Abstract:
Conventional transceivers do provide some compensation for in-phase/quadrature (I/Q) imbalance. However, these techniques do not separately compensate for I/Q imbalance for the transmitter and receiver sides of the transceiver. Here, a transceiver is provided that allows for compensation of I/Q imbalance in the transmitter and receiver irrespective of the other to allow for a more accurate transceiver.

Description:
TECHNICAL FIELD 
       [0001]    The invention relates generally to a radio frequency (RF) transceiver and, more particularly, to in-phase/quadrature (I/Q) mismatch compensation for an RF transceiver. 
       BACKGROUND 
       [0002]    The direct conversion radio architecture is a popular choice for radio frequency (RF) transceiver design. It uses fewer components and is easier to integrate compared to the conventional heterodyne architecture. See, e.g., Mirabbasi et al., “Classical and modern receiver architectures,”  IEEE Commun. Mag ., vol. 38, no. 11, pp. 132-139, 2000. Those benefits also come with several well-known impairments, namely in-phase/quadrature (I/Q) imbalance and direct current (DC) offset. 
         [0003]    In a direct conversion transceiver, a quadrature modulator (QM) implements I/Q modulation and RF upconversion, while a quadrature demodulator (QDM) realizes I/Q demodulation and RF downconversion. Because of the limitation in analog circuit precision, the quadrature carriers used in QM and QDM do not have exactly the same amplitude and a perfect 90° phase difference. Similarly, the analog reconstruction filters on the I and Q paths may not match exactly. These imperfections are called I/Q imbalance, which causes cross talk between I and Q channels and which creates undesired images of the original signal. Addition, because of limited isolation in analog components, some of the local oscillator (LO) power can leak into RF output, which creates LO spikes in transmitted signal and DC offset in received signal. 
         [0004]    Most existing compensation techniques treat transmit and receive I/Q imbalance separately. To compensate for I/Q imbalance in the transmitter, the receive side is either assumed to have dedicated feedback loops or digital demodulators with perfect quadrature carriers. See, e.g., Cavers et al., “Adaptive compensation for imbalance and offset losses in direct conversion transceivers,”  IEEE Trans. Veh. Technol ., vol. 42, no. 4, pp. 581-588, 1993; Faulkner et al., “Automatic adjustment of quadrature modulators,”  Electron. Lett ., vol. 27, no. 3, pp. 214-216, 1991; Ding et al., “Compensation of Frequency-Dependent Gain/Phase imbalance in predistortion linearization systems,”  IEEE Trans. Circuits Syst ., vol. 55, no. 1, pp. 390-397, 2008; and Anttila et al., “Frequency-Selective I/Q mismatch calibration of wideband Direct-Conversion transmitters,”  IEEE Trans. Circuits Syst ., vol. 55, no. 4, pp. 359-363, 2008. To compensate for I/Q imbalance in the receiver, the transmit side is usually assumed to be free of I/Q imbalance. See, e.g., Anttila et al., “Circularity-Based I/Q imbalance compensation in wideband Direct-Conversion receivers,”  IEEE Trans. Veh. Technol ., vol. 57, no. 4, pp. 2099-2113, 2008. Each of these different approaches, however, has undesirable effects and drawbacks. 
         [0005]    Some other conventional techniques are as follows: U.S. Pat. No. 7,463,866; U.S. Pat. No. 7,567,611; and U.S. Pat. No. 7,567,783. 
       SUMMARY 
       [0006]    A preferred embodiment of the present invention, accordingly, provides an apparatus. The apparatus comprises a transmit path that is coupled to radio frequency (RF) circuitry, wherein the transmit path includes a modulation compensator that compensates for in-phase/quadrature (I/Q) mismatch in the transmit path; a switch that is coupled to the RF circuitry; a delay element that is coupled to the RF circuitry and that is coupled to the switch so as to actuate and deactuate the switch; and a receive path that is coupled to switch, wherein the receive path includes a demodulation compensator that compensates for I/Q mismatch in the receive path. 
         [0007]    In accordance with a preferred embodiment of the present invention, each of the receive and transmit paths further comprise an in-phase path and a quadrature path. 
         [0008]    In accordance with a preferred embodiment of the present invention, the in-phase path for the transmit path further comprises: an in-phase filter that is coupled to the modulation compensator; an in-phase digital-to-analog converter (DAC) that is coupled to the in-phase filter; an in-phase low pass filter (LPF) that is coupled to the in-phase DAC; and an in-phase mixer coupled to the in-phase LPF. 
         [0009]    In accordance with a preferred embodiment of the present invention, the quadrature path for the transmit path further comprises: a quadrature filter that is coupled to the modulation compensator; a quadrature DAC that is coupled to the quadrature filter; a quadrature LPF that is coupled to the quadrature DAC; and a quadrature mixer coupled to the quadrature LPF. 
         [0010]    In accordance with a preferred embodiment of the present invention, the transmit path further comprises: an adder that is coupled to the in-phase mixer and the quadrature mixer; a local oscillator that is coupled to the in-phase mixer; and a phase delay circuit that is coupled to the local oscillator and the quadrature mixer. 
         [0011]    In accordance with a preferred embodiment of the present invention, the in-phase path for the receive path further comprises: an in-phase mixer that is coupled to the switch; an in-phase LPF that is coupled to the in-phase mixer; and an in-phase analog-to-digital converter (ADC) that is coupled to the in-phase LPF and the demodulation compensator. 
         [0012]    In accordance with a preferred embodiment of the present invention, the quadrature path for the receive path further comprises: a first quadrature mixer that is coupled to the switch; a quadrature LPF that is coupled to the first quadrature mixer; a quadrature ADC that is coupled to the quadrature LPF; and a second quadrature mixer that is coupled to the quadrature ADC and the demodulation compensator. 
         [0013]    In accordance with a preferred embodiment of the present invention, the transmit path further comprises: a local oscillator that is coupled to the in-phase mixer; and a phase delay circuit that is coupled to the local oscillator and the first quadrature mixer. 
         [0014]    In accordance with a preferred embodiment of the present invention, the relationship between an input signal (u(n)) and an output signal (x(n)) for the modulation compensator is: 
         [0000]    
       
