NCC 2011 presentation

1. Mutual Information with ﬁlterbank equalization for MIMO frequency selective Mutual Information with ﬁlterbank channels Vijaya Krishna equalization for MIMO frequency selective A, Shashank V channels Vijaya Krishna A Shashank V Department of ECE P E S Institute of Technology, Bangalore NCC 2011

2. Outline Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels 1 Motivation Vijaya Krishna A, Shashank V 2 Signal model 3 Block processing 4 Filterbank framework 5 Mutual information with ﬁlterbank equalization 6 Conclusion

3. Motivation Mutual Information with filterbank equalization for MIMO frequency MIMO systems: Higher rate, more reliability selective channels Vijaya Krishna Frequency selectivity: Equalization required at receiver A, Shashank V Typically, block processing used: Motivation Zero padding or cyclic prefixing: Convert frequency Signal model selective fading to flat fading Block processing Redundancy of the order of channel length required Filterbank framework Mutual Lower data rates information Simulations Additional processing required: coding, etc Conclusion

7. Mutual Information with filterbank equalization Filterbank equalizers: for MIMO frequency selective Instead of converting to flat fading, view the channel as channels FIR filter Vijaya Krishna A, Shashank V Equalization: Inverse filtering Motivation Signal model Block processing Filterbank framework Mutual information By adding no/minimal redundancy, we can find FIR Simulations inverse filters Conclusion

10. Mutual Information with ﬁlterbank equalization for MIMO frequency Mutual information: Acheivable data rate selective channels I(X ; Y ) = H(X ) − H(X |Y ) Vijaya Krishna A, Shashank V Aim: Quantify data rate: Mutual information for Filter Motivation bank case Signal model Block Our Contribution: processing 1 Derivation of expression for MI with ﬁlterbank Filterbank framework equalization for the MMSE criterion Mutual 2 MI expression for the case of symbol by symbol information detection Simulations Conclusion

13. Signal model Mutual Information with ﬁlterbank equalization Consider M×N frequency selective LH tap MIMO for MIMO frequency channel selective channels Signal model: Vijaya Krishna LH −1 A, Shashank V y [n] = H(k ) x(n − k ) + v (n) Motivation k =0 Signal model Block Y(ejω ) = H(ejω )X(ejω ) + V(ejω ) processing Filterbank Mutual information of channel: framework ˆ π 1 p0 log IN + 2 H∗ (ejω )H(ejω ) dω Mutual information I(H) = I(X ; Y ) = 2πN −π σv Simulations Conclusion Difﬁcult to evaluate

17. Block processing Mutual Information Block processing: Zero padding scheme with ﬁlterbank equalization ˜ ˜ ˜ y (n) = HP x (n) + v (n) for MIMO frequency selective channels   H(0) . . . H(LH − 1) 0 . . . 0 Vijaya Krishna A, Shashank  .. .. .. . .  V   0 . . . .   HP =   .. .. .. . .  Motivation 0 . . . .  . .   Signal model  . .. .. .. . . . .  Block  . .  processing 0 ... H(0) · · · H(LH − 1) Filterbank framework M(P+LH -1) by NP Block Toeplitz matrix Mutual information P: no of input symbols per block Simulations x (n) = [x T (Pn), x T (Pn − 1), ....., x T (P(n − 1) − 1)]T ˜ Conclusion Results of ﬂat fading channels can be used for block processing

20. Mutual Information with ﬁlterbank equalization For ﬂat fading channel with channel matrix H, mutual for MIMO frequency information: selective channels 1 P0 Vijaya Krishna I(H) = log2 I + 2 H∗ H A, Shashank N σv V Motivation Mutual information with zero padding: Signal model 1 P0 ∗ Block IB (H) = log2 I + 2 HP HP processing N(P + LH − 1) σv Filterbank framework lim IB (HP ) = I(H) Mutual P→∞ information Simulations Can be realized using joint ML detection at receiver Conclusion

23. Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Optionally, Vijaya Krishna A, Shashank 1. Successive interference cancellation (MMSE-SIC) V Motivation 2. Eigenmode precoding Signal model Block processing May not be feasible. Suboptimal MMSE with symbol by Filterbank symbol detection used. framework Mutual information Simulations Conclusion

