Abstract:
The present invention discloses a method and apparatus to provide, effectively and robustly, a phase reference in phase-domain for digital phase-modulated signals. Not only the first-order but also higher order PLLs are delineated for robust and fast tracking of frequency errors and time-varying frequency errors between the transmitter and the receiver. This invention can be applied to any phase-modulated signal such as PSK, DPSK, π/4-DPSK, and CPM. The decoders with this invention can achieve close to the performance of coherent detection. 
     Reference 
     [1] D. Divsalar and M. K. Simon, “Multiple-symbols differential detection of MPSK,”  IEEE Trans. Commun ., vol. 38, pp. 300-308, March 1990. 
     [2] Specification of the Bluetooth System, 2.0+EDR, 4 Nov. 2004.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to digital communication systems, and more particularly to a methods and apparatus for phase reference tracking of digital phase modulated signals in the receiver. 
     2. Background 
     Digital phase modulation is one of the popular digital modulations due to its simplicity and robustness. Source information is transmitted by selecting phases of the signal according to the information bits. Continuous phase modulation (CPM), Phase-shift keying (PSK) and differential phase-shift keying (DPSK) are examples of digital phase modulation. 
     In the receiver, it is necessary to detect an accurate phase reference for decoding the transmitted information bits. Otherwise, phase reference errors may cause significant performance degradation. For differentially encoded digital phase modulations such as DPSK, DQPSK and D8PSK, a phase reference can be derived from the previous symbol to facilitate the demodulation. For simple receivers, DPSK signals may be differentially decoded. That means, previous phase is used as a reference for the current symbol. However, since this reference is noisy, the performance can degrade up to 3 dB, compared to the performance with perfect phase reference. Phase reference tracking is also useful for DPSK signals. 
     To facilitate such a phase reference estimate, a training sequence is often transmitted at the beginning of a data packet. The phase reference may be easily estimated with the training sequence known at the receiver, but the throughput may be slightly decreased as the training sequence does not contain source information. Moreover, the phase reference may be time-varying due to the imperfect oscillators at the transmitter (TX) or the receiver (RX). In this case, phase reference tracking will be necessary for the receiver to maintain best performance while receiving information bits. Phase references may be heavily time-varying due to the mismatching between TX and RX oscillators. This mismatching is so-called frequency offset (FO). Moreover frequency drift may cause difficulty in tracking accurate phase reference. By estimating and/or tracking this FO, phase reference may be kept accurate. 
     For this phase reference tracking, multiple symbol detection [1] based on maximum likelihood sequence detection (MLSD) was proposed, but its complexity exponentially increases with the number of observation symbols. Furthermore, U.S. Pat. No. 7,245,672 issued to Smit et al., entitled “Method and apparatus for phase-domain semi-coherent demodulation” disclosed that a first-order IIR filter in phase-domain, of which complexity further decreases due to the phase operations instead of the complex signal operations as shown in  FIG. 3 , However, this first order phase tracking is not sufficient to handle heavy phase variations. For example, Bluetooth spec [2] allows frequency to drift up to 400 Hz/μs. Also, double errors (decoded symbol error propagation) are un-avoidable in this invention due to the proposed differential decoding. 
     According to above problems, the related field needs a simple and robust phase tracking method, where the phase reference is tracked in phase-domain with a high order digital phase-locked loop, for general phase-modulated signals. Also, related field suggests a better way to track phase for differentially encoded digital phase modulations, such as DBPSK, (π/4) DQPSK and DBPSK modulated signals. 
     BRIEF SUMMARY OF THE INVENTION 
     It is an objective of the present invention to provide an effective and robust method for phase reference tracking of digital phase modulated signals in the receiver. It tracks the phase errors and generates reliable phase reference to estimate. 
     To achieve the above objective, the present invention provides a method used for phase reference tracking of digital phase modulated signals in the receiver, comprising the steps of: converting a received complex signal to the received phase r n , feeding the received phase r n  to a phase reference tracking unit, producing an estimated transmit phase {tilde over (s)} n  from the phase reference tracking unit, feeding the estimated transmit phase {tilde over (s)} n  to a coherent decoder, and producing a decoded symbol â n  from the coherent decoder. 
     According to one aspect of the present invention, the received complex signal can be encoded by BPSK, MPSK, PSK and DPSK modulation systems. 
     According to one aspect of the present invention, the received phase r n  can be converted to different forms according the received complex signal. 
     Another objective of the present invention is to provide an effective and robust apparatus for phase reference tracking of digital phase modulated signals in the receiver. The phase reference tracking unit takes the received phase and the decoded symbols as its input, generates reliable phase reference to estimate by tracking phase errors due to frequency offset and the variation of frequency offset. A gradient algorithm in phase-domain based on the measured phase error is utilized in the phase reference tracking unit. 
     To achieve the above objective, the present invention provides an apparatus used for phase reference tracking of digital phase modulated signals in the receiver, comprising: a complex-to-phase converter, used for converting the in-phase (I n ) and the quadrature (Q n ) components of a received complex signal to a received phase r n , the phase reference tracking unit, which is electrically connected to the complex-to-phase converter, used for producing an estimated transmit phase {tilde over (s)} n , and the coherent decoder, which is electrically connected to the phase reference tracking unit, used for producing a decoded symbol â n  and sending a re-modulation phase signal ŝ n  to the phase reference tracking unit. 
     According to one aspect of the present invention, the apparatus used for phase reference tracking of digital phase modulated signals in the receiver can be applied in BPSK, MPSK, PSK and DPSK modulation systems. 
