-filter tap q 
                     p   s  is the vector from the antenna origin to the phase centre of receive sub array p 
         D is the power attenuation between transmission and reception 
       
     
         [0084]    For a target  15  the main beam is D equal to D T , as defined in the following equation 2.3.2: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     T 
                   
                   = 
                   
                     
                       
                         
                           c 
                           0 
                           2 
                         
                         · 
                         
                           σ 
                           T 
                         
                       
                       
                         16 
                         · 
                         π 
                         · 
                         
                           ω 
                           cw 
                           2 
                         
                         · 
                         
                           r 
                           T 
                           4 
                         
                       
                     
                     = 
                     
                       
                         
                           σ 
                           T 
                         
                         π 
                       
                       · 
                       
                         
                           ( 
                           
                             
                               c 
                               0 
                             
                             
                               4 
                               · 
                               
                                 ω 
                                 cw 
                               
                               · 
                               
                                 r 
                                 T 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.3 
                      
                     .2 
                   
                   ) 
                 
               
             
           
         
       
       
         Where: 
         σ T  is the radar cross section of the target 
         r T  is the range to the target 
         ω cw  is the angular carrier frequency of the waveform 
       
     
         [0089]    For jammer number j is D equal to D J     j   , as defined in equation 2.3.3. This definition ensures jamming with constant power independent of the power received by the jammer. 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     
                       J 
                       j 
                     
                   
                   = 
                   
                     
                       
                         
                           PG 
                           
                             J 
                             j 
                           
                         
                         · 
                         
                           c 
                           0 
                           2 
                         
                       
                       
                         
                           
                             P 
                             Tx 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   J 
                                   j 
                                 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 MB 
                               
                             
                             ) 
                           
                         
                         · 
                         4 
                         · 
                         
                           ω 
                           cw 
                           2 
                         
                         · 
                         
                           r 
                           
                             J 
                             j 
                           
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           PG 
                           
                             J 
                             j 
                           
                         
                         
                           
                             P 
                             Tx 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   J 
                                   j 
                                 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 MB 
                               
                             
                             ) 
                           
                         
                       
                       · 
                       
                         
                           ( 
                           
                             
                               c 
                               0 
                             
                             
                               2 
                               · 
                               
                                 ω 
                                 cw 
                               
                               · 
                               
                                 r 
                                 
                                   J 
                                   j 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.3 
                      
                     .3 
                   
                   ) 
                 
               
             
           
         
       
       
         Where: 
         PG J     j    is the power-gain product of repetition jammer number j (equal to the jammer strength) 
         P Tx ({circumflex over (r)} J     j   ,{circumflex over (r)} MB ) is the transmitted power in the direction of repetition jammer number j 
         r J     j    is the range to repetition jammer number j 
         J is the number of jammers 
         j is the jammers counting index jε{0 . . . (J−1)}, not to be confused with the imaginary unit 
       
     
       Down Conversion to Baseband 
       [0000]    
       
         The next step is to split the transmit waveforms V {right arrow over (n)}     s   (t) in one carrier part and one baseband part, ν {right arrow over (n)}     s   (t), as indicated in the following equation 2.4.1. 
       
     
         [0000]        V   {right arrow over (n)}     s   ( t )= e   −j·ω     cw     ·t ·ν {right arrow over (n)}     s     (Equation 2.4.1)
 
         [0097]    The angular carrier frequency of the waveform ω cw  is assumed to be equal for all transmit waveforms V {right arrow over (n)}     s   (t). After reception is the waveform, V Rx , ({circumflex over (r)},{circumflex over (r)} MB ,t), down converted to baseband by multiplication with e j·ω     cw     ·t . Since the convolution operator is both commutative and associative the resulting down converted waveform, ν Rx ({circumflex over (r)},{circumflex over (r)} MB ,t) can be expressed as indicated in the following equation 2.4.2. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       Rx 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       D 
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               → 
                             
                             s 
                           
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               → 
                             
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 ← 
                               
                               s 
                             
                             = 
                             0 
                           
                           
                             
                               
                                 N 
                                 ← 
                               
                               s 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 m 
                                 → 
                               
                               = 
                               0 
                             
                             
                               
                                 M 
                                 → 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   m 
                                   ← 
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   M 
                                   ← 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   
                                     w 
                                     → 
                                   
                                   
                                     
                                       
                                         n 
                                         → 
                                       
                                       s 
                                     
                                     , 
                                     
                                       m 
                                       → 
                                     
                                   
                                 
                                 · 
                                 
                                   
                                     w 
                                     ← 
                                   
                                   
                                     
                                       
                                         n 
                                         ← 
                                       
                                       s 
                                     
                                     , 
                                     
                                       m 
                                       ← 
                                     
                                   
                                 
                                 · 
                                 
                                   
                                      
                                     
                                       j 
                                       · 
                                       
                                         ω 
                                         cw 
                                       
                                       · 
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     m 
                                                     ↔ 
                                                   
                                                   + 
                                                   
                                                     m 
                                                     ← 
                                                   
                                                 
                                                 ) 
                                               
                                               · 
                                               Δ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             t 
                                           
                                           - 
                                           
                                             
                                               
                                                 τ 
                                                 → 
                                               
                                               
                                                 
                                                   n 
                                                   → 
                                                 
                                                 s 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                               
                                               ) 
                                             
                                           
                                           - 
                                           
                                             
                                               
                                                 τ 
                                                 ← 
                                               
                                               
                                                 
                                                   n 
                                                   ← 
                                                 
                                                 s 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   . 
                                   
                                     
 
                                   
                                    
                                   
                                     
                                       
                                         h 
                                         → 
                                       
                                       
                                         
                                           n 
                                           → 
                                         
                                         s 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           r 
                                           ^ 
                                         
                                         , 
                                         
                                           
                                             r 
                                             ^ 
                                           
                                           MB 
                                         
                                         , 
                                         
                                           t 
                                           - 
                                           
                                             
                                               
                                                 m 
                                                 ← 
                                               
                                               · 
                                               Δ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             
                                               
                                                 τ 
                                                 → 
                                               
                                               
                                                 
                                                   n 
                                                   → 
                                                 
                                                 s 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 τ 
                                                 ← 
                                               
                                               
                                                 
                                                   n 
                                                   ← 
                                                 
                                                 s 
                                               
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               * 
                               
                                 
                                   
                                     h 
                                     ← 
                                   
                                   
                                     
                                       n 
                                       ← 
                                     
                                     s 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     , 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       MB 
                                     
                                     , 
                                     
                                       t 
                                       - 
                                       
                                         
                                           
                                             m 
                                             ← 
                                           
                                           · 
                                           Δ 
                                         
                                          
                                         
                                             
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         
                                           
                                             τ 
                                             ← 
                                           
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             , 
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               MB 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               * 
                               
                                 
                                   v 
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               m 
                                               → 
                                             
                                             + 
                                             
                                               m 
                                               ← 
                                             
                                           
                                           ) 
                                         
                                         · 
                                         Δ 
                                       
                                        
                                       
                                           
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       
                                         
                                           τ 
                                           → 
                                         
                                         
                                           
                                             n 
                                             → 
                                           
                                           s 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             ^ 
                                           
                                           , 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             MB 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           τ 
                                           ← 
                                         
                                         
                                           
                                             n 
                                             ← 
                                           
                                           s 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             r 
                                             ^ 
                                           
                                           , 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             MB 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.4 
                      
                     .2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0098]    Matched Filtering 
         [0099]    The matched filter is usually optimized for the waveform arriving in the main beam direction.            c  is the mutual centre frequency of the transmitter and receiver.            B  is the mutual bandwidth of the transmitter and receiver. 
         [0000]    
       
         
           
             
               
                 
                   ω 
                    
                   
                       
                   
                 
                 ↔ 
               
               c 
             
             = 
             
               
                 
                   
                     max 
                     [ 
                     
                       
                         
                           
                             
                               
                                 ω 
                                 → 
                               
                               c 
                             
                             - 
                             
                               
                                 
                                   ω 
                                   → 
                                 
                                 B 
                               
                               2 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ω 
                                     ← 
                                   
                                   c 
                                 
                                 - 
                                 
                                   
                                     
                                       ω 
                                       ← 
                                     
                                     B 
                                   
                                   2 
                                 
                               
                               ] 
                             
                             + 
                             
                               min 
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           ω 
                                           → 
                                         
                                         c 
                                       
                                       + 
                                       
                                         
                                           
                                             ω 
                                             → 
                                           
                                           B 
                                         
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             ω 
                                             ← 
                                           
                                           c 
                                         
                                         + 
                                         
                                           
                                             
                                               ω 
                                               ← 
                                             
                                             B 
                                           
                                           2 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   2 
                 
                  
                 
                   
 
                 
                  
                 and 
                  
                 
                   : 
                 
                  
                 
                   
 
                 
                  
                 
                   
                     ω 
                     ↔ 
                   
                   B 
                 
               
               = 
               
                 min 
                 [ 
                 
                   
                     
                       
                         
                           
                             ω 
                             → 
                           
                           c 
                         
                         + 
                         
                           
                             
                               ω 
                               → 
                             
                             B 
                           
                           2 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ω 
                                 ← 
                               
                               c 
                             
                             + 
                             
                               
                                 
                                   ω 
                                   ← 
                                 
                                 B 
                               
                               2 
                             
                           
                           ] 
                         
                         - 
                         
                           max 
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       ω 
                                       → 
                                     
                                     c 
                                   
                                   - 
                                   
                                     
                                       
                                         ω 
                                         → 
                                       
                                       B 
                                     
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ω 
                                         ← 
                                       
                                       c 
                                     
                                     - 
                                     
                                       
                                         
                                           ω 
                                           ← 
                                         
                                         B 
                                       
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0100]    The superscript * will onwards represent complex conjugation and u g  [V/√{square root over (rad/s)}] is equal to the effective value per square root of angular frequency of each waveform generator. Thus is the power density from each waveform generator, P g [W/rad/s], equal to u g   2 . The transmit and the receive element amplitude pattern at the waveform centre frequency in the main beam direction is respectively denoted {right arrow over (a)} e ({circumflex over (r)} MB ,{circumflex over (r)} e     MB   ,ω cw ) and            e ({circumflex over (r)} MB ,{circumflex over (r)} e     MB   ,ω cw ). {circumflex over (r)} e     MB    is the unit vector pointing in the main beam direction of each element. Onwards will these vectors be assumed to be equal for all array elements. The effective values, u g , are assumed to be equal in all waveform generators and thus also the power, P g . This assumption does not restrict the validity of the theory as the magnitude of the voltage feed to each sub array can be adjusted with the magnitude of the transmit FIR-filter coefficient, {right arrow over (w)}. 
         [0101]    Thus is the matched filter impulse response assumed to be equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           ↔ 
                         
