PATENT DOCUMENT

Abstract:
Secure communication of information is effected from a first party to a second party when the first party knows its own global location and the global location of the second party, and employs what essentially is an undiscoverable code signal that is broadcast to, and received by, both the first and the second parties. The first party securely communicates information to the second party by modifying the code signal with the information that is to be communicated and sends the modified code signal to the second party. Illustratively, the code signal is related to the Y component of a GPS signal.

Full Description:
RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 12/231,094, which was filed on Aug. 29, 2008 which is a continuation in part of U.S. patent application, Ser. No. 12/012,327, which was filed on Feb. 2, 2008. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    When information needs to be communicated in a secure manner one typically turns to cryptographic techniques. It is generally recognized that with many cryptographic techniques the encrypted data can be recovered by an adversary, but only if the adversary has sufficient resources (e.g., computing power) and sufficient time. Most users are satisfied when a method is secure “enough,” meaning that the time, effort, or expense to recover the data embedded in an encrypted message is too great to make the data useful to an adversary. 
         [0003]    With the above in mind, cryptographic techniques usually depend on encryption and decryption keys being in possession of the communicating parties. Aside from the concern about the inherent security of message encrypted with a particular method, the biggest concern is with the secure creation, distribution and maintenance of the keys. 
       SUMMARY OF THE INVENTION 
       [0004]    An advance in the art is achieved with a method that implements secure transmission of information from one party to another without the need for cryptographic keys but, rather, based on unique geographic attributes such as position as well as time. More specifically, secure communication of information is effected from a first party to a second party when the first party knows its own global location and the global location of the second party, and employs a code signal that is broadcast to, and received by, both the first and the second parties. The first party securely communicates information to the second party by modifying the code signal with the information that is to be communicated and sends the modified code signal to the second party. The code signal that is received by the first party and is used to convey information to the second party need not to be actually known to either of the parties, and from the standpoint of secure communication it is advantageous for the broadcasted code signal (and the corresponding related received signals) to not be known to either of the parties and to be essentially impossible for the parties to discover. The signal that is employed in the disclosed illustrative example is related to the Y component of a GPS signal. Other wireless sources that are modified to include a signal like the Y component of the GPS signal can also be employed. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1  depicts a first unit that securely transmits information without the use of cryptographic keys; 
           [0006]      FIG. 2  shows some of the processing within processor  40  of  FIG. 1 ; and 
           [0007]      FIG. 3  depicts some of the processing within a unit that receives the signal transmitted by the  FIG. 1  unit. 
       
    
    
     DETAILED DESCRIPTION 
       [0008]    U.S. patent application Ser. No. 12/012,327 discloses an approach whereby a first device, at location x, can verify an assertion by a second device, at location y, as to the global position of the second device. The disclosure presented an illustrative embodiment that is based on the Global Positioning System (GPS) but the principles disclosed therein are broader and are not limited to the GPS. For examples, they can be readily applied to the other global navigation satellite systems being developed and deployed worldwide. 
         [0009]    To assist the reader in understanding the instant invention without having to read the aforementioned application, the following repeats a significant portion of the mathematical underpinnings presented in the 12/012,327 application. It should be kept in mind that here, as well, the principles of the disclosed invention are broader than the illustrative example that uses the GPS. 
         [0010]      FIG. 1  shows unit  100  that belongs to Remote Device B, which simultaneously receives a number of GPS satellite signals on frequency L 1 , where the signal transmitted by satellite n can be expressed as 
         [0000]        S   transmitted   n   =A   n   D   n ( t ) x   C   n ( t )cos(2π( f   L1 ) t+φ   1 )+ B   n   D   n ( t ) x   Y   n ( t )sin(2π( f   L1 ) t+φ   1 )  (1)
 
         [0000]    where D n (t) is the data signal of satellite n, x C   n (t) is a code signal assigned to satellite n that is publicly known, x Y   n (t) is a code signal assigned to satellite n that is not publicly known, f L1  is the frequency of the carrier, and φ 1  is the phase of the carrier relative to the beginning of the data and code signals. Unit  100  is the party that wishes to send information to a remote unit  200  without the use of cryptographic keys. 
         [0011]    A GPS receiver receives a signal corresponding to the sum of the signals of the individual satellites. The receiver can engage in the processing of signals as if all of the possible satellites are present but, of course, some of the satellites are not within range of the GPS receiver&#39;s antenna (i.e., not detectible) so the processing results for those satellites are not viable. In other words, the signal arriving at the  FIG. 1  antenna corresponds to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       K 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   A 
                                   n 
                                 
                                  
                                 
                                   
                                     D 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     x 
                                     C 
                                     n 
                                   
                                    
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           ( 
                                           
                                             f 
                                             
                                               L 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                           
                                           ) 
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         ϕ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 B 
                                 n 
                               
                                
                               
                                 
                                   D 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                 
                                   x 
                                   Y 
                                   n 
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             f 
                                             
                                               L 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       ϕ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   Noise 
                 
               
               
