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Presentation on Mutual information using filterbank equalization for MIMO frequency selective channels presented at the National Conference on Communications 2011 held at IISc, Bangalore

Presentation on Mutual information using filterbank equalization for MIMO frequency selective channels presented at the National Conference on Communications 2011 held at IISc, Bangalore

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  • 1. Mutual Informationwith filterbank equalization for MIMO frequency selective Mutual Information with filterbank channelsVijaya 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 Informationwith filterbank equalization for MIMO frequency selective channels 1 MotivationVijaya Krishna A, Shashank V 2 Signal model 3 Block processing 4 Filterbank framework 5 Mutual information with filterbank equalization 6 Conclusion
  • 3. Motivation Mutual Informationwith filterbank equalization for MIMO frequency MIMO systems: Higher rate, more reliability selective channelsVijaya Krishna Frequency selectivity: Equalization required at receiver A, Shashank V Typically, block processing used:Motivation Zero padding or cyclic prefixing: Convert frequencySignal model selective fading to flat fadingBlockprocessing Redundancy of the order of channel length requiredFilterbankframeworkMutual Lower data ratesinformationSimulations Additional processing required: coding, etcConclusion
  • 4. Motivation Mutual Informationwith filterbank equalization for MIMO frequency MIMO systems: Higher rate, more reliability selective channelsVijaya Krishna Frequency selectivity: Equalization required at receiver A, Shashank V Typically, block processing used:Motivation Zero padding or cyclic prefixing: Convert frequencySignal model selective fading to flat fadingBlockprocessing Redundancy of the order of channel length requiredFilterbankframeworkMutual Lower data ratesinformationSimulations Additional processing required: coding, etcConclusion
  • 5. Motivation Mutual Informationwith filterbank equalization for MIMO frequency MIMO systems: Higher rate, more reliability selective channelsVijaya Krishna Frequency selectivity: Equalization required at receiver A, Shashank V Typically, block processing used:Motivation Zero padding or cyclic prefixing: Convert frequencySignal model selective fading to flat fadingBlockprocessing Redundancy of the order of channel length requiredFilterbankframeworkMutual Lower data ratesinformationSimulations Additional processing required: coding, etcConclusion
  • 6. Motivation Mutual Informationwith filterbank equalization for MIMO frequency MIMO systems: Higher rate, more reliability selective channelsVijaya Krishna Frequency selectivity: Equalization required at receiver A, Shashank V Typically, block processing used:Motivation Zero padding or cyclic prefixing: Convert frequencySignal model selective fading to flat fadingBlockprocessing Redundancy of the order of channel length requiredFilterbankframeworkMutual Lower data ratesinformationSimulations Additional processing required: coding, etcConclusion
  • 7. Mutual Informationwith filterbank equalization Filterbank equalizers: for MIMO frequency selective Instead of converting to flat fading, view the channel as channels FIR filterVijaya Krishna A, Shashank V Equalization: Inverse filteringMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformation By adding no/minimal redundancy, we can find FIRSimulations inverse filtersConclusion
  • 8. Mutual Informationwith filterbank equalization Filterbank equalizers: for MIMO frequency selective Instead of converting to flat fading, view the channel as channels FIR filterVijaya Krishna A, Shashank V Equalization: Inverse filteringMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformation By adding no/minimal redundancy, we can find FIRSimulations inverse filtersConclusion