                         B 
                       
                     
                     · 
                     
                       
                         v 
                         Rx 
                         * 
                       
                        
                       
                         ( 
                         
                           
                             
                               r 
                               ^ 
                             
                             MB 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             MB 
                           
                           , 
                           
                             - 
                             t 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 N 
                                 → 
                               
                               s 
                             
                             · 
                             
                               u 
                               g 
                             
                             · 
                             
                               
                                 D 
                                 T 
                               
                             
                             · 
                             
                               
                                 a 
                                 → 
                               
                               e 
                               * 
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   
                                     e 
                                     MB 
                                   
                                 
                                 , 
                                 
                                   ω 
                                   cw 
                                 
                               
                               ) 
                             
                             · 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               a 
                               ← 
                             
                             e 
                             * 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 MB 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   e 
                                   MB 
                                 
                               
                               , 
                               
                                 ω 
                                 cw 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.5 
                      
                     .0 
                   
                   ) 
                 
               
             
           
         
       
     
         [0102]    Normalization 
         [0103]    with 
         [0000]    
       
         
           
             
               
                 N 
                 → 
               
               s 
             
             · 
             
               u 
               g 
             
             · 
             
               
                 D 
                 T 
               
             
             · 
             
               
                 
                   a 
                   → 
                 
                 e 
                 * 
               
                
               
                 ( 
                 
                   
                     
                       r 
                       ^ 
                     
                     MB 
                   
                   , 
                   
                     
                       r 
                       ^ 
                     
                     
                       e 
                       MB 
                     
                   
                   , 
                   
                     ω 
                     cw 
                   
                 
                 ) 
               
             
             · 
             
               
                 
                   
                     a 
                     ← 
                   
                   e 
                   * 
                 
                  
                 
                   ( 
                   
                     
                       
                         r 
                         ^ 
                       
                       MB 
                     
                     , 
                     
                       
                         r 
                         ^ 
                       
                       
                         e 
                         MB 
                       
                     
                     , 
                     
                       ω 
                       cw 
                     
                   
                   ) 
                 
               
               / 
               
                 
                   
                     ω 
                     ↔ 
                   
                   B 
                 
               
             
           
         
       
     
         [0000]    is required to make the matched filter independent of all these factors and to give it the correct dimension [rad/s]. The spectral domain transfer function of the matched filter is equal to the complex conjugate of the used waveform. Thus is the impulse response of the matched filter equal to the complex conjugate of the used waveform with time reversal. The resulting baseband waveform, u B ({circumflex over (r)},{circumflex over (r)} MB ,t), after matched filtering can thus be calculated as indicated in the following equation 2.5.1. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                       B 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ω 
                               ↔ 
                             
                             B 
                           
                         
                         · 
                         
                           
                             v 
                             Rx 
                             * 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 MB 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 MB 
                               
                               , 
                               
                                 - 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     N 
                                     → 
                                   
                                   s 
                                 
                                 · 
                                 
                                   u 
                                   g 
                                 
                                 · 
                                 
                                   
                                     D 
                                     T 
                                   
                                 
                                 · 
                                 
                                   
                                     a 
                                     → 
                                   
                                   e 
                                   * 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       MB 
                                     
                                     , 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       
                                         e 
                                         MB 
                                       
                                     
                                     , 
                                     
                                       ω 
                                       cw 
                                     
                                   
                                   ) 
                                 
                                 · 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   a 
                                   ← 
                                 
                                 e 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     MB 
                                   
                                   , 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     
                                       e 
                                       MB 
                                     
                                   
                                   , 
                                   
                                     ω 
                                     cw 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     * 
                     
                       
                         v 
                         Rx 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             ^ 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             MB 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.5 
                      
                     .1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0104]    Baseband Waveform after Matched Filtering 
         [0105]    The baseband waveform after matched filtering is calculated by inserting equation 2.4.2 twice into equation 2.5.1. The result, after some rearrangements, is found in the following equation 2.6.1. Observe that all indexes equipped with a prime originate from the matched filter. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                       B 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ω 
                               ↔ 
                             
                             B 
                           
                           · 
                           D 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     N 
                                     → 
                                   
                                   s 
                                 
                                 · 
                                 
                                   u 
                                   g 
                                 
                                 · 
                                 
                                   
                                     a 
                                     → 
                                   
                                   e 
                                   * 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       MB 
                                     
                                     , 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       
                                         e 
                                         MB 
                                       
                                     
                                     , 
                                     
                                       ω 
                                       cw 
                                     
                                   
                                   ) 
                                 
                                 · 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   a 
                                   ← 
                                 
                                 e 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     MB 
                                   
                                   , 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     
                                       e 
                                       MB 
                                     
                                   
                                   , 
                                   
                                     ω 
                                     cw 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               → 
                             
                             ′ 
                           
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               → 
                             
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 → 
                               
                               s 
                             
                             = 
                             0 
                           
                           
                             
                               
                                 N 
                                 → 
                               
                               s 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                                 ′ 
                               
                               = 
                               0 
                             
                             
                               
                                 
                                   N 
                                   ← 
                                 
                                 s 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     ← 
                                   
                                   s 
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   
                                     N 
                                     ← 
                                   
                                   s 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 
                                   
                                     M 
                                     → 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       m 
                                       → 
                                     
                                     = 
                                     0 
                                   
                                   
                                     
                                       M 
                                       → 
                                     
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         
                                           m 
                                           ← 
                                         
                                         ′ 
                                       
                                       = 
                                       0 
                                     
                                     
                                       
                                         M 
                                         ← 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           m 
                                           ← 
                                         
                                         = 
                                         0 
                                       
                                       
                                         
                                           M 
                                           ← 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         
                                           
                                             w 
                                             → 
                                           
                                           
                                             
                                               
                                                 n 
                                                 → 
                                               
                                               s 
                                               ′ 
                                             
                                             , 
                                             
                                               
                                                 m 
                                                 → 
                                               
                                               ′ 
                                             
                                           
                                           * 
                                         
                                         · 
                                         
                                           
                                             w 
                                             → 
                                           
                                           
                                             
                                               
                                                 n 
                                                 → 
                                               
                                               s 
                                             
                                             , 
                                             
                                               m 
                                               → 
                                             
                                           
                                         
                                         · 
                                         
                                           
                                             w 
                                             ← 
                                           
                                           
                                             
                                               
                                                 n 
                                                 ← 
                                               
                                               s 
                                               ′ 
                                             
                                             , 
                                             
                                               
                                                 m 
                                                 ← 
                                               
                                               ′ 
                                             
                                           
                                           * 
                                         
                                         · 
                                         
                                           
                                             w 
                                             ← 
                                           
                                           
                                             
                                               
                                                 n 
                                                 ← 
                                               
                                               s 
                                             
                                             , 
                                             
                                               m 
                                               ← 
                                             
                                           
                                         
                                         · 
                                         
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                               · 
                                               
                                                 ω 
                                                 cw 
                                               
                                               · 
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                           
                                                           - 
                                                           
                                                             m 
                                                             → 
                                                           
                                                           + 
                                                           
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                           
                                                           - 
                                                           
                                                             m 
                                                             ← 
                                                           
                                                         
                                                         ) 
                                                       
                                                       · 
                                                       Δ 
                                                     
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     t 
                                                   
                                                   + 
                                                   
                                                     
                                                       
                                                         τ 
                                                         → 
                                                       
                                                       
                                                         
                                                           n 
                                                           → 
                                                         
                                                         s 
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         , 
                                                         
                                                           
                                                             r 
                                                             ^ 
                                                           
                                                           MB 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       
                                                         τ 
                                                         ← 
                                                       
                                                       
                                                         
                                                           n 
                                                           ← 
                                                         
                                                         s 
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         , 
                                                         
                                                           
                                                             r 
                                                             ^ 
                                                           
                                                           MB 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                             
                                           
                                           . 
                                           
                                             
 
                                           
                                            
                                           
                                             
                                               
                                                 h 
                                                 → 
                                               
                                               
                                                 
                                                   n 
                                                   → 
                                                 
                                                 s 
                                                 ′ 
                                               
                                               * 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                                 , 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   MB 
                                                 
                                                 , 
                                                 
                                                   
                                                     - 
                                                     t 
                                                   
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           m 
                                                           ← 
                                                         
                                                         ′ 
                                                       
                                                       · 
                                                       Δ 
                                                     
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     t 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       * 
                                       
                                         
                                           
                                             h 
                                             → 
                                           
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             , 
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               MB 
                                             
                                             , 
                                             
                                               t 
                                               - 
                                               
                                                 
                                                   
                                                     m 
                                                     ← 
                                                   
                                                   · 
                                                   Δ 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 t 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     τ 
                                                     → 
                                                   
                                                   
                                                     
                                                       n 
                                                       → 
                                                     
                                                     s 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       r 
                                                       ^ 
                                                     
                                                     , 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       MB 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     τ 
                                                     ← 
                                                   
                                                   
                                                     
                                                       n 
                                                       ← 
                                                     
                                                     s 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       r 
                                                       ^ 
                                                     
                                                     , 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       MB 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       * 
                                       
                                         
                                           
                                             h 
                                             ← 
                                           
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                             ′ 
                                           
                                           * 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               MB 
                                             
                                             , 
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               MB 
                                             
                                             , 
                                             
                                               
                                                 - 
                                                 t 
                                               
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       m 
                                                       ← 
                                                     
                                                     ′ 
                                                   
                                                   · 
                                                   Δ 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 t 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       * 
                                       
                                         
                                           
                                             h 
                                             ← 
                                           
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               r 
                                               ^ 
                                             
                                             , 
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               MB 
                                             
                                             , 
                                             
                                               t 
                                               - 
                                               
                                                 
                                                   
                                                     m 
                                                     ← 
                                                   
                                                   · 
                                                   Δ 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 t 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     τ 
                                                     ← 
                                                   
                                                   
                                                     
                                                       n 
                                                       ← 
                                                     
                                                     s 
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       r 
                                                       ^ 
                                                     
                                                     , 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       MB 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       * 
                                       
                                         
                                           v 
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                             ′ 
                                           