                 
                   ( 
                   
                     1 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where K is the number of satellites that are within view of the antenna. 
         [0012]    The following analysis follows the signal of only one satellite and, for sake of simplicity superscript n is omitted from the equations. The fact that other satellite signals exist is addressed later. 
         [0013]    The transmitted signal is subjected to transit time delay before reaching the receiver, and the signal that is received by a first receiver&#39;s antenna experiences a Doppler frequency shift, f D , due to the satellite&#39;s movement in its orbit and possible receiver motion. Also, the transmitter and the receiver do not have a common clock, which means that even when the transmitter and the receiver clocks are at identical frequency, there is a phase difference between them. To make the equations more general, one might assume that there is a time shift (the transitions are not fully aligned) between the AD(t)x C (t) and the BD(t)x Y (t) , so the signal received at the first receiver can be expressed as 
         [0000]        S   received,1   =A   1   D ( t −τ C,1 ) x   C ( t −τ C,1 )cos(2π( f   L1   +f   D,1 )( t−τ   1 )+φ 1,1 )+ B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 )sin(2π( f   L1   +f   D,1 )( t−τ   1 )+φ 1,1 )  (2)
 
         [0000]    or simplified to 
         [0000]        S   received,1   =A   1   D ( t −τ C,1 ) x   C ( t −τ C,1 )cos(2π( f   L1   +f   D,1 ) t+φ   1,1   −φ   2,1 )+ B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 )sin(2π( f   L1   +f   D,1 ) t )+φ 1,1 −φ 2,1 )  (3)
 
         [0000]    It may be noted that for GPS, the C/A code and Y code are aligned, so this generalization is not needed for a GPS-based embodiment. 
         [0014]    As shown in  FIG. 1 , the received signal is detected and amplified in element  10 , conventionally downshifted in element  12  to a preselected intermediate frequency (IF) by multiplying the received signal by signal 
         [0000]      sin(2π( f   L1   −f   IF ) t+φ   3,1 )  (4)
 
         [0000]    and low pass filtered by element  15 . The signal of equation (4) is generated from reference oscillator  20  by frequency synthesizer  22 , where φ 3  is the phase of the locally generated signal (relative to the beginning of the data and code signals at the transmitting satellite which, of course, is unknown). The result at the output of the low pass filter is 
         [0000]        S   downshifted,1   =A   1   D ( t−τ   C,1 ) x   C ( t−τ   C,1 )cos(2π( f   IF   +f   D,1 ) t+φ   1,1 −φ 2,1 −φ 3,1 )+ B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 )sin(2π( f   IF   +f   D,1 ) t+φ   1,1 −φ 2,1 −φ 3,1 )  (5)
 
         [0000]    or simplified to 
         [0000]        S   downshifted,1   =A   1   D ( t−τ   C,1 ) x   C ( t−τ   C,1 )cos(2π( f   IF   +f   D,1 ) t+θ   1 )+ B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 )sin(2π( f   IF   +f   D,1 ) t+θ   1 ).  (6)
 
         [0015]    It may be noted that the above-described use of downshifting by use of an IF modulator and low pass filter is illustrative, and that the A/D can be connected directly to the amplifier, and controlled to generate a digital signal as if it were downshifted as shown in  FIG. 1 . 
         [0016]    As depicted, the output signal of the low pass filter is digitized in A/D converter  18  and applied to a combination of processor  40  and associated memory  41 , where the remainder of the processing takes place. 
         [0017]    The processing in accord with the instant disclosure, shown in  FIG. 2 , begins with a carrier generator module  31  creating the signal 
         [0000]      cos(2π( f   IF   +{circumflex over (f)}   D ) t+{circumflex over (θ)}   1 )− i sin(2π( f   IF   +{circumflex over (f)}   D ) t+{circumflex over (θ)}   1 ),  (7)
 
         [0000]    where {circumflex over (f)} D  is an estimate of the Doppler frequency shift f D , and {circumflex over (θ)} 1  is an estimate of the phase θ 1 . To be clear, the Doppler frequency shift and the phase estimates are estimates for a particular satellite. Multiplying the received (and downshifted) signal of equation (6) by the phasor of equation (7) in element  32  yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       1 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               1 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         C 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               1 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 + 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 f 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   f 
                                                   ^ 
                                                 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         θ 
                                         1 
                                       
                                       - 
                                       
                                         
                                           θ 
                                           ^ 
                                         
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       θ 
                                       1 
                                     
                                     - 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                   + 
                   
                     
                       B 
                       1 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               1 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         Y 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               1 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       { 
                       
                         
                           
                             
                               + 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       θ 
                                       1 
                                     
                                     - 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 i 
                               
                                
                               
                                   
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       t 
                                     
                                     + 
                                     
                                       θ 
                                       1 
                                     
                                     - 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be viewed as a real or inphase component (which is not shown in  FIG. 2 ) 
         [0000]        S   I,1   =A   1   D ( t−τ   C,1 ) x   C ( t−τ   C,1 ){+cos(2π( f   D,1   −{circumflex over (f)}   D,1 ) t+θ   1 −{circumflex over (θ)} 1 )}+ B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 ){+sin(2π( f   D,1   −{circumflex over (f)}   D,1 ) t+θ   1 −{circumflex over (θ)} 1 )}  (9)
 
