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 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 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 4 • 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: √ 2 Pe jφ n , s n = (1) where P = E s /T denotes the constant signal power, E s is the signal energy, T denotes the symbol duration, and φ n = { 2 mπ/M ; m = 0 , 1 , . . . , M − 1 } is the modulation phase. The received signal will be r n = s n e jθ n + n n , (2) where n n is a sample of AWGN with power σ 2 = 2 N 0 /T , and θ n is a uniformly distributed phase introduced by the channel in the interval ( − π, π ) . • When the received sequence of length N is observed r = s e jθ + n , (3) Elham Torabi : Differential Detection Schemes for M -ary DPSK

2. Multiple-Symbol Differential Detection for M -ary DPSK Signals 5 • For differentially encoded PSK signals, where φ n = φ n − 1 + ∆ φ n , the decision rule statistic is given as choose ∆ˆ φ if 2 � � i =0 r n − i e − j � N − i − 2 � r n − N +1 + � N − 2 η � ∆ φ n − i − m is maximum . � m =0 � � (4) • For N = 3 decision rule is simplified to choose ∆ˆ φ n and ∆ˆ φ n − 1 if � n − 1 e − j ∆ˆ n − 2 e − j ∆ˆ n − 2 e − j (∆ˆ φ n +ˆ φ n − 1 ) � φ n + r n − 1 r ∗ φ n − 1 + r n r ∗ r n r ∗ ℜ 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 7 • A simple upper bound on the average bit error probability P b has been obtained for MS-DD, using the union bound: 1 � P b ≤ w ( u , ˆ u ) × Pr { ˆ η > η | ∆ φ } , (6) ( N − 1) log 2 M ∆ˆ φ � = ∆ φ where u is the sequence of ( N − 1) log 2 M information bits that produces u denotes bits corresponding to detected ∆ˆ ∆ φ at the transmitter, 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 �� � P b ≤ 1 E b . 2 erfc (7) N 0 Elham Torabi : Differential Detection Schemes for M -ary DPSK

2. Multiple-Symbol Differential Detection for M -ary DPSK Signals 8 4−DPSK −1 10 −2 10 −3 10 BER −4 10 N = 2 Theoretical upper bound; N = 2 N = 3 Theoretical upper bound; N = 3 −5 10 N = 5 Theoretical upper bound; N = 5 N = Infinity −6 10 6 6.5 7 7.5 8 8.5 9 9.5 10 Eb/N0 (dB) Figure 2: Bit error probability versus E b /N 0 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 9 • 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 10 • Considering M -ary DPSK signal transmission over AWGN channel: – The output of symbol detector of order j is given as z ( j ) n = r n r ∗ n − j = 2 Pa n a n − 1 · · · a n − j +1 + n ( j ) n , (8) where a n = e jφ n . – The metric, which represents the quadratic error sum of the detector out- puts is given by 2 2 � � � � � z (1) � z (2) η = n − 2 S ˜ a n + n − 2 S ˜ a n ˆ a n − 1 + · · · � � � � � � 2 � � � z ( L ) + − 2 S ˜ a n ˆ a n − 1 · · · ˆ a n − L +1 , L > 1 . (9) � � n � Elham Torabi : Differential Detection Schemes for M -ary DPSK

3. Multiple Differential Feedback Detection for M -ary DPSK Signals 11 – Simplifying the metric gives the decision rule as L � a n = max ℜ { ˜ a n M j } , ˆ (10) a n ˜ j =1 with � z (1) n ∗ j=1 M j = n ∗ j=2, 3, . . . , L. a n − j +1 z ( j ) a n − 1 ˆ ˆ a n − 2 · · · ˆ Elham Torabi : Differential Detection Schemes for M -ary DPSK

3. Multiple Differential Feedback Detection for M -ary DPSK Signals 12 • 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 �� � P b → 1 E b . (11) 2 erfc N 0 Elham Torabi : Differential Detection Schemes for M -ary DPSK

3. Multiple Differential Feedback Detection for M -ary DPSK Signals 13 4−DPSK −1 10 −2 10 −3 10 BER −4 10 L = 1 Theoretical; L = 1 L = 2 Theoretical; L = 2 −5 10 L = 3 Theoretical; L=3 Theoretical; Coherent −6 10 3 4 5 6 7 8 9 10 Eb/N0 (dB) Figure 4: Bit error probability versus E b /N 0 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 14 • 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 15 • Considering M -ary DPSK signals through AWGN channel, and assuming transmission of N -symbol sequence ∆ φ = { ∆ φ N , ∆ φ N − 1 , . . . , ∆ φ 2 , ∆ φ 1 } , the path metric in Viterbi algorithm is found to be N � Λ = λ n , (12) n =1 where the branch metric λ n is � n � ∗   l − 1 � �  . λ n = ℜ  r n r n − l exp j ∆ φ n − k exp − j ∆ φ n (13) l =1 k =1 • The optimal decision rule is then obtained as ∆ˆ φ = max over ∆ 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 n l − 1 � � � � µ l − 1 η n − 1 (∆ φ n − 1 ) = r n − 1 + r n − 1 exp j ∆ φ n − k l =2 k =1 = r n − 1 + µη n − 2 (∆ φ n − 2 ) exp j ∆ φ n − 1 , (15) where η n − 1 (∆ φ n − 1 ) is an estimate of r n − 1 , which is computed by feeding back the sequence of symbols along the surviving path ending in state ∆ φ n − 1 . • And, the branch metric for transition from state ∆ φ n − 1 to state ∆ φ n is cal- culated as r n η ∗ � � λ (∆ φ n − 1 → ∆ φ n ) = ℜ n − 1 (∆ φ n − 1 ) exp − j ∆ φ n . (16) Elham Torabi : Differential Detection Schemes for M -ary DPSK

4. Reduced-state Viterbi Differential Detection for M -ary DPSK Signals 17 • 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 , ∆ φ n − D is detected. Figure 6: Trellis diagram for 4-DPSK. Elham Torabi : Differential Detection Schemes for M -ary DPSK