     According to one aspect of the present invention, the types of the coherent decoder can be selected according to BPSK, MPSK and DPSK modulation systems. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       All the objects, advantages, and novel features of the invention will become more apparent from the following detailed descriptions when taken in conjunction with the accompanying drawings. 
         FIG. 1  is a block diagram for a general decoder and the invented phase reference tracking; 
         FIG. 2  is a block diagram for the PSK decoder and the invented phase reference tracking; 
         FIG. 3  is a DPSK receiver block diagram proposed by Smit (Prior Art); 
         FIG. 4  is a block diagram for a DPSK decoder and the invented phase reference tracking; and 
         FIG. 5  is an alternative block diagram for a DPSK decoder and the invented phase reference tracking. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Although the invention has been explained in relation to several preferred embodiments, the accompanying drawings and the following detailed descriptions are the preferred embodiment of the present invention. It is to be understood that the following disclosed descriptions will be examples of present invention, and will not limit the present invention into the drawings and the special embodiment. 
     Phase reference tracking is not necessary for some phase-modulated signals such as Gaussian frequency shift-keying (GFSK) and DPSK. However, it is well-known that coherent detection may help to improve performances up to 3 dB. Here, a simple, robust and generalized method for phase reference tracking in phase-domain is provided. 
     To understand the spirit of the present invention, referring to  FIG. 1 , it shows the block diagram for a general decoder and the invented phase reference tracking. An apparatus for phase reference tracking of digital phase modulated signals in the receiver  100  comprises a complex-to-phase converter  110 , a phase reference tracking unit  120 , a coherent decoder  130 . The complex-to-phase converter  110  is used for converting the in-phase (I n ) and the quadrature (Q n ) components of a received complex signal  101  to a received phase r n    111 . The phase reference tracking unit  120 , which is electrically connected to the complex-to-phase converter  110 , is used for producing an estimated transmit phase {tilde over (s)} n    141 . The coherent decoder  130 , which is electrically connected to the phase reference tracking unit  120 , is used for producing a decoded symbol â n    212  and sending a re-modulation phase signal ŝ n    149  to the phase reference tracking unit  120 . 
     The apparatus used for phase reference tracking of digital phase modulated signals in the receiver can be applied in BPSK, MPSK, PSK and DPSK modulation systems. The received complex signal  101  can be encoded by BPSK, MPSK, PSK and DPSK modulation systems. The received phase r n    111  can be converted to different forms according to the received complex signal  101 . The types of the coherent decoder  130  can be selected according to BPSK, MPSK and DPSK modulation systems. 
     Now, referring to  FIG. 1 , the phase reference tracking unit further comprises a first subtracter  121 , a second subtracter  122 , a first multiplier  123 , a first adder  124 , a first sample delay unit  125 , a second multiplier  126 , a second adder  127 , a third adder  128 , and a second sample delay unit  129 . The first subtracter  121  is used for subtracting the previous phase reference estimate {tilde over (θ)} n-1    142  from the received phase r n    111  and producing the estimated transmit phase {tilde over (s)} n    141 . The second subtracter  122 , which is electrically connected to the coherent decoder  130 , is used for subtracting the re-modulation phase signal ŝ n-d    131  from the estimated transmit phase {tilde over (s)} n    141  and producing a tracking error ε n    145 . The first multiplier  146 , which is electrically connected to the second subtracter  122 , is used for scaling the tracking error ε n    145  by a value of β  146 . The first adder  124 , which is electrically connected to the first multiplier  123 , is used for adding a scaled tracking error βε n  and a previous phase correction factor {tilde over (θ)}′ n-1 . The first sample delay unit  125 , which is electrically connected to the first adder  124 , is used for taking the phase correction factor {tilde over (θ)}′ n    147  to the previous state of {tilde over (θ)}′ n-1  and providing a feedback signal of previous phase correction factor {tilde over (θ)}′ n-1  to the first adder  124 . The second multiplier  126 , which is electrically connected to the second subtracter  122 , is used for scaling the tracking error ε n    145  by a value of a  143 . The second adder  127 , which is electrically connected to the second multiplier  126 , is used for adding a scaled tracking error αε n  and the phase correction factor {tilde over (θ)}′ n    147 . The third adder  128 , which is electrically connected to the second adder  127 , is used for adding the previous phase reference estimate {tilde over (θ)} n-1 , the scaled tracking error αε n  and the phase correction factor {tilde over (θ)}′ n    147 . The second sample delay unit  129 , which is electrically connected to the second adder, is used for the taking the phase reference estimate {tilde over (θ)} n    147  to the previous state {tilde over (θ)} n-1  and providing a feedback signal of previous reference estimate {tilde over (θ)} n-1  to the third adder  128  and the first subtracter  121 . 
     Moreover, to compensate the delay (d) caused in the coherent decoder and to generate the estimated transmit phase {tilde over (s)} n    149  with correct timing, a coherent decoder  140 , which is electrically connected to the first subtracter  121 , is provided. Therefore, the estimated transmit phase {tilde over (s)} n    149  and the decoded symbol â n    212  turn into a estimated transmit phase with a delay (d) {tilde over (s)} n-d    144  and decoded symbol with a delay (d) â n-d    132  which are also denoted as decoded symbols  133 . 
     The construction of the block diagram of the apparatus according the present invention may be modified and/or simplified with combining the phase reference tracking units and the coherent decoder units by removing redundant units and/or re-organizing the block diagrams. 
     Besides, the procedure of the present invention can further described as the following steps:
         Step1: converting a received complex signal to the received phase r n    111 ;   Step2: feeding the received phase r n    111  to a phase reference tracking unit;   Step3: producing an estimated transmit phase ŝ n    141  from the phase reference tracking unit;   Step4: feeding the estimated transmit phase {tilde over (s)} n    141  to a coherent decoder; and   Step5: producing a decoded symbol â n    212  from the coherent decoder.       