         [0000]    and a quadrature component (which is shown in  FIG. 2 ) 
         [0000]        S   Q,1   =A   1   D ( t−τ   C,1 ) x   C ( t−τ   C,1 ){sin(2π( f   D,1   −{circumflex over (f)}   D,1 ) t+θ   1 −{circumflex over (θ)} 1 )}− B   1   D ( t−τ   Y,1 ) x   Y ( t−τ   Y,1 ){cos(2π( f   D,1   −{circumflex over (f)}   D,1 ) t+θ   1 −{circumflex over (θ)} 1 )}  (10)
 
         [0000]    Integrating this signal in element  33  over a preselected interval that is long enough to filter out the 2f IF  signal component yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         
                           A 
                           1 
                         
                          
                         
                           D 
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 τ 
                                 
                                   C 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             x 
                             C 
                           
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 τ 
                                 
                                   C 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     f 
                                                     IF 
                                                   
                                                   + 
                                                   
                                                     
                                                       f 
                                                       ^ 
                                                     
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             
                                               θ 
                                               ^ 
                                             
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     f 
                                                     IF 
                                                   
                                                   + 
                                                   
                                                     f 
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             θ 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   IF 
                                                 
                                                 + 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   IF 
                                                 
                                                 + 
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                     + 
                     
                       ∫ 
                       
                         
                           B 
                           1 
                         
                          
                         
                           D 
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 τ 
                                 
                                   Y 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             x 
                             Y 
                           
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 τ 
                                 
                                   Y 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     f 
                                                     IF 
                                                   
                                                   + 
                                                   
                                                     
                                                       f 
                                                       ^ 
                                                     
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             
                                               θ 
                                               ^ 
                                             
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     f 
                                                     IF 
                                                   
                                                   + 
                                                   
                                                     f 
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             θ 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   IF 
                                                 
                                                 + 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   IF 
                                                 
                                                 + 
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                    
                   
                     
 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be written as 
         [0000]    
       
         
           
             
               
                 
                   
                     ∫ 
                     
                       
                         A 
                         1 
                       
                        
                       
                         D 
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               τ 
                               
                                 C 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           x 
                           C 
                         
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               τ 
                               
                                 C 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         { 
                         
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   f 
                                                   IF 
                                                 
                                               
                                               + 
                                               
                                                 f 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   f 
                                                   ^ 
                                                 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         θ 
                                         1 
                                       
                                       + 
                                       
                                         
                                           θ 
                                           ^ 
                                         
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 f 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   f 
                                                   ^ 
                                                 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         θ 
                                         1 
                                       
                                       - 
                                       
                                         
                                           θ 
                                           ^ 
                                         
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     f 
                                                     IF 
                                                   
                                                 
                                                 + 
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                         + 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 i 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               
                                                 f 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   f 
                                                   ^ 
                                                 
                                                 
                                                   D 
                                                   , 
                                                   1 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         t 
                                       
                                       + 
                                       
                                         θ 
                                         1 
                                       
                                       - 
                                       
                                         
                                           θ 
                                           ^ 
                                         
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   + 
                   
                     ∫ 
                     
                       
                         B 
                         1 
                       
                        
                       
                         D 
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               τ 
                               
                                 Y 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           x 
                           Y 
                         
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               τ 
                               
                                 Y 
                                 , 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           { 
                           
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     f 
                                                     IF 
                                                   
                                                 
                                                 + 
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                                 + 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                         + 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                         - 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     i 
                                      
                                     
                                         
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       f 
                                                       IF 
                                                     
                                                   
                                                   + 
                                                   
                                                     f 
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       f 
                                                       ^ 
                                                     
                                                     
                                                       D 
                                                       , 
                                                       1 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           
                                             θ 
                                             1 
                                           
                                           + 
                                           
                                             
                                               θ 
                                               ^ 
                                             
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   f 
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     f 
                                                     ^ 
                                                   
                                                   
                                                     D 
                                                     , 
                                                     1 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           t 
                                         
                                         + 
                                         
                                           θ 
                                           1 
                                         
                                         - 
                                         
                                           
                                             θ 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0018]    In accord with one implementation of the principles disclosed herein, unit  100  knows its own global position and it also knows the global position of a particular remote unit  200 . 
         [0019]    Illustratively, unit  100  is situated on the roof of a corporate headquarters building in one city, and unit  200  is situated on the roof of a corporate building in another city. If unit  100  does not know the location of unit  200 , it may obtain it from unit  200  and verify that the location is bona fide in the manner disclosed in the parent application identified above. Armed with this knowledge, unit  100  is able to send information to unit  200  by generating and processing a signal that is a close facsimile of the signal that unit  100  knows is received by unit  200 . Specifically, unit  100  may modulate its signal with a time delay (delay-based encoding), a frequency shift (frequency-based encoding), or both, in order to effect secure information transmittal to unit  200 . To realize frequency-based encoding, unit  100  may proceed as follows: from readily available information, unit  100  obtains a fairly good estimate of the Doppler frequency shift, {circumflex over (f)} D,2 , of the signal arriving at the unit  200  location from a particular satellite (which is a fairly good estimate), generates a reference signal with this frequency in element  35 , frequency shifts that signal by a chosen frequency value, f datum  in element  36 , to obtain a frequency {tilde over (f)} D,2  (i.e., {tilde over (f)} D,2 ={circumflex over (f)} D,2 +f datum ), and thus creates the signal 
         [0000]      cos[2π( f   IF   +{tilde over (f)}   D,2 )( t )+{circumflex over (θ)} 2   ]−i sin[2π( f   IF   +{tilde over (f)}   D,2 )( t )+{circumflex over (θ)} 2 ]  (13)
 
