Differential Detection Schemes for M -ary DPSK Elham Torabi - - PowerPoint PPT Presentation

Differential Detection Schemes for M-ary DPSK Elham Torabi Wireless Communications Course Project April 2006 Department of Electrical & Computer Engineering The University of British Columbia elhamt@ece.ubc.ca

Outline

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• 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

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• M-ary phase shift keying (M-ary PSK) is a bandwidth efficient, and therefore

an 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 makes the 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 results

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

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• The key to this approach is extending the observation interval of two symbols,

as in conventional differential detection (DD), to several symbol intervals, and performing joint decision on several symbols simultaneously.

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

sn = √ 2Pejφn, (1) where P = Es/T denotes the constant signal power, Es is the signal energy, T denotes the symbol duration, and φn = {2mπ/M; m = 0, 1, . . . , M − 1} is the modulation phase. The received signal will be rn = snejθn + nn, (2) where nn is a sample of AWGN with power σ2 = 2N0/T, and θn is a uniformly distributed phase introduced by the channel in the interval (−π, π).

• When the received sequence of length N is observed

r = sejθ + n, (3)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• For differentially encoded PSK signals, where φn = φn−1 + ∆φn, the decision

rule statistic is given as choose ∆ˆ φ if η

• rn−N+1 + N−2

i=0 rn−ie−j N−i−2

m=0

∆φn−i−m

• 2

is maximum. (4)

• For N = 3 decision rule is simplified to

choose ∆ˆ φn and ∆ˆ φn−1 if ℜ

• rnr∗

n−1e−j∆ˆ φn + rn−1r∗ n−2e−j∆ˆ φn−1 + rnr∗ n−2e−j(∆ˆ φn+ˆ φn−1)

is maximum. (5)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

6 Figure 1: Implementation of multiple bit differential detector; N = 3. Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• A simple upper bound on the average bit error probability Pb has been obtained

for MS-DD, using the union bound: Pb ≤ 1 (N − 1) log2 M

• ∆ˆ

φ=∆φ

w (u, ˆ u) × Pr {ˆ η > η|∆φ} , (6) where u is the sequence of (N − 1) log2 M information bits that produces ∆φ at the transmitter, ˆ u denotes bits corresponding to detected ∆ˆ φ. w (u, ˆ u) is the Hamming distance between u and ˆ u, and Pr {ˆ η > η|∆φ} denotes the pairwise probability.

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

Pb ≤ 1 2erfc

• Eb

N0

• .

(7)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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6 6.5 7 7.5 8 8.5 9 9.5 10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) BER 4−DPSK N = 2 Theoretical upper bound; N = 2 N = 3 Theoretical upper bound; N = 3 N = 5 Theoretical upper bound; N = 5 N = Infinity

Figure 2: Bit error probability versus Eb/N0 for MS-DD of 4-DPSK. Simulation results along with theoretical upper bounds are shown. Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• The goal is to improve the BER performance by using symbol detectors with

delays larger than a symbol period and feeding back the detected symbols to the detection unit.

• This approach is based on minimizing the quadratic errors of the outputs of

L symbol detectors of orders j = 1, 2, . . . , L.

Figure 3: Conventional and decision-feedback detector for M-ary DPSK signals; L=3. Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• Considering M-ary DPSK signal transmission over AWGN channel:

– The output of symbol detector of order j is given as z(j)

n = rnr∗ n−j = 2Panan−1 · · · an−j+1 + n(j) n ,

(8) where an = ejφn. – The metric, which represents the quadratic error sum of the detector out- puts is given by η =

• z(1)

n − 2S˜

an

• 2

+

• z(2)

n − 2S˜

anˆ an−1

• 2

+ · · · +

• z(L)

n

− 2S˜ anˆ an−1 · · · ˆ an−L+1

• 2

, L > 1. (9)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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– Simplifying the metric gives the decision rule as ˆ an = max

˜ an L

• j=1

ℜ {˜ anMj} , (10) with Mj =

• z(1)

n ∗

j=1 ˆ an−1ˆ an−2 · · · ˆ an−j+1z(j)

n ∗ j=2, 3, . . . , L.

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• Exact formulas for BER for M-ary DPSK, when correct symbols are fed back

to decision unit have been obtained.

• For 4-DPSK, as L → ∞ the BER is given as

Pb → 1 2 erfc

• Eb

N0

• .

