Patent Publication Number: US-8126082-B2

Title: Gain weighted code combining system and method for combining BPSK codes

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation in part of U.S. application Ser. No. 11/831,378, filed Jul. 31, 2007, and Ser. No. 11/831,405, also filed on Jul. 31, 2007, which are both hereby incorporated by reference into the present application. 
    
    
     STATEMENT OF U.S. GOVERNMENT RIGHTS 
     The subject matter of the present disclosure was developed at least in part pursuant to a contract with the U. S. Air Force pursuant to contract number FA8807-04-C-0002. The U.S. Government has certain rights in the subject matter of the present disclosure. 
    
    
     FIELD 
     The present disclosure relates to binary phase shift keying (BPSK) code combining systems and methods, and more particularly to a combining method and system of a plurality of BPSK codes to enable a single, composite BPSK code to be generated that is representative of plurality of separate BPSK codes. 
     BACKGROUND 
     The statements in this section merely provide background information related to the present disclosure and may not constitute prior art. 
     Currently, the global positioning system GPS IIF transmits three binary phase shift keying (BPSK) codes using a modulation scheme known in the industry as “Interplex” modulation. The Interplex modulation scheme modulates three BPSK codes such that the modulated signal has a constant power. The GPS III system needs to transmit two additional BPSK codes. However, the transmitted signal still needs to have a constant power. 
     One option for combining the five signals involves using the Interlace combining scheme. The Interlace combining scheme can be used to combine the three BPSK codes into one, so the original total of five BPSK codes are reduced to three. These three codes can then be modulated using Interplex just like what is presently being done in GPS IIF. 
     The above technique suffers from certain limitations and drawbacks because it requires complex logic. For one, the existing Interlace combining technique requires a uniform random number generator. Optimum performance of the Interlace combining technique depends on the authenticity (i.e., flatness) of the uniform random number generator (i.e., the degree to which the generated random numbers are uniformly distributed). In addition, when a uniform random number generator is utilized, a mapping table is required to make a selection for the current chip of the composite code between either the majority voted code (the code that is formed by the three component codes on the majority-vote basis) or one of the two BPSK component codes of higher code powers, depending on the magnitude of the random number. This process repeats when a random number is generated. 
     SUMMARY 
     The present disclosure relates to a method and system for generating a composite BPSK code from a plurality of potentially different component BPSK codes. In one implementation the method involves combining three BPSK codes into one BPSK code. In this implementation the first, second and third component BPSK codes are gain weighted by their respective code power ratios to form first, second and third gain weighted codes respectively. The first, second and third gain weighted codes are processed in accordance with an algorithm to form a composite BPSK code. The composite BPSK code has a probability of more than fifty percent of matching each one of the component BPSK codes. 
     In the above exemplary implementation, gain weightings (also known as code power ratios) of the first, second and third BPSK codes involve the operations of: 
     assigning a(t) to represent the first component BPSK code, where a(t) is a random BPSK code equally likely to be +1 or −1, and has the highest code power of the three component BPSK codes; 
     assigning b(t) to represent the second component BPSK code, where b(t) is a random BPSK code equally likely to be +1 or −1, and has the second highest code power of the three component BPSK codes; 
     assigning c(t) to represent the third component BPSK code, where c(t) is a random BPSK code equally likely to represent +1 or −1, and has a code power no more than that of the other two component BPSK codes; 
     determining code power ratios using the formulas: 
     
       
         
           
             
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     and where g a , g b  and g c  are arranged such that:
 
 g   a   ≧g   b   ≧g   c =1
 
     As used herein, the term “code power” means the power associated with a specific component BPSK code. 
     The composite BPSK code is represented by a term x(t), and determined in a chip-synchronous manner by the formula (where * denotes multiplication):
 
 x ( t )=sign{[√{square root over ( g   a   a )}( t )+√{square root over ( g   b )} b ( t )+ c ( t )−√{square root over ( g   a )} a ( t )*√{square root over ( g   b )} b ( t )* c ( t )]}.
 
     A system for generating a composite BPSK code from three component BPSK codes is also disclosed. Advantageously, the system does not require the use of a uniform random number generator or a mapping table. 
     This present methodology has its unique combining efficiency related to the correlations between the composite BPSK code and each of the component BPSK codes. This combining efficiency, being less than 100%, can be practically interpreted as a reduction of the effective code power for each of the component codes. The correlation between the composite BPSK code and a component code may lie between −1 to 1 inclusive, and shows the resemblance of one with the other. For example, when the correlation is close to 1, it means almost all the chips of the composite BPSK code and the respective chips of the component BPSK code are the same. Likewise when the correlation is close to −1, except that the two are 180-degree out of phase. When the correlation is close to zero, a very low percentage of the composite BPSK code chips and the respective component code BPSK chips are the same. 
     In another implementation a method and system for generating a single, composite BPSK code from a pair of potentially different component BPSK codes is disclosed. In one implementation the method involves gain weighting first and second component BPSK codes by its respective code power ratio to form first and second gain weighted codes. The first and second gain weighted codes are processed in accordance with an algorithm to form a composite BPSK code. The composite BPSK code has a probability of fifty percent or greater of matching each one of the component BPSK codes. Determining the gain weightings (i.e., code power ratios) of the first and second component BPSK codes may involve the operations of: 
     assigning a(t) to represent the first component BPSK code, where a(t) is a random BPSK code equally likely to be +1 or −1, and has the higher code power of the two component BPSK codes; 
     assigning b(t) to represent the second component BPSK code, where b(t) is a random BPSK code equally likely to be +1 or −1, and has a code power no more than the code power of the other component BPSK code; 
     determining code power ratios using the formulas: 
     
       
         
           
             
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     and where wherein g a  and g b  are arranged such that:
 
 g   a   ≧g   b =1.
 