         [0000]    that is applied to element  37 . To realize delay-based encoding, unit  100  may proceed as follows: element  37  multiplies the signal of equation (13) by the quadrature signal of equation (12), delays it by δ datum  in element  38 , and outputs the signal 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     
                       Q 
                       , 
                       1 
                       , 
                       2 
                       , 
                       δ 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 A 
                                 1 
                               
                                
                               
                                 D 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         C 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       δ 
                                       datum 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   x 
                                   C 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         C 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       δ 
                                       datum 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                  
                                 
                                   [ 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                             - 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 1 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             δ 
                                             datum 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       θ 
                                       1 
                                     
                                     - 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 B 
                                 1 
                               
                                
                               
                                 D 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         Y 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       δ 
                                       datum 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   x 
                                   Y 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         Y 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       δ 
                                       datum 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                 [ 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       π 
                                        
                                       
                                         ( 
                                         
                                           
                                             f 
                                             
                                               D 
                                               , 
                                               1 
                                             
                                           
                                           - 
                                           
                                             
                                               f 
                                               ^ 
                                             
                                             
                                               D 
                                               , 
                                               1 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           δ 
                                           datum 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     θ 
                                     1 
                                   
                                   - 
                                   
                                     
                                       θ 
                                       ^ 
                                     
                                     1 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                     × 
                     2 
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   [ 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               IF 
                                             
                                             + 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 2 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             δ 
                                             datum 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   [ 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         π 
                                          
                                         
                                           ( 
                                           
                                             
                                               f 
                                               IF 
                                             
                                             + 
                                             
                                               
                                                 f 
                                                 ^ 
                                               
                                               
                                                 D 
                                                 , 
                                                 2 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             δ 
                                             datum 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         θ 
                                         ^ 
                                       
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be expressed more compactly as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       
                         Q 
                         , 
                         1 
                         , 
                         2 
                         , 
                         δ 
                       
                     
                     = 
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     A 
                                     1 
                                   
                                    
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             C 
                                             , 
                                             1 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           datum 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       x 
                                       C 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             C 
                                             , 
                                             1 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           datum 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       α 
                                       ) 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   B 
                                   1 
                                 
                                  
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           Y 
                                           , 
                                           1 
                                         
                                       
                                       - 
                                       
                                         δ 
                                         datum 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     x 
                                     Y 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           Y 
                                           , 
                                           1 
                                         
                                       
                                       - 
                                       
                                         δ 
                                         datum 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     α 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       × 
                       2 
                        
                       
                         { 
                         
                           
                             cos 
                              
                             
                               ( 
                               β 
                               ) 
                             
                           
                           - 
                           
                             i 
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   where 
                    
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     α 
                     = 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 f 
                                 
                                   D 
                                   , 
                                   1 
                                 
                               
                               - 
                               
                                 
                                   f 
                                   ^ 
                                 
                                 
                                   D 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               δ 
                               datum 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         θ 
                         1 
                       
                       - 
                       
                         
                           
                             θ 
                             ^ 
                           
                           1 
                         
                          
                         
                             
                         
                          
                         and 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     β 
                     = 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 f 
                                 IF 
                               
                               + 
                               
                                 
                                   f 
                                   ~ 
                                 
                                 
                                   D 
                                   , 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               δ 
                               datum 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             θ 
                             ^ 
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0020]    It might be remembered that the analysis above focuses on the signal of one satellite while recognizing that signals from a number of satellites are concurrently processed, and it also should be remembered that the signal of equation (13) that is created within unit  100  pertains, relative to the Doppler shift and delay, to a single satellite that is chosen by unit  100 . Thus, it should be realized that the equation (15) signal is really a sum of signals of the form found within the brackets {} that are all multiplied by the 2{cos(β)−isin(β)} term that is adjusted to parameters for the one chosen satellite; to wit, the output signal of unit  100 , S 100 , is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       100 
                     
                     = 
                     
                       2 
                        
                       
                         { 
                         
                           
                             cos 
                              
                             
                               ( 
                               β 
                               ) 
                             
                           
                           - 
                           
                             i 
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           K 
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       A 
                                       1 
                                       n 
                                     
                                      
                                     
                                       
                                         D 
                                         n 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             τ 
                                             
                                               C 
                                               , 
                                               1 
                                             
                                           
                                           - 
                                           
                                             δ 
                                             datum 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         x 
                                         C 
                                         n 
                                       
                                        
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             τ 
                                             
                                               C 
                                               , 
                                               1 
                                             
                                           
                                           - 
                                           
                                             δ 
                                             datum 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           i 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     1 
                                     n 
                                   
                                    
                                   
                                     
                                       D 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             Y 
                                             , 
                                             1 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           datum 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       x 
                                       Y 
                                       n 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             Y 
                                             , 
                                             1 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           datum 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         α 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   where 
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       α 
                       i 
                     
                     = 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 f 
                                 
                                   D 
                                   , 
                                   1 
                                 
                                 
                                   ( 
                                   i 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   f 
                                   ^ 
                                 
                                 
                                   D 
                                   , 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               δ 
                               datum 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         θ 
                         1 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       - 
                       
                         
                           θ 
                           ^ 
                         
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    f D,1   (i)  is the Doppler of satellite i as measured at unit  100 
 
θ 1   (i)  is the phase shift of the signal from satellite i as measured at unit  100 .
 
