-ary DPSK M Diﬀerential Detection Schemes for - UBC ECEelhamt/index_files/563 Project...

Differential Detection Schemes for M-ary DPSK

Elham TorabiWireless Communications Course Project

April 2006

Department of Electrical & Computer EngineeringThe University of British Columbia

[email protected]

Outline 2

1. Overview

2. Multiple-Symbol Differential Detection (MS-DD)

3. Multiple Differential Feedback Detection (MD-FD)

4. Reduced-state Viterbi Differential Detection (RSV-DD)

5. Conclusions

Elham Torabi : Differential Detection Schemes for M-ary DPSK

1. Overview 3

M -ary phase shift keying (M -ary PSK) is a bandwidth efficient, and thereforean attractive candidate for wireless communications, where available band-width is limited.

Coherent detection of PSK signals achieves good bit error rate (BER) perfor-mance in additive white Gaussian noise (AWGN) channel, but

Coherent detection requires carrier acquisition and tracking, which makesthe receiver implementation complex.

Carrier recovery is difficult, if not impossible, in fading environments.

Therefore, in applications where simplicity and robustness is important, dif-ferentially PSK (DPSK) modulation becomes a preferable alternative.

Since, conventional differential detection (DD) of M -ary DPSK signals resultsin inferior bit error rate (BER) performance compared to ideal coherent de-tection, different detection schemes for enhancing the performance of M -aryDPSK signals have been introduced in the literature.


2. Multiple-Symbol Differential Detection for M -ary DPSK Signals 4

The key to this approach is extending the observation interval of two symbols,as in conventional differential detection (DD), to several symbol intervals, andperforming joint decision on several symbols simultaneously.

Considering M -ary PSK signal transmission over an AWGN channel:sn =

2Pejn, (1)

where P = Es/T denotes the constant signal power, Es is the signal energy,T denotes the symbol duration, and n = {2m/M ; m = 0, 1, . . . ,M 1}is the modulation phase. The received signal will be

rn = snejn + nn, (2)

where nn is a sample of AWGN with power 2 = 2N0/T , and n is a uniformly

distributed phase introduced by the channel in the interval (, ). When the received sequence of length N is observed

r = sej + n, (3)



For differentially encoded PSK signals, where n = n1 + n, the decisionrule statistic is given as

choose if

rnN+1 +N2i=0 rniejNi2m=0 nim2 is maximum.

(4)

For N = 3 decision rule is simplified to

choose n and n1 if

{rnr

n1e

jn + rn1rn2ejn1 + rnrn2e

j(n+n1)}

is maximum.

(5)



Figure 1: Implementation of multiple bit differential detector; N = 3.



A simple upper bound on the average bit error probability Pb has been obtainedfor MS-DD, using the union bound:

Pb 1(N 1) log2 M

=

w (u, u) Pr { > |} , (6)

where u is the sequence of (N 1) log2 M information bits that produces at the transmitter, u denotes bits corresponding to detected .

w (u, u) is the Hamming distance between u and u, and Pr { > |}denotes the pairwise probability.

The upper bound for 4-DPSK, when N is obtained as

Pb 12erfc

(EbN0

). (7)



6 6.5 7 7.5 8 8.5 9 9.5 1010

6

105

104

103

102

101

Eb/N0 (dB)

BE

R

4DPSK

N = 2Theoretical upper bound; N = 2N = 3Theoretical upper bound; N = 3N = 5Theoretical upper bound; N = 5N = Infinity

Figure 2: Bit error probability versus Eb/N0 for MS-DD of 4-DPSK. Simulation results along with theoretical upper bounds areshown.


3. Multiple Differential Feedback Detection for M -ary DPSK Signals 9

The goal is to improve the BER performance by using symbol detectors withdelays larger than a symbol period and feeding back the detected symbols tothe detection unit.

This approach is based on minimizing the quadratic errors of the outputs ofL symbol detectors of orders j = 1, 2, . . . , L.

Figure 3: Conventional and decision-feedback detector for M-ary DPSK signals; L=3.



Considering M -ary DPSK signal transmission over AWGN channel:

The output of symbol detector of order j is given as

z(j)n = rnrnj = 2Panan1 anj+1 + n(j)n , (8)

where an = ejn.