     The composite BPSK code is represented by a term x(t), and determined in a chip-synchronous manner by the formula (where * denotes multiplication): 
     
       
         
           
             
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     A system for generating a composite BPSK code from two component BPSK codes is also disclosed. Advantageously, the system does not require the use of a uniform random number generator or a mapping table. 
     A method for determining the reduction of the effective code power for each of the component BPSK codes (whether two or three component BPSK codes are being combined) is also disclosed, together with a method for compensating for these reductions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIG. 1  is a simplified block diagram of one embodiment of a system in accordance with the present disclosure; 
         FIG. 2  is a flowchart setting forth a plurality of exemplary operations of a method of the present disclosure for determining a composite BPSK code from a plurality of three potentially different component BPSK codes; 
         FIG. 3  is a table illustrating the probability of matching chips between the composite BPSK code and the component BPSK codes; 
         FIG. 4  is a table illustrating the probability of matching chips between the composite BPSK code and component BPSK codes; 
         FIG. 4A  illustrates certain ones of major formulas associated with the implementation of the present disclosure; 
         FIG. 5  is a table illustrating the probability of matching chips between the composite BPSK code and the component BPSK codes; 
         FIG. 6  is a graph illustrating the matching probability of each component BPSK code against the composite BPSK code over the ranges of g a  and g b ; 
         FIG. 7  is a table illustrating the probability of matching chips between the composite BPSK code and the component BPSK codes; 
         FIGS. 8A and 8B  are graphs illustrating the correlations of the composite BPSK code with itself and with the component BPSK codes; 
         FIGS. 9A and 9B  are graphs illustrating the correlations between the composite BPSK code and the component BPSK codes but for different code power ratios; 
         FIGS. 10A and 10B  are graphs illustrating the correlations between the composite BPSK code and the component BPSK codes for different code power ratios, and highlighting the reduction in correlation between each of the component BPSK codes and the composite BPSK code when the code power ratios are close to the border between the cross-hatched and stippled areas of  FIG. 6 ; and 
         FIGS. 11A and 11B  are additional graphs illustrating the correlation between the composite BPSK code and the component BPSK codes, and how the correlations drop virtually to zero when the code power ratios fall within the stippled area of the graph of  FIG. 6 ; and 
         FIG. 12  is a diagram illustrating how the effective power of a component BPSK code is formed. 
         FIG. 13  is a simplified block diagram of a different embodiment of the present disclosure for combining two component BPSK codes into a single, composite BPSK code; 
         FIG. 14  is a flowchart setting forth a plurality of exemplary operations of a method of operation corresponding to the system in  FIG. 13 , for combining two potentially different component BPSK codes into a single composite BPSK code; 
         FIG. 15  is a table illustrating the probability of matching chips between the composite BPSK code and the two component BPSK codes; 
         FIG. 16  is a table illustrating the probability of matching chips between the composite BPSK code and two component BPSK codes for different code power ratios; 
         FIG. 17  is a graph illustrating the matching probability of each component BPSK code over the range of g a ; 
         FIG. 18  is a table illustrating the probability of matching chips between the composite BPSK code and the two component BPSK codes for different code power ratios; 
         FIGS. 19A and 19B  are graphs illustrating the correlations of the composite BPSK code with itself and with the two component BPSK codes. 
         FIGS. 20A and 20B  are graphs illustrating the correlation between the composite BPSK code and the two component BPSK codes but for different code power ratios; 
         FIGS. 21A and 21B  are graphs illustrating the correlation between the composite BPSK code and the two component BPSK codes for different code power ratios; 
         FIGS. 22A and 22B  are additional graphs illustrating the correlation between the composite BPSK code and the two component BPSK codes, and how the correlations drop virtually to zero when the code power ratios exceed a predetermined interval; and 
         FIG. 23  is a diagram of the variables and operations that determine the effective power of a component BPSK code. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     The present disclosure relates to methods and systems for generating a composite binary phase shift keying (BPSK) code from a plurality of independent component BPSK codes. In one specific implementation a method and system for generating a single, composite BPSK code from three potentially different component BPSK codes is disclosed. In another implementation a system and method for combining two potentially different component BPSK codes to form a single, composite BPSK code is disclosed. In either implementation, the composite BPSK code can then be transmitted as one BPSK code and is a representative of the plurality of component BPSK codes. The two above-mentioned implementations will be discussed separately in the following paragraphs, with the implementation for combining three component BPSK codes being discussed first. 
     First Exemplary Implementation for Combining Three Component BPSK Codes 
     In the first exemplary implementation for combining three component BPSK codes, the three component BPSK codes are initially defined by assigning a(t) to represent the first component BPSK code, where a(t) is a random BPSK code equally likely to be +1 or −1, and has the highest code power of the three component BPSK codes. The term b(t) is assigned to represent the second component BPSK code, where b(t) is a random BPSK code equally likely to be +1 or −1, and has the second highest code power of the three component BPSK codes. The term c(t) is assigned to represent the third component BPSK code, where c(t) is a random component BPSK code equally likely to represent +1 or −1, and has a code power no more than that of the other two component BPSK codes. 
     Code power ratios (also known as gain weightings) associated with each of the component BPSK codes g a , g b  and g c  are then determined by the following formulas: 
                     g   a     =       code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     a   ⁡     (   t   )           code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     c   ⁡     (   t   )                   Equation   ⁢           ⁢   1                 g   b     =       code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     b   ⁡     (   t   )           code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     c   ⁡     (   t   )                   Equation   ⁢           ⁢   2                 g   c     =         code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     c   ⁡     (   t   )           code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     c   ⁡     (   t   )           =   1             Equation   ⁢           ⁢   3               
and the code powers are in descending order (i.e., power of a(t)≧power of b(t)≧power of c(t), and g c  is set equal to 1 (i.e., g a ≧g b ≧g c =1).
 