                                           * 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               t 
                                             
                                             - 
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         m 
                                                         → 
                                                       
                                                       ′ 
                                                     
                                                     + 
                                                     
                                                       
                                                         m 
                                                         ← 
                                                       
                                                       ′ 
                                                     
                                                   
                                                   ) 
                                                 
                                                 · 
                                                 Δ 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               t 
                                             
                                           
                                           ) 
                                         
                                       
                                       * 
                                       
                                         
                                           v 
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             t 
                                             - 
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       m 
                                                       → 
                                                     
                                                     + 
                                                     
                                                       m 
                                                       ← 
                                                     
                                                   
                                                   ) 
                                                 
                                                 · 
                                                 Δ 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               t 
                                             
                                             + 
                                             
                                               
                                                 
                                                   τ 
                                                   → 
                                                 
                                                 
                                                   
                                                     n 
                                                     → 
                                                   
                                                   s 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   , 
                                                   
                                                     
                                                       r 
                                                       ^ 
                                                     
                                                     MB 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               
                                                 
                                                   τ 
                                                   ← 
                                                 
                                                 
                                                   
                                                     n 
                                                     ← 
                                                   
                                                   s 
                                                 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     r 
                                                     ^ 
                                                   
                                                   , 
                                                   
                                                     
                                                       r 
                                                       ^ 
                                                     
                                                     MB 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   
                     ( 
                     
                       2.6 
                        
                       .1 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0106]    Remove the convolutions between all sub array impulse responses in equation 2.6.1, returning to the time domain results in equation 2.6.2. 
         [0107]    All other new parameters in equation 2.6.2 combined with the new notation that a right hand pointing arrow (→) indicates transmit direction and that a left hand pointing arrow (←) indicating receive direction. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       u 
                       B 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           ^ 
                         
                         , 
                         
                           
                             r 
                             ^ 
                           
                           MB 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ω 
                               ↔ 
                             
                             B 
                           
                           · 
                           D 
                         
                       
                       
                         
                           
                             N 
                             → 
                           
                           s 
                         
                         · 
                         
                           u 
                           g 
                         
                       
                     
                     · 
                     
                       
                         
                           a 
                           → 
                         
                         e 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             ^ 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             
                               e 
                               MB 
                             
                           
                           , 
                           
                             ω 
                             cw 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       
                         
                           a 
                           ← 
                         
                         e 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             ^ 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             
                               e 
                               MB 
                             
                           
                           , 
                           
                             ω 
                             cw 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       
                         
                           ω 
                           ↔ 
                         
                         B 
                       
                       
                         2 
                         · 
                         π 
                       
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               → 
                             
                             s 
                             ′ 
                           
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               → 
                             
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 → 
                               
                               s 
                             
                             = 
                             0 
                           
                           
                             
                               
                                 N 
                                 → 
                               
                               s 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                                 ′ 
                               
                               = 
                               0 
                             
                             
                               
                                 
                                   N 
                                   ← 
                                 
                                 s 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     ← 
                                   
                                   s 
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   
                                     N 
                                     ← 
                                   
                                   s 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 
                                   
                                     M 
                                     → 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       m 
                                       → 
                                     
                                     = 
                                     0 
                                   
                                   
                                     
                                       M 
                                       → 
                                     
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         
                                           m 
                                           ← 
                                         
                                         ′ 
                                       
                                       = 
                                       0 
                                     
                                     
                                       
                                         M 
                                         ← 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           m 
                                           ← 
                                         
                                         = 
                                         0 
                                       
                                       
                                         
                                           M 
                                           ← 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         
                                           w 
                                           → 
                                         
                                         
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                             ′ 
                                           
                                           , 
                                           
                                             
                                               m 
                                               → 
                                             
                                             ′ 
                                           
                                         
                                         * 
                                       
                                       · 
                                       
                                         
                                           w 
                                           → 
                                         
                                         
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                           
                                           , 
                                           
                                             m 
                                             → 
                                           
                                         
                                       
                                       · 
                                       
                                         
                                           w 
                                           ← 
                                         
                                         
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                             ′ 
                                           
                                           , 
                                           
                                             
                                               m 
                                               ← 
                                             
                                             ′ 
                                           
                                         
                                         * 
                                       
                                       · 
                                       
                                         
                                           w 
                                           ← 
                                         
                                         
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                           
                                           , 
                                           
                                             m 
                                             ← 
                                           
                                         
                                       
                                       · 
                                       
                                         
                                            
                                           
                                             
                                               - 
                                               j 
                                             
                                             · 
                                             
                                               ω 
                                               cw 
                                             
                                             · 
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             m 
                                                             → 
                                                           
                                                           ′ 
                                                         
                                                         - 
                                                         
                                                           m 
                                                           → 
                                                         
                                                         + 
                                                         
                                                           
                                                             m 
                                                             ← 
                                                           
                                                           ′ 
                                                         
                                                         - 
                                                         
                                                           m 
                                                           ← 
                                                         
                                                       
                                                       ) 
                                                     
                                                     · 
                                                     Δ 
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   t 
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       τ 
                                                       → 
                                                     
                                                     
                                                       
                                                         n 
                                                         → 
                                                       
                                                       s 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       , 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         MB 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       τ 
                                                       ← 
                                                     
                                                     
                                                       
                                                         n 
                                                         ← 
                                                       
                                                       s 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       , 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         MB 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         . 
                                         
                                           
 
                                         
                                          
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 
                                                   n 
                                                   → 
                                                 
                                                 e 
                                                 ′ 
                                               
                                               = 
                                               0 
                                             
                                             
                                               
                                                 
                                                   
                                                     N 
                                                     → 
                                                   
                                                   e 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       n 
                                                       → 
                                                     
                                                     s 
                                                     ′ 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               1 
                                             
                                           
                                            
                                           
                                             
                                               ∑ 
                                               
                                                 
                                                   
                                                     n 
                                                     → 
                                                   
                                                   e 
                                                 
                                                 = 
                                                 0 
                                               
                                               
                                                 
                                                   
                                                     
                                                       N 
                                                       → 
                                                     
                                                     e 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         → 
                                                       
                                                       s 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 1 
                                               
                                             
                                              
                                             
                                               
                                                 ∑ 
                                                 
                                                   
                                                     
                                                       n 
                                                       ← 
                                                     
                                                     e 
                                                     ′ 
                                                   
                                                   = 
                                                   0 
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         N 
                                                         ← 
                                                       
                                                       e 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           n 
                                                           ← 
                                                         
                                                         s 
                                                         ′ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   - 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 
                                                   ∑ 
                                                   
                                                     
                                                       
                                                         n 
                                                         ← 
                                                       
                                                       e 
                                                     
                                                     = 
                                                     0 
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           N 
                                                           ← 
                                                         
                                                         e 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                         
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     1 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         a 
                                                         → 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           e 
                                                           ′ 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           s 
                                                           ′ 
                                                         
                                                       
                                                       * 
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         → 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           e 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           s 
                                                         
                                                       
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         ← 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           e 
                                                           ′ 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                           ′ 
                                                         
                                                       
                                                       * 
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         ← 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           e 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                         
                                                       
                                                     
                                                     · 
                                                     
                                                       
                                                          
                                                         
                                                           j 
                                                           · 
                                                           
                                                             ω 
                                                             cw 
                                                           
                                                           · 
                                                           
                                                             [ 
                                                             
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                           
                                                         
                                                       
                                                       . 
                                                       
                                                         
 
                                                       
                                                        
                                                       sin 
                                                     
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     c 
                                                     [ 
                                                     
                                                       
                                                         
                                                           
                                                             ω 
                                                             ↔ 
                                                           
                                                           B 
                                                         
                                                         2 
                                                       
                                                       · 
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           + 
                                                           
                                                             
                                                             2 
                                                             · 
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             - 
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             ) 
                                                             
                                                             · 
                                                             Δ 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             t 
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                               
                                                             
                                                             
                                                             2 
                                                             · 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                             ( 
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                             * 
                                                             
                                                             
                                                             v 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             * 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             t 
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             + 
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             ) 
                                                             
                                                             · 
                                                             Δ 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             t 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             * 
                                                             
                                                             
                                                             v 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             t 
                                                             - 
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             + 
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             ) 
                                                             
                                                             · 
                                                             Δ 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             t 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.6 
                      
                     .2 
                   
                   ) 
                 
               
             
           
         
       
     
         [0108]    If we further assume a perfect point target and ideal repetition jammers combined with a perfect matched filter the equality in equation 2.6.3 is true. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                           ′ 
                         
                         * 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             t 
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                     * 
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                         
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       v 
                       
                         
                           n 
                           → 
                         
                         s 
                         ′ 
                       
                       * 
                     
                     ⊙ 
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                         
                       
                        
                       
                         ( 
                         
                           t 
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.6 
                      
                     .3 
                   
                   ) 
                 
               
             
           
         
       
     
         [0109]    Where ⊙ is the correlation operator, defined as: 
         [0000]    
       
         
           
             
               
                 g 
                 * 
               
               ⊙ 
               
                 h 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
             
             = 
             
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       g 
                       * 
                     
                      
                     
                       ( 
                       
                         τ 
                         - 
                         t 
                       
                       ) 
                     
                   
                   · 
                   
                     h 
                      
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   · 
                   
                      
                     t 
                   
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   ∞ 
                 
                  
                 
                   
                     
                       g 
                       * 
                     
                      
                     
                       ( 
                       τ 
                       ) 
                     
                   
                   · 
                   
                     h 
                      
                     
                       ( 
                       
                         τ 
                         + 
                         t 
                       
                       ) 
                     
                   
                   · 
                   
                      
                     τ 
                   
                 
               
             
           
         
       
     
         [0110]    For the two special, but very important, cases when all the waveforms V {right arrow over (n)}     s   (t) are correlated or uncorrelated wideband white noise equation 2.6.3 could be further simplified as indicated in equation 2.6.4 for the case with correlated wideband white noise, and as indicated in equation 2.6.5 for the case with uncorrelated wideband white noise. Only these two examples are treated in the analysis to follow. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                           ′ 
                         
                         * 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             t 
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                     * 
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                         
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       u 
                       g 
                       2 
                     
                     · 
                     
                       δ 
                        
                       
                         ( 
                         
                           t 
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.6 
                      
                     .4 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                           ′ 
                         
                         * 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             t 
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                     * 
                     