     The received complex signal can be encoded by BPSK, MPSK, PSK and DPSK modulation systems. The received phase r n    111  can be converted to different forms according to the received complex signal. 
     Moreover, the procedure of producing an estimated transmit phase {tilde over (s)} n    141  further comprising the steps of:
         Step1: subtracting a previous phase reference estimate {tilde over (θ)} n-1    142  from a received phase r n    111 ;   Step2: producing a tracking error ε n    145  by subtracting a re-modulation phase signal ŝ n    149  from an estimated transmit phase {tilde over (s)} n    141 ;   Step3: scaling the tracking error ε n    145  by a value of β  146 ;   Step4: adding the scaled of tracking error βε n  with a previous phase correction factor {tilde over (θ)}′ n-1  and derive a phase correction factor {tilde over (θ)}′ n ;   Step5: scaling the tracking error ε n    145  by a value of α  143 ;   Step6: adding the scaled of tracking error αε n  with the phase correction factor {tilde over (θ)}′ n ; and   Step7: adding the scaled of tracking error αε n  with the phase correction factor {tilde over (θ)}′ n  with the previous phase reference estimate {tilde over (θ)} n-1  and derive the phase reference estimate {tilde over (θ)} n    148 .       

     The procedure of producing an estimated transmit phase {tilde over (s)} n  further comprising the steps of:
         Step1: subtracting a previous phase reference estimate {tilde over (θ)} n-1  from a received phase r n    111 ;   Step2: adding a delay (d) to an estimated transmit phase {tilde over (s)} n    141 ;   Step3: producing a tracking error ε n    145  by subtracting a re-modulation phase with a delay (d){tilde over (s)} n-d    131  from a re-modulation phase with a delay (d){tilde over (s)} n-d    131 ;   Step4: scaling the tracking error ε n  by a value of β;   Step5: adding the scaled of tracking error βε n  with a previous phase correction factor {tilde over (θ)}′ n-1  and derive a phase correction factor {tilde over (θ)}′ n ;   Step6: scaling the tracking error ε n    145  by a value of α  143 ;   Step7: adding the scaled of tracking error αε n  with the phase correction factor {tilde over (θ)}′ n ; and   Step8: adding the scaled of tracking error αε n  with the phase correction factor {tilde over (θ)}′ n  with the previous phase reference estimate {tilde over (θ)} n-1  and derive the phase reference estimate {tilde over (θ)} n    148 .       