     Once the code power ratios have been determined, and in view of the fact that g c  is set equal to 1, the following algorithm can be used to determine the composite BPSK code:
 
 x ( t )=sign{[√{square root over ( g   a )} a ( t )+√{square root over ( g   b )} b ( t )+ c ( t )−√{square root over ( g   a )} a ( t )*√{square root over ( g   b )} b ( t )* c *( t )]}  Equation 4
 
     A system  10  in accordance with one embodiment of the present disclosure for implementing Equation 4 is illustrated in  FIG. 1 . The system  10  makes use of two square root determining circuits  12  and  14 , four multiplier circuits  16 ,  18 ,  20  and  22 , three summing circuits  24 ,  26  and  28 , and a zero threshold comparing circuit  30 . Alternatively, some or all of the functions of system  10  may be implemented by a suitable programmed computer processor. For convenience, portions of various equations that are performed by their corresponding components in system  10  are reproduced adjacent to their respective components. 
     With reference to flowchart  50  of  FIG. 2  and the system  10  shown in  FIG. 1 , the operation of the system  10  will be described. At operation  52 , determine the code power ratios g a  and g b  in accordance with the Equations set forth in Equations 1 and 2 above. At operation  54 , the square root of each code ratio g a  and g b  is determined using square root determining circuits  12  and  14 . At operation  56 , multiply √{square root over (g a )} and a(t) to form √{square root over (g a )}a(t) using multiplier circuit  16 . At operation  58 , multiply √{square root over (g b )} and b(t) to form and √{square root over (g b )}b(t) using multiplier circuit  18 . At operation  60 , multiply c(t) with √{square root over (g b )}b(t) to form √{square root over (g b )}b(t)c(t) using multiplier circuit  22 . At operation  62 , multiply the result of operation  60  by √{square root over (g a )}a(t) to form √{square root over (g a g b )}a(t)b(t)c(t) using multiplier circuit  20 . At operation  64 , subtract the result of operation  62  from √{square root over (g a )}a(t) to form √{square root over (g a )}a(t)-√{square root over (g a g b )}a(t)b(t)c(t) using summing circuit  24 . At operation  66 , sum √{square root over (g b )}b(t) and c(t) to form √{square root over (g b )}b(t)+c(t) using summing circuit  26 . At operation  68 , sum the results of operation  64  and  66  to form √{square root over (g a )}a(t)+√{square root over (g b )}b(t)+c(t)−√{square root over (g a g b )}a(t)b(t) c(t) using summing circuit  28 . At operation  70 , perform a zero-reference threshold comparison using zero threshold comparing circuit  30  on the result of operation  68  to give x(t), the composite BPSK code. 
     The composite code x(t) has the matching probabilities illustrated in  FIG. 3  if g a =g b =g c =1. From the graphs of  FIG. 3 , it will be appreciated that each component BPSK code matches the composite BPSK code 75% (i.e., on average each component code matches the composite code 6 out of 8 chips). By a “match”, it is meant that the product of a component code chip and its corresponding composite code chip is “1”; a mismatch is “−1”. When all 8 chips are matched (i.e., matching probability of 100%), the correlation between a component BPSK code and the composite BPSK code would be: 
     
       
         
           
             
               
                 
                   
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     This is perfect or “full” correlation. When the matching probability drops below 100%, perfect correlation is no longer achievable. For a matching probability of 75%, the impact on the correlation is shown below: 
     
       
         
           
             
               
                 
                   
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     Hence, 75% matching probability translates to 3 dB correlation loss. Note that each matching or mismatching chip increases or decreases the correlation, respectively. 
     If g a  and g b  vary, the matching probabilities will also vary. For code power ratios g a =10, g b =4, and g c =1, the matching probabilities are tabulated at  FIG. 4 . 
     Note that from the tables in  FIGS. 3 and 4 , x(t) changes from 1 to −1 when a(t)=b(t)=c(t)=1 (or changes from −1 to 1 when a(t)=b(t)=c(t)=−1). Referring to  FIG. 5 , when swapping g a  and g b , the same result is achieved. This is expected since √{square root over (g a )}a(t) and √{square root over (g b )}b(t) satisfy the communicative law. The communicative law holds that when one adds or multiplies numbers, the order doesn&#39;t matter. Fifty percent matching means on average each component code matches the composite code 4 out of 8 chips. The correlation is reduced to zero: 
     
       
         
           
             
               
                 
                   
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     Since the code power ratios among the component BPSK codes change the correlation between the composite BPSK code and each component BPSK code, it is then essential to ascertain the ranges of g a  and g b  and the corresponding matching probabilities for the component BPSK codes. The development is shown below from Equation 4:
 
 x ( t )=sign{[√{square root over ( g   a )} a ( t )+√{square root over ( g   b )} b ( t )+ c ( t )−√{square root over ( g   a )} a ( t )*√{square root over ( g   b )} b ( t )* c ( t )]}  Equation 8
 
Since x(t) changes from 1 to −1 when a(t)=b(t)=c(t)=1, it can be shown that
 
                     x   ⁡     (   t   )       =       sign   ⁡     [         g   a       +       g   b       +   1   -         g   a     ⁢     g   b           ]       =     {         1           0             -   1                       Equation   ⁢           ⁢   9               
depends on the gain ratios, and it can be interpreted as
 
                 g   a       +       g   b       +   1   -           g   a     ⁢     g   b         ⁢     &gt;   &lt;     ⁢   0                     g   a       +       g   b       +     1   ⁢     &gt;   &lt;     ⁢         g   a     ⁢     g   b                 
Hence, the formula:
 