                       
                         v 
                         
                           
                             n 
                             → 
                           
                           s 
                         
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       u 
                       g 
                       2 
                     
                     · 
                     
                       δ 
                       
                         
                           
                             n 
                             → 
                           
                           s 
                           ′ 
                         
                         , 
                         
                           
                             n 
                             → 
                           
                           s 
                         
                       
                     
                     · 
                     
                       δ 
                        
                       
                         ( 
                         
                           t 
                           + 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     → 
                                   
                                   + 
                                   
                                     
                                       m 
                                       ← 
                                     
                                     ′ 
                                   
                                   - 
                                   
                                     m 
                                     ← 
                                   
                                 
                                 ) 
                               
                               · 
                               Δ 
                             
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                           
                             
                               
                                 τ 
                                 → 
                               
                               
                                 
                                   n 
                                   → 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 τ 
                                 ← 
                               
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 , 
                                 
                                   
                                     r 
                                     ^ 
                                   
                                   MB 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.6 
                      
                     .5 
                   
                   ) 
                 
               
             
           
         
       
     
         [0111]    In equation 2.6.4 and 2.6.5 are 
         [0000]    
       
         
           
             δ 
             
               
                 
                   n 
                   → 
                 
                 s 
                 ′ 
               
               , 
               
                 
                   n 
                   → 
                 
                 s 
               
             
           
         
       
     
         [0000]    equal to the Kronecker delta function and δ(t) equal to the Dirac delta function introduced. Observe that the transmitter bandwidth is determined by the transmit sub array impulse response, {right arrow over (h)} {right arrow over (n)}     s   ({circumflex over (r)},{circumflex over (r)} MB ,t), and that the waveform characteristics is determined by the transmit FIR-filter coefficients, {right arrow over (w)} {right arrow over (n)}     s     ,{right arrow over (m)} , for each transmit sub array, in the special but very important cases, when equation 2.6.4 or equation 2.6.5 is valid. We are now in a position where we can calculate the received baseband waveform after matched filtering for the special cases when the waveforms V {right arrow over (n)}     s   (t) are correlated or uncorrelated wideband white noise. The result is given in equation 2.6.6. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         u 
                         B 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             ^ 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             MB 
                           
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             u 
                             g 
                           
                           · 
                           
                             
                               
                                 
                                   ω 
                                   ↔ 
                                 
                                 B 
                               
                               · 
                               D 
                             
                           
                         
                         
                           2 
                           · 
                           π 
                           · 
                           
                             
                               N 
                               → 
                             
                             s 
                           
                         
                       
                       · 
                       
                         
                           
                             a 
                             → 
                           
                           e 
                         
                          
                         
                           ( 
                           
                             
                               r 
                               ^ 
                             
                             , 
                             
                               
                                 r 
                                 ^ 
                               
                               
                                 e 
                                 MB 
                               
                             
                             , 
                             
                               ω 
                               cw 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     · 
                     
                       
                         
                           a 
                           ← 
                         
                         e 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             ^ 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             
                               e 
                               MB 
                             
                           
                           , 
                           
                             ω 
                             cw 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               → 
                             
                             s 
                             ′ 
                           
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               → 
                             
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 → 
                               
                               s 
                             
                             = 
                             0 
                           
                           
                             
                               
                                 N 
                                 → 
                               
                               s 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 
                                   n 
                                   ← 
                                 
                                 s 
                                 ′ 
                               
                               = 
                               0 
                             
                             
                               
                                 
                                   N 
                                   ← 
                                 
                                 s 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     ← 
                                   
                                   s 
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   
                                     N 
                                     ← 
                                   
                                   s 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 
                                   
                                     M 
                                     → 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       m 
                                       → 
                                     
                                     = 
                                     0 
                                   
                                   
                                     
                                       M 
                                       → 
                                     
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         
                                           m 
                                           ← 
                                         
                                         ′ 
                                       
                                       = 
                                       0 
                                     
                                     
                                       
                                         M 
                                         ← 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           m 
                                           ← 
                                         
                                         = 
                                         0 
                                       
                                       
                                         
                                           M 
                                           ← 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                         
                                           w 
                                           → 
                                         
                                         
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                             ′ 
                                           
                                           , 
                                           
                                             
                                               m 
                                               → 
                                             
                                             ′ 
                                           
                                         
                                         * 
                                       
                                       · 
                                       
                                         
                                           w 
                                           → 
                                         
                                         
                                           
                                             
                                               n 
                                               → 
                                             
                                             s 
                                           
                                           , 
                                           
                                             m 
                                             → 
                                           
                                         
                                       
                                       · 
                                       
                                         
                                           w 
                                           ← 
                                         
                                         
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                             ′ 
                                           
                                           , 
                                           
                                             
                                               m 
                                               ← 
                                             
                                             ′ 
                                           
                                         
                                         * 
                                       
                                       · 
                                       
                                         
                                           w 
                                           ← 
                                         
                                         
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                           
                                           , 
                                           
                                             m 
                                             ← 
                                           
                                         
                                       
                                       · 
                                       
                                         { 
                                         
                                           
                                             
                                               1 
                                             
                                             
                                               
                                                 
                                                   if 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   η 
                                                 
                                                 = 
                                                 0 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 δ 
                                                 
                                                   
                                                     
                                                       n 
                                                       → 
                                                     
                                                     s 
                                                     ′ 
                                                   
                                                   , 
                                                   
                                                     
                                                       n 
                                                       → 
                                                     
                                                     s 
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   if 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   η 
                                                 
                                                 = 
                                                 1 
                                               
                                             
                                           
                                         
                                         } 
                                       
                                       · 
                                       
                                         
                                            
                                           
                                             
                                               - 
                                               j 
                                             
                                             · 
                                             
                                               ω 
                                               cw 
                                             
                                             · 
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             m 
                                                             → 
                                                           
                                                           ′ 
                                                         
                                                         - 
                                                         
                                                           m 
                                                           → 
                                                         
                                                         + 
                                                         
                                                           
                                                             m 
                                                             ← 
                                                           
                                                           ′ 
                                                         
                                                         - 
                                                         
                                                           m 
                                                           ← 
                                                         
                                                       
                                                       ) 
                                                     
                                                     · 
                                                     Δ 
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   t 
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       τ 
                                                       → 
                                                     
                                                     
                                                       
                                                         n 
                                                         → 
                                                       
                                                       s 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       , 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         MB 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     
                                                       τ 
                                                       ← 
                                                     
                                                     
                                                       
                                                         n 
                                                         ← 
                                                       
                                                       s 
                                                     
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         r 
                                                         ^ 
                                                       
                                                       , 
                                                       
                                                         
                                                           r 
                                                           ^ 
                                                         
                                                         MB 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         . 
                                         
                                           
 
                                         
                                          
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 
                                                   n 
                                                   → 
                                                 
                                                 e 
                                                 ′ 
                                               
                                               = 
                                               0 
                                             
                                             
                                               
                                                 
                                                   
                                                     N 
                                                     → 
                                                   
                                                   e 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       n 
                                                       → 
                                                     
                                                     s 
                                                     ′ 
                                                   
                                                   ) 
                                                 
                                               
                                               - 
                                               1 
                                             
                                           
                                            
                                           
                                             
                                               ∑ 
                                               
                                                 
                                                   
                                                     n 
                                                     → 
                                                   
                                                   e 
                                                 
                                                 = 
                                                 0 
                                               
                                               
                                                 
                                                   
                                                     
                                                       N 
                                                       → 
                                                     
                                                     e 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         n 
                                                         → 
                                                       
                                                       s 
                                                     
                                                     ) 
                                                   
                                                 
                                                 - 
                                                 1 
                                               
                                             
                                              
                                             
                                               
                                                 ∑ 
                                                 
                                                   
                                                     
                                                       n 
                                                       ← 
                                                     
                                                     e 
                                                     ′ 
                                                   
                                                   = 
                                                   0 
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         N 
                                                         ← 
                                                       
                                                       e 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           n 
                                                           ← 
                                                         
                                                         s 
                                                         ′ 
                                                       
                                                       ) 
                                                     
                                                   
                                                   - 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 
                                                   ∑ 
                                                   
                                                     
                                                       
                                                         n 
                                                         ← 
                                                       
                                                       e 
                                                     
                                                     = 
                                                     0 
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           N 
                                                           ← 
                                                         
                                                         e 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                         
                                                         ) 
                                                       
                                                     
                                                     - 
                                                     1 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         a 
                                                         → 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           e 
                                                           ′ 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           s 
                                                           ′ 
                                                         
                                                       
                                                       * 
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         → 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           e 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             → 
                                                           
                                                           s 
                                                         
                                                       
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         ← 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           e 
                                                           ′ 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                           ′ 
                                                         
                                                       
                                                       * 
                                                     
                                                     · 
                                                     
                                                       
                                                         a 
                                                         ← 
                                                       
                                                       
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           e 
                                                         
                                                         , 
                                                         
                                                           
                                                             n 
                                                             ← 
                                                           
                                                           s 
                                                         
                                                       
                                                     
                                                     · 
                                                     
                                                       
                                                          
                                                         
                                                           j 
                                                           · 
                                                           
                                                             ω 
                                                             cw 
                                                           
                                                           · 
                                                           
                                                             [ 
                                                             
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             α 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                           
                                                         
                                                       
                                                       . 
                                                       
                                                         
 
                                                       
                                                        
                                                       sin 
                                                     
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     c 
                                                     [ 
                                                     
                                                       
                                                         
                                                           
                                                             ω 
                                                             ↔ 
                                                           
                                                           B 
                                                         
                                                         2 
                                                       
                                                       · 
                                                       
                                                         ( 
                                                         
                                                           t 
                                                           + 
                                                           
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             - 
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                             ) 
                                                             
                                                             + 
                                                             
                                                             3 
                                                             · 
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             - 
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             · 
                                                             Δ 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             t 
                                                           
                                                           + 
                                                           
                                                             2 
                                                             · 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                           + 
                                                           
                                                             3 
                                                             · 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.6 
                      
                     .6 
                   
                   ) 
                 
               
             
           
         
       
     
         [0112]    Where η=0 if V {right arrow over (n)}     s   (t) are correlated wideband white noise waveforms and η=1 if V {right arrow over (n)}     s   (t) are uncorrelated wideband white noise waveforms. It should be observed that there is effectively one summation less on the second row in equation 2.6.6 if η=1, due to the 
         [0000]    
       
         
           