     The delay (d) is used for generating the estimated transmit phase {tilde over (s)} n    141  with correct timing. 
     The procedure of producing a decoded symbol â n    212  further comprising the steps of:
         Step1: feeding an estimated transmit phase {tilde over (s)} n    141  to a coherent decoder;   Step2: de-mapping the estimated transmit phase {tilde over (s)} n    141 ;   Step3: producing a decoded symbol â n    212 ;   Step4: mapping the decoded symbol â n    212 ; and   Step5: producing a re-modulation phase signal ŝ n    141 ;       

     The re-modulation phase signal ŝ n    141  is feed to a phase reference tracking unit and used for the calculation of a tracking error ε n    145 . The method used for phase reference tracking of digital phase modulated signals in the receiver as described above, the method is generalized with n-th order tracking. 
     A complex-to-phase converter  110  converts the incoming received complex signal  101 , consisting of the in-phase (I n ) and the quadrature (Q n ) components, to a received phase r n    111  using the following equation: 
     
       
         
           
             
               
                 
                   
                     
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     where n represents the symbol time index. Note that the operations on phase are based on modular 2π. The received phase r n    111  can be converted to different forms according the received complex signal and also can be converted to different forms according the received complex signal  101 . 
     This received phase r n    111  is fed to a phase reference tracking unit  120 , which produces an estimated transmit phase {tilde over (s)} n    141 . This estimated transmit phase {tilde over (s)} n    141  is an estimated transmit phase and is fed to a coherent decoder  130 . Since the allowed transmit phase is quantized for a digital phase modulation, the coherent decoder  130  decodes the estimated transmit phase {tilde over (s)} n    141  based on a de-mapping table to produce the decoded symbol with a delay (d) â n-d    132 . For example, the coherent decoder  130  decodes for BPSK can be found in TABLE 1 below. If required, the coherent decoder  130  may utilize the received complex signal  101 . The coherent decoder  130  also uses a “mapping” table to reconstruct the phase (also known as re-modulation) for the decoded symbol â n    132 , denoted ŝ n-d    131 , and sends it to the phase reference tracking unit  120 . An example for the mapping table for BPSK modulated signal is shown in Table 2 below. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 De-Mapping Table for BPSK 
               