  • 9. Mutual Informationwith filterbank equalization Filterbank equalizers: for MIMO frequency selective Instead of converting to flat fading, view the channel as channels FIR filterVijaya Krishna A, Shashank V Equalization: Inverse filteringMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformation By adding no/minimal redundancy, we can find FIRSimulations inverse filtersConclusion
  • 10. Mutual Informationwith 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 FilterMotivation bank caseSignal modelBlock Our Contribution:processing 1 Derivation of expression for MI with filterbankFilterbankframework equalization for the MMSE criterionMutual 2 MI expression for the case of symbol by symbolinformation detectionSimulationsConclusion
  • 11. Mutual Informationwith 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 FilterMotivation bank caseSignal modelBlock Our Contribution:processing 1 Derivation of expression for MI with filterbankFilterbankframework equalization for the MMSE criterionMutual 2 MI expression for the case of symbol by symbolinformation detectionSimulationsConclusion
  • 12. Mutual Informationwith 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 FilterMotivation bank caseSignal modelBlock Our Contribution:processing 1 Derivation of expression for MI with filterbankFilterbankframework equalization for the MMSE criterionMutual 2 MI expression for the case of symbol by symbolinformation detectionSimulationsConclusion
  • 13. Signal model Mutual Informationwith 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 =0Signal modelBlock Y(ejω ) = H(ejω )X(ejω ) + V(ejω )processingFilterbank Mutual information of channel:framework ˆ π 1 p0 log IN + 2 H∗ (ejω )H(ejω ) dωMutualinformation I(H) = I(X ; Y ) = 2πN −π σvSimulationsConclusion Difficult to evaluate
  • 14. Signal model Mutual Informationwith 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 =0Signal modelBlock Y(ejω ) = H(ejω )X(ejω ) + V(ejω )processingFilterbank Mutual information of channel:framework ˆ π 1 p0 log IN + 2 H∗ (ejω )H(ejω ) dωMutualinformation I(H) = I(X ; Y ) = 2πN −π σvSimulationsConclusion Difficult to evaluate
  • 15. Signal model Mutual Informationwith 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 =0Signal modelBlock Y(ejω ) = H(ejω )X(ejω ) + V(ejω )processingFilterbank Mutual information of channel:framework ˆ π 1 p0 log IN + 2 H∗ (ejω )H(ejω ) dωMutualinformation I(H) = I(X ; Y ) = 2πN −π σvSimulationsConclusion Difficult to evaluate
  • 16. Signal model Mutual Informationwith 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 =0Signal modelBlock Y(ejω ) = H(ejω )X(ejω ) + V(ejω )processingFilterbank Mutual information of channel:framework ˆ π 1 p0 log IN + 2 H∗ (ejω )H(ejω ) dωMutualinformation I(H) = I(X ; Y ) = 2πN −π σvSimulationsConclusion Difficult to evaluate
  • 17. Block processing Mutual Information Block processing: Zero padding schemewith filterbank equalization ˜ ˜ ˜ y (n) = HP x (n) + v (n) for MIMO frequency selective channels   H(0) . . . H(LH − 1) 0 . . . 0Vijaya Krishna A, Shashank  .. .. .. . .  V   0 . . . .   HP =   .. .. .. . . Motivation 0 . . . .  . .  Signal model  . .. .. .. . . . . Block  . . processing 0 ... H(0) · · · H(LH − 1)Filterbankframework M(P+LH -1) by NP Block Toeplitz matrixMutualinformation P: no of input symbols per blockSimulations 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
  • 18. Block processing Mutual Information Block processing: Zero padding schemewith filterbank equalization ˜ ˜ ˜ y (n) = HP x (n) + v (n) for MIMO frequency selective channels   H(0) . . . H(LH − 1) 0 . . . 0Vijaya Krishna A, Shashank  .. .. .. . .  V   0 . . . .   HP =   .. .. .. . . Motivation 0 . . . .  . .  Signal model  . .. .. .. . . . . Block  . . processing 0 ... H(0) · · · H(LH − 1)Filterbankframework M(P+LH -1) by NP Block Toeplitz matrixMutualinformation P: no of input symbols per blockSimulations 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