26. Mutual Information with ﬁlterbank equalization for MIMO Symbol by symbol detection: frequency selective channels For the k th symbol, rate is Vijaya Krishna   A, Shashank V B 1  1  Ik ,MMSE = log2   −1  Motivation N(P + LH − 1)  p0 ∗ Signal model I+ 2H H σv P P k ,k Block processing Filterbank Total rate is framework MP−1 Mutual B 1 B information IMMSE = Ik ,MMSE Simulations N(P + LH − 1) k =0 Conclusion

29. Filterbank framework Mutual Information with ﬁlterbank equalization y(z) = H(z)x(z) + v(z) for MIMO frequency selective channels ˜ ˜ ˜ y (n) = Hx (n) + v (n) Vijaya Krishna A, Shashank V   H(0) . . . H(LH − 1) 0 . . . 0 Motivation  .. .. .. . .  Signal model   0 . . . .   Block H=  .. .. .. . .  processing 0 . . . .  . .   Filterbank  . .. .. .. . . . .  framework  . .  Mutual 0 ... H(0) · · · H(LH − 1) information Simulations MLF by N(LF +LH -1) block Toeplitz matrix Conclusion LF : Length of FIR ﬁlter used for equalization

30. Filterbank framework Mutual Information with ﬁlterbank equalization z −d X(z) = F(z)Y(z) for MIMO frequency selective channels ˆ ˜ ˜ x (n − d) = FHx (n) + Fv (n) Vijaya Krishna A, Shashank V MMSE inverse: FMMSE = Rxx Jd H∗ (HRx x H∗ + Rv v )−1 ¯¯ ¯¯ Motivation Jd = [0N×Nd IN×N 0N×N(LH +LF −d−2) ] Signal model Block   processing 0 ··· 0 1 ··· 0 0 ··· 0  . .. . .. .. .. .. .. .  Filterbank framework Jd =  . . . . . . . . . . .  . Mutual 0 ··· 0 0 ··· 1 0 ··· 0 information Simulations 2 If Rxx = I and Rv v = σv I Conclusion ¯¯ FMMSE = Jd H∗ (HH∗ + σv I)−1 2

33. Mutual information Mutual Information with ﬁlterbank Idea is that error vector is orthogonal to the estimate equalization for MIMO and frequency selective ˆ X = Axy Y + X⊥Y = X + E channels Vijaya Krishna A, Shashank V ˆ −1 X = X|Y = Axy Y = Rxy Ryy Y Motivation Signal model |Rxx | Block IF (H) = log2 processing |Ree | Filterbank framework Mutual Theorem information Simulations 1 |Rxx | Conclusion IF (H) = log2 ∗ (HR H∗ + R )−1 HJ R | N |Rxx − Rxx Jd H ¯¯ xx ¯¯ vv d xx

34. Mutual information Mutual Information with ﬁlterbank Idea is that error vector is orthogonal to the estimate equalization for MIMO and frequency selective ˆ X = Axy Y + X⊥Y = X + E channels Vijaya Krishna A, Shashank V ˆ −1 X = X|Y = Axy Y = Rxy Ryy Y Motivation Signal model |Rxx | Block IF (H) = log2 processing |Ree | Filterbank framework Mutual Theorem information Simulations 1 |Rxx | Conclusion IF (H) = log2 ∗ (HR H∗ + R )−1 HJ R | N |Rxx − Rxx Jd H ¯¯ xx ¯¯ vv d xx

35. Proof Mutual Information with ﬁlterbank equalization for MIMO frequency selective I(X ; Y ) = h(X ) − h(X |Y ) channels Vijaya Krishna A, Shashank For the MMSE equalizer, h(X |Y ) = h(E), the entropy of V the error vector Motivation 1 |Rxx | IF (H) = h(X ) − h(E) = log2 Signal model N |Ree | Block processing Filterbank Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ } ˜ ˜˜ ˜ framework Mutual information Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯ Simulations Conclusion

