Soft input/output LMS Equalizer

Transcription

1 Soft input/output Equalizer Jacob H. Gunther 1 Todd K. Moon 1 Aleksey Orekhov 2 Dan Monroe 3 1 Utah State University 2 Cooper Union 3 Bradley University August 12, 29

2 Equalizer Block Diagram Equalizer Overview Noise Bits x x Symbols Channel Σ Equalizer Recovered Symbols Constellation Vector q = [q, q 1,..., q M 1 ]

3 What is Really Going On Equalizer Overview Channel Equalizer Noise Bits x x Symbols Σ Recovered Symbols

4 What We Want to Achieve Equalizer Overview

5 Adaptive Filter Configuration General Adaptive Filter Block Diagram d(k) χ(k) 1/z χ(k-1) χ(k-2) χ(k-3) 1/z 1/z h h1 h2 h3 x x x x Σ Σ Σ y(k) Error Computation Error Signal Update Algorithm y(k) = L 1 m= h mx(k m) = h H k x k

6 Inverse System Identification Adaptive Filter Configuration Delay d Signal Unknown System x Adaptive Filter y + Σ - e

7 Soft Inverse System Identification Adaptive Filter Configuration d Delay Probability Function Signal Unknown System x Adaptive Filter y Probability Function ɸ Error Calculation e

8 Standard Update Rules Adaptive Filter Configuration Error Calculation e(k) = d(k) y(k) Coefficients Update h k+1 = h k + µe (k)x k

9 Adaptive Filter Configuration A Posteriori Symbol Probability Function φ i (y) = = exp ( 1 y(k) q 2σ 2 i 2) P(q i ) M 1 j= exp ( 1 y(k) q 2σ 2 j 2) P(q j ) j i exp ( 1 σ 2 R [ y(k) ( q i q j )]) P(qj ) P(q i ) φ i (y) = 1 2σ 2 ( j i ( ) qi qj exp ( 1 R [ y(k) ( )]) P(qj ) q σ 2 i q j 1 + j i exp ( 1 σ 2 R [ y(k) ( q i q j )]) P(qj ) P(q i ) P(q i ) ) 2

10 Euclidean Update Rules Adaptive Filter Configuration Error Calculation e k = d k φ k Coefficients Update h k+1 = h k + µe T k φ yx k

11 Adaptive Filter Configuration Kullback-Leibler Update Rules Error Calculation ɛ i (k) = d i(k) φ i (y) Coefficients Update h k+1 = h k + µɛ T k φ yx k

12 Standard Update Rules Adaptive Filter Configuration y(k + 1) = h H k x k+1 e(k + 1) = d(k + 1) y(k + 1) P k x k+1 K k+1 = w + x H k+1 P kx k+1 h k+1 = h k + K k+1 e(k + 1) P k+1 = 1 [ ] P k K k+1 x H k+1 w P k

13 A Posteriori Function Derivative Adaptive Filter Configuration exp ( 1 y(k) q 2σ φ i (y) = 2 i 2) P(q i ) M 1 j= exp ( 1 y(k) q 2σ 2 j 2) P(q j ) φ i (y) = f (y) g(y) φ i (y) = f (y) g(y) f (y) g (y) g(y) g(y) f (y) = kf (y) φ f (y) i (y) = g(y) φ i (y) = 1 2σ 2 φ i(y) ( k g (y) g(y) ) kf (y) g(y) ) (x k qi x k x H k h

14 VITAL!!!!!! The Problem Adaptive Filter Configuration q... q 1 Q = q M 1

15 Euclidean Update Rules Adaptive Filter Configuration y(k + 1) = h H k x k+1 e k+1 = d k+1 φ y δ(k + 1) = e T k+1 Q φ y γ(k + 1) = e T k+1 φ y P k x k+1 K k+1 = w γ(k+1) + xh k+1 P kx k+1 [ ] δ(k + 1) h k+1 = h k + K k+1 y(k + 1) γ(k + 1) P k+1 = 1 [ ] P k K k+1 x H k+1 w P k

16 Adaptive Filter Configuration Kullback-Leibler Update Rules y(k + 1) = h H k x k+1 ɛ i (k + 1) = d i(k + 1) φ i (y) δ(k + 1) = ɛ T k+1 Q φ y γ(k + 1) = ɛ T k+1 φ y P k x k+1 K k+1 = w γ(k+1) + xh k+1 P kx k+1 [ ] δ(k + 1) h k+1 = h k + K k+1 y(k + 1) γ(k + 1) P k+1 = 1 [ ] P k K k+1 x H k+1 w P k