             
          
           
               
                   
                 {tilde over (s)} n   
                 Decoded Symbol 
               
               
                   
                   
               
               
                   
                 −π/2 &lt; {tilde over (s)} n  &lt; π/2 
                 0 
               
               
                   
                 Otherwise 
                 1 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Mapping Table for BPSK 
               
             
          
           
               
                   
                 Decoded Symbol 
                 Re-modulated Phase 
               
               
                   
                   
               
               
                   
                 0 
                 0 
               
               
                   
                 1 
                 π 
               
               
                   
                   
               
             
          
         
       
     
     Inside the phase reference tracking unit  120 , the estimated transmit phase {tilde over (s)} n    141  at the receiver is calculated by subtracting the previous phase reference estimate {tilde over (θ)} n-1    142  from r n    111 . A tracking error ε n    145  is calculated by subtracting ŝ n-d    131  from {tilde over (s)} n-d    144 , where d is a delay introduced by the coherent decoder  130 . Then, a phase correction factor due to frequency error, {tilde over (θ)}′ n    147 , and a phase reference estimate, {tilde over (θ)} n    148 , are updated with the well-known gradient method:
 
{tilde over (θ)}′ n ={tilde over (θ)}′ n-1 +βε n ,  Eq. (2a)
 
{tilde over (θ)} n ={tilde over (θ)} n-1 +αε n +{tilde over (θ)}′ n ,  Eq. (2b)
 
     where 0≦α≦1 and 0≦β≦1. Note that {tilde over (θ)}′ n    147  is a phase-error correction factor based on an estimated frequency-offset between the TX and the RX. Such a phase tracking loop is traditionally known as a second order phase-locked-loop (PLL). This tracking scheme can be easily generalized to a third order PLL as follows:
 
{tilde over (θ)}″ n ={tilde over (θ)}″ n-1 +γε n ,  Eq. (3a)
 
{tilde over (θ)}′ n ={tilde over (θ)}′ n-1 +βε n +{tilde over (θ)}′ n ,  Eq. (3b)
 
{tilde over (θ)} n ={tilde over (θ)} n-1 +αε n +{tilde over (θ)}′ n ,  Eq. (3c)
 
     where 0≦α≦1, 0≦β≦1 and 0≦γ≦1. In the same manner, this tracking can be further generalized to an n-th order PLL. Note that this 3-rd order PLL can track not only static frequency errors but also time-varying frequency errors. 
     Note that the above n-th order phase reference tracking algorithm may be applied to any phase-modulated signals. In general, the inputs of the phase reference tracking unit  120  are the received phase r n    111  and the re-modulation phase signal ŝ n-d    131 . The output of the phase reference tracking unit  120  is the estimated transmit phase {tilde over (s)} n    141 , after proper phase/frequency error correction, at the receiver. If required, the overall block diagram may be re-organized to save computational power and/or hardware size. 
     For clearer explanations, consider an M-ary PSK signal. In the transmitter (TX), k (=log 2  M) information bits are mapped to one of the M phases. Let a n  and s n  be the n-th symbol with k information bits and its corresponding mapped phase, respectively. This, the transmit phase, s n  may be represented as
 
s n =           (a n ), n=0, 1, . . . , M−1,   Eq. (4)

     where            (•) denotes the phase-mapping function. Note the phase mapping for M=2 is shown in TABLE 2. In the proposed MPSK receiver (RX) shown in  FIG. 2 , the phase of a received phase r n    111  may be represented as
 
 r   n   =s   n +θ n ,  Eq. (5)

     where θ n  is the phase mismatching caused by the phase mismatching between the TX and the RX. The proposed second order PLL for decoding s n  and tracking θ n  for a received MPSK signal is as follows: 
     Decoding/Phase-Tracking algorithm for MPSK signals ( FIG. 2 ) 
     For n=0 to N−1
 