             δ 
             
               
                 
                   n 
                   → 
                 
                 s 
                 ′ 
               
               , 
               
                 
                   n 
                   → 
                 
                 s 
               
             
           
         
       
     
         [0000]    factor. The influence of the 
         [0000]    
       
         
           
             δ 
             
               
                 
                   n 
                   → 
                 
                 s 
                 ′ 
               
               , 
               
                 
                   n 
                   → 
                 
                 s 
               
             
           
         
       
     
         [0000]    factor could be accounted for by removing the summation over {right arrow over (n)}′ s  and replacing {right arrow over (n)}′ s  with {right arrow over (n)} s  in the remaining part of the equation. 
         [0113]    Normalised Signal to Noise-Jammer Ratio 
         [0114]    The signal power can be estimated as the expected value, of |u B ({circumflex over (r)} MB ,{circumflex over (r)} MB ,t)| 2 , since u B ({circumflex over (r)} MB ,{circumflex over (r)} MB ,t) is an effective value, as indicated in equation 2.7.1. 
         [0000]        S=E[|u   B ( {circumflex over (r)}   MB   ,{circumflex over (r)}   MB   ,t )| 2   ]=E[u*   B ( {circumflex over (r)}   MB   ,{circumflex over (r)}   MB   ,t )· u   B ( {circumflex over (r)}   MB   ,{circumflex over (r)}   MB   ,t )]  (Equation 2.7.1)
 
         [0115]    Where E[f(x)] is the expected value of f(x). The final expression for the signal power, found in equation 2.7.2, is obtained by inserting equation 2.6.6 twice into equation 2.7.1. Observe that all indexes equipped with a back prime originate from the complex conjugated factor in all power calculations. {right arrow over (EP)}({circumflex over (r)} MB ,{circumflex over (r)} e     MB   ,ω cw ) and          ({circumflex over (r)} MB ,{circumflex over (r)} e     MB   ,ω cw ) is respectively the transmit and receive element power pattern. Since equation 2.7.2 is an extensive equation it is found at the end of this document under Equations. 
         [0116]    If η=1 could the influence of the 
         [0000]    
       
         
           
             
               δ 
               
                 
                   ` 
                    
                   
                       
                   
                    
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                   
                 
                 , 
                 
                   ` 
                    
                   
                       
                   
                    
                   
                     
                       n 
                       → 
                     
                     s 
                   
                 
               
             
             · 
             
               δ 
               
                 
                     
                 
                  
                 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                   
                   , 
                   
                       
                   
                    
                   
                     
                       n 
                       → 
                     
                     s 
                   
                 
               
             
           
         
       
     
         [0000]    factor be accounted for by removing the summations over ‘{right arrow over (n)}′ s  and {right arrow over (n)}′ s  and replacing ‘{right arrow over (n)}′ s  with ‘{right arrow over (n)} s  and {right arrow over (n)}′ s  with {right arrow over (n)} s  in the remaining part of the equation. The jammer power, J, not to be confused with the total number of jammers, can be estimated in a similar way, as the signal power, with equation 2.7.3. 
         [0000]    
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         
                           J 
                           - 
                           1 
                         
                       
                        
                       
                         E 
                          
                         
                           [ 
                           
                             
                                
                               
                                 
                                   u 
                                   B 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       Jj 
                                     
                                     , 
                                     
                                       
                                         r 
                                         ^ 
                                       
                                       MB 
                                     
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         
                           J 
                           - 
                           1 
                         
                       
                        
                       
                         E 
                          
                         
                           [ 
                           
                             
                               
                                 u 
                                 B 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     Jj 
                                   
                                   , 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     MB 
                                   
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             · 
                             
                               
                                 u 
                                 B 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     Jj 
                                   
                                   , 
                                   
                                     
                                       r 
                                       ^ 
                                     
                                     MB 
                                   
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.7 
                      
                     .3 
                   
                   ) 
                 
               
             
           
         
       
     
         [0117]    The final expression for the jammer power, found in equation 2.7.4, is obtained by inserting equation 2.6.6 twice into equation 2.7.3. Since equation 2.7.4 is an extensive equation it is found at the end of this document under Equations. 
         [0118]    The receiver noise power can, be estimated with equation 2.7.5, since ν Rx ({circumflex over (r)} mB ,{circumflex over (r)} MB ,t) is an effective value. Since equation 2.7.5 is an extensive equation it is found at the end of this document under Equations. 
         [0119]    Where υ p,q (t) is the noise voltage in element number p in sub array number q. It is further assumed that the noise powers in all antenna elements are uncorrelated with equal power density, onwards denoted as N 0  [W/rad/s] (N 0 =k·T/(2·π). Where k and T respectively is equal to Boltzmanns constant and the equivalent noise temperature of each antenna element. The final expression for the receiver noise power, found in equation 2.7.6, is obtained by inserting equation 2.4.2 twice into equation 2.7.5. Since equation 2.7.6 is an extensive equation it is found at the end of this document under Equations. 
         [0120]    The influence of the 
         [0000]    
       
         
           
             δ 
             
               
                 ` 
                  
                 
                     
                 
                  
                 
                   
                     n 
                     ← 
                   
                   s 
                   ′ 
                 
               
               , 
               
                   
               
                
               
                 
                   n 
                   ← 
                 
                 s 
               
             
           
         
       
     
         [0000]    factor could be accounted for by removing the summation over           and replacing           with           in the remaining part of the equation. The 
         [0000]    
       
         
           
             δ 
             
               
                 ` 
                  
                 
                     
                 
                  
                 
                   
                     n 
                     ← 
                   
                   e 
                 
               
               , 
               
                 
                   n 
                   ← 
                 
                 e 
               
             
           
         
       
     
         [0000]    factor could be accounted for in an analogous manner. 
         [0121]    Next we introduce the “Normalised Signal to Noise-Jammer Ratio”, NSNJR defined according to equation 2.7.7. This ratio describes the SNR (Signal Noise Ration) reduction due to the jamming and is always less than unity (0 dB). 
         [0000]    
       
         
           
             
               
                 
                   NSNJR 
                   = 
                   
                     
                       
                         S 
                         
                           N 
                           + 
                           J 
                         
                       
                       
                         
                           S 
                           
                             m 
                              
                             
                                 
                             
                              
                             ax 
                           
                         
                         N 
                       
                     
                     = 
                     
                       
                         S 
                         
                           S 
                           
                             ma 
                              
                             
                                 
                             
                              
                             x 
                           
                         
                       
                       · 
                       
                         1 
                         
                           1 
                           + 
                           
                             J 
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.7 
                      
                     .7 
                   
                   ) 
                 
               
             
           
         
       
     
         [0122]    In equation (2.7.7) is S max  defined as the maximum signal power. The maximum signal power is obtained when {right arrow over (α)}=         =0, {right arrow over (β)}=         ={right arrow over (γ)}=         =1, see the nomenclature chapter. S max  can be expressed as indicated in equation 2.7.8. Since equation 2.7.8 is an extensive equation it is found at the end of this document under Equations. 
         [0123]    Transmitted Power 
         [0124]    The transmitted power in the {circumflex over (r)} direction, P Tx ({circumflex over (r)},{circumflex over (r)} MB ), can be calculated with equation 2.8.1, since V Tx ({circumflex over (r)},{circumflex over (r)} MB ,t) is an effective value. 
         [0000]        P   Tx ( {circumflex over (r)},{circumflex over (r)}   MB )= E[|V   Tx ( {circumflex over (r)},{circumflex over (r)}   MB   ,t )| 2   ]=E[V*   Tx ( {circumflex over (r)},{circumflex over (r)}   MB   ,t )· V   Tx ( {circumflex over (r)},{circumflex over (r)}   MB   ,t )]  (Equation 2.8.1)
 
         [0125]    The final expression for the transmitted power in the {circumflex over (r)} direction, P Tx ({circumflex over (r)},{circumflex over (r)} MB ), is found equation 2.8.2. Since equation 2.8.2 is an extensive equation it is found at the end of this document under Equations. Equation 2.8.2 can be used in equation 2.3.3 to calculate P Tx ({circumflex over (r)} J     j   ,{circumflex over (r)} MB ), it can also be used to calculate the wideband transmit antenna pattern as well as the main beam power for a given set of transmit FIR-filter coefficients. 
         [0126]    And finally, equation 2.7.2, 2.7.8, 2.7.4, and 2.7.6 can be inserted into equation 2.7.7 for calculation of NSNJR. The values of the {right arrow over (w)}-matrix and the          -matrix that maximizes the NSNJR are to prefer. The {right arrow over (w)}-matrix shapes the transmit waveform and the          -matrix shapes the receive antenna pattern. 
         [0127]    Auto Correlation Function of the Transmitted Waveform in the Main Beam Direction 
         [0128]    The auto correlation function of the transmitted waveform in the main beam direction, R Tx (t), determines the range resolution. This important function can be calculated in a way similar to equation 2.8.2, the result is found in equation 2.9.1. Since equation 2.9.1 is an extensive equation it is found at the end of this document under Equations. 
         [0129]    Transmitted Power Spectrum in the Main Beam Direction 
         [0130]    The resulting transmitted power spectrum in the main beam direction, S Tx (ω), is equal to the Fourier transform of the auto correlation function of the transmitted waveform in the main beam direction, R Tx (τ). It is thus possible to calculate the resulting transmitted power spectrum in the main beam direction by Fourier transformation of equation 2.9.1, the result is found in equation 2.10.1. 
       Examples of the Present Invention 
       [0131]    Examples of the theory previously described will now follow. 
         [0132]    Assumptions and input values
       Uniform linear array antenna geometry, common for both transmit and receive
           Three equal linear sub arrays ({right arrow over (N)} s =           s =N s =3) equipped with five antenna
               elements each (∀{right arrow over (N)} e ({right arrow over (n)} s )=∀           e (           s )=N e =5)   
               Element separation: d=14.276 mm   
           Element data, common for both transmit and receive
           Element main beam direction: {circumflex over (r)} e     MB   ={circumflex over (z)}=(0 0 1)   Element power pattern:   
               
 
         [0000]        {right arrow over (EP)} ( {circumflex over (r)},{circumflex over (r)}   e     MB   ,ω cw )=         ( {circumflex over (r)},{circumflex over (r)}   e     MB   ,ω cw )≈ {circumflex over (r)}   T   ·{circumflex over (z)} =cos(θ)
           Uniform element excitation:           
 
         [0000]    
       
         
           