(11)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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3 4 5 6 7 8 9 10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) BER 4−DPSK L = 1 Theoretical; L = 1 L = 2 Theoretical; L = 2 L = 3 Theoretical; L=3 Theoretical; Coherent

Figure 4: Bit error probability versus Eb/N0 of MD-FD for 4-DPSK. Simulation results along with theoretical exact BERs when correct symbols have been fed back, are shown. Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• This approach is a reduced-state Viterbi algorithm that incorporates feedback

into the structure of path metric computations.

• As a result, the number of states in the trellis is reduced to M, and for each

state, the phase reference is estimated recursively in the trellis, along the surviving path ending in each state.

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• Considering M-ary DPSK signals through AWGN channel, and assuming

transmission of N-symbol sequence ∆φ = {∆φN, ∆φN−1, . . . , ∆φ2, ∆φ1}, the path metric in Viterbi algorithm is found to be Λ =

N

• n=1

λn, (12) where the branch metric λn is λn = ℜ  rn n

• l=1

rn−l exp j

l−1

• k=1

∆φn−k ∗ exp −j∆φn   . (13)

• The optimal decision rule is then obtained as

∆ˆ φ = max

• ver∆φ Λ.

(14)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

16 Figure 5: Block diagram of RSV-DD.

• The phase reference can be recursively estimated by

ηn−1 (∆φn−1) = rn−1 +

n

• l=2

µl−1

• rn−1 exp j

l−1

• k=1

∆φn−k

• = rn−1 + µηn−2(∆φn−2) exp j∆φn−1,

(15) where ηn−1(∆φn−1) is an estimate of rn−1, which is computed by feeding back the sequence of symbols along the surviving path ending in state ∆φn−1.

• And, the branch metric for transition from state ∆φn−1 to state ∆φn is cal-

culated as λ(∆φn−1 → ∆φn) = ℜ

• rnη∗

n−1(∆φn−1) exp −j∆φn

• .

(16)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• RSV-DD allows M paths to survive and traces back the most likely path D

symbols back to output the decoded symbol. Therefore, there is a D-symbol decision delay, i.e., at each time n, ∆φn−D is detected.

Figure 6: Trellis diagram for 4-DPSK. Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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• If only one single path is allowed to survive and symbol-by-symbol decision is

performed, 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+µ

1−µ.

• Approximate BER expressions of MF-DD for M-ary DPSK have been provided

in the literature. The result for 4-DPSK, when µ = 1 is as follows Pb = erfc Eb N0  1 − 1 2erfc Eb N0   . (17)

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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

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3 4 5 6 7 8 9 10 10

−6

10

−5

10

−4

10

−3

10

−2

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−1

10 Eb/N0 (dB) BER 4−DPSK u = 0 u = 0.5 u = 0.6 u = 0.9 u = 1 Theoretical; u = 0 Theoretical; u = 0.5 Theoretical; u = 0.6 Theoretical; u = 0.9 Theoretical; u = 1

Figure 7: Bit error probability versus Eb/N0 of RSV-DD for 4-DPSK. Simulation results along with theoretical approximated BERs of DF-DD with same µ are shown. Elham Torabi : Differential Detection Schemes for M-ary DPSK

• 5. Conclusions

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• The BER performance of MS-DD improves by increasing the observation in-

terval N, and can practically achieve the coherent detection performance by extending the observation interval to only a few symbols. However, decision should be performed over MN hypothesis symbols, which is rather compli- cated.

• MD-FD is less complex than MS-DD, but it achieves slightly inferior BER

performance compared to MS-DD with the same total signal delay.

• RSV-DD reduces the number of states in trellis of Viterbi algorithm to M

states, which significantly reduces the computational complexity. Comparing to Viterbi DD that requires ML−1-state Viterbi decoder, and ML branch met- ric computations.

• RSV-DD requires only M2 branch metric computations, and therefore it is

M L−2 times less complex than Viterbi DD.

Elham Torabi : Differential Detection Schemes for M-ary DPSK

• 5. Conclusions

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• The performance of RSV-DD is slightly inferior to Viterbi DD.
• The performance of RSV-DD can be simply improved by increasing µ, and

without increasing the complexity.

• DF-DD achieves similar BER performance as MD-FD with compatible µ and

L.

• RSV-DD is superior to MD-DF, and DF-DD (compared at the same µ). For

µ ≥ 0.8 the performance difference between RSV-DD and DF-DD becomes very small.

Elham Torabi : Differential Detection Schemes for M-ary DPSK

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• Elham Torabi : Differential Detection Schemes for M-ary DPSK