# Patent

## Title
Algorithm for multiple-symbol differential detection

## Abstract

A method for differential phase evaluation of M-ary communication data is employed in which the data consists of N sequential symbols r1 . . . rN, each having one of M transmitted phases. Selected sequences of N−1 elements that represent possible sequences of phase differentials are evaluated using multiple-symbol differential detection. Using r1 as the reference for each phase differential estimate, sN−1 phase differential sequences are selected in the form (P2i, P3i, . . . , PNi) for i=1 to s for evaluating said symbol set, where s is predetermined and 1<s<M. Each set of s phase differential estimate values are chosen based on being the closest in value to the actual transmitted phase differential value. These s phase differential estimates can be determined mathematically as those which produce the maximum results using conventional differential detection.

## Background
FIELD OF THE INVENTION
This invention relates to differential detection of multiple-phase shift keying (MPSK) modulation in communication systems. This invention generally relates to digital communications, including, but not limited to CDMA systems.
BACKGROUND OF THE INVENTION
Conventionally, communication receivers use two types of MPSK modulated signal detection: coherent detection and differential detection. In coherent detection, a carrier phase reference is detected at the receiver, against which subsequent symbol phases are compared to estimate the actual information phase. Differential detection processes the difference between the received phases of two consecutive symbols to determine the actual phase. The reference phase is the phase of the first of the two consecutive symbols, against which the difference is taken. Although differential detection eliminates the need for carrier phase reference processing in the receiver, it requires a higher signal-to-noise ratio at a given symbol error rate.
Differential detection in an Additive White Gaussian Noise (AWGN) channel is preferred over coherent detection when simplicity of implementation and robustness take precedence over receiver sensitivity performance. Differential detection is also preferred when it is difficult to generate a coherent demodulation reference signal. For differential detection of multiple-phase shift keying (MPSK) modulation, the input phase information is differentially encoded at the transmitter, then demodulation is implemented by comparing the received phase between consecutive symbol intervals. Therefore, for proper operation, the received carrier reference phase should be constant over at least two symbol intervals.
Multiple-symbol differential detection (MSDD) uses more than two consecutive symbols and can provide better error rate performance than conventional differential detection (DD) using only two consecutive symbols. As in the case of DD, MSDD requires that the received carrier reference phase be constant over the consecutive symbol intervals used in the process.
Detailed discussions of MSDD and Multiple Symbol Detection (MSD) are found in, “Multiple—Symbol Differential Detection of MPSK” (Divsalar et al., IEEE TRANSACTIONS ON COMMUNICATIONS, Vol. 38, No. 3, March 1990) and “Multiple-Symbol Detection for Orthogonal Modulation in CDMA System” (Li et al., IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Vol. 50, No. 1, January 2001).
Conventional MPSK MSDD is explained in conjunction with FIGS. 1 and 2 below. FIG. 1 shows an AWGN communication channel 101 with an MPSK signal sequence r that comprises N consecutive symbols r1 . . . rN received by receiver 110. Symbol rk represents the kth component of the N length sequence r, where 1≦k≦N. The value for rk is a vector represented by Equation (1): 



  
    
      
        
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having symbol energy ES, symbol interval TS and transmitted phase φk where j=√{square root over (−1)}. Value nk is a sample taken from a stationary complex white Gaussian noise process with zero mean. Value θk is an arbitrary random channel phase shift introduced by the channel and is assumed to be uniformly distributed in the interval (−π, π). Although channel phase shift θk is unknown, differential detection conventionally operates assuming θk is constant across the interval of observed symbols r1 to rN. For differential MPSK (DMPSK), phase information is differentially encoded at the transmitter, and transmitted phase φk is represented by:

φk=φk−1+Δφk  Eq. (2)

where Δφk is the transmitted information phase differential corresponding to the kth transmission interval that takes on one of M uniformly distributed values within the set Ω={2 πm/M, m=0, 1, . . . , M−1} around the unit circle, as in a Gray mapping scheme. For example, for QPSK, M=4 and Δφk=0, π/2, π, or 3π/2 for each k from 1 to N.