                           g   b       +       g   b       +   1           g   a     ⁢     g   b           ⁢     &gt;   &lt;     ⁢   1.           Equation   ⁢           ⁢   10               
The implication is that:
 
     
       
         
           
             
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                               1 
                             
                             
                               
                                 
                                   g 
                                   a 
                                 
                                 ⁢ 
                                 
                                   g 
                                   b 
                                 
                               
                             
                           
                         
                         &gt; 
                         1 
                       
                     
                   
                   
                     
                       
                         0 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         50 
                         ⁢ 
                         % 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         matching 
                       
                     
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               + 
                               
                                 
                                   g 
                                   b 
                                 
                               
                               + 
                               1 
                             
                             
                               
                                 
                                   g 
                                   a 
                                 
                                 ⁢ 
                                 
                                   g 
                                   b 
                                 
                               
                             
                           
                         
                         = 
                         1 
                       
                     
                   
                   
                     
                       
                         
                           - 
                           1 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         50 
                         ⁢ 
                         % 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         matching 
                       
                     
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               + 
                               
                                 
                                   g 
                                   b 
                                 
                               
                               + 
                               1 
                             
                             
                               
                                 
                                   g 
                                   a 
                                 
                                 ⁢ 
                                 
                                   g 
                                   b 
                                 
                               
                             
                           
                         
                         &lt; 
                         1 
                       
                     
                   
                 
               
             
           
         
       
     
     The above formulas are also presented in  FIG. 4A  for convenience. 
       FIG. 6  illustrates the ranges and matching probabilities of each component BPSK code against the composite BPSK code x(t) where cross-hatched area  80  represents an area where 75% matching probability is achieved, and stippled area  82  represents an area where 50% matching probability is achieved. When g a =9 and g b =4, then: 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         sign 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 g 
                                 a 
                               
                             
                             + 
                             
                               
                                 g 
                                 b 
                               
                             
                             + 
                             1 
                             - 
                             
                               
                                 
                                   g 
                                   a 
                                 
                                 ⁢ 
                                 
                                   g 
                                   b 
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       0 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   if 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             g 
                             a 
                           
                         
                         + 
                         
                           
                             g 
                             b 
                           
                         
                         + 
                         1 
                       
                       
                         
                           
                             g 
                             a 
                           
                           ⁢ 
                           
                             g 
                             b 
                           
                         
                       
                     
                     = 
                     1. 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     The occurrence of Equation 11 is on the border between areas  80  and  82 . Thus, the probability of occurrence is very small and is negligible. The probabilities of the component BPSK codes matching the composite BPSK code x(t) are as illustrated in the table of  FIG. 7 . Thus, the ranges for g a  and g b  and the corresponding matching probabilities can mathematically be expressed as 
     
       
         
           
             
               
                 
                   
                     g 
                     a 
                   
                 
                 + 
                 
                   
                     g 
                     b 
                   
                 
                 + 
                 1 
               
               
                 
                   
                     g 
                     a 
                   
                   ⁢ 
                   
                     g 
                     b 
                   
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         &gt;1 
                       
                     
                     
                       
                         75 
                         ⁢ 
                         % 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         matching 
                       
                     
                   
                   
                     
                       
                         &lt;1 
                       
                     
                     
                       
                         50 
                         ⁢ 
                         % 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         matching 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
       FIGS. 8A ,  8 B,  9 A,  9 B,  10 A,  10 B and  11 A,  11 B demonstrate the relationship between the matching probability and the associated correlation at various code power ratios. These Figures assume infinite bandwidth. Had a filter of a finite bandwidth been applied before correlation takes place, the correlation will be reduced due to out-of-band loss. The amount of reduction depends on the filter bandwidth and the chip rates of the component BPSK codes.  FIGS. 8A and 8B  are graphs illustrating the correlations of the composite BPSK code with itself and with the component BPSK codes; the correlation of the composite BPSK code with itself has a peak of unity since it matches itself perfectly as it should. The correlations of the composite BPSK code with the component BPSK codes have peaks roughly 0.5 indicating the 75% matching between the composite BPSK code and the component BPSK codes (see Equation 6 below).  FIG. 8B  is an enlarged version of  FIG. 8A  focusing on the details of the peaks of these correlations. 
     The correlations shown in  FIGS. 8B and 9B  have code power ratios in the cross-hatched area  80  of  FIG. 6  and have a matching probability of roughly 75% as stated in Equation 6. The correlations in  FIG. 11B  are virtually zero since the code power ratios are well within the stippled area  82  of  FIG. 6 . The correlations in  FIG. 10B  are approaching zero since the code power ratios are on the border between cross-hatched  80  and stippled  82  areas in  FIG. 6 . 
     The matching probability for each component BPSK code determines the correlatable power (also known as effective code power) of the component code received at the output of a correlator of each component code where code z(t)=a(t), b(t) or c(t) and P x  is the power of the composite BPSK code x(t), P x =P a +P b +P c =(g a +g b +1)*P c ; as indicated in  FIG. 12 . 
     For matching probabilities equal to 75%, the P z,effective  is 0.25 of P x  for each component BPSK code. This means a total of 0.75=3(0.25) of the total power of composite BPSK code can be recovered and the combining efficiency is equal to 0.75. 
     