         
           
             
               
                 x 
                  
                 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 
                   u 
                    
                   
                     ( 
                     n 
                     ) 
                   
                 
                 - 
                 
                   
                     
                       h 
                       m 
                     
                     
                       h 
                       d 
                     
                   
                    
                   
                     
                       u 
                       * 
                     
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                 
                 - 
                 
                   c 
                   
                     h 
                     d 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    wherein c, h m , and h d  are parameters for the apparatus. 
         [0015]    In accordance with a preferred embodiment of the present invention, the relationship between an input signal (y(n)) and an output signal (v(n)) for the demodulation compensator is: 
         [0000]    
       
         
           
             
               
                 v 
                  
                 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       y 
                        
                       
                         ( 
                         n 
                         ) 
                       
                     
                     - 
                     d 
                   
                   ] 
                 
                 - 
                 
                   
                     
                       g 
                       m 
                     
                     
                       g 
                       d 
                       * 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           y 
                           * 
                         
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                       - 
                       d 
                     
                     ] 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    wherein d, g m , and h d  are parameters for the apparatus. 
         [0016]    In accordance with a preferred embodiment of the present invention, an apparatus is provided. The apparatus comprises a transmit path having: a modulation compensator that compensates for I/Q mismatch in the transmit path; an in-phase filter that is coupled to the modulation compensator; a quadrature filter that is coupled to the modulation compensator; a first DAC that is coupled to the in-phase filter; a second DAC that is coupled to the quadrature filter; a first LPF that is coupled to the first DAC; a second LPF that is coupled to the second DAC; a first local oscillator; a first mixer that is coupled to the first local oscillator and the first LPF; a first phase delay circuit that is coupled to the first local oscillator; a second mixer that is coupled to first phase delay circuit and the second LPF; and an adder that is coupled to the first and second mixers; RF circuitry that is coupled to the adder; a switch that is coupled to the RF circuitry; a delay element that is coupled to the RF circuitry and that is coupled to the switch so as to actuate and deactuate the switch; and a receive path having: a third mixer that is coupled to the switch; a fourth mixer that is coupled to the switch; a second local oscillator that is coupled to the third mixer; a second phase delay circuit that is coupled to the second local oscillator and the fourth mixer; a third LPF that is coupled to the third mixer; a fourth LPF that is coupled to the fourth mixer; a first analog-to-digital converter (ADC) that is coupled to the third LPF; a second ADC that is coupled to the fourth LPF; a fifth mixer that is coupled to the second ADC; and a demodulation compensator is coupled to the first ADC and the fifth mixer, wherein the demodulation compensator compensates for I/Q mismatch in the receive path. 
         [0017]    The foregoing has outlined rather broadly the features and technical advantages of the present invention in order that the detailed description of the invention that follows may be better understood. Additional features and advantages of the invention will be described hereinafter which form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and the specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0018]    For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
           [0019]      FIG. 1  is a radio frequency (RF) transceiver in accordance with a preferred embodiment of the present invention; 
           [0020]      FIGS. 2A and 2B  are graphs depicting the operation of the transceiver of  FIG. 1  without in-phase/quadrature (I/Q) imbalance compensation; and 
           [0021]      FIGS. 3A and 3B  are graphs depicting the operation of the transceiver of  FIG. 1  with I/Q imbalance compensation. 
       