It is assumed for simplicity that arbitrary phase value θk is constant (θk =θ) over the N-length of the observed sequence.
At the receiver, optimum detection using multiple-symbol differential detection (MSDD) is achieved by selecting an estimated sequence of phase differentials {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} which maximizes the following decision statistic: 



  
    
      
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By Equation (3), the received signal is observed over N symbol time intervals while simultaneously selecting the optimum estimated phase sequence {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1}. The maximized vector sum of the N-length signal sequence rk, provides the maximum-likelihood detection, where estimated phase differential d{circumflex over (φ)}m is the difference between estimated phase {circumflex over (φ)}m+1 and the estimate of the first phases {circumflex over (φ)}1.

d{circumflex over (φ)}m={circumflex over (φ)}m+1−{circumflex over (φ)}1.  Eq.(4)

The estimate of transmitted information phase sequence {Δ{circumflex over (φ)}1, Δ{circumflex over (φ)}2, . . . , Δ{circumflex over (φ)}N−1} is obtained from the estimated phase sequence {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} using Equation (5). 



  
    
      
        
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Value Δ{circumflex over (φ)}k is an estimate of transmitted phase differential Δφk. Since d{circumflex over (φ)}k (1≦k≦N−1) takes on one of M uniformly distributed Ω values {2 πm/M, m=0, 1, . . . , M−1}, the conventional MSDD detection searches all possible phase differential sequences and there are MN−1 such phases. The error rate performance improves by increasing the observed sequence length N, which preferably is selected to be N=4 or N=5. As an example, for 16PSK modulation with N=5, the number of phase differential sequences to search is 164=65536. As evident by this considerably large number of sequences, simplicity in the search sequence is sacrificed in order to achieve a desirable error rate performance.

FIG. 2 shows the process flow diagram for algorithm 200, which performs conventional MSDD. It begins with step 201 where N consecutive symbols rk for k=1 to N are observed. Next, the possible sets of phase differential sequences {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} where each d{circumflex over (φ)}k, for k=1 to N−1, is one from the set of M uniformly distributed phase values in the set Ω={2 πm/M, m=0, 1, . . . , M−1}. There are MN−1 possible sets. FIG. 5 shows an example of an array of such sets, where N=4 and M=4, which illustrates the 44−1=64 possible sets of phase differential sequences. In step 203, each possible phase sequence is attempted in the expression 



  
    
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giving a total of MN−1 values. Next, in step 204, the maximum value is found for step 203, which indicates the best estimate phase differential sequence. Finally, in step 205, the final information phase sequence {Δ{circumflex over (φ)}1, Δ{circumflex over (φ)}2, . . . , Δ{circumflex over (φ)}N−1} is estimated from {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} using Equation (5) and the information bits are obtained from Gray de-mapping between phase and bits.

Although MSDD provides much better error performance than conventional DD (symbol-by-symbol), MSDD complexity is significantly greater. Therefore, it is desirable to provide an improved method and system for MSDD with less complexity.
SUMMARY
A method for multiple-symbol differential detection phase evaluation of M-ary communication data is employed in which the data consists of N sequential symbols r1 . . . rN, each having one of M transmitted phases. Selected sequences of N−1 elements that represent possible sequences of phase differentials are evaluated using multiple-symbol differential detection. Next, using r1 as the reference for each phase differential estimate, sN−1 phase differential sequences are selected in the form (P2i, P3i, . . . , PNi) for i=1 to s for evaluating said symbol set, where s is predetermined and 1<s<M. Rather than attempting every one of the M possible phase differential values during the estimation, the reduced subset of s phase differential estimate values are chosen based on being the closest in value to the actual transmitted phase differential value. These s phase differential estimates can be determined mathematically as those which produce the maximum results in the differential detection expression |r1+rk+1e−jβk|2.
Each of the sN−1 phase differential sequences are then evaluated using MSDD to determine the final maximum likelihood phase sequence. The resulting final phase sequence can be used to determine the information phase estimates and the phase information bits by Gray de-mapping.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows a representation of a channel symbol stream for a receiver.
FIG. 2 shows a process flow diagram of an algorithm 200 for conventional MSDD.
FIG. 3A shows a process flow diagram of an algorithm 300 for reduced complexity MSDD.
FIG. 3B shows a detailed process flow diagram for step 302 of FIG. 3A.
FIGS. 4A, 4B, 4C show a block diagram of an implementation of the reduced complexity MSDD algorithm.
FIG. 5 shows a table of possible phase sequences processed by a conventional MSDD algorithm.
FIG. 6 graphically shows a comparison of the symbol error rate performances for the conventional and simplified MSDD algorithms.