 {tilde over (s)}   n   =r   n −{tilde over (θ)} n-1   Eq. (6a)
 
 â   n =             −1 ( {tilde over (s)}   n )  Eq. (6b)
 
 ŝ   n =         ( â   n )  Eq. (6c)
 
ε n   ={tilde over (s)}   n   −ŝ   n   Eq. (6d)
 
{tilde over (θ)}′ n ={circumflex over (θ)}′ n-1 +βε n   Eq. (6e)
 
{tilde over (θ)} n ={tilde over (θ)} n-1 +αε n +{tilde over (θ)}′ n   Eq. (6f)

     An PSK coherent decoder  210  comprises a de-mapping unit and a mapping unit. The function            (•) is the de-mapping unit  213 , i.e., the inverse function of          (•). This de-mapping unit is for decoding an MPSK signal to produce a decoded symbol â n    212 . This decoded symbol â n    212  is mapped again to generate a re-modulation phase signal ŝ n    149  with the mapping unit  211 . Note an example de-mapping table is given in TABLE 1 and the corresponding mapping table is given in TABLE 2 for a BPSK modulated signal.
     Inside the phase reference tracking unit  220 , the n-th estimated transmit phase {tilde over (s)} n    141 , is calculated by subtracting the previous phase reference estimate {tilde over (θ)} n-1    142  from r n    111 . Initial phase reference {tilde over (θ)} −1  is assumed to be estimated with the help of a training sequence which is known to both TX and RX. Even if this initial phase reference {tilde over (θ)} −1  is well-estimated, this reference may be further tracked for better Rx performance. Moreover, this invention may help to track phase reference with the phase variations during receiving due to imperfection in the Tx or the Rx path. 
     Then, an error ε n    145  is calculated by subtracting ŝ n    149  from {tilde over (s)} n    141 . Note that ε n    145  tends to be smaller with a more accurate {tilde over (θ)} n-1    142 . A phase correction factor due to FO between the TX and the RX, {tilde over (θ)}′ n    147 , is obtained with ε n    145  and β  146  from the previous estimate {tilde over (θ)}′ n-1 . Note: Units  125  and  129  represent “sample delays” and the circuitry shown in  220  implements Eq. (6e). The initial estimate {tilde over (θ)}′ 1  may be set to zero or previous estimate based on a training sequence. Finally, a phase reference estimate {tilde over (θ)} n    148  is updated with ε n    145 , α  143  and {tilde over (θ)}′ n    147  from {tilde over (θ)} n-1    142  using Eq. (6f). This process shall be repeated until every symbol is decoded. 
     This invention can be also applied to DPSK signals. DPSK signals are popular for many communication systems due to the simple non-coherent detections even though coherent detections outperform non-coherent detections by up to 3 dB. Those non-coherent detection losses may be reduced by reliable phase reference tracking. 
     U.S. Pat. No. 7,245,672 disclosed the so-called ‘semi-coherent demodulation for DPSK signals’ ( FIG. 3 ) which is similar to the phase tracking algorithm for PSK signals with a first-order PLL, but his algorithm does not track the higher-order phase variations. Moreover, the phase error measurement is based on the transmit phase constellations. That means, the phase error measurement may be not as reliable as that of the present invention (shown later) since the number of constellations may be larger than M for M-ary DPSK. For example, Bluetooth adopts π/4 DQPSK of which number of constellations is not four, but eight. Another disadvantage of the algorithm is that single error in PSK decoder (unit  310 ) causes double errors after differential decoding (shown later). 
     Smit&#39;s decoding algorithm for DPSK signals ( FIG. 3 ) 
     For n=0 to N−1
 
 {tilde over (s)}   n   =r   n −{tilde over (θ)} n-1   Eq. (7a)
 
 ŝ   n   =               D ( {tilde over (s)}   n )  Eq. (7b)
 