The metric, which represents the quadratic error sum of the detector out-puts is given by

=z(1)n 2San2 + z(2)n 2Sanan12 +

+z(L)n 2Sanan1 anL+12 , L > 1. (9)



Simplifying the metric gives the decision rule as

an = maxan

Lj=1

{anMj} , (10)

with

Mj =

{z

(1)n

j=1an1an2 anj+1z(j)n j=2, 3, . . . , L.



Exact formulas for BER for M -ary DPSK, when correct symbols are fed backto decision unit have been obtained.

For 4-DPSK, as L the BER is given as

Pb 12

erfc

(EbN0

). (11)



3 4 5 6 7 8 9 1010

6

105

104

103

102

101

Eb/N0 (dB)

BE

R

4DPSK

L = 1Theoretical; L = 1L = 2Theoretical; L = 2L = 3Theoretical; L=3Theoretical; Coherent

Figure 4: Bit error probability versus Eb/N0 of MD-FD for 4-DPSK. Simulation results along with theoretical exact BERs whencorrect symbols have been fed back, are shown.


4. Reduced-state Viterbi Differential Detection for M -ary DPSK Signals 14

This approach is a reduced-state Viterbi algorithm that incorporates feedbackinto the structure of path metric computations.

As a result, the number of states in the trellis is reduced to M , and for eachstate, the phase reference is estimated recursively in the trellis, along thesurviving path ending in each state.



Considering M -ary DPSK signals through AWGN channel, and assumingtransmission of N -symbol sequence = {N, N1, . . . , 2, 1},the path metric in Viterbi algorithm is found to be

=N

n=1

n, (12)

where the branch metric n is

n = rn

(n

l=1

rnl exp jl1k=1

nk

)expjn

. (13)

The optimal decision rule is then obtained as = max

over. (14)



Figure 5: Block diagram of RSV-DD.

The phase reference can be recursively estimated by

n1 (n1) = rn1 +n

l=2

l1(

rn1 exp jl1k=1

nk

)

= rn1 + n2(n2) exp jn1, (15)

where n1(n1) is an estimate of rn1, which is computed by feeding backthe sequence of symbols along the surviving path ending in state n1.

And, the branch metric for transition from state n1 to state n is cal-culated as

(n1 n) = [rn

n1(n1) expjn

]. (16)



RSV-DD allows M paths to survive and traces back the most likely path Dsymbols back to output the decoded symbol. Therefore, there is a D-symboldecision delay, i.e., at each time n, nD is detected.

Figure 6: Trellis diagram for 4-DPSK.



If only one single path is allowed to survive and symbol-by-symbol decision isperformed, RSV-DD reduces to its simplest version, known as decision feed-back DD (DF-DD).

DF-DD is equivalent to MD-FD using L = 1+1.

Approximate BER expressions of MF-DD for M -ary DPSK have been providedin the literature. The result for 4-DPSK, when = 1 is as follows

Pb = erfc

(EbN0

)1 12erfc

(EbN0

) . (17)



3 4 5 6 7 8 9 1010

6

105

104

103

102

101

100

Eb/N0 (dB)

BE

R

4DPSK

u = 0u = 0.5u = 0.6u = 0.9u = 1Theoretical; u = 0Theoretical; u = 0.5Theoretical; u = 0.6Theoretical; u = 0.9Theoretical; u = 1

Figure 7: Bit error probability versus Eb/N0 of RSV-DD for 4-DPSK. Simulation results along with theoretical approximatedBERs of DF-DD with same are shown.


5. Conclusions 20

The BER performance of MS-DD improves by increasing the observation in-terval N , and can practically achieve the coherent detection performance byextending the observation interval to only a few symbols. However, decisionshould be performed over MN hypothesis symbols, which is rather compli-cated.

MD-FD is less complex than MS-DD, but it achieves slightly inferior BERperformance compared to MS-DD with the same total signal delay.

RSV-DD reduces the number of states in trellis of Viterbi algorithm to Mstates, which significantly reduces the computational complexity. Comparingto Viterbi DD that requires ML1-state Viterbi decoder, and ML branch met-ric computations.

RSV-DD requires only M 2 branch metric computations, and therefore it isML2 times less complex than Viterbi DD.


5. Conclusions 21

The performance of RSV-DD is slightly inferior to Viterbi DD.

The performance of RSV-DD can be simply improved by increasing , andwithout increasing the complexity.

DF-DD achieves similar BER performance as MD-FD with compatible andL.

RSV-DD is superior to MD-DF, and DF-DD (compared at the same ). For 0.8 the performance difference between RSV-DD and DF-DD becomesvery small.


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