DESCRIPTION OF THE PREFERRED EMBODIMENTS
FIG. 3A shows a MSDD algorithm 300 that reduces the search complexity of the MSDD of algorithm 200, using a subset search concept. First, in step 301, N consecutive symbols rk are observed for 1≦k≦N−1. In step 302, sN−1 sets of phase differential estimate sequences {β1, β2, . . . , βN−1 } are selected as optimum estimates from among the full set of MN−1 phase estimates attempted in algorithm 200. Turning to FIG. 3B, step 302 is broken down in further detail. In step 302A, the initial received signal r1 is selected as a preferred reference for determining phase differentials between r1 and each subsequent rk. In step 302B, a small candidate subset of s phase differential estimates {βk1, βk2, . . . , βks}(1≦k ≦N−1), among all M possible phases {2 πm/M, m=0, 1, . . . , M−1 } where 1<s<M and s is predetermined. The s phase estimates that are selected are the closest in value to the actual phase differential Δφk. In order to obtain the closest values for the phase differential estimates, each βk is applied to the conventional DD expression |r1 +rk+1 e−jβk|2 from which the s phase differential estimates {βk1, βk2, . . . , βks } that produce the maximum resulting value are selected. With the inclusion of this symbol-by-symbol DD process step (302B), it can be seen that algorithm 300 is a combination of MSDD and DD processing. In step 302C, there are now sN−1 sets of optimum phase differential sequences, where Pk={βk1, βk2, . . . , βks}. Returning to FIG. 3A, the result of step 302 is sN−1 sequences of phases (P1, P2, . . . PN−1). These are the maximum-likelihood phase differential candidates. That is, the s values for P1 are the closest in value to the actual phase differential Δφ1, the s values for P2 are the closest to actual phase differential Δφ2, and so on.
In step 303, all sN−1 possible phase sequences (P1, P2, . . . PN−1) are attempted within the expression 



  
    
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These sets of phase candidates are significantly reduced in number compared with algorithm 200 since s<M and sN−1<MN−1. When s is very small, the number of phase differential sequences to search becomes much smaller, which leads to significant complexity savings. As an example, for s=2, N=4 and M=4, there will be eight (8) sets of phase differential sequences that will result. This is a much smaller subset of the sixty-four (64) phase differential sequences shown in FIG. 5, which would be processed in a conventional MSDD algorithm, such as algorithm 200.

In step 304, the maximum resulting vectors from step 303 determine the optimum phase differential sequence {β1, β2, . . . , βN−1} Steps 303 and 304 in combination can be expressed by the following decision statistic: 



  
    
      
        
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When s=M, the statistic is simply ηnew=η.

In step 305, the final information phase sequence {Δ{circumflex over (φ)}1, Δ{circumflex over (φ)}2, . . . , Δ{circumflex over (φ)}N−1} is estimated from the optimum phase differential sequence {β1, β2, . . . βN−1} using Equation (7) and the phase information bits are obtained by Gray de-mapping. 



  
    
      
        
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FIG. 4 shows a block diagram of MSDD parallel implementation 400, where N=4, s=2. Since N=4, there are N−1=3 parallel selection circuits 401, 402, 403, for determining sN−1 (i.e., 8) subsets (P1, P2, P3) of candidate phases. Selection circuit 401 comprises delay blocks 410, 411; conjugator 412, multiplier 413, multiplier 415k (k=0 to N−1), amplitude blocks 416k (k=0 to N−1), decision block 417 multipliers 418, 419 and switch 450. Input symbol rk+3 passes through delays 410, 411 for establishing rk as the reference symbol and rk+1 as the consecutive symbol against which the phase differential is to be estimated. The output of conjugator 412 produces conjugate rk*, which when multiplied with consecutive symbol rk+1 by multiplier 413, produces a phase difference value. Next, the phase difference is multiplied by multipliers 415k to each phase in the set βk, where βk=(2 πk/M, k=0, 1, . . . , M−1). Next, the products are passed through amplitude blocks 415k and input to decision block 417, which selects the maximum s=2 inputs for the subset P1=[βk1, βk2]. The outputs of block 401 are the products rk+1 e−jβk1 and rk+1 e−jβk2 output by multipliers 418, 419.
Decision circuits 402 and 403 comprise parallel sets of similar elements as described for block 401. Decision circuit 402 includes delay blocks 420, 421, which allow processing of reference symbol rk with rk+2, whereby decision block 427 chooses candidate phases P2=[βk3, βk4]. Likewise, block 403 includes delay block 431 to allow decision block 437 to select phase differential candidates P3=[βk5, βk6] for reference symbol rk and symbol rk+3. Summer 404 adds alternating combinations of outputs from blocks 401, 402 and 403 alternated by switches 450, 451, 452, respectively, plus reference symbol rk. Since s=2, there are 23=8 combinations of phase differential sequence (P1, P2, P3) produced by switches 450, 451, 452. Decision block 405 selects the optimum phase differential sequence {β1, β2, β3}, which is the phase differential sequence (P1, P2, P3) that produces the maximum sum.
FIG. 6 shows the symbol error rate (SER) performance of the MSDD algorithm for 16PSK, where s=2 for different symbol observation lengths N=3, 4 and 5. As shown in FIG. 6, reduced-complexity MSDD algorithm 300 with s=2 provides almost the same performance as the original MSDD algorithm 200 where s=M. This is because the MSDD algorithm 300 selects one of the two closest phases between the vector rk+1e−jβk 1≦k≦N−1) and r1 in order to maximize the statistic of Equation (6). Therefore, for 2<s<M, the performance is essentially the same as for s=2, which means there is no benefit to increasing the complexity of algorithm 300 to s>2. Therefore, the optimum results are gained using the simplest possible choice for s, that is s=2.
Table 1 shows the complexity comparison of algorithm 300 with s=2 for symbol observation length N=5 against algorithm 200. The number of phase differential sequences to search is reduced significantly, resulting in faster processing speeds.