         [0021]    Unit 100 sends the signal of equation (15a) to unit  200 , which also receives its own GPS signal, s 2 . Within unit  200  signal s 2  is downshifted to develop the signal 
         [0000]        s   2   ·s   mix,2   =A   2   D ( t−τ   C,2 ) x   C ( t−τ   C,2 )cos[2π( f   IF   +f   D,2 ) t+θ′   2   ]+B   2   D ( t−τ   Y,2 ( x   P ( t−τ   Y,2 )sin[2π( f   IF   +f   D,2 ) t+θ′   2 ],  (16)
 
         [0000]    where τ c,2 , τ c,2  and f D,2  are the delays and Doppler frequency shift experienced by the signal that reaches unit  200 . It is noted that in a conventional manner the Doppler frequency f D,2  may be determined, for example by tracking in a frequency-lock or phase-lock loop. 
         [0022]    In accord with one embodiment of the principles disclosed herein and depicted in  FIG. 3 , unit  200  receives a signal by means of elements that correspond to elements  10 ,  12 ,  15 ,  18 ,  20  and  22  of  FIG. 1 , but which are not shown in  FIG. 2  for sake of clarity, and within the processor of unit  200  (which corresponds to processor  40  of  FIG. 1 ) performs what effectively is a two-dimensional correlation between the signal received from the satellites and the signal received from unit  100 , by shifting the signal received from the satellites (i.e., the signal of equation (16)) by f *  in element  53 , delaying the shifted signal by δ *  in element  54 , multiplying the output of element  54  by the signal received from unit  100  (i.e., the signal of equation (15) in element  55 ), integrating in element  56 , and repeating the process with different values of f *  and δ *  to find a peak, all under management of controller  57 . 
         [0023]    The frequency shifting and the time delaying of the equation (16) signal yields 
         [0000]        s   2,δ   =A   2   D ( t−τ   C,2 −δ * ) x   C ( t−τ   C,2 −δ * )cos[2π( f   IF   +{hacek over (f)}   D,2 )( t−δ   * )+θ 2   ′]+B   2   D ( t−τ   Y,2 −δ * ) x   P ( t−τ   Y,2 −δ * )sin[2π( f   IF   +{hacek over (f)}   D,2 )( t −δ * )+θ 2 ′]  (17)
 
         [0000]    where {hacek over (f)} D,2 =f D,2 +f * . Labeling 2π(f IF +{hacek over (f)} D,2 )(t−δ * )+θ 2 ′ as γ yields 
         [0000]        s   2,δ   =A   2   D ( t−τ   C,2 −δ * ) x   C ( t−τ   C,2 −δ * )cos(γ) + B   2   D ( t−τ   Y,2 −δ * ) x   P ( t−τ   Y,2 −δ * )sin(γ)  (18)
 
         [0000]    The multiplication of the equation (18) signal by the signal of equation (15) yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       
                         Q 
                         , 
                         1 
                         , 
                         2 
                         , 
                         δ 
                       
                     
                     · 
                     
                       s 
                       
                         2 
                         , 
                         
                           δ 
                           * 
                         
                       
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   A 
                                   1 
                                 
                                  
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           C 
                                           , 
                                           1 
                                         
                                       
                                       - 
                                       δ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     x 
                                     C 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           C 
                                           , 
                                           1 
                                         
                                       
                                       - 
                                       δ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     α 
                                     ) 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 B 
                                 1 
                               
                                
                               
                                 D 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         Y 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     δ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   x 
                                   P 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       
                                         Y 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     δ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   α 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                     × 
                     2 
                      
                     
                       { 
                       
                         
                           cos 
                            
                           
                             ( 
                             β 
                             ) 
                           
                         
                         - 
                         
                           i 
                            
                           
                               
                           
                            
                           
                             sin 
                              
                             
                               ( 
                               β 
                               ) 
                             
                           
                         
                       
                       } 
                     
                     × 
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     A 
                                     2 
                                   
                                    
                                   
                                     D 
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             C 
                                             , 
                                             2 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           * 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       x 
                                       C 
                                     
                                      
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           
                                             C 
                                             , 
                                             2 
                                           
                                         
                                         - 
                                         
                                           δ 
                                           * 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       γ 
                                       ) 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   B 
                                   2 
                                 
                                  
                                 
                                   D 
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           Y 
                                           , 
                                           2 
                                         
                                       
                                       - 
                                       
                                         δ 
                                         * 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     x 
                                     P 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         τ 
                                         
