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.

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 
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         [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 
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         [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 
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                                 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 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                                
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                   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]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                   ( 
                   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 .