       
         
           
             
               
                 
                   
                     η 
                     z 
                   
                   = 
                   
                     
                       
                         P 
                         
                           z 
                           , 
                           effective 
                         
                       
                       
                         P 
                         x 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               R 
                               
                                 x 
                                 , 
                                 z 
                               
                             
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           
                             | 
                             
                               τ 
                               = 
                               0 
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
             
               
                 
                   0.75 
                   = 
                   
                     η 
                     = 
                     
                       
                         
                           η 
                           a 
                         
                         + 
                         
                           η 
                           b 
                         
                         + 
                         
                           η 
                           c 
                         
                       
                       = 
                       
                         0.25 
                         + 
                         0.25 
                         + 
                         0.25 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   13 
                 
               
             
           
         
       
     
     This demonstrates via Equation 10 that maintaining the code power ratios g a  and g b  will maintain component BPSK code power efficiencies η a , η b  and η c  as well as the composite code efficiency η. 
     Due to the combining efficiency in Equation 13 not being 100%, there is a combining loss. This means the effective code power at the correlator output for each component code will not be PZ, but something less P z,effective &lt;P z . If P z  is desired at the correlator output of each component code (i.e., P z  is expected as the effective code power at the correlator output), then the difference between P z,effective  (the effective code power before component code power compensation) and P z  (the effective code power after component code compensation) needs to be made up by some power compensation to the code. The code power compensation can be done for each component code. 
     For the composite code, the additional power needed for compensation is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     P 
                     compensation 
                   
                   = 
                   
                     
                       
                         1 
                         η 
                       
                       ⁢ 
                       
                         P 
                         x 
                       
                     
                     - 
                     
                       P 
                       x 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     This is equivalent to boosting the composite BPSK code power by a gain (1/η), and the compensated composite BPSK code power may be represented by the equation: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       η 
                     
                     ⁢ 
                     
                       P 
                       x 
                     
                   
                   = 
                   
                     
                       
                         P 
                         x 
                       
                       + 
                       
                         P 
                         compensation 
                       
                     
                     = 
                     
                       
                         P 
                         x 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               
                                 1 
                                 η 
                               
                               ⁢ 
                               
                                 P 
                                 x 
                               
                             
                             - 
                             
                               P 
                               x 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     Note that adding compensation power and maintaining code power ratios do not change the efficiencies shown in Equations 12 and 13. The effective code power for component code a(t) after compensation can be calculated to be: 
     
       
         
           
             
               
                 
                   
                     
                       η 
                       a 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           x 
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 1 
                                 η 
                               
                               ⁢ 
                               
                                 P 
                                 x 
                               
                             
                             - 
                             
                               P 
                               x 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         η 
                         a 
                       
                       ⁢ 
                       
                         
                           P 
                           x 
                         
                         η 
                       
                     
                     = 
                     
                       
                         
                           P 
                           
                             a 
                             , 
                             effective 
                           
                         
                         η 
                       
                       = 
                       
                         
                           P 
                           a 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         Equation 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         16 
                       
                     
                   
                   
                     
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     Likewise for the other two component codes: 
     
       
         
           
             
               
                 
                   
                     
                       η 
                       b 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           x 
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 1 
                                 η 
                               
                               ⁢ 
                               
                                 P 
                                 x 
                               
                             
                             - 
                             
                               P 
                               x 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         η 
                         b 
                       
                       ⁢ 
                       
                         
                           P 
                           x 
                         
                         η 
                       
                     
                     = 
                     
                       
                         
                           P 
                           
                             b 
                             , 
                             effective 
                           
                         
                         η 
                       
                       = 
                       
                         P 
                         b 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         Equation 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         17 
                       
                     
                   
                   
                     
                       
                         ( 
                         2 
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       η 
                       c 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           x 
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 1 
                                 η 
                               
                               ⁢ 
                               
                                 P 
                                 x 
                               
                             
                             - 
                             
                               P 
                               x 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         η 
                         c 
                       
                       ⁢ 
                       
                         
                           P 
                           x 
                         
                         η 
                       
                     
                     = 
                     
                       
                         
                           P 
                           
                             c 
                             , 
                             effective 
                           
                         
                         η 
                       
                       = 
                       
                         
                           P 
                           c 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         Equation 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         18 
                       
                     
                   
                   
                     
                       
                         ( 
                         3 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     If the composite code can be made available as the local replica in the correlator (i.e., z(t)=x(t)), the total power of the composite BPSK code x(t) can be recovered. For matching probabilities equal to 50%, no code power can be recovered. This feature can be used to identify that gain weighted combining of three component BPSK Codes is the option being used among a number of different code combining methodologies. Note that the code combining above may also be applied to component BPSK codes of different chip rates. 
     Second Implementation For Combining Two Component BPSK Codes 
     Initially the two component BPSK codes are defined by assigning a(t) to represent the first component BPSK code, where a(t) is a random BPSK code equally likely to be +1 or −1, and has the higher code power of the two component BPSK codes. The term b(t) is assigned to represent the second component BPSK code, where b(t) is a random BPSK code equally likely to be +1 or −1, and has a code power no more than that of the first component BPSK code. 
     A code power ratio (i.e., gain) associated with each of the component BPSK codes a(t) and b(t) is then determined by the following formulas: 
                     g   a     =       code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     a   ⁡     (   t   )           code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     b   ⁡     (   t   )                   Equation   ⁢           ⁢   19                 g   b     =         code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     b   ⁡     (   t   )           code   ⁢           ⁢   power   ⁢           ⁢   of   ⁢           ⁢     b   ⁡     (   t   )           =   1             Equation   ⁢           ⁢   20               
and the code powers are in the descending order (i.e., power of a(t)≧power of b(t), and g b  is set equal to 1 (i.e., g a ≧g b =1).
 