    
    
     DETAILED DESCRIPTION 
       [0022]    Refer now to the drawings wherein depicted elements are, for the sake of clarity, not necessarily shown to scale and wherein like or similar elements are designated by the same reference numeral through the several views. 
         [0023]    Referring to  FIG. 1  of the drawings, the reference numeral  100  generally designates a radio frequency (RF) transceiver in accordance with a preferred embodiment of the present invention. Transceiver  100  generally comprises a transmit path  102 , RF circuitry  150 , a receive path  104 , a delay element  152 , and switch  51 . The transmit path  102  also generally comprises modulation compensator  106 , in-phase (I) filter  108 , quadrature (Q) filter  114 , digital-to-analog converters (DACs)  110  and  116 , low pass filters (LPFs)  112  and  118 , mixers  122  and  126 , local oscillator (LO)  120 , 90° phase delay circuit  124 , and adder  128 . The receive path  104  generally comprises a demodulation compensator  130 , analog-to-digital converters (ADCs)  132  and  138 , LPFs  134  and  140 , LO  144 , mixers  136 ,  142 , and  148 , and 90° phase delay circuit  146 . 
         [0024]    Look first to the transmit path  102 , in operation, the transmit path  102  receives a baseband signal u(n) and outputs an RF signal through the RF circuitry  150 . Initially, the modulation compensator  106  generates a compensated signal x(n) from the baseband signal u(n). I filter  108  and Q filter  114  generate the I or real portion x i (n) and Q or imaginary portion x q (n) of the compensated signal x(n), respectively. DACs  110  and  116  convert the I portion x i (n) and Q portion x q (n) of the compensated signal x(n), respectively, to digital signals, which are then filtered by LPFs  122  and  126 , converted to an RF signal by mixers  122  and  125 , phase delay circuit  124 , and adder  128 , and transmitted through RF circuitry  150 . 
         [0025]    Now turning to the receive path  104 , in operation, the receive path  104  receives a signal through the RF circuitry  150  to generate a baseband signal v(n). Simply put, ADCs  132  and  138 , LPFs  134  and  140 , LO  144 , mixers  136 ,  142 , and  148 , and phase delay circuit  146  demodulate the RF signal from the RF circuitry  150  to generate a demodulated signal y(n). This demodulated signal y(n) is adjusted by the demodulation compensator  130  to generate a compensated baseband signal v(n). 
         [0026]    Of interest, however, is the operation of the modulation compensator  106  and demodulation circuit  130 . Each of these circuits  130  and  106  is able to compensate for I/Q imbalance within its respect path irrespective of the other path. These circuits  130  and  106  are able to accomplish this as a result of the delay element  152  (which can be linear or non-linear) and switch  51  (which is actuated and deactuated by the delay element  152 . As a result, greatly improved performance can be observed. Additionally, delay element  152  may be comprised of a phase rotation circuit so as to introduce different phase rotations for each path  102  and  104 . 
         [0027]    For a given baseband signal z t (n), signal x(n) can be represented as follows: 
         [0000]        z   t ( n )= h   d   x ( n )+ h   m   x *( n )+ c   (1)
 
         [0000]    where h d , h m , and c are numeral constants. Additionally, because the switch Si can represent two different paths with different delays or the same paths with variable delay settings, phase rotations can be introduced to the baseband signal z t (n) for two paths or channels A and B becoming: 
         [0000]        z   r ( n )= Az   t ( n )  (2)
 