                                           Y 
                                           , 
                                           2 
                                         
                                       
                                       - 
                                       
                                         δ 
                                         * 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     γ 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Carrying out the multiplication, grouping terms, and dropping the terms involving cos(γ+β) and sin(γ+β) because subsequent integration acts as low pass filtering, yields 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               s 
                               
                                 Q 
                                 , 
                                 1 
                                 , 
                                 2 
                                 , 
                                 δ 
                               
                             
                             · 
                             
                               s 
                               
                                 2 
                                 , 
                                 
                                   δ 
                                   * 
                                 
                               
                             
                           
                           = 
                           
                             
                               ( 
                               
                                 U 
                                 - 
                                 V 
                               
                               ) 
                             
                             × 
                             
                               ( 
                               
                                 W 
                                 + 
                                 iX 
                                 + 
                                 Y 
                                 - 
                                 iZ 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             { 
                             
                               
                                 
                                   
                                     UW 
                                     + 
                                     iUX 
                                     + 
                                     UY 
                                     - 
                                     iUZ 
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     VW 
                                     - 
                                     iVX 
                                     - 
                                     VY 
                                     + 
                                     iVZ 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   U 
                   = 
                   
                     
                       A 
                       1 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               1 
                             
                           
                           - 
                           δ 
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         C 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               1 
                             
                           
                           - 
                           δ 
                         
                         ) 
                       
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         α 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   V 
                   = 
                   
                     
                       B 
                       1 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               1 
                             
                           
                           - 
                           δ 
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         Y 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               1 
                             
                           
                           - 
                           δ 
                         
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         α 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   W 
                   = 
                   
                     
                       A 
                       2 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         C 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           γ 
                           - 
                           β 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   X 
                   = 
                   
                     
                       A 
                       2 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         C 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               C 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           γ 
                           - 
                           β 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       B 
                       2 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         Y 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           γ 
                           - 
                           β 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   Z 
                   = 
                   
                     
                       B 
                       2 
                     
                      
                     
                       D 
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         x 
                         Y 
                       
                        
                       
                         ( 
                         
                           t 
                           - 
                           
                             τ 
                             
                               Y 
                               , 
                               2 
                             
                           
                           - 
                           
                             δ 
                             * 
                           
                         
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           γ 
                           - 
                           β 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0024]    The signal of equation (19) is integrated for various values of the delay δ *  and frequency offset f *  to develop 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           S 
                           
                             δ 
                             , 
                             
                               δ 
                               * 
                             
                           
                         
                         = 
                         
                           ∫ 
                           
                             
                               s 
                               
                                 Q 
                                 , 
                                 1 
                                 , 
                                 2 
                                 , 
                                 δ 
                               
                             
                             · 
                             
                               s 
                               
                                 2 
                                 , 
                                 
                                   δ 
                                   * 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           ∫ 
                           
                             
                               { 
                               
                                 UW 
                                 + 
                                 iUX 
                                 + 
                                 UY 
                                 - 
                                 iUZ 
                                 - 
                                 VW 
                                 - 
                                 iVX 
                                 - 
                                 VY 
                                 + 
                                 iVZ 
                               
                               } 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0025]    At this point it may be noted that although the signal of equation (18) shows the signal of one satellite, unit  200  actually develops a signal that includes a contribution from all visible satellites. It can be shown that each of the integration results therefore may result in a plurality of peaks, but the one that pertains to the chosen satellite is the peak with highest energy. The energy of this peak, relative to the peaks for the other satellites commonly visible to units  100  and  200 , may be further enhanced through signal processing; for example, with a directional antenna or a beamsteering antenna array focused to enhance the signal of the satellite of interest. 
         [0026]    The Doppler frequency estimate {circumflex over (f)} D,1  is very close to f D,1 , and the expression θ 1  is very close to {circumflex over (θ)} 1  (as a result of the capture and tracking operations). Consequently, all terms containing the factor “U” (which includes sin(α)≈α≈0) drop out. Also, the cos(α) can be replaced by 1. Likewise, it is noted that the code sequence x C  is orthogonal to the code sequence x Y  (meaning that following integration the sum of their product is zero). Consequently, the “VW” and “VX” terms drop out. Additionally, the “V” term, which contains the cos(α) reduces to V=B 1  D(t−τ Y,1 −δ) x Y  (t−τ Y,1 −δ), leaving 
         [0000]        S   δ,δ     *     =∫{−VY+iVZ}=−∫VY+i∫VZ.    (28)
 
         [0027]    As indicated above, the expression of equation (27) is evaluated for different values of δ *  and f *  (effectively a two-dimensional correlation), and values δ best   *  and f best   *  are found that yield the maximum magnitude; i.e., 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                            
                           
                             
                               S 
                               
                                 δ 
                                 , 
                                 
                                   δ 
                                   * 
                                 
                               
                             
                              
                             
                               | 
                               max 
                             
                           
                            
                         
                         2 
                       
                       = 
                       
                         max 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         ∫ 
                                         
                                           
                                             Ψ 
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 , 
                                                 
                                                   δ 
                                                   * 
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               α 
                                               ) 
                                             
                                           
                                            