     Once the code powers ratios have been determined, and in view of the fact that g b  is set equal to 1, the following algorithm can be used in a chip-synchronous manner to determine the composite BPSK code (where * denotes multiplication): 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     sign 
                     ⁢ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 g 
                                 a 
                               
                             
                             ⁢ 
                             
                               a 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               
                                 g 
                                 b 
                               
                             
                             ⁢ 
                             
                               b 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               g 
                               a 
                             
                             ⁢ 
                             
                               
                                 a 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               g 
                               b 
                             
                             ⁢ 
                             
                               
                                 b 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               
                                 g 
                                 a 
                               
                             
                             ⁢ 
                             
                               a 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             * 
                             
                               
                                 g 
                                 b 
                               
                             
                             ⁢ 
                             
                               b 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   21 
                 
               
             
           
         
       
     
     By Equation 21, the component codes are respectively weighted by their code power ratios then multiplied, summed and thresholded to form the composite BPSK code x(t). Since the composite BPSK code may not fully represent both of the two component codes at any given time, the receiver of each component BPSK code may experience some correlation loss. The implementation of this combining and the amount of possible correlation loss will be discussed further in the following paragraphs. 
     Since g b =1, and since:
 
 a   2 ( t )= b   2 ( t )=1, the Equation 21 is simplified to:   Equation 22
 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     sign 
                     ⁢ 
                     
                       
                         { 
                         
                           [ 
                           
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               ⁢ 
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             + 
                             
                               b 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 a 
                               
                             
                             - 
                             
                               1 
                               2 
                             
                             + 
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               ⁢ 
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               * 
                               
                                 b 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   23 
                 
               
             
           
         
       
     
     A system  100  in accordance with one embodiment of the present disclosure for implementing Equation 21 is illustrated in  FIG. 13 . The system  100  makes use of one square root determining circuit  102 , three multiplier circuits  104 ,  106  and  108 , four summing circuits  110 ,  112 ,  114  and  116 , and a zero threshold comparing circuit  118 . Alternatively, some or all of the functions may be implemented by a suitably programmed computer processor. For convenience, portions of various equations that are performed by their corresponding components in system  100  are reproduced adjacent to their respective components. 
     With reference to the flowchart  150  of  FIG. 14  and the system  100  shown in  FIG. 13 , the operation of the system  100  will be described. At operation  152 , first determine the code power ratio (g a ) of component BPSK code a(t), then take the square root of it using square root circuit  102 . At operation  154 , multiply √{square root over (g a )} and a(t) to form √{square root over (g a )}a(t) using multiplier circuit  104 . At operation  156 , multiply b(t) with √{square root over (g a )}a(t) to form √{square root over (g a )}a(t)b(t) using multiplier circuit  106 . At operation  158 , sum √{square root over (g a )}a(t) and √{square root over (g a )}a(t)b(t) using summing circuit  112 . At operation  160 , negatively sum g a  and unity (i.e., 1.0), using summing circuit  110 . At operation  162 , multiply result of operation  160  by a factor of 0.5 to form −0.5 (g a +1), using multiplier  108 . At operation  164 , sum the result of operation  162  and b(t) to form −0.5 (g a +1)+b(t) using summing circuit  114 . At operation  166 , sum the results of operations  158  and  164  to form √{square root over (g a )}a(t)+b(t)−0.5 (g a +1)+√{square root over (g a )}a(t)b(t) using summing circuit  116 . At operation  168  perform a zero-reference threshold comparison on the result of operation  166 , using zero threshold comparing circuit  118 , to generate x(t), the composite BPSK code. 
     Referring to the table of  FIG. 15 , the composite BPSK code x(t) has the matching probabilities shown in the table if g a =g b =1 (x(t) is the logical AND of a(t) and b(t)). By a “match”, it is meant that the product of a component code chip and its corresponding composite code chip is “1”; a mismatch is “−1”. Each component BPSK code matches the composite code 75% (i.e., on average each component BPSK code matches the composite BPSK code 3 out of 4 chips). When all 4 chips are matched, the correlation between a component BPSK code and the composite BPSK code would be 
                     R     x   ,   a       =       R     x   ,   b       =           4   ⁢     (   1   )       +     0   ⁢     (     -   1     )         4     =     1   =     0   ⁢           ⁢     dB   .                     Equation   ⁢           ⁢   24               
This is a perfect or “full” correlation. When the matching probability drops below 100%, perfect correlation is no longer achievable. For a matching probability of 75%, the impact on the correlation is shown below:
 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       x 
                       , 
                       a 
                     
                   
                   = 
                   
                     
                       R 
                       
                         x 
                         , 
                         b 
                       
                     
                     = 
                     
                       
                         
                           
                             3 
                             ⁢ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                           + 
                           
                             1 
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         4 
                       
                       = 
                       
                         0.50 
                         = 
                         
                           
                             - 
                             3 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             dB 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   25 
                 
               
             
           
         
       
     
     Hence, a 75% matching probability translates to a 3 dB correlation loss. Note that each matching or mismatching chip increases or decreases the correlation, respectively. 
     If g a  varies, the matching probabilities will also vary. For code power ratios g a =18 and g b =1, the matching probabilities are shown in the table of  FIG. 16 . A matching probability of 50% means on average each component BPSK code matches the composite BPSK code two out of four chips. The correlation is reduced to zero: 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       x 
                       , 
                       a 
                     
                   
                   = 
                   
                     
                       R 
                       
                         x 
                         , 
                         b 
                       
                     
                     = 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         4 
                       
                       = 
                       
                         0 
                         = 
                         
                           
                             - 
                             ∞ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             dB 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   26 
                 
               
             
           
         
       
     
     Since the code power ratio g a  between the component BPSK codes changes the correlation between the composite BPSK code and each component BPSK code, one will need to ascertain the range of g a  and the corresponding matching probabilities for the component codes. This development is shown below from Equation 27: 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     sign 
                     ⁢ 
                     