         [0000]        z   r ( n )= Bz   t ( n )  (3)
 
         [0000]    Preferably, the phase rotations between the two paths can have at least a 10% difference. For example, the phase difference can be between 36° and 324°. Similar to equation (1), signal y(n) also can be represented as follows: 
         [0000]        y ( n )= g   d   z   r ( n )+ g   m   z   r *( n )+ d   (4)
 
         [0000]    where g d , g m , and d are numerical constants; numerical constant d is specifically related to the length of the delay of delay element  152 . Now substituting equations (1), (2), and (3) into equation (4), it becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           y 
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             A 
                              
                             
                                 
                             
                              
                             
                               h 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                              
                             
                               x 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   A 
                                    
                                   
                                       
                                   
                                    
                                   
                                     h 
                                     m 
                                   
                                    
                                   
                                     g 
                                     d 
                                   
                                 
                                 + 
                                 
                                   
                                     A 
                                     * 
                                   
                                    
                                   
                                     h 
                                     d 
                                     * 
                                   
                                    
                                   
                                     g 
                                     m 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 x 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               Ag 
                               d 
                             
                              
                             c 
                           
                           + 
                           
                             
                               A 
                               * 
                             
                              
                             
                               g 
                               m 
                             
                              
                             
                               c 
                               * 
                             
                           
                           + 
                           d 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               A 
                               d 
                             
                              
                             
                               x 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               A 
                               m 
                             
                              
                             
                               
                                 x 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             A 
                             o 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           y 
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               Bh 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                              
                             
                               x 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   
                                     Bh 
                                     m 
                                   
                                    
                                   
                                     g 
                                     d 
                                   
                                 
                                 + 
                                 
                                   
                                     B 
                                     * 
                                   
                                    
                                   
                                     h 
                                     d 
                                     * 
                                   
                                    
                                   
                                     g 
                                     m 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 x 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               Bg 
                               d 
                             
                              
                             c 
                           
                           + 
                           
                             
                               B 
                               * 
                             
                              
                             
                               g 
                               m 
                             
                              
                             
                               c 
                               * 
                             
                           
                           + 
                           d 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               B 
                               d 
                             
                              
                             
                               x 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               B 
                               m 
                             
                              
                             
                               
                                 x 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             B 
                             o 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Writing each of equations (5) and (6), each (respectively) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           y 
                           → 
                         
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   y 
                                    
                                   
                                     ( 
                                     0 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 … 
                               
                             
                             
                               
                                 
                                   y 
                                    
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       x 
                                       * 
                                     
                                      
                                     
                                       ( 
                                       0 
                                       ) 
                                     
                                   
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   … 
                                 
                                 
                                   … 
                                 
                                 
                                   … 
                                 
                               
                               
                                 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       x 
                                       * 
                                     
                                      
                                     
                                       ( 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     A 
                                     d 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     m 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     o 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     x 
                                     → 
                                   
                                 
                                 
                                   
                                     
                                       x 
                                       * 
                                     
                                     → 
                                   
                                 
                                 
                                   
                                     1 
                                     → 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     A 
                                     d 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     m 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     o 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             X 
                             → 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     A 
                                     d 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     m 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     o 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     y 
                     → 
                   
                   = 
                     
                    
                   
                     
                       X 
                       → 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               B 
                               d 
                             
                           
                         
                         
                           
                             
                               B 
                               m 
                             
                           
                         
                         
                           
                             
                               B 
                               o 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The least square solutions of equations (7) and (8) are (respectively), then: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               X 
                               → 
                             
                             + 
                           
                            
                           
                             X 
                             → 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         X 
                         → 
                       
                       + 
                     
                      
                     
                       y 
                       → 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 X 
                                 → 
                               
                               + 
                             
                              
                             
                               X 
                               → 
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                        
                       
                         
                           X 
                           → 
                         
                         + 
                       
                        
                       
                         
                           X 
                           → 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   A 
                                   d 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   m 
                                 
                               
                             
                             
                               
                                 
                                   A 
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               A 
                               d 
                             
                           
                         
                         
                           
                             
                               A 
                               m 
                             
                           
                         
                         
                           
                             
                               A 
                               o 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               X 
                               → 
                             