  • 19. Block processing Mutual Information Block processing: Zero padding schemewith filterbank equalization ˜ ˜ ˜ y (n) = HP x (n) + v (n) for MIMO frequency selective channels   H(0) . . . H(LH − 1) 0 . . . 0Vijaya Krishna A, Shashank  .. .. .. . .  V   0 . . . .   HP =   .. .. .. . . Motivation 0 . . . .  . .  Signal model  . .. .. .. . . . . Block  . . processing 0 ... H(0) · · · H(LH − 1)Filterbankframework M(P+LH -1) by NP Block Toeplitz matrixMutualinformation P: no of input symbols per blockSimulations 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 Informationwith filterbank equalization For flat fading channel with channel matrix H, mutual for MIMO frequency information: selective channels 1 P0Vijaya Krishna I(H) = log2 I + 2 H∗ H A, Shashank N σv VMotivation Mutual information with zero padding:Signal model 1 P0 ∗Block IB (H) = log2 I + 2 HP HPprocessing N(P + LH − 1) σvFilterbankframework lim IB (HP ) = I(H)Mutual P→∞informationSimulations Can be realized using joint ML detection at receiverConclusion
  • 21. Mutual Informationwith filterbank equalization For flat fading channel with channel matrix H, mutual for MIMO frequency information: selective channels 1 P0Vijaya Krishna I(H) = log2 I + 2 H∗ H A, Shashank N σv VMotivation Mutual information with zero padding:Signal model 1 P0 ∗Block IB (H) = log2 I + 2 HP HPprocessing N(P + LH − 1) σvFilterbankframework lim IB (HP ) = I(H)Mutual P→∞informationSimulations Can be realized using joint ML detection at receiverConclusion
  • 22. Mutual Informationwith filterbank equalization For flat fading channel with channel matrix H, mutual for MIMO frequency information: selective channels 1 P0Vijaya Krishna I(H) = log2 I + 2 H∗ H A, Shashank N σv VMotivation Mutual information with zero padding:Signal model 1 P0 ∗Block IB (H) = log2 I + 2 HP HPprocessing N(P + LH − 1) σvFilterbankframework lim IB (HP ) = I(H)Mutual P→∞informationSimulations Can be realized using joint ML detection at receiverConclusion
  • 23. Mutual Informationwith filterbank equalization for MIMO frequency selective channels Optionally,Vijaya Krishna A, Shashank 1. Successive interference cancellation (MMSE-SIC) VMotivation 2. Eigenmode precodingSignal modelBlockprocessing May not be feasible. Suboptimal MMSE with symbol byFilterbank symbol detection used.frameworkMutualinformationSimulationsConclusion
  • 24. Mutual Informationwith filterbank equalization for MIMO frequency selective channels Optionally,Vijaya Krishna A, Shashank 1. Successive interference cancellation (MMSE-SIC) VMotivation 2. Eigenmode precodingSignal modelBlockprocessing May not be feasible. Suboptimal MMSE with symbol byFilterbank symbol detection used.frameworkMutualinformationSimulationsConclusion
  • 25. Mutual Informationwith filterbank equalization for MIMO frequency selective channels Optionally,Vijaya Krishna A, Shashank 1. Successive interference cancellation (MMSE-SIC) VMotivation 2. Eigenmode precodingSignal modelBlockprocessing May not be feasible. Suboptimal MMSE with symbol byFilterbank symbol detection used.frameworkMutualinformationSimulationsConclusion
  • 26. Mutual Informationwith filterbank equalization for MIMO Symbol by symbol detection: frequency selective channels For the k th symbol, rate isVijaya 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 ,kBlockprocessingFilterbank Total rate isframework MP−1Mutual B 1 Binformation IMMSE = Ik ,MMSESimulations N(P + LH − 1) k =0Conclusion
  • 27. Mutual Informationwith filterbank equalization for MIMO Symbol by symbol detection: frequency selective channels For the k th symbol, rate isVijaya 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 ,kBlockprocessingFilterbank Total rate isframework MP−1Mutual B 1 Binformation IMMSE = Ik ,MMSESimulations N(P + LH − 1) k =0Conclusion
  • 28. Mutual Informationwith filterbank equalization for MIMO Symbol by symbol detection: frequency selective channels For the k th symbol, rate isVijaya 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 ,kBlockprocessingFilterbank Total rate isframework MP−1Mutual B 1 Binformation IMMSE = Ik ,MMSESimulations N(P + LH − 1) k =0Conclusion