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 γ 
                                                 - 
                                                 β 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       ∫ 
                                       
                                         
                                           Ψ 
                                            
                                           
                                             ( 
                                             
                                               t 
                                               , 
                                               
                                                 δ 
                                                 * 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             α 
                                             ) 
                                           
                                         
                                          
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               γ 
                                               - 
                                               β 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     where 
                      
                     
                       
 
                     
                      
                     
                       
                         Ψ 
                          
                         
                           ( 
                           
                             t 
                             , 
                             
                               δ 
                               * 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               B 
                               1 
                             
                              
                             
                               D 
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     τ 
                                     
                                       Y 
                                       , 
                                       1 
                                     
                                   
                                   - 
                                   δ 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               B 
                               2 
                             
                              
                             
                               D 
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     τ 
                                     
                                       Y 
                                       , 
                                       2 
                                     
                                   
                                   - 
                                   
                                     δ 
                                     * 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             x 
                             Y 
                           
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 τ 
                                 
                                   Y 
                                   , 
                                   1 
                                 
                               
                               - 
                               δ 
                             
                             ) 
                           
                         
                          
                         
                           
                             
                               x 
                               Y 
                             
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   τ 
                                   
                                     Y 
                                     , 
                                     2 
                                   
                                 
                                 - 
                                 
                                   δ 
                                   * 
                                 
                               
                               ) 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0028]    When the approximations are good; that is, f D,1 ≈{circumflex over (f)} D,1  and θ 1 ≈{circumflex over (θ)} 1  then cos(α)≈1, and reinstating what γ and β stand for, and looking only within the brackets, we have 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ∫ 
                                         
                                           
                                             Ψ 
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 , 
                                                 
                                                   δ 
                                                   * 
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               x 
                                               Y 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 
                                                   τ 
                                                   
                                                     Y 
                                                     , 
                                                     1 
                                                   
                                                 
                                                 - 
                                                 δ 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               x 
                                               Y 
                                             
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 
                                                   τ 
                                                   
                                                     Y 
                                                     , 
                                                     2 
                                                   
                                                 
                                                 - 
                                                 
                                                   δ 
                                                   * 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               2 
                                                
                                               
                                                 π 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         f 
                                                         ⋓ 
                                                       
                                                       
                                                         D 
                                                         , 
                                                         2 
                                                       
                                                     
                                                     - 
                                                     
                                                       
                                                         f 
                                                         ⋒ 
                                                       
                                                       
                                                         D 
                                                         , 
                                                         2 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               t 
                                             
                                             + 
                                             Γ 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       ∫ 
                                       
                                         
                                           Ψ 
                                            
                                           
                                             ( 
                                             
                                               t 
                                               , 
                                               
                                                 δ 
                                                 * 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             x 
                                             Y 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               - 
                                               
                                                 τ 
                                                 
                                                   Y 
                                                   , 
                                                   1 
                                                 
                                               
                                               - 
                                               δ 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             x 
                                             Y 
                                           
                                            
                                           
                                             ( 
                                             
                                               t 
                                               - 
                                               
                                                 τ 
                                                 
                                                   Y 
                                                   , 
                                                   2 
                                                 
                                               
                                               - 
                                               
                                                 δ 
                                                 * 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       f 
                                                       ⋓ 
                                                     
                                                     
                                                       D 
                                                       , 
                                                       2 
                                                     
                                                   
                                                   - 
                                                   
                                                     
                                                       f 
                                                       ⋒ 
                                                     
                                                     
                                                       D 
                                                       , 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             t 
                                           
                                           + 
                                           Γ 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ) 
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   Γ 
                   = 
                   
                     
                       2 
                        
                       
                         π 
                          
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     f 
                                     IF 
                                   
                                   + 
                                   
                                     
                                       f 
                                       ~ 
                                     
                                     
                                       D 
                                       , 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                                
                               δ 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     f 
                                     IF 
                                   
                                   + 
                                   
                                     
                                       f 
                                       ⋓ 
                                     
                                     
                                       D 
                                       , 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 δ 
                                 * 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       θ 
                       2 
                       ′ 
                     
                     - 
                     
                       
                         
                           θ 
                           ^ 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
         [0029]    Under the assumption that the code and the data take on value of only +1 or −1, and because the autocorrelation of x Y (t) is close to zero at all but t=0, it follows that equation (29) is essentially 0 except when (−τ Y,1 −δ)=(−τ Y,2 −δ * ), or 
         [0000]      δ=δ best   * −τ Y,1 +τ Y,2 .  (32)
 
         [0000]    at which point it degenerates to 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             
                               δ 
                               , 
                               
                                 δ 
                                 * 
                               
                             
                           
                            
                           
                             | 
                             max 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   B 
                                   1 
                                 
                                  
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           ∫ 
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                     π 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           
                                                             f 
                                                             ⋓ 
                                                           
                                                           
                                                             D 
                                                             , 
                                                             2 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             f 
                                                             ~ 
                                                           
                                                           
                                                             D 
                                                             , 
                                                             2 
                                                           
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   t 
                                                 
                                                 + 
                                                 Γ 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         ∫ 
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 
                                                   π 
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           f 
                                                           ⋓ 
                                                         