                             + 
                           
                            
                           
                             X 
                             → 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       
                         X 
                         → 
                       
                       + 
                     
                      
                     
                       y 
                       → 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 X 
                                 → 
                               
                               + 
                             
                              
                             
                               X 
                               → 
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                        
                       
                         
                           X 
                           → 
                         
                         + 
                       
                        
                       
                         
                           X 
                           → 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   B 
                                   d 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   m 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   o 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               B 
                               d 
                             
                           
                         
                         
                           
                             
                               B 
                               m 
                             
                           
                         
                         
                           
                             
                               B 
                               o 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Additionally, a relationship between terms of equations (5) and (6) can be established as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             m 
                           
                           
                             A 
                             d 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 m 
                               
                                
                               
                                 g 
                                 d 
                               
                             
                             + 
                             
                               
                                 A 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                           
                           
                             A 
                              
                             
                                 
                             
                              
                             
                               h 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               m 
                             
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 A 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               m 
                             
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 A 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 A 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             B 
                             m 
                           
                           
                             B 
                             d 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 Bh 
                                 m 
                               
                                
                               
                                 g 
                                 d 
                               
                             
                             + 
                             
                               
                                 B 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                           
                           
                             
                               Bh 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               m 
                             
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 B 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                             
                               
                                 Bh 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               m 
                             
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 B 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                             
                               
                                 B 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             o 
                           
                           
                             A 
                             d 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 Ag 
                                 d 
                               
                                
                               c 
                             
                             + 
                             
                               
                                 A 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                                
                               
                                 c 
                                 * 
                               
                             
                             + 
                             d 
                           
                           
                             A 
                              
                             
                                 
                             
                              
                             
                               h 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 A 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                                
                               
                                 c 
                                 * 
                               
                             
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                           + 
                           
                             d 
                             
                               A 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 A 
                                 d 
                                 * 
                               
                                
                               
                                 c 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                             
                               
                                 A 
                                 d 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                           + 
                           
                             d 
                             
                               A 
                               d 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             B 
                             o 
                           
                           
                             B 
                             d 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 Bg 
                                 d 
                               
                                
                               c 
                             
                             + 
                             
                               
                                 B 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                                
                               
                                 c 
                                 * 
                               
                             
                             + 
                             d 
                           
                           
                             
                               Bh 
                               d 
                             
                              
                             
                               g 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 B 
                                 * 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                                
                               
                                 c 
                                 * 
                               
                             
                             
                               
                                 Bh 
                                 d 
                               
                                
                               
                                 g 
                                 d 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                           + 
                           
                             d 
                             
                               B 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             c 
                             
                               h 
                               d 
                             
                           
                           + 
                           
                             
                               
                                 B 
                                 d 
                                 * 
                               
                                
                               
                                 c 
                                 * 
                               
                                
                               
                                 g 
                                 m 
                               
                             
                             
                               
                                 B 
                                 d 
                               
                                
                               
                                 h 
                                 d 
                                 * 
                               
                                
                               
                                 g 
                                 d 
                                 * 
                               
                             
                           
                           + 
                           
                             d 
                             
                               B 
                               d 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equations (11) and (12) and equations (13) and (14) can be combined into the following systems: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     A 
                                     m 
                                   
                                   
                                     A 
                                     d 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     m 
                                   
                                   
                                     B 
                                     d 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         = 
                           
                          
                         
                           
                             
                               M 
                               → 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         h 
                                         m 
                                       
                                       
                                         h 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         g 
                                         m 
                                       
                                       
                                         g 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ⇒ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       M 
                                       → 
                                     
                                     + 
                                   
                                    
                                   
                                     M 
                                     → 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 M 
                                 → 
                               
                               + 
                             
                              
                             
                               
                                 M 
                                 → 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           h 
                                           m 
                                         
                                         
                                           h 
                                           d 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           g 
                                           m 
                                         
                                         
                                           g 
                                           d 
                                           * 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     M 
                                     → 
                                   
                                   + 
                                 
                                  
                                 
                                   M 
                                   → 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 M 
                                 → 
                               
                               + 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         A 
                                         m 
                                       
                                       
                                         A 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         B 
                                         m 
                                       
                                       
                                         B 
                                         d 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     h 
                                     m 
                                   