  • 29. Filterbank framework Mutual Informationwith 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 . . . 0Motivation  .. .. .. . . Signal model   0 . . . .  Block H=  .. .. .. . . processing 0 . . . .  . .  Filterbank  . .. .. .. . . . . framework  . . Mutual 0 ... H(0) · · · H(LH − 1)informationSimulations MLF by N(LF +LH -1) block Toeplitz matrixConclusion LF : Length of FIR filter used for equalization
  • 30. Filterbank framework Mutual Informationwith 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 modelBlock  processing 0 ··· 0 1 ··· 0 0 ··· 0  . .. . .. .. .. .. .. . Filterbankframework Jd =  . . . . . . . . . . .  .Mutual 0 ··· 0 0 ··· 1 0 ··· 0informationSimulations 2 If Rxx = I and Rv v = σv IConclusion ¯¯ FMMSE = Jd H∗ (HH∗ + σv I)−1 2
  • 31. Filterbank framework Mutual Informationwith 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 modelBlock  processing 0 ··· 0 1 ··· 0 0 ··· 0  . .. . .. .. .. .. .. . Filterbankframework Jd =  . . . . . . . . . . .  .Mutual 0 ··· 0 0 ··· 1 0 ··· 0informationSimulations 2 If Rxx = I and Rv v = σv IConclusion ¯¯ FMMSE = Jd H∗ (HH∗ + σv I)−1 2
  • 32. Filterbank framework Mutual Informationwith 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 modelBlock  processing 0 ··· 0 1 ··· 0 0 ··· 0  . .. . .. .. .. .. .. . Filterbankframework Jd =  . . . . . . . . . . .  .Mutual 0 ··· 0 0 ··· 1 0 ··· 0informationSimulations 2 If Rxx = I and Rv v = σv IConclusion ¯¯ FMMSE = Jd H∗ (HH∗ + σv I)−1 2
  • 33. Mutual information Mutual Informationwith 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 channelsVijaya Krishna A, Shashank V ˆ −1 X = X|Y = Axy Y = Rxy Ryy YMotivationSignal model |Rxx |Block IF (H) = log2processing |Ree |FilterbankframeworkMutual TheoreminformationSimulations 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 Informationwith 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 channelsVijaya Krishna A, Shashank V ˆ −1 X = X|Y = Axy Y = Rxy Ryy YMotivationSignal model |Rxx |Block IF (H) = log2processing |Ree |FilterbankframeworkMutual TheoreminformationSimulations 1 |Rxx |Conclusion IF (H) = log2 ∗ (HR H∗ + R )−1 HJ R | N |Rxx − Rxx Jd H ¯¯ xx ¯¯ vv d xx
  • 35. Proof Mutual Informationwith filterbank equalization for MIMO frequency selective I(X ; Y ) = h(X ) − h(X |Y ) channelsVijaya Krishna A, Shashank For the MMSE equalizer, h(X |Y ) = h(E), the entropy of V the error vectorMotivation 1 |Rxx | IF (H) = h(X ) − h(E) = log2Signal model N |Ree |BlockprocessingFilterbank Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ } ˜ ˜˜ ˜frameworkMutualinformation Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯SimulationsConclusion
  • 36. Proof Mutual Informationwith 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 channelsVijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d vMotivation 1 1Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dBlock σvprocessingFilterbankframework For the case of symbol by symbol detection,Mutual N−1information 1 1Simulations IF (H) = log2 N p0 ∗ −1 J∗Conclusion k =0 Jd (I + 2 H H) σv d k ,k
  • 37. Proof Mutual Informationwith 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 channelsVijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d vMotivation 1 1Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dBlock σvprocessingFilterbankframework For the case of symbol by symbol detection,Mutual N−1information 1 1Simulations IF (H) = log2 N p0 ∗ −1 J∗Conclusion k =0 Jd (I + 2 H H) σv d k ,k
  • 38. Mutual Informationwith filterbank equalization for MIMO 2 If Rxx = p0 I and Ree = σv I then frequency selective channelsVijaya Krishna 1 1 A, Shashank V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d σvMotivationSignal modelBlock For the case of symbol by symbol detection,processing N−1Filterbank 1 1framework IF (H) = log2 N p0 ∗ −1 J∗Mutual k =0 Jd (I + 2 H H) σv d k ,kinformationSimulationsConclusion
  • 39. Mutual Informationwith filterbank equalization for