             
               ∀ 
               
                 a 
                 → 
               
             
             = 
             
               
                 ∀ 
                 
                   a 
                   → 
                 
               
               = 
               
                 1 
                 
                   N 
                   e 
                 
               
             
           
         
       
       
         
           
             Array antenna control, common for both transmit and receive
           Main beam azimuth angle: Az MB =30°   Main beam elevation angle: El MB =0°
               Resulting main beam unit vector: {circumflex over (r)} MB =[0,500 0,000 0,866] T      
               TTD-control of each element in each sub array: γ=1   TTD-control of each sub array in the array antenna: α=0, β=1   
         
             Transmit FIR-filter
           Number taps: {right arrow over (M)}=4   Delay between FIR-filter taps: Δt=1 ns   
         
             Receive FIR-filter
           Number taps:          =1,          =1   
         
             Transmitter and receiver
           Centre angular frequency: {right arrow over (ω)} c =           c =           c =ω c =2·π·9.0 GHz   Bandwidth: {right arrow over (ω)} B =           B =           B =ω B =2·π·1.0 GHz   
         
             Waveform
           White bandwidth limited Gaussian noise
               Correlated (η=0) between transmit sub arrays   Uncorrelated (η=1) between transmit sub arrays   
               Centre angular frequency: ω cw =ω c =2·π·9.0 GHz   Bandwidth: ω Bw =ω B =2·π·1.0 GHz   
         
             Jammer data
           Number of jammers: J=1   Jammer number 1
               Repetition jamming   Jammer power normalised to the equivalent receiver noise level i n the antenna element:   
               
         
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     NJP 
                     1 
                   
                   = 
                   
                     
                       
                         
                           PG 
                           
                             J 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           
                             N 
                             0 
                           
                           · 
                           
                             
                               ω 
                               ← 
                             
                             B 
                           
                         
                       
                       · 
                       
                         
                           ( 
                           
                             
                               c 
                               0 
                             
                             
                               2 
                               · 
                               
                                 ω 
                                 cw 
                               
                               · 
                               
                                 r 
                                 
                                   J 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     
                       20 
                        
                       
                           
                       
                        
                       dB 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Azimuth angle: Az J1 =−45.0° 
                     Elevation angle: El J1 =0.0°
                   Resulting unit vector: {circumflex over (r)} J1 =[−0.707 0,000 0.707] T  (Observe the index in all equations; Jammer number 1 has index number 0, and so on)   
                 
                   
                 
               
             
           
         
       
     
         [0169]    With these assumptions and input values we can make these observations:
       S=S max      ω c ·Δt=2·π·int (Where int is an integer value)   ω B ·Δt=2·π       
 
         [0173]    Based on these observations following simplifications are possible:
       Equation 2.8.2, transmitted power, P Tx  result in equation 3.2.1. Since equation 3.2.1 is an extensive equation it is found at the end of this document under Equations.   Equation 2.7.2, received signal power, S result in equation 3.2.2. Since equation 3.2.2 is an extensive equation it is found at the end of this document under Equations.   Equation 2.7.6, received noise power, N:       
 
         [0000]    
       
         
           
             
               
                 
                   N 
                   = 
                   
                     
                       
                         
                           N 
                           0 
                         
                         · 
                         
                           ω 
                           B 
                         
                         · 
                         
                           N 
                           s 
                         
                       
                       
                         N 
                         e 
                       
                     
                     · 
                     
                       
                         ∑ 
                         
                           
                             
                               n 
                               → 
                             
                             s 
                             ′ 
                               
                               
                             ‵ 
                           
                           = 
                           0 
                         
                         
                           
                             N 
                             s 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             
                               
                                 n 
                                 → 
                               
                               s 
                               ′ 
                             
                             = 
                             0 
                           
                           
                             
                               N 
                               s 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               
                                 
                                   m 
                                   → 
                                 
                                   
                                 ′ 
                                   
                                   
                                 ‵ 
                               
                               = 
                               0 
                             
                             
                               
                                 M 
                                 → 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   
                                     m 
                                     → 
                                   
                                   ′ 
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   M 
                                   → 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   w 
                                   → 
                                 
                                 
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                     ′ 
                                       
                                       
                                     ‵ 
                                   
                                   , 
                                   
                                     
                                       m 
                                       → 
                                     
                                       
                                     ′ 
                                       
                                       
                                     ‵ 
                                   
                                 
                               
                               · 
                               
                                 
                                   w 
                                   → 
                                 
                                 
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                     ′ 
                                   
                                   , 
                                   
                                     
                                       m 
                                       → 
                                     
                                     ′ 
                                   
                                 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     3.2 
                      
                     .3 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             Equation 2.7.4, received jammer power, J, result in equation 3.2.4. Since equation 3.2.4 is an extensive equation it is found at the end of this document under Equations. 
             Equation 2.7.7, “Normalised Signal to Noise-Jammer Ratio”, NSNJR: 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   NSNJR 
                   = 
                   
                     1 
                     
                       1 
                       + 
                       
                         J 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     3.2 
                      
                     .5 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             Equation 2.9.1, radar range resolution, R Tx (τ), result in equation 3.2.6. Since equation 3.2.6 is an extensive equation it is found at the end of this document under Equations. 
             Equation 2.10.1, transmitted power spectrum in the main beam direction, P Tx (ω) result in equation 3.2.7. Since equation 3.2.7 is an extensive equation it is found at the end of this document under Equations. 
           
         
       
     
         [0181]    If equation 3.2.4 and 3.2.3 are inserted into equation 3.2.5 and usage is made of equation 2.3.3 together with equation 3.2.1 it is possible to formulate the final expression for the “Normalised Signal to Noise-Jammer Ratio”, The result is found in equation 3.2.8. Since equation 3.2.8 is an extensive equation it is found at the end of this document under Equations. 
         [0182]    The remaining problem is to find the transmit FIR-filter coefficients, {right arrow over (w)}, that maximises NSNJR. This will be done by a numerical search. Observe 
         [0000]    
       
         
           
             
               that 
                
               
                   
               
                
               
                 
                   Λ 
                   
                     
                       
                         n 
                         → 
                       
                       s 
                       ′ 
                         
                         
                       ‵ 
                     
                     , 
                     
                       
                         n 
                         → 
                       
                       s 
                         
                         
                         
                       ‵ 
                     
                     , 
                     
                       
                         n 
                         → 
                       
                       s 
                       ′ 
                     
                     , 
                     
                       
                         n 
                         → 
                       
                       s 
                     
                   
                 
                  
                 
                   ( 
                   η 
                   ) 
                 
               
             
             , 
             
               
                 Λ 
                 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                   
                 
                 ′ 
               
                
               
                 ( 
                 η 
                 ) 
               
             
             , 
             
               
                 Ψ 
                 
                   
                     
                       n 
                       → 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                   
                   , 
                   j 
                 
               
                
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   MB 
                 
                 ) 
               
             
             , 
             
               
 
             
              
             
               
                 
                   ψ 
                   ↔ 
                 
                 
                   
                     
                       m 
                       → 
                     
                       
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                         
                       ‵ 
                     
                      
                     
                       m 
                       → 
                     
                   
                   , 
                   
                     
                       m 
                       → 
                     
                     ′ 
                   
                   , 
                   
                     m 
                     → 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                     ′ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                   
                   , 
                   j 
                 
               
                
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   MB 
                 
                 ) 
               
             
             , 
             
               A 
               
                 
                   
                     m 
                     → 
                   
                     
                   ′ 
                     
                     
                   ‵ 
                 
                 , 
                 
                   
                       
                     ‵ 
                   
                    
                   
                     m 
                     → 
                   
                 
                 , 
                 
                   
                     m 
                     → 
                   
                   ′ 
                 
                 , 
                 
                   m 
                   → 
                 
               
             
             , 
             
               
 
             
              
             
               
                 B 
                 
                   
                     
                       n 
                       → 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     e 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     e 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     e 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     e 
                   
                   , 
                   j 
                 
               
                
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   MB 
                 
                 ) 
               
             
             , 
             and 
           
         
       
       
         
           
             
               Γ 
               
                 
                   
                     n 
                     → 
                   
                   s 
                   ′ 
                     
                     
                   ‵ 
                 
                 , 
                 
                   
                     n 
                     → 
                   
                   s 
                     
                     
                     
                   ‵ 
                 
                 , 
                 
                   
                     n 
                     → 
                   
                   s 
                   ′ 
                 
                 , 
                 
                   
                     n 
                     → 
                   
                   s 
                 
                 , 
                 
                   
                     n 
                     ← 
                   
                   s 
                   ′ 
                     
                     
                   ‵ 
                 
                 , 
                 
                   
                     n 
                     ← 
                   
                   s 
                     
                     
                     
                   ‵ 
                 
                 , 
                 
                   
                     n 
                     ← 
                   
                   s 
                   ′ 
                 
                 , 
                 
                   
                     n 
                     ← 
                   
                   s 
                 
               
             
              
             
               ( 
               
                 
                   r 
                   ^ 
                 
                 MB 
               
               ) 
             
           
         
       
     
         [0000]    theoretically all could be calculated prior to the numerical search, to reduce the computation time, since they all are independent of {right arrow over (ω)}. However in the practical implementation on a standard PC the working memory is not sufficient to hold for instance 
         [0000]    
       
         
           
             
               
                 
                   ψ 
                   ↔ 
                 
                 
                   
                     
                       m 
                       → 
                     
                       
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                         
                       ‵ 
                     
                      
                     
                       m 
                       → 
                     
                   
                   , 
                   
                     
                       m 
                       → 
                     
                     ′ 
                   
                   , 
                   
                     m 
                     → 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                     ′ 
                   
                   , 
                   
                     
                       n 
                       → 
                     
                     s 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                     ′ 
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                       
                       
                       
                     ‵ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                     ′ 
                   
                   , 
                   
                     
                       n 
                       ← 
                     
                     s 
                   
                   , 
                   j 
                 
               
                
               
                 ( 
                 
                   
                     r 
                     ^ 
                   
                   MB 
                 
                 ) 
               
             
             . 
           