ε n   ={tilde over (s)}   n   −ŝ   n   Eq. (7c)
 
{tilde over (θ)} n ={tilde over (θ)} n-1 +αε n   Eq. (7d)
 
 â   n =         ( ŝ   n   −ŝ   n-1 )  Eq. (7e)

     where            D(a) is a function that gives out the phase of the closest constellation to a. For π/4 DQPSK, the number of the possible ŝ n    149  values is eight, not four due to the π/4 shifting. In this case, a less reliable phase error estimate, ε n    145 , is generated per Eq. (7c). A PSK decoder unit  310  decodes a PSK signal with the first-order PLL in phase-domain, generating the re-modulation phase signal ŝ n    149 . Then, a differential decoder unit  320  differentially the re-modulation phase signal ŝ n    149 , generating â n    212 . Due to the differential decoding in Eq. (7e), single error in ŝ n    149  causes double errors in â n    212  for a DPSK signal.
     Here, we propose a method for a DPSK signal to overcome the disadvantages of the prior invention such as the first-order PLL tracking limitation, unreliable phase error estimate for π/4 DQPSK, and the double errors. Let&#39;s consider an M-ary DPSK signal similar to a MPSK signal. In the TX, k (=log 2  M) information bits are mapped to one of the M phases. Let a n  and x n  be the n-th symbol with k information bits and its corresponding mapped phase, respectively. This phase, x n  may be represented as
 
 x   n =           ( a   n ),  n= 0, 1, . . . ,  N− 1.  Eq. (8)

     Those mapped phases are accumulated before transmitting. In the RX, the phase of the received phase r n    111  may be represented as 
                       r   n     =       s   n     +     θ   n         ,       where   ⁢           ⁢     s   n       =       ∑     m   =   0     n     ⁢     x   n                 Eq   .           ⁢     (   9   )                 
where θ n  is the phase mismatching between the TX and the RX as previous explained. The proposed algorithm of θ n  estimation for DPSK is as follows:
 
     A Phase Tracking and Decoding algorithm for DPSK Signals ( FIG. 4 ) 
     For n=1 to N−1
 
 {tilde over (s)}   n   =r   n −{tilde over (θ)} n-1   Eq. (10a)
 
 {tilde over (x)}   n   ={tilde over (s)}   n   −ŝ   n-1   Eq. (10b)
 
 â   n   =M   −1 ( {tilde over (x)}   n )  Eq. (10c)
 
 {circumflex over (x)}   n =           ( â   n )  Eq. (10d)
 
 ŝ   n   =ŝ   n-1   +{circumflex over (x)}   n   Eq. (10e)
 
ε n   ={tilde over (s)}   n   −ŝ   n   Eq. (10f)
 
{tilde over (θ)}′ n ={tilde over (θ)}′ n-1 +βε n   Eq. (10g)
 
{tilde over (θ)} n ={tilde over (θ)} n-1 +αε n +{tilde over (θ)}′ n   Eq. (10h)

     This algorithm is similar to that for MPSK signals except the coherent decoder  410 . Because the mapped phase x n  is accumulated in the TX, ŝ n-1  is subtracted from {tilde over (s)} n    141  before de-mapping, as is shown in Eq. (10b) and illustrated in coherent decoder  410 . The initial phase reference {tilde over (θ)} 0  may be set to r 0  or a previous estimate. The other initial estimate {tilde over (θ)}′ 0  may be set to zero or a previous estimate. 
     This algorithm can be re-written without {tilde over (s)} n    141  and ŝ n    149  as follows: 
     An Alternative implementation of phase-tracking and decoding algorithm for DPSK Signals 
     For n=1 to N−1 
     
       
         
           
             
               
                 
                   
                     
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     The algorithm for DPSK signals is further simplified by introducing {tilde over (φ)} n . Let 
                       ϕ   ~     n     =         ∑     m   =   0     n     ⁢       x   ^     m       +         θ   ~     n     .               Eq   .           ⁢     (   12   )                 
Then,
 