36. Proof Mutual Information with ﬁlterbank 2 If Rxx = p0 I and Ree = σv I then equalization for MIMO frequency selective Ree = p0 I − p0 Jd H∗ (p0 HH∗ + σv I)−1 HJd p0 2 channels Vijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d v Motivation 1 1 Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Block σv processing Filterbank framework For the case of symbol by symbol detection, Mutual N−1 information 1 1 Simulations IF (H) = log2 N p0 ∗ −1 J∗ Conclusion k =0 Jd (I + 2 H H) σv d k ,k

37. Proof Mutual Information with ﬁlterbank 2 If Rxx = p0 I and Ree = σv I then equalization for MIMO frequency selective Ree = p0 I − p0 Jd H∗ (p0 HH∗ + σv I)−1 HJd p0 2 channels Vijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d v Motivation 1 1 Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Block σv processing Filterbank framework For the case of symbol by symbol detection, Mutual N−1 information 1 1 Simulations IF (H) = log2 N p0 ∗ −1 J∗ Conclusion k =0 Jd (I + 2 H H) σv d k ,k

38. Mutual Information with ﬁlterbank equalization for MIMO 2 If Rxx = p0 I and Ree = σv I then frequency selective channels Vijaya Krishna 1 1 A, Shashank V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d σv Motivation Signal model Block For the case of symbol by symbol detection, processing N−1 Filterbank 1 1 framework IF (H) = log2 N p0 ∗ −1 J∗ Mutual k =0 Jd (I + 2 H H) σv d k ,k information Simulations Conclusion

39. Mutual Information with ﬁlterbank equalization for MIMO 2 If Rxx = p0 I and Ree = σv I then frequency selective channels Vijaya Krishna 1 1 A, Shashank V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d σv Motivation Signal model Block For the case of symbol by symbol detection, processing N−1 Filterbank 1 1 framework IF (H) = log2 N p0 ∗ −1 J∗ Mutual k =0 Jd (I + 2 H H) σv d k ,k information Simulations Conclusion

40. Mutual Information with ﬁlterbank equalization Observation for MIMO frequency 1 1 selective IF (H) = log2 p0 ∗ −1 J∗ | channels N |Jd (I + 2 H H) d Vijaya Krishna σv A, Shashank V 1 1 IB (H) = log2 −1 Motivation N(P + LH − 1) p0 ∗ I+ 2H H σv P P Signal model Block processing Filterbank Remark framework The MI for ﬁlterbank equalization depends on the Mutual p information determinant of N by N submatrix of (I + σ0 H∗ H)−1 . So we 2 v Simulations can choose the delay so as to maximize MI Conclusion

41. Mutual Information with ﬁlterbank equalization Observation for MIMO frequency 1 1 selective IF (H) = log2 p0 ∗ −1 J∗ | channels N |Jd (I + 2 H H) d Vijaya Krishna σv A, Shashank V 1 1 IB (H) = log2 −1 Motivation N(P + LH − 1) p0 ∗ I+ 2H H σv P P Signal model Block processing Filterbank Remark framework The MI for ﬁlterbank equalization depends on the Mutual p information determinant of N by N submatrix of (I + σ0 H∗ H)−1 . So we 2 v Simulations can choose the delay so as to maximize MI Conclusion

42. Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank V Motivation Signal model Block processing Filterbank framework Mutual information Simulations Conclusion p0 ∗ −1 Choose submatrix of (I + 2 H H) σv with lowest determinant

43. Simulations Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank 4×3 Rayleigh fading channels of length LH = 8 V Block processing case: no of inputs symbols per block Motivation P = 20 Signal model Block Filterbank case: Length of equalizer LF = 21 processing Filterbank framework Mutual information Simulations Conclusion

44. Simulations Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank V Motivation Signal model Block processing Filterbank framework Mutual information Simulations Conclusion Figure: Comparison between block processing and Filterbank equalizers

45. Simulations Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank V Motivation Signal model Block processing Filterbank framework Mutual information Simulations Conclusion Figure: MI with variation in delay. SNR=15 dB

46. Simulations Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank V Motivation Signal model Block processing Filterbank framework Mutual information Simulations Conclusion Figure: MI for different LF ’s. SNR=15 dB