                                                         
                                                           D 
                                                           , 
                                                           2 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           f 
                                                           ~ 
                                                         
                                                         
                                                           D 
                                                           , 
                                                           2 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 t 
                                               
                                               + 
                                               Γ 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   B 
                                   1 
                                 
                                  
                                 
                                   B 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           ∫ 
                                           
                                             sin 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                     π 
                                                      
                                                     
                                                       ( 
                                                       
                                                         Δ 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         f 
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   t 
                                                 
                                                 + 
                                                 Γ 
                                               
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         ∫ 
                                         
                                           cos 
                                            
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 
                                                   π 
                                                    
                                                   
                                                     ( 
                                                     
                                                       Δ 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       f 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 t 
                                               
                                               + 
                                               Γ 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Δf={hacek over (f)} D,2 −{tilde over (f)} D,2  which leads to 
         [0000]      Δf=( f   D,2   −{circumflex over (f)}   D,2 )+( f   best   *   −f   secret )  (34)
 
         [0030]    The peak in the value of S δ,δ     *   | max  occurs when Δf is very small. Since the estimate {circumflex over (f)} D,2  is very close to f D,2 , equation (34) degenerates to f secret =f * , and that equation (33) reduces to: 
         [0000]      S δ,δ     *   | max =(B 1 B 2 ) 2 .  (35)
 
         [0031]    What we have, then, is that when the transit delay and Doppler frequency shift information derived from published tables, geometric considerations, etc. are accurate, the autocorrelation has a peak only when 
         [0032]    (a) equation (32) condition holds; i.e., δ datum =δ * −τ Y,1 +τ Y,2 , and 
         [0033]    (b) equation (34) condition holds; i.e., f datum =f * . 
         [0034]    Since unit  200  can compute the transit delay difference (τ Y,2 −τ Y,1 ) using, for example, published tables describing the satellite orbits, the δ datum  information injected into the signal by unit  100  is easily recovered at unit  200  (equation 32). Conversely, when unit  100  wishes unit  200  to recover a particular value δ * , unit  100  accounts for the transit delay difference and computes the δ datum  that it needs to send. Also, when δ datum =0, equation (32) yields a value that corresponds to (τ Y,1 −τ Y,2 ), and from the above-mentioned tables, unit  200  can determine the location of unit  100 . 
         [0035]    The same capability exists in connection with the frequencies, in that information can be communicated from unit  100  to unit  200  via the f datum  value. 
         [0036]    Going back to equation (12), it is noted that it includes a signal component that is modulated by the x C  code, which is publicly known. The chip rate of the x C  code has a bandwidth of about 2 MHz. (2 MHz main lobe). 
         [0037]    Based on this observation, an alternative embodiment in accord with the principles disclosed herein passes the signal of equation (12) through a bandstop filter that is adjusted to remove the publicly known x C  code-modulated component (alternatively one can pass the signal of equations (13) or (14) through the bandstop filter). Passing the signal though the such a filter alters the equation (15) signal to 
         [0000]      s Q,1,2,δ =−2[ B   1   D ( t−τ   Y,1 −δ) x   Y ( t−τ   Y,1 −δ)cos(α)][cos(β)− i sin(β)]  (36)
 
         [0000]    but that does NOT change equation (28). The difference, of course, is that the embodiment that includes the bandstop filter does not send a signal that includes a knowable signal component that perhaps might be used by an adversary, and yet accomplishes the same result as an embodiment that does not use the bandstop filter. 
         [0038]    In yet another embodiment in accord with the principles disclosed herein, the signal that is processed by unit  100  and sent to unit  200  can be the signal of just a selected subset of the visible satellites; perhaps just one of the satellites. Illustratively, this is accomplished by having the input antenna of unit  100  be steerable, though other more complicated techniques that would work as well. Any of the known designs or techniques for creating a steerable antenna is acceptable. There are also other techniques that may be used which work as well. This method reduces the number of peaks that are achievable at an adversary unit, as well as at unit  200 , and further it obscures the satellite signal which was used for the delay-based and/or frequency-based encoding, making reverse-engineering of the values of δ datum  and/or f datum  substantially harder (or impracticable) for an adversary. 
         [0039]    The above embodiments do not specify the duration of the signal that unit  100  sends to unit  200 . A short duration results in a smaller correlation peak. A smaller peak is more difficult to detect in the presence of peaks that result from spurious signals (noise). It is, therefore, useful to limit the duration of the signal that unit  100  sends. 
         [0040]    The above discloses the notion that two pieces of information can be send by unit  100  to unit  200 , in a secure manner, with each transmission of a signal segment: one embedded in δ datum  and the other embedded in f datum . Data can be communicated continuously, of course, by sending {δ datum , f datum }-tuples in successive frames. 
         [0041]    It may be noted that although the above discloses the principles of this invention in connection with GPS signals, that is not a limitation of this invention. Alternate sources that can create signal like the Y code include device that operate in a WiFi protocol, Blue tooth protocol, cellular telephony protocols, etc. 
         [0042]    It may be further noted that the illustrative embodiment disclosed above is adapted to a situation where the location of unit  100  is known to unit  200 .