 {tilde over (x)}   n   =r   n −{tilde over (φ)} n-1 ,  Eq. (13)
 
Since {tilde over (θ)} n ={tilde over (θ)} n-1 +αε n +{tilde over (f)} n , φ n  can be derived as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                     ( 
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     Therefore, the algorithm for DPSK signals may be written as follows: 
     An Alternative Algorithm for Phase-Tracking and Decoding of DPSK Signals ( FIG. 5 ) 
     For n=1 to N−1
 
 {tilde over (x)}   n   =r   n −{tilde over (φ)} n-1   Eq. (15a)
 
 â   n =           ( {tilde over (x)}   n )  Eq. (15b)
 
 {circumflex over (x)}   n =         ( â   n )  Eq. (15c)
 
ε n   ={circumflex over (x)}   n   −{circumflex over (x)}   n   Eq. (15d)
 
{tilde over (θ)}′ n ={tilde over (θ)}′ n-1 +βε n   Eq. (15e)
 
{tilde over (φ)} n   =r   n −(1−α)·ε n +{tilde over (θ)}′ n   Eq. (15f)

       FIG. 5  shows the corresponding implementation, compared to the previous algorithm, accumulation of {circumflex over (x)} n    414  is no longer required in this algorithm. In addition, this algorithm becomes the commonly used non-coherent detection when setting α=1 and β={tilde over (θ)}′ 0 =0. As shown in this alternative algorithm for DPSK signals, the phase reference tracking unit  120  and coherent decoder  130  may be combined to save computation power and/or hardware size by sharing units and/or re-organizing units. 
     This DPSK phase-tracking and decoding algorithm is simpler than the Smit&#39;s algorithm if a higher-order PLL for phase-tracking is disabled. Moreover, this is more robust for π/4 DPSK signals than the Smit&#39;s because the hard-decisional error probability is smaller with a greater distance among a four-phase constellation set than an eight-phase constellation set. In Smit&#39;s algorithm, ŝ n    149  is set to the closest constellation from {tilde over (s)} n    141  (Eq. (7b)). Since the number of constellations is eight, the minimum phase distance among constellations is only π/4. For the current invention, the minimum phase distance to decide {circumflex over (x)} n    414  is π/2. Note that this algorithm is also good for heavy phase variations caused by frequency errors thanks to the higher order tracking. The double errors are also avoidable with this invention. Current error in â n    212  may cause phase tracking degraded but not necessarily cause the next symbol error. In Smit&#39;s, an error in ŝ n    149  causes double errors for sure with a DPSK signal which is not shifted. Note that single error is still possible with Smit&#39;s for a π/4 shifted DQPSK signal. 
     Even though the proposed algorithm shown in the above are all 1st or 2nd order PLL&#39;s, one can easily generalize it to a 3rd order PLL as follows: 
     For n=1 to N−1
 
 {tilde over (x)}   n   =r   n −{tilde over (φ)} n-1   Eq. (16a)
 
 â   n =           ( {tilde over (x)}   n )  Eq. (16b)
 
 {circumflex over (x)}   n =         ( â   n )  Eq. (16c)
 
ε n   ={tilde over (x)}   n   −{circumflex over (x)}   n   Eq. (16d)
 
{tilde over (θ)}″ n ={tilde over (θ)}″ n-1 +γε n  (where 0≦γ≦1)  Eq. (16e)
 
{tilde over (θ)}′ n ={tilde over (θ)}′ n-1 +βε n +{tilde over (θ)}′ n   Eq. (16f)
 
{tilde over (φ)} n   =r   n −(1−α)·ε n +{tilde over (θ)}′ n   Eq. (16g)

     When compared to a 2nd order PLL as shown in Eq. (15), the only difference in the above is the addition of {tilde over (θ)}″ n , which can be used to track the FO variations. For Bluetooth applications, one found the 3rd order PLL, proposed in the above, offers the best performance against dirty packets, for which a FO and a sine-wave based frequency variation are both added to the transmitted BT EDR packets.