                                   
                                     h 
                                     d 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   
                                     g 
                                     d 
                                     * 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     A 
                                     o 
                                   
                                   
                                     A 
                                     d 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     o 
                                   
                                   
                                     B 
                                     d 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     A 
                                     o 
                                     * 
                                   
                                   
                                     A 
                                     d 
                                     * 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     o 
                                     * 
                                   
                                   
                                     B 
                                     d 
                                     * 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   
                                     
                                       
                                         A 
                                         d 
                                         * 
                                       
                                        
                                       
                                         g 
                                         m 
                                       
                                     
                                     
                                       
                                         A 
                                         d 
                                       
                                        
                                       
                                         g 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     1 
                                     
                                       A 
                                       d 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     
                                       
                                         B 
                                         d 
                                         * 
                                       
                                        
                                       
                                         g 
                                         m 
                                       
                                     
                                     
                                       
                                         B 
                                         d 
                                       
                                        
                                       
                                         g 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     1 
                                     
                                       B 
                                       d 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         A 
                                         d 
                                       
                                        
                                       
                                         g 
                                         m 
                                         * 
                                       
                                     
                                     
                                       
                                         A 
                                         d 
                                         * 
                                       
                                        
                                       
                                         g 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     1 
                                     
                                       A 
                                       d 
                                       * 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         B 
                                         d 
                                       
                                        
                                       
                                         g 
                                         m 
                                         * 
                                       
                                     
                                     
                                       
                                         B 
                                         d 
                                         * 
                                       
                                        
                                       
                                         g 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     1 
                                     
                                       B 
                                       d 
                                       * 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     c 
                                     
                                       h 
                                       d 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       c 
                                       * 
                                     
                                     
                                       h 
                                       d 
                                       * 
                                     
                                   
                                 
                               
                               
                                 
                                   d 
                                 
                               
                               
                                 
                                   
                                     d 
                                     * 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               N 
                               → 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       c 
                                       
                                         h 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         c 
                                         * 
                                       
                                       
                                         h 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     d 
                                   
                                 
                                 
                                   
                                     
                                       d 
                                       * 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ⇒ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       → 
                                     
                                     + 
                                   
                                    
                                   
                                     N 
                                     → 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 N 
                                 → 
                               
                               + 
                             
                              
                             
                               
                                 N 
                                 → 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         c 
                                         
                                           h 
                                           d 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           c 
                                           * 
                                         
                                         
                                           h 
                                           d 
                                           * 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     N 
                                     → 
                                   
                                   + 
                                 
                                  
                                 
                                   N 
                                   → 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 N 
                                 → 
                               
                               + 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         A 
                                         o 
                                       
                                       
                                         A 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         B 
                                         o 
                                       
                                       
                                         B 
                                         d 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         A 
                                         o 
                                         * 
                                       
                                       
                                         A 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         B 
                                         o 
                                         * 
                                       
                                       
                                         B 
                                         d 
                                         * 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   c 
                                   
                                     h 
                                     d 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     c 
                                     * 
                                   
                                   
                                     h 
                                     d 
                                     * 
                                   
                                 
                               
                             
                             
                               
                                 d 
                               
                             
                             
                               
                                 
                                   d 
                                   * 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, equations (9), (10), (15), and (16) allow one to estimate the parameters (numerical constants) of the transceiver  100  accurately. 
         [0028]    As a result of the equations (9), (10), (15), and (16), the structures of compensators  106  and  130  can also be chosen. In particular, values x(n) and v(n) can be chosen as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     x 
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       u 
                        
                       
                         ( 
                         n 
                         ) 
                       
                     
                     - 
                     
                       
                         
                           h 
                           m 
                         
                         
                           h 
                           d 
                         
                       
                        
                       
                         
                           u 
                           * 
                         
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                     - 
                     
                       c 
                       
                         h 
                         d 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           y 
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         - 
                         d 
                       
                       ] 
                     
                     - 
                     
                       
                         
                           g 
                           m 
                         
                         
                           g 
                           d 
                           * 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               y 
                               * 
                             
                              
                             
                               ( 
                               n 
                               ) 
                             
                           
                           - 
                           d 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Now substituting equation (17) into equation (1), z t (n) for compensator  106  can be reduced as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             z 
                             t 
                           
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               h 
                               d 
                             
                              
                             
                               x 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               h 
                               m 
                             