         
       
     
         [0000]    The amount of calculations prior to the numerical search has to be matched against the available amount working memory on the computer used for the calculations. 
       Results for Examples of the Present Invention 
       [0183]    An iterative search is performed for the transmit FIR-filter coefficient, {right arrow over (w)} p,q , minimizing the “Normalised Signal to Noise-Jammer Ratio”, NSNJR. Initial values according to  FIG. 5  is used, both for an example with correlated white bandwidth limited Gaussian noise and for an example with uncorrelated white bandwidth limited Gaussian noise, in analogy with a fourth order Barker code. 
         [0184]    An example is for the correlated white bandwidth limited Gaussian noise, where the convergence of the iterative process is schematically illustrated in  FIG. 6 . This example comprises one noise generator connected to several FIR-filters which are connected to sub array antenna elements.  FIG. 6  schematically illustrate the minimizing of NSNJR in relation to the number of iterations. 
         [0185]    The wideband antenna pattern, the transmitted power spectrum in the main beam direction, and the autocorrelation function of the waveform transmitted in the main beam direction are plotted in  FIG. 7 ,  FIG. 8 , and  FIG. 9  respectively for the transmit FIR-filter coefficients obtained after one, two, five, ten, and twenty iterations. In  FIG. 8  and  FIG. 9 , the Fourier transform pair formed by the power spectrum and the autocorrelation function in the main bean direction is visualized. 
         [0186]    An example is for the uncorrelated white bandwidth limited Gaussian noise, where the convergence of the iterative process is schematically illustrated in  FIG. 10 . This example comprises several uncorrelated noise generators each connected to the input of one FIR-filter. The output of each FIR-filter is connected to one sub array.  FIG. 10  schematically illustrate the minimizing of NSNJR in relation to the number of iterations. 
         [0187]    The wideband antenna pattern, the transmitted power spectrum in the main beam direction, and the autocorrelation function of the waveform transmitted in the main beam direction are plotted in  FIG. 11 ,  FIG. 12 , and  FIG. 13  respectively for the transmit FIR-filter coefficients obtained after one, two, five, ten, twenty, fifty, and one hundred iterations. The wideband transmit antenna pattern illustrated in  FIG. 11  has a lower gain compared to the example with correlated white bandwidth limited Gaussian noise illustrated in  FIG. 8  due to the much wider main beam. The wide main beam, for the example with uncorrelated white bandwidth limited Gaussian noise, reflects the size of the sub array rather than the size of the antenna. This might at a first glance appear as a loss, but it is possible to regain this loss by extension of the integration time on the receive side if a cluster of receive beams covering the total transmit main beam is used. This non-loss both improves the LPI, Low Probability of Intercept, performance and decreases the reaction time of the system. One receiver is needed for each receive main beam. 
         [0188]    The side lobe level of the wideband antenna pattern for a few side lobes, particularly for the side lobes adjacent to the main beam and also in the direction of the repetition jammer, increases during the convergence of the iterative process for the example with correlated white bandwidth limited Gaussian noise while the wideband antenna pattern for the example with uncorrelated white bandwidth limited Gaussian noise remains stable, even in the direction of the repetition jammer, during the convergence of the iterative process. The behavior of both the power spectrum and the autocorrelation function in the main beam direction is very similar for both examples. The power spectrum becomes more narrowbanded with a smooth maximum during the convergence of the iterative process and this results naturally in a wider peak of the auto correlation function with reduced side lobes. In both examples are the minima on each side of the peak of the auto correlation function gradually filled out during the convergence of the iterative process. Values of NSNJR close to 0 dB are obtained after convergence of the iterative process. 
         [0189]    An antenna operating under wideband conditions radiates directional dependent waveforms. It is thus not possible to separate spatial and temporal filtering, as is usually done in narrow band systems, if the antenna operates under these conditions. The waveform radiated towards the target and the waveform radiated towards the jammer is thus different. The jammer may be a repetition jammer or a strong clutter reflection. In the examples of the present invention the matched filter are matched to the waveform radiated and received in the main beam direction and demonstrate that considerable attenuation, in the matched filter, of the waveform returned from the jammed direction can be obtained by proper waveform selection. This is equivalent to reception of nearly orthogonal waveforms in the direction from the jammer and in the direction from the target. 
         [0190]    It should further be noticed that no spatial receive cancellation is used in the examples of the present invention to ensure that all attenuation of the jammer is achieved by proper waveform selection only. The combination of proper waveform selection, deterministic spatial transmit cancellation, and adaptive spatial receive cancellation is to prefer in an operational system. The above experimental results indicate that an uncorrelated noise source preceding each transmit FIR-filter combined with a cluster of receive beams covering the total transmit main beam suppress repetition jammer and clutter signals. The gain loss due to the wide transmit main beam is fully regained by a proportional extension of the integration time. The main beam width reflects the size of the sub array rather than the size of the antenna. One signal processing chain is needed for each receive main beam. 
         [0191]    The present invention enables suppression of repetition jammers by proper waveform selection. Further, the present invention also enables clutter suppression by proper waveform selection 
         [0192]    The invention is not limited to the example described above, but may be modified without departing from the scope of the claims below. 
         [0193]    The terminology used herein is for the purpose of describing particular examples only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” “comprising,” “includes” and/or “including” when used herein, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
         [0194]    Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms used herein should be interpreted as having a meaning that is consistent with their meaning in the context of this specification and the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein. 
         [0195]    The foregoing has described the principles, preferred examples and modes of operation of the present invention. However, the invention should be regarded as illustrative rather than restrictive, and not as being limited to the particular examples discussed above. The different features of the various examples of the invention can be combined in other combinations than those explicitly described. It should therefore be appreciated that variations may be made in those examples by those skilled in the art without departing from the scope of the present invention as defined by the following claims. 
         [0196]    Nomenclature 
         [0197]    Constants
   c 0  is the speed of light   γ is Euler&#39;s constant, approximately equal to: 0.577215664901532860606512   
 
         [0200]    Unary Operators
   g* is the complex conjugate operator acting on g   ℑ[g(t)] is the Fourier transform operator acting on g(t)   ℑ −1 [G(ω)] is the inverse Fourier transform operator acting on G(ω)   M T  is the transpose operator acting on the matrix M   
 
         [0205]    Binary Operators
   • is the scalar product operator   · is the multiplication operator   ⊙ is the correlation operator   * is the convolution operator   
 
         [0210]    Integers Variables
   {right arrow over (M)} is equal to the number of taps in each transmit FIR-filter              is equal to the number of taps in each receive FIR-filter   {right arrow over (N)} s  number of transmit sub arrays               s  number of receive sub arrays   {right arrow over (N)} e ({right arrow over (n)} s ) number of transmit elements in transmit sub array {right arrow over (n)} s                  e (           s ) number of receive elements in receive sub array            s      J is equal to the total number of jammers   j is the jammers counting index jε{0 . . . (J−1)}, not to be confused with the imaginary unit   {right arrow over (α)} is equal to unity if the transmit sub arrays are phase shifter controlled and zero otherwise              is equal to unity if the receive sub arrays are phase shifter controlled and zero otherwise   {right arrow over (β)} is equal to unity if the transmit sub arrays are TTD controlled and zero otherwise              is equal to unity if the receive sub arrays are TTD controlled and zero otherwise   {right arrow over (γ)} is equal to unity if the transmit elements are TTD controlled and zero if the elements are phase shifter controlled, not to be confused with Euler&#39;s constant              is equal to unity if the receive elements are TTD controlled and zero if the elements are phase shifter controlled, not to be confused with Euler&#39;s constant   η is equal to unity if V {right arrow over (n)}     s   (t) are uncorrelated wideband white noise waveforms and zero if V {right arrow over (n)}     s   (t) are correlated wideband white noise waveforms   p is an arbitrary integer index   q is another arbitrary integer index   
 
         [0228]    Geometrical Variables
   {right arrow over (r)} {right arrow over (n)}     s     s  is the vector from the transmit antenna origin to the phase centre of transmit sub array {right arrow over (n)} s                 is the vector from the receive antenna origin to the phase centre of receive sub array            s      {right arrow over (r)} {right arrow over (n)}     s     ,{right arrow over (n)}     e     e  is the vector from the phase centre of transmit sub array {right arrow over (n)} s  to the phase centre of transmit element {right arrow over (n)} e  in sub array, if all transmit sub arrays are identical the transmit sub array index, {right arrow over (n)} s , can be omitted              is the vector from the phase centre of receive sub array            s  to the phase centre of receive element            e  in sub array            s , if all receive sub arrays are identical the receive sub array index,            s , can be omitted   {circumflex over (r)} MB  is the unit vector in the main beam direction (θ MB ,φ MB )   {circumflex over (r)} e     MB    is the unit vector in the main beam direction of each element, assumed to be equal for all elements   {circumflex over (r)} is the unit vector in the observation direction (θ,φ)   r T  is the range to the target   r J     j    is the range to repetition jammer number j   
 
         [0238]    Electrical Variables
   ω Bw  is the bandwidth of the used waveform   ω cw  is the centre frequency of the used waveform   {right arrow over (ω)} B  is the transmitter bandwidth   {right arrow over (ω)} c  is the transmitter centre frequency               B  is the receiver bandwidth               c  is the receiver centre frequency               B  is the mutual bandwidth of the transmitter and receiver               c  is the mutual centre frequency of the transmitter and receiver   Δt is equal to the delay between each tap in the FIR-filter   {right arrow over (a)}{right arrow over (n)}   e     ,{right arrow over (n)}     s    is the amplitude weight of transmit element {right arrow over (n)} e  in transmit sub array {right arrow over (n)} s                 is the amplitude weight of receive element            e  in receive sub array            s      V {right arrow over (n)}     s   (t) is the waveform connected to transmit sub array {right arrow over (n)} s      ν {right arrow over (n)}     s   (t) is the baseband part of V {right arrow over (n)}     s   (t)   D is the power attenuation between transmission and reception   D T  is the power attenuation between transmission and reception in the main beam direction equal to the target direction   D J     j    is the power attenuation between transmission and reception in the direction of jammer number j   σ T  is the radar cross section of the target   NSNJR Normalised Signal to Noise-Jammer Ratio   PG J     j    is the power-gain product of repetition jammer number j (equal to the jammer strength)   NJP j  is the jammer power normalized to the equivalent receiver noise level for repetition jammer number j   u g  is the effective value per square root of angular frequency of each waveform generator   P g  is the power density from each waveform generator, equal to u g   2      
 
         [0261]    Filter Coefficients
   {right arrow over (w)} p,q  is the transmit FIR-filter coefficient for transmit sub array p and transmit FIR-filter tap q.               p,q  is the receive FIR-filter coefficient for receive sub array p and receive FIR-filter tap q.   
 