                       
                         { 
                         
                           [ 
                           
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               ⁢ 
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             + 
                             
                               b 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             - 
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 a 
                               
                             
                             - 
                             
                               1 
                               2 
                             
                             + 
                             
                               
                                 
                                   g 
                                   a 
                                 
                               
                               ⁢ 
                               
                                 a 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               * 
                               
                                 b 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   27 
                 
               
             
           
         
       
     
     Since the composite BPSK code x(t) changes from 1 to −1 at some gain g a  when a(t)=b(t)=1, it can be shown that: 
                     x   ⁡     (   t   )       =       sign   ⁢     {     [       2   ⁢       g   a         -       1   2     ⁢     g   a       +     1   2       ]     }       =     {         1           0             -   1                       Equation   ⁢           ⁢   28               
depends on the gain ratios and it can be interpreted as
 
                     2   ⁢       g   a         -       1   2     ⁢     g   a       +       1   2     ⁢     &gt;   &lt;     ⁢   0             Equation   ⁢           ⁢   29                   4   ⁢       g   a         +     1   ⁢     &gt;   &lt;     ⁢     g   a         ⁢     
     ⁢   Hence           Equation   ⁢           ⁢   30                     4   ⁢       g   a         +   1       g   a       ⁢     &gt;   &lt;     ⁢   1           Equation   ⁢           ⁢   31               
The implication is that
 
     
       
         
           
               
             
               
                 x 
                 ⁢ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
                 
               ⁢ 
               
                 
                   sign 
                   ⁡ 
                   
                     [ 
                     
                       
                         2 
                         ⁢ 
                         
                           
                             g 
                             a 
                           
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           g 
                           a 
                         
                       
                       + 
                       
                         1 
                         2 
                       
                     
                     ] 
                   
                 
                 = 
                   
                 ⁢ 
                 
                   { 
                   
                     
                       
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           and 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           75 
                           ⁢ 
                           % 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           matching 
                         
                       
                       
                         if 
                       
                       
                         
                           
                             
                               
                                 4 
                                 ⁢ 
                                 
                                   
                                     g 
                                     a 
                                   
                                 
                               
                               + 
                               1 
                             
                             
                               g 
                               a 
                             
                           
                           &gt; 
                           1 
                         
                       
                     
                     
                       
                         
                           0 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           and 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           50 
                           ⁢ 
                           % 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           matching 
                         
                       
                       
                         if 
                       
                       
                         
                           
                             
                               
                                 4 
                                 ⁢ 
                                 
                                   
                                     g 
                                     a 
                                   
                                 
                               
                               + 
                               1 
                             
                             
                               g 
                               a 
                             
                           
                           = 
                           1 
                         
                       
                     
                     
                       
                         
                           
                             - 
                             1 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           and 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           50 
                           ⁢ 
                           % 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           matching 
                         
                       
                       
                         if 
                       
                       
                         
                           
                             
                               
                                 4 
                                 ⁢ 
                                 
                                   
                                     g 
                                     a 
                                   
                                 
                               
                               + 
                               1 
                             
                             
                               g 
                               a 
                             
                           
                           &lt; 
                           1 
                         
                       
                     
                   
                 
               
             
           
         
       
     
       FIG. 17  shows a graph  180  indicating ranges and matching probabilities for the composite BPSK code x(t) and each of the two component BPSK codes. For g a &lt;(2+√{square root over (5)}) 2  the matching probability is 75%; otherwise the matching probability is 50%. The derivation is shown below: 
                       x   ⁡     (   t   )       =       sign   ⁡     [       2   ⁢       g   a         -       1   2     ⁢     g   a       +     1   2       ]       =       0   ⁢           ⁢   if   ⁢           ⁢         4   ⁢       g   a         +   1       g   a         =   1         ⁢     
     ⁢           4   ⁢       g   a         +   1       g   a       =   1     ⁢     
     ⁢         g   a     -     4   ⁢       g   a         -   1     =   0             Equation   ⁢           ⁢   32               
This is a quadratic equation and its solution is:
 
                       g   a       =         4   ±       16   +   4         2     .             Equation   ⁢           ⁢   33               
Since g a  can only be positive, then
 
√{square root over ( g   a )}=2+√{square root over (5)}
 
 g   a =(2+√{square root over (5)}) 2  
 
     The occurrence of this equation is on the border between matching probabilities 75% and 50%, and the probability of its occurrence is very small and is negligible. The matching probabilities are shown in the table of  FIG. 18 . Hence the ranges for g a  and the corresponding matching probabilities can mathematically be expressed as 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             x 
                             ⁢ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             sign 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     
                                       g 
                                       a 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     g 
                                     a 
                                   
                                 
                                 + 
                                 
                                   1 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     1 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     and 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     75 
                                     ⁢ 
                                     % 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     matching 
                                   
                                 
                                 
                                   if 
                                 
                                 
                                   
                                     
                                       
                                         
                                           4 
                                           ⁢ 
                                           
                                             
                                               g 
                                               a 
                                             
                                           
                                         
                                         + 
                                         1 
                                       
                                       
                                         g 
                                         a 
                                       
                                     
                                     &gt; 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     and 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     50 
                                     ⁢ 
                                     % 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     matching 
                                   
                                 
                                 
                                   if 
                                 
                                 
                                   
                                     
                                       
                                         
                                           4 
                                           ⁢ 
                                           
                                             
                                               g 
                                               a 
                                             
                                           
                                         
                                         + 
                                         1 
                                       
                                       
                                         g 
                                         a 
                                       
                                     
                                     &lt; 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     34 
                   
                 
               
             
           
         
       