                              
                             
                               
                                 x 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           c 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             u 
                              
                             
                               ( 
                               n 
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 h 
                                 m 
                               
                               
                                 h 
                                 d 
                               
                             
                              
                             
                               
                                 u 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           - 
                           
                             c 
                             
                               h 
                               d 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               d 
                             
                              
                             
                               ( 
                               
                                 
                                   u 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       h 
                                       m 
                                     
                                     
                                       h 
                                       d 
                                     
                                   
                                    
                                   
                                     
                                       u 
                                       * 
                                     
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   c 
                                   
                                     h 
                                     d 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 h 
                                 m 
                               
                                
                               
                                 ( 
                                 
                                   
                                     u 
                                      
                                     
                                       ( 
                                       n 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         h 
                                         m 
                                       
                                       
                                         h 
                                         d 
                                       
                                     
                                      
                                     
                                       
                                         u 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         n 
                                         ) 
                                       
                                     
                                   
                                   - 
                                   
                                     c 
                                     
                                       h 
                                       d 
                                     
                                   
                                 
                                 ) 
                               
                             
                             * 
                           
                           + 
                           c 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               h 
                               d 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               h 
                               m 
                             
                              
                             
                               
                                 u 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           - 
                           c 
                           + 
                           
                             
                               h 
                               m 
                             
                              
                             
                               
                                 u 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               
                                 
                                   h 
                                   m 
                                 
                                  
                                 
                                   h 
                                   m 
                                   * 
                                 
                               
                               
                                 h 
                                 d 
                                 * 
                               
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 h 
                                 m 
                               
                                
                               
                                 c 
                                 * 
                               
                             
                             
                               h 
                               d 
                               * 
                             
                           
                           + 
                           c 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 h 
                                 d 
                               
                                
                               
                                 u 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     h 
                                     m 
                                   
                                    
                                   
                                     h 
                                     m 
                                     * 
                                   
                                 
                                 
                                   h 
                                   d 
                                   * 
                                 
                               
                                
                               
                                 u 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   h 
                                   m 
                                 
                                  
                                 
                                   c 
                                   * 
                                 
                               
                               
                                 h 
                                 d 
                                 * 
                               
                             
                           
                           ≈ 
                           
                             
                               h 
                               d 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Since h m  and c are on the order of 30 dB lower in amplitude than h d , the last two terms of equation (19) can be ignored. Thus, compensator  106  effectively compensates for imbalance in the transmit path  102  irrespective of the receive path  104 . Now assuming that the phased rotation is introduced by channel A, equation (5) when combined with equations (2) and (19) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           y 
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               g 
                               d 
                             
                              
                             
                               
                                 z 
                                 r 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               g 
                               m 
                             
                              
                             
                               
                                 z 
                                 r 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           d 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               Ag 
                               d 
                             
                              
                             
                               z 
                               t 
                             
                           
                           + 
                           
                             
                               A 
                               * 
                             
                              
                             
                               g 
                               m 
                             
                              
                             
                               
                                 z 
                                 t 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           d 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               Ag 
                               d 
                             
                              
                             
                               h 
                               d 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               A 
                               * 
                             
                              
                             
                               g 
                               m 
                             
                              
                             
                               h 
                               d 
                               * 
                             
                              
                             
                               
                                 u 
                                 * 
                               
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                           + 
                           d 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    When equation (20) is substituted into equation (18), equation (18) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           v 
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 y 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               - 
                               d 
                             
                             ] 
                           
                           - 
                           
                             
                               
                                 g 
                                 m 
                               
                               
                                 g 
                                 d 
                                 * 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     y 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                                 - 
                                 d 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 Ag 
                                 d 
                               
                                
                               
                                 h 
                                 d 
                               
                                
                               
                                 u 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   Ag 
                                   m 
                                   2 
                                 
                                  
                                 
                                   h 
                                   d 
                                   * 
                                 
                                  
                                 
                                   u 
                                    
                                   
                                     ( 
                                     n 
                                     ) 
                                   
                                 
                               
                               
                                 g 
                                 d 
                                 * 
                               
                             
                             + 
                             
                               
                                 
                                   g 
                                   m 
                                 
                                 
                                   g 
                                   d 
                                   * 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     d 
                                     * 
                                   