17 Parameters The Problem Coefficients Error Output Probability of Bit Error 1 Channel Amplitude Response Magnitude, db Normalized Frequency Filter Length: 5 Channel Length: 15 BPSK setting with complex AWGN SNR: 1

18 The Problem Coefficients Error Output Probability of Bit Error 1, µ =.1 1 Euclidean, µ =.1 1 Kullback Leibler, µ =.1 Amplitude.5 Amplitude.5 Amplitude Samples 2 x 1 4, µ = Samples x 1 4 Euclidean, µ = Samples 2 x 1 4 Kullback Leibler, µ =.1.4 Amplitude.2 Amplitude.1 Amplitude Samples 2 x 1 4, µ = Samples x 1 4 Euclidean, µ = Samples 2 x 1 4 Kullback Leibler, µ =.1.2 Amplitude.2 Amplitude.1 Amplitude Samples 2 x Samples x Samples 2 x 1 4

19 The Problem Coefficients Error Output Probability of Bit Error.4 Amplitude Samples Euclidean Euclidean 2 2 Amplitude Amplitude Amplitude Samples Kullback Leibler Samples Samples

20 The Problem Coefficients Error Output Probability of Bit Error MSE µ =.1 µ=.1 µ=.1 MSE Samples Euclidean 1 µ =.1 µ=.1.5 µ=.1 MSE Samples Kullback Leibler 1 µ =.1 µ=.1.5 µ= Samples

21 The Problem Coefficients Error Output Probability of Bit Error 1 MSE Samples Euclidean 1 MSE.5 K L Divergence Samples Kullback Leibler Samples

22 The Problem Coefficients Error Output Probability of Bit Error, µ =.1 Euclidean, µ =.1 Kullback Leibler, µ =.1 Imaginary Real, µ =.1 Imaginary Real Euclidean, µ =.1 Imaginary Real Kullback Leibler, µ =.1 Imaginary Imaginary Real, µ = Real Imaginary Imaginary Real Euclidean, µ = Real Imaginary Imaginary Real Kullback Leibler, µ = Real

23 The Problem Coefficients Error Output Probability of Bit Error Imaginary Real Imaginary 2 2 Euclidean Real Kullback Leibler Imaginary Real

24 Coefficients Error Output Probability of Bit Error Probability of Bit Error Vs. µ Values, 1 hard prop. of error 1 1 db 1dB 2dB 3dB 4dB 5dB 6dB 7dB 8dB 9dB 1dB x 1 4

25 Coefficients Error Output Probability of Bit Error Probability of Bit Error Vs µ Values, Euclidean 1 soft prop. of error 1 1 db 1dB 2dB 3dB 4dB 5dB 6dB 7dB 8dB 9dB 1dB x 1 4

26 Coefficients Error Output Probability of Bit Error Probability of Bit Error Vs µ Values, K-L 1 softkl prop. of error 1 1 db 1dB 2dB 3dB 4dB 5dB 6dB 7dB 8dB 9dB 1dB x 1 4

27 Coefficients Error Output Probability of Bit Error Probability of Bit Error Versus Signal to Noise Ratio 1 Mu = Mu = Probability of bit error EUC KL KL Eb/No, db Probability of bit error EUC 1 4 KL KL Eb/No, db 1 1 Mu = e Mu = 1e 5 Probability of bit error EUC 1 4 KL KL Eb/No, db Probability of bit error EUC 1 4 KL KL Eb/No, db

28 Coefficients Error Output Probability of Bit Error P biterror Vs SNR After Waiting 1 Million Samples 1 Mu = Mu = Probability of bit error EUC KL KL Eb/No, db Probability of bit error EUC 1 4 KL KL Eb/No, db 1 1 Mu = e Mu = 1e 5 Probability of bit error EUC 1 4 KL KL Eb/No, db Probability of bit error EUC 1 4 KL KL Eb/No, db

29 Noise Bits x x Symbols Encoder Scramble Channel x x Soft Decoder Unscramble Soft Equalizer - Scramble + Prior Probabilities