Asymptotic Analysis And Design Of Iterative Receivers For Non Linear ISI Channels

Transcription

1 Asymptotic Analysis And Design Of Iterative Receivers For Non Linear ISI Channels Bouchra Benammar 1 Nathalie Thomas 1, Charly Poulliat 1, Marie-Laure Boucheret 1 and Mathieu Dervin 2 1 University of Toulouse (ENSEEIHT/IRIT) 2 Thales Alenia Space August 21, /23

2 Table of Contents 1 Context 2 Non linear satellite channel 3 Iterative Volterra receiver 4 Receiver analysis and design 5 Results and conclusions 2/23

3 Context Non linear satellite channel with high order modulation (16 and 32 APSK). 3/23

4 Context Non linear satellite channel with high order modulation (16 and 32 APSK). Iterative symbol optimal equalizer. 3/23

5 Context Non linear satellite channel with high order modulation (16 and 32 APSK). Iterative symbol optimal equalizer. LDPC code design for an iterative receiver for non linear satellite channels. 3/23

6 State of the Art State of the Art Iterative receiver optimization for Linear ISI channels: Static linear ISI channel using density evolution: Kavcic et al 1. Partial response channel: Varnica et al 2. MIMO with CSI at the receiver, using EXIT curve fitting technique: Ten Brink et al 3. 1 A. Kavcic, Xiao Ma, and M. Mitzenmacher, Binary intersymbol interference channels: Gallager codes, density evolution, and code performance bounds, IEEE Trans. on Inf. Theory, N. Varnica and A. Kavcic, Optimized low-density parity-check codes for partial response channels, IEEE Comm. Letters, S. ten Brink, G. Kramer, and A. Ashikhmin, Design of low-density parity-check codes for modulation and detection, IEEE Trans. on Comm., /23

7 State of the Art State of the Art-2 Iterative receiver design using factor graphs: Worthen et al 4 Receiver optimization for satellite modulation/coding over AWGN: De Gaudenzi et al 5. 4 Andrew P. Worthen and W.E. Stark, Unified design of iterative receivers using factor graphs, IEEE Trans. on Inf. Theory, R. De Gaudenzi, A. Guillen i Fabregas, and A. Martinez, Performance analysis of turbo-coded APSK modulations over nonlinear satellite channels, IEEE Trans on Wireless Comm /23

8 Effects of the nonlinear satellite channel Effect of the nonlinear satellite channel Root raised cosine pulse shaping and memoryless HPA: Figure: System description 1.5 Scatterplot 16APSK modulation 1.5 IBO = 3dB roll off Im 0 Im Re Re Figure: Scatterplot for 16APSK Figure: Scatterplots for 16APSK with IBO=3dB 6/23

9 Time domain Volterra model Non linear Volterra channel model The time domain Volterra model writes as a nonlinear ISI channel as follows: z n = M 1 i=0 h i x n i + M 1 i=0 M 1 j=0 M 1 k=0 h ijk x n i x n j x n k + w n = F (x n k ) k I, x n + w n }{{} (1) σ n 1 Trellis representation of the Volterra model: Figure: Trellis of the non linear Volterra channel 7/23

10 System description System description Figure: System model description 8/23

11 Low Density Parity Check Code Low Density Parity Check Code Parity check matrix H. Degree-edge polynomials λ(x) and ρ(x) : λ(x) = d v d c λ i X i 1 ρ(x) = ρ j X j 1 (2) i=2 j=2 Design rate: R = 1 Belief propagation decoder dc j=2 ρ j/j dv i=2 λ i/i (3) 9/23

12 Objective Problem formulation Problem formulation Design an iterative receiver for the non linear Volterra channel using the EXIT chart technique. Maximize the design rate R with concentrated distribution of ρ(x). 10/23

13 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

14 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

15 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

16 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

17 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

18 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

19 Scheduling Scheduling The following scheduling is used: Figure: Receiver scheduling 11/23

20 LLR distributions Extrinsic LLR distribution for the 16-APSK BICM The distribution of the Extrinsic LLRs is supposed to be a consistent Gaussian Mixture: P DF (LLR) = M G m=1 π m N ( β m 2 σ2, β m σ 2 ) (4) 12/23