                                   - 
                                   d 
                                 
                                 ) 
                               
                             
                           
                           ≈ 
                           
                             
                               Ag 
                               d 
                             
                              
                             
                               h 
                               d 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Alternatively, if that the phased rotation is introduced by channel B, equation (5) when combined with equations (3) and (19) becomes: 
         [0000]        y ( n )= Bg   d   h   d   u ( n )+ B*g   m   h   d   *u *( n )+ d   (22)
 
         [0000]    When equation (22) is substituted into equation (18), equation (18) becomes: 
         [0000]        v ( n )≈ Bg   d   h   d   u ( n )  (23)
 
         [0000]    Thus, from equations (21) and (23), it can be seen that compensator  130  effectively compensates for imbalance in the receive path  104  irrespective of the transmit path  102 . 
         [0029]    Compensators  106  and  130  can also be extended to compensate for frequency-dependent I/Q distortion. Assuming that A m , A d , B m , and B d  are N-tap digital filters (i.e., finite impulse response filters) instead of constant values, equations (11) and (12) become (in the frequency domain): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         A 
                         m 
                       
                        
                       
                         ( 
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             ω 
                           
                         
                         ) 
                       
                     
                     
                       
                         A 
                         d 
                       
                        
                       
                         ( 
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             ω 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           h 
                           m 
                         
                          
                         
                           ( 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               ω 
                             
                           
                           ) 
                         
                       
                       
                         
                           h 
                           d 
                         
                          
                         
                           ( 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               ω 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             A 
                             d 
                             * 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             g 
                             m 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             A 
                             d 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             g 
                             d 
                             * 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         B 
                         m 
                       
                        
                       
                         ( 
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             ω 
                           
                         
                         ) 
                       
                     
                     
                       
                         B 
                         d 
                       
                        
                       
                         ( 
                         
                            
                           
                              
                              
                             
                                 
                             
                              
                             ω 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           h 
                           m 
                         
                          
                         
                           ( 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               ω 
                             
                           
                           ) 
                         
                       
                       
                         
                           h 
                           d 
                         
                          
                         
                           ( 
                           
                              
                             
                                
                                
                               
                                   
                               
                                
                               ω 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             B 
                             d 
                             * 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             g 
                             m 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             B 
                             d 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             g 
                             d 
                             * 
                           
                            
                           
                             ( 
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 ω 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Once the solutions for 
         [0000]    
       
         
           
             
               
                 
                   h 
                   m 
                 
                  
                 
                   ( 
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       ω 
                     
                   
                   ) 
                 
               
               
                 
                   h 
                   d 
                 
                  
                 
                   ( 
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       ω 
                     
                   
                   ) 
                 
               
             
              
             
                 
             
              
             and 
              
             
                 
             
              
             
               
                 
                   g 
                   m 
                 
                  
                 
                   ( 
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       ω 
                     
                   
                   ) 
                 
               
               
                 
                   g 
                   d 
                   * 
                 
                  
                 
                   ( 
                   
                      
                     
                        
                        
                       
                           
                       
                        
                       ω 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    are obtained, the solutions can be converted back to the time-domain and used in a compensators  106  and  130  in a similar manner to equations (17)-(23). In some applications, model coefficients h m , h d , g m , and g d  can be filters, and at least some multiplications between signals and coefficients in equations (1)-(24) can be replaced with convolutions. 
         [0030]    Turning now to  FIGS. 2A ,  2 B,  3 A, and  3 B, the effects of compensators can be seen. In  FIGS. 2A and 2B , the operation of the transmit path  102  and receive path  104  without compensation can be seen. Additionally, in  FIGS. 3A and 3B , the operation of the transmit path  102  and receive path  104  can be seen with compensation. As a result of the compensation, it can clearly be seen that the second lower peak for each of the paths  102  and  104  is substantially eliminated, indicating that transceiver  100  is more accurate than conventional transceivers. 
         [0031]    Having thus described the present invention by reference to certain of its preferred embodiments, it is noted that the embodiments disclosed are illustrative rather than limiting in nature and that a wide range of variations, modifications, changes, and substitutions are contemplated in the foregoing disclosure and, in some instances, some features of the present invention may be employed without a corresponding use of the other features. Accordingly, it is appropriate that the appended claims be construed broadly and in a manner consistent with the scope of the invention.