         [0264]    Functions
   Π(x) is the rectangle function, defined as:   
 
         [0000]    
       
         
           
             
               Π 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     1 
                   
                   
                     
                       
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                         1 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         {right arrow over (a)} e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) is the transmit element amplitude pattern at the waveform centre frequency in the observation direction (θ,φ) 
                     e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) is the receive element amplitude pattern at the waveform centre frequency in the observation direction (θ,φ) 
         {right arrow over (EP)}({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) is the transmit element power pattern at the waveform centre frequency in the observation direction (θ,φ), equal to {right arrow over (a)}* e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw )·{right arrow over (a)} e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) 
                   ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) is the receive element power pattern at the waveform centre frequency in the observation direction (θ,φ), equal to          * e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw )·           e ({circumflex over (r)},{circumflex over (r)} e     MB   ,ω cw ) 
         {right arrow over (h)} p ({circumflex over (r)},{circumflex over (r)} MB ,t) is the transmit sub array impulse response for transmit sub array p 
                     p ({circumflex over (r)},{circumflex over (r)} MB ,t) is the receive sub array impulse response for receive sub array p 
         {right arrow over (τ)} p ({circumflex over (r)},{circumflex over (r)} MB ) is equal to: 
       
     
         [0000]    
       
         
           
             
               
                 
                   r 
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                 ) 
               
             
             
               c 
               0 
             
           
         
       
       
                     p ({circumflex over (r)},{circumflex over (r)} MB ) is equal to: 
       
     
         [0000]    
       
         
           
             
               
                 
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         [0000]    is equal to: 
         [0000]    
       
         
           
             
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                   · 
                   
                     
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         P Tx ({right arrow over (r)} J     i   ,{right arrow over (r)} MB ) is the transmitted power in the direction of repetition jammer number j 
         R(τ) is the cross-correlation between the waveform returned from the target and the superposition of the waveforms returned from all clutter and repetition jammers 
         S Tx (ω) is the transmitted power spectrum in the main beam direction 
         V Tx ({right arrow over (r)},{right arrow over (r)} MB ,t) is the transmitted waveform 
         V Rx ({right arrow over (r)},{right arrow over (r)} MB ,t) is the received waveform ν Rx ({right arrow over (r)},{right arrow over (r)} MB ,t) is the baseband part of V Rx ({right arrow over (r)},{right arrow over (r)} MB ,t) 
         u B ({right arrow over (r)},{right arrow over (r)} MB ,t) is the is the resulting baseband waveform after matched filtering 
         Si(x) is the Sine integral 
         Ci(x) is the Cosine integral 
       
     
       Equations 
       [0282]    
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       
                         
                           P 
                           g 
                         
                         · 
                         
                           D 
                           T 
                         
                         · 
                         
                           
                             ω 
                             ↔ 
                           
                           B 
                         
                       
                       
                         
                           N 
                           → 
                         
                         s 
                         2 
                       
                     
                     · 
                     
                       
                         EP 
                         → 
                       
                        
                       
                         ( 
                         
                           
                             
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                               ^ 
                             
                             MB 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             
                               e 
                               MB 
                             
                           
                           , 
                           
                             ω 
                             cw 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       
                         EP 
                         ← 
                       
                        
                       
                         ( 
                         
                           
                             
                               r 
                               ^ 
                             
                             MB 
                           
                           , 
                           
                             
                               r 
                               ^ 
                             
                             
                               e 
                               MB 
                             
                           
                           , 
                           
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                             cw 
                           
                         
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                                                              
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             - 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             + 
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                             ) 
                                                             
                                                             + 
                                                             
                                                             3 
                                                             · 
                                                             
                                                             ( 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             - 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             + 
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             · 
                                                             Δ 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             t 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.7 
                      
                     .2 
                   
                   ) 
                 
               
             
             
               
                 
                   J 
                   = 
                   
                     
                       
                         
                           P 
                           g 
                         
                         · 
                         
                           
                             ω 
                             ↔ 
                           
                           B 
                         
                       
                       
                         
                           N 
                           → 
                         
                         s 
                         2 
                       
                     
                     · 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           0 
                         
                         
                           J 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           D 
                           
                             J 
                             j 
                           
                         
                         · 
                         
                           
                             EP 
                             → 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   J 
                                   j 
                                 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   e 
                                   MB 
                                 
                               
                               , 
                               
                                 ω 
                                 cw 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           
                             EP 
                             ← 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   J 
                                   j 
                                 
                               
                               , 
                               
                                 
                                   r 
                                   ^ 
                                 
                                 
                                   e 
                                   MB 
                                 
                               
                               , 
                               
                                 ω 
                                 cw 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           
                             ∑ 
                             
                               
                                 ` 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     n 
                                     → 
                                   
                                   s 
                                   ′ 
                                 
                               
                               = 
                               0 
                             
                             
                               
                                 
                                   N 
                                   → 
                                 
                                 s 
                               
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   ` 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                   
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   
                                     N 
                                     → 
                                   
                                   s 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     
                                       n 
                                       → 
                                     
                                     s 
                                     ′ 
                                   
                                   = 
                                   0 
                                 
                                 
                                   
                                     
                                       N 
                                       → 
                                     
                                     s 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         → 
                                       
                                       s 
                                     
                                     = 
                                     0 
                                   
                                   
                                     
                                       
                                         N 
                                         → 
                                       
                                       s 
                                     
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       
                                         ` 
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             n 
                                             ← 
                                           
                                           s 
                                           ′ 
                                         
                                       
                                       = 
                                       0 
                                     
                                     
                                       
                                         
                                           N 
                                           ← 
                                         
                                         s 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       ∑ 
                                       
                                         
                                           ` 
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                           
                                         
                                         = 
                                         0 
                                       
                                       
                                         
                                           
                                             N 
                                             ← 
                                           
                                           s 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           
                                             
                                               n 
                                               ← 
                                             
                                             s 
                                             ′ 
                                           
                                           = 
                                           0 
                                         
                                         
                                           
                                             
                                               N 
                                               ← 
                                             
                                             s 
                                           
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           ∑ 
                                           
                                             
                                               
                                                 n 
                                                 ← 
                                               
                                               s 
                                             
                                             = 
                                             0 
                                           
                                           
                                             
                                               
                                                 N 
                                                 ← 
                                               
                                               s 
                                             
                                             - 
                                             1 
                                           
                                         
                                          
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 ` 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   
                                                     m 
                                                     → 
                                                   
                                                   ′ 
                                                 
                                               
                                               = 
                                               0 
                                             
                                             
                                               
                                                 M 
                                                 → 
                                               
                                               - 
                                               1 
                                             
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               ∑ 
                                               
                                                 
                                                   ` 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     m 
                                                     → 
                                                   
                                                 
                                                 = 
                                                 0 
                                               
                                               
                                                 
                                                   M 
                                                   → 
                                                 
                                                 - 
                                                 1 
                                               
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               
                                                 ∑ 
                                                 
                                                   
                                                     
                                                       m 
                                                       → 
                                                     
                                                     ′ 
                                                   
                                                   = 
                                                   0 
                                                 
                                                 
                                                   
                                                     M 
                                                     → 
                                                   
                                                   - 
                                                   1 
                                                 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   ∑ 
                                                   
                                                     
                                                       m 
                                                       → 
                                                     
                                                     = 
                                                     0 
                                                   
                                                   
                                                     
                                                       M 
                                                       → 
                                                     
                                                     - 
                                                     1 
                                                   
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   
                                                     ∑ 
                                                     
                                                       
                                                         ` 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           
                                                             m 
                                                             ← 
                                                           
                                                           ′ 
                                                         
                                                       
                                                       = 
                                                       0 
                                                     
                                                     
                                                       
                                                         M 
                                                         ← 
                                                       
                                                       - 
                                                       1 
                                                     
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     
                                                       ∑ 
                                                       
                                                         
                                                           ` 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             m 
                                                             ← 
                                                           
                                                         
                                                         = 
                                                         0 
                                                       
                                                       
                                                         
                                                           M 
                                                           ← 
                                                         
                                                         - 
                                                         1 
                                                       
                                                     
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       
                                                         ∑ 
                                                         
                                                           
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                           
                                                           = 
                                                           0 
                                                         
                                                         
                                                           
                                                             M 
                                                             ← 
                                                           
                                                           - 
                                                           1 
                                                         
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         
                                                           ∑ 
                                                           
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             = 
                                                             0 
                                                           
                                                           
                                                             
                                                             M 
                                                             ← 
                                                             
                                                             - 
                                                             1 
                                                           
                                                         
                                                          
                                                         
                                                           
                                                             { 
                                                             
                                                             
                                                             
                                                             1 
                                                             
                                                             
                                                             
                                                             
                                                             if 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             η 
                                                             
                                                             = 
                                                             0 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             δ 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                             
                                                             
                                                             · 
                                                             
                                                             δ 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             if 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             η 
                                                             
                                                             = 
                                                             1 
                                                             
                                                             
                                                             
                                                             
                                                             } 
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             → 
                                                             
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             → 
                                                             
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                             
                                                             * 
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             → 
                                                             
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             * 
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             → 
                                                             
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             ← 
                                                             
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             ← 
                                                             
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                             
                                                             * 
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             ← 
                                                             
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             * 
                                                           
                                                           · 
                                                           
                                                             
                                                             w 
                                                             ← 
                                                             
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             m 
                                                             ← 
                                                             
                                                             
                                                           
                                                           · 
                                                           
                                                             
                                                              
                                                             
                                                             j 
                                                             · 
                                                             
                                                             ω 
                                                             cw 
                                                             
                                                             · 
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             
                                                             - 
                                                             
                                                             ` 
                                                              
                                                             
                                                             m 
                                                             → 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             + 
                                                             
                                                             m 
                                                             → 
                                                             
                                                             + 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             m 
                                                             → 
                                                             
                                                             ′ 
                                                             
                                                             
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                                                             · 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             Jj 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             Jj 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             → 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             → 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             Jj 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             ` 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             
                                                             , 
                                                             
                                                             ` 
                                                              
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             Jj 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             ′ 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             ′ 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             τ 
                                                             ← 
                                                             
                                                             
                                                             β 
                                                             
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             s 
                                                             
                                                             , 
                                                             
                                                             
                                                             n 
                                                             ← 
                                                             
                                                             e 
                                                             
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             Jj 
                                                             
                                                             , 
                                                             
                                                             
                                                             r 
                                                             ^ 
                                                             
                                                             MB 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2.7 
                      
                     .4 
                   
                   )