     
       FIGS. 19A ,  19 B,  20 A,  20 B,  21 A,  21 B,  22 A and  22 B demonstrate the relationship between the matching probability and the associated correlation at various code power ratios.  FIGS. 19-22  assume infinite bandwidth. Had a low-pass filter of a finite bandwidth been applied before correlation takes place, the correlation would be reduced due to out-of-band loss. The amount of reduction depends on the filter bandwidth and the chip rates of the component codes.  FIGS. 19A and 19B  are graphs illustrating the correlations of the composite BPSK code with itself and with the component BPSK codes. The correlation of the composite BPSK code with itself has a peak of unity since it matches itself perfectly as it should (Equation 24). The correlations of the composite BPSK code with the component BPSK codes have peaks roughly 0.5 indicating the 75% matching between the composite BPSK code and the component BPSK codes (see Equation 25).  FIG. 19B  is an enlarged version of  FIG. 19A  focusing on the details of the peaks of these correlations. 
     The correlations shown in  FIGS. 19-21  have gain ratios in the interval of [1 . . . (2+√{square root over (5)}) 2 ) in  FIG. 17 , and have a matching probability of roughly 75% as stated in Equation 19. The correlations in  FIG. 22B  are virtually zero since the code power ratios are well beyond the interval [1 . . . (2+√{square root over (5)}) 2 ) in  FIG. 17 . The correlation of composite BPSK code x(t) in  FIGS. 19 through 21  do not have zero noise floor since the composite BPSK code is not equally probable between −1 and +1. 
     The matching probability for each component BPSK code determines the correlatable power (i.e., “effective code power”) of the component code received at the output of the correlator of each component BPSK code where code z(t)=a(t) or b(t), and P x  is the power of the composite BPSK code x(t), P x =P a +P b =(g a +1)*P b . This is shown in  FIG. 23 . 
     For matching probabilities equal to 75%, the R x,a (τ=0)=R x,b (τ=0)=0.5 and the P z,effective  is 0.25 of P x  for each component BPSK code. This means a total of 0.5=2(0.25) of the total power of the composite BPSK code can be recovered and the power efficiency=0.5. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           η 
                           z 
                         
                         = 
                         
                           
                             
                               P 
                               
                                 z 
                                 , 
                                 effective 
                               
                             
                             
                               P 
                               x 
                             
                           
                           = 
                           
                             
                               [ 
                               
                                 
                                   
                                     R 
                                     
                                       x 
                                       , 
                                       z 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     τ 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   | 
                                   
                                     τ 
                                     = 
                                     0 
                                   
                                 
                               
                               ] 
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         0.5 
                         = 
                         
                           η 
                           = 
                           
                             
                               η 
                               a 
                             
                             + 
                             
                               η 
                               b 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         Equation 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         35 
                       
                     
                   
                   
                     
                       
                         ( 
                         4 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     This demonstrates via Equation 34 that maintaining the code power ratio g a  in a certain range will maintain each component code power efficiency η z  as well as the composite code efficiency η. 
     Due to the combining efficiency in Equation 35 not being 100%, there is a combining loss. This means the effective code power at the correlator output for each component BPSK code will not be P z , but something less P z,effective &lt;P z . If P z  is desired at the correlator output of each BPSK component code (i.e., P z  is expected as the effective code power at the correlator output), then the difference between P z,effective  (the effective code power before component BPSK code power compensation) and P z  (the effective code power after component BPSK code compensation) needs to be made up by some power compensation to the code. The code power compensation can be done for each component BPSK code. 
     For the composite BPSK code, the additional power needed for the compensation is calculated as 
     
       
         
           
             
               
                 
                   
                     P 
                     compensation 
                   
                   = 
                   
                     
                       
                         1 
                         η 
                       
                       ⁢ 
                       
                         P 
                         x 
                       
                     
                     - 
                     
                       
                         P 
                         x 
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   36 
                 
               
             
           
         
       
     
     It will be noted that adding compensation power and maintaining code power ratios does not change the efficiencies shown in Equation 35. The compensation power defined by Equation 36 is equivalent to boosting the composite code power by a gain (1/η) and the compensated composite code power is: 
                       1   η     ⁢     P   x       =         P   x     +     P   compensation       =       P   x     +       (         1   η     ⁢     P   x       -     P   x       )     .                 Equation   ⁢           ⁢   37               
The effective code power for a(t) after compensation can be calculated to be:
 
                       η   a     ⁡     [       P   x     +     (         1   η     ⁢     P   x       -     P   x       )       ]       =         η   a     ⁢       P   x     η       =         P     a   ,   effective       η     =       P   a     .                 Equation   ⁢           ⁢   38               
Likewise, for the other component BPSK code, the effective code power after compensation can be calculated to be:
 
     
       
         
           
             
               
                 
                   
                     
                       η 
                       b 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           x 
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 1 
                                 η 
                               
                               ⁢ 
                               
                                 P 
                                 x 
                               
                             
                             - 
                             
                               P 
                               x 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         η 
                         b 
                       
                       ⁢ 
                       
                         
                           P 
                           x 
                         
                         η 
                       
                     
                     = 
                     
                       
                         
                           P 
                           
                             b 
                             , 
                             effective 
                           
                         
                         η 
                       
                       = 
                       
                         
                           P 
                           b 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   39 
                 
               
             
           
         
       
     
     It will be appreciated that a code power and chip rate for each component BPSK code (in both of the two and three code combining implementations) could also be remotely programmed to tailor the system  10  to meet the needs of a specific application. 
     While various embodiments have been described, those skilled in the art will recognize modifications or variations which might be made without departing from the present disclosure. The examples illustrate the various embodiments and are not intended to limit the present disclosure. Therefore, the description and claims should be interpreted liberally with only such limitation as is necessary in view of the pertinent prior art.