TABLE 1







No. of
No. of






phase
phase




differential
differential




sequences
sequences

Speed




to search
to search

factor




for MSDD 200
for MSDD 300
Reduction
(x times


M
Modulation
(MN−1)
(sN−1)
factor
faster)




















4
 4 PSK
256
16
16
12


8
 8 PSK
4096
16
256
229


16
16 PSK
65536
16
4096
3667

## Claims

1. A system for performing multiple-symbol differential detection in a communication receiver for a received set of N sequential rk symbols (r1 to rN), each pair of consecutive symbols having a phase differential from among M phases where M>2, comprising:
N−1 parallel selection circuits, each for producing a set of S candidate phases from the set (βk=2πk/M, k=0, 1, . . . , M−1), where 1<S<M, each selection circuit comprising: 
delay elements for allowing evaluation of the sequential symbols; 
M multipliers for producing parallel products (rk)(e−jβk); 
a decision element for choosing S candidate phases which produce the S maximum values within the expression |r1+rk+1e31 jβk|2; and 
S multipliers for producing the products (rk+i)(e−jβ′k), where (i=1 to N−1) and β′k represents one of S selected phase candidates; 

a summer for adding the product outputs of the N−1 selection circuits; and 
a decision element for evaluating the SN−1 phase sequences from the N−1 parallel selection circuits and selecting the phase sequence which produces the maximum product value.

2. A system for performing multiple-symbol differential detection in a communication receiver for a received set of N sequential rk symbols (r1 to rN), each pair of consecutive symbols having a phase differential from among M phases where M>2, comprising:
N−1 parallel selection circuits, each for producing a set of S candidate phases from the set (βk=2πk/M, k=0, 1, . . . , M−1), where 1<S<M, each selection circuit comprising: 
delay elements for allowing evaluation of the sequential symbols; 
M multipliers for producing parallel products (rk)(e−jβk); 
a selection decision element for choosing S candidate phases βk1 to βks which are closest phases of the M phases for each k, selected from the set (2πk/M, k=0, 1, . . . , M−1); and 
S multipliers for producing the products (r+i)(e−jβ′′k), where (i=1 to N−1) and β′k represents one of S selected phase candidates; 

a summer for adding the product outputs of the N−1 selection circuits; and 
an output decision element for evaluating the SN−1 phase sequences from the N−1 parallel selection circuits and selecting the phase sequence which produces the maximum product value.

3. The system of claim 2 wherein the selection decision element chooses S candidate phases βk1 to βks which produce the S largest values using the expression |r1+rk+1e−jβk|2.

4. The system of claim 2 wherein the output decision element evaluates each of the SN−1 phase sequences in the expression 



  
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and selects the phase sequence that produces the maximum result.

5. The system of claim 2, wherein S=2 and the two closest phases βk1 and βks are determined.

6. The system of claim 2, wherein M=4, N=4 and S=2, said phases βk1 to βks are selected from the set (0, π/2, π, 3 π/2), to produce SN−1 phase sets:
(P21, P31, P41) 
(P21, P31, P42) 
(P21, P32, P41) 
(P21, P32, P42) 
(P22, P31, P41) 
(P22, P31, P42) 
(P22, P32, P41) 
(P22, P32, P42), 
where Pk={βk1, βk1, βk2, . . . , βks}.