47. Conclusion Mutual Information with filterbank equalization for MIMO frequency selective Filterbank equalization achieves significantly higher channels information rate when compared to block processing Vijaya Krishna A, Shashank V We have the flexibility of choosing the delay so as to Motivation maximize MI Signal model Block Disadvantage of this scheme: Processing complexity, processing Filterbank similar to BP framework Mutual Future: Mutual information using zero forcing equalizers information Simulations Conclusion

51. References Mutual Information with ﬁlterbank equalization for MIMO frequency Vijaya Krishna. A, A ﬁlterbank precoding framework for selective channels MIMO frequency selective channels, PhD thesis, Indian Vijaya Krishna Institute of Science, 2006. A, Shashank V G. D. Forney Jr., “Shannon meets Wiener II: On MMSE Motivation estimation in successive decoding schemes,” In Proc. Signal model Allerton Conf., Sep. 2004. Block processing (http://arxiv.org/abs/cs/0409011) Filterbank framework X. Zhang and S.-Y. Kung, “Capacity analysis for parallel Mutual and sequential MIMO equalizers,” IEEE Trans on Signal information Processing, vol. 51, pp. 2989- 3002, Nov. 2003. Simulations Conclusion

52. References Mutual Information with ﬁlterbank equalization for MIMO frequency P. P. Vaidyanathan, Multirate systems and ﬁlter banks, selective channels Englewood Cliffs, NJ: Prentice-Hall, 1993. Vijaya Krishna A, Shashank Vijaya Krishna. A, K. V. S. Hari, ”Filterbank precoding for V FIR equalization in high rate MIMO communications,” Motivation IEEE Trans. Signal Processing, vol. 54, No. 5, pp. Signal model 1645-1652, May 2006. Block processing A. Scaglione, S. Barbarossa, and G. B, Giannakis, Filterbank framework “Filterbank transceivers optimizing information rate in Mutual block transmissions over dispersive channels,” IEEE information Trans. Info. Theory, Vol. 45, pp. 1019-1032, Apr. 1999. Simulations Conclusion

53. Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Vijaya Krishna A, Shashank V Motivation THANK YOU Signal model Block processing Filterbank framework Mutual information Simulations Conclusion

54. Mutual information Mutual Information with ﬁlterbank equalization for MIMO frequency I(X ; Y ) = h(X ) − h(X |Y ) selective channels Vijaya Krishna For the MMSE equalizer, h(X |Y ) = h(E), the entropy of A, Shashank V the error vector Motivation Signal model 1 |Rxx | IF (H) = h(X ) − h(E) = log2 Block N |Ree | processing Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ } Filterbank framework ˜ ˜˜ ˜ Mutual information Simulations Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯ Conclusion

58. Mutual information Mutual Information with ﬁlterbank 2 If Rxx = p0 I and Ree = σv I then equalization for MIMO frequency selective Ree = p0 I − p0 Jd H∗ (p0 HH∗ + σv I)−1 HJd p0 2 channels Vijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d v Motivation 1 1 Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Block σv processing Filterbank framework For the case of symbol by symbol detection, Mutual N−1 information 1 1 Simulations IF (H) = log2 N p0 ∗ −1 J∗ Conclusion k =0 Jd (I + 2 H H) σv d k ,k

59. Mutual information Mutual Information with ﬁlterbank 2 If Rxx = p0 I and Ree = σv I then equalization for MIMO frequency selective Ree = p0 I − p0 Jd H∗ (p0 HH∗ + σv I)−1 HJd p0 2 channels Vijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d v Motivation 1 1 Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Block σv processing Filterbank framework For the case of symbol by symbol detection, Mutual N−1 information 1 1 Simulations IF (H) = log2 N p0 ∗ −1 J∗ Conclusion k =0 Jd (I + 2 H H) σv d k ,k

60. Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Observation Vijaya Krishna A, Shashank 1 1 V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Motivation σv Signal model 1 1 Block IB (H) = log2 N(P + LH − 1) p0 ∗ I+ 2H H processing σv P P Filterbank framework Mutual information Simulations Conclusion

61. Mutual Information with ﬁlterbank equalization for MIMO frequency selective channels Observation Vijaya Krishna A, Shashank 1 1 V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d Motivation σv Signal model 1 1 Block IB (H) = log2 N(P + LH − 1) p0 ∗ I+ 2H H processing σv P P Filterbank framework Mutual information Simulations Conclusion

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