21 LLR distributions Extrinsic LLR distribution for the 16-APSK BICM The distribution of the Extrinsic LLRs is supposed to be a consistent Gaussian Mixture: P DF (LLR) = M G m=1 π m N ( β m 2 σ2, β m σ 2 ) (4) Histogram Approximated pdf 0.01 PDF bins Figure: 16APSK LLR soft demapper SNR = 14dB 12/23

22 LLR distributions Mutual information output of the 16APSK demapper The mutual information of a 16APSK soft demapper writes as: where σ 2 = 2/σ 2 w Ψ(σ) = M G m=1 π m J( β m σ) (5) 1 16 APSK Mutual Information Approximations Mutual Information Measured Mutual Information Gaussian Approximation Gaussian Mixture approximation E b /N 0 [db] 13/23

23 LLR distributions Mutual information output of the 16APSK demapper-2 The inverse function Ψ 1 (I) σ is approximated as : Ψ 1 (I) = { a1 I 4 + a 2 I 3 + a 3 I 2 + a 4 I if I < b 1 log( b 2 (I 1)) + b 3 I if I (6) 14/23

24 LLR distributions LLR distributions throughout the scheduling Extrinsic LLRs follow Gaussian (G) and Gaussian Mixture (GM) distributions. Figure: Distribution of the Extrinsic LLRs between the receiver components 15/23

25 LLR distributions Mutual information evolution in the scheduling Average mutual information from VN to CN: I t V,C = d v i=2 λ i I t V,C(i) (7) where: IV t C (i) = ( 2 ( MG m=1 π mj β m Ψ (I 1 MAP,V (i)) t + (i 1)J 1 ( ) IMAP,V t (i) = T I t 1 V,MAP (i) I t 1 V,MAP (i) = J ( ij 1 ( I t 1 C,V )) I t 1 C,V ) 2 ) 16/23

26 LLR distributions Mutual information evolution in the scheduling-2 Average mutual information from CN to VN: d c ( IC,V t = 1 ρ j J j 1J 1 ( 1 IV,C) ) t (8) j=2 with application of the reciprocal channel approximation 6. Combining the previous equations yields the following relationship: ( ) IV,C t = G λ(x), I t 1 V,C, ρ(x), T () (9) 6 S. ten Brink, G. Kramer, and A. Ashikhmin, Design of low-density parity-check codes for modulation and detection, IEEE Transactions on Comm /23

27 Optimization Optimization problem formulation Problem formulation Maximizing the rate R for concentrated ρ(x) subject to the constraints: Convergence G( λ, x, ρ, T (.)) x Proportions d v i=2 λ i = 1 Stability λ 2 MG m=0 π me βmψ 1 (T (1))/8 < 1 dc j=2 ρ j(j 1) 18/23

28 Optimization 1 dv = Output mutual information dv = CN trajectory Combined VN and equalizer trajectory Input mutual information Figure: Curve fitting of the Gaussian Mixture equalizer approximation 19/23

29 Designed rates Rates MAP equalizer Gaussian: degree max = 10 Gaussian Mixture: degree max = 10 ISI free 16APSK BICM Non iterative receiver for Volterra channel E b /N 0 [db] Figure: Designed rates compared to the MAP optimal detection and the 16-APSK ISI-free rates for a maximum d v = 10 20/23

30 BER results 10 0 Optimized Code AWGN Code Non iterative Optimized Code 10 1 BER E b /N 0 Figure: Bit error rate for the iterative receiver using a 4K LDPC code where λ 2 = λ 3 = λ 4 = λ 10 = and ρ 6 = 0.25 ρ 7 = /23

31 Conclusion Design of an iterative receiver for the non linear Volterra channel Approximation of the equalizer output as a Gaussian Mixture Improved information rate and BER achieved with the designed code 22/23

32 Questions Thank you for your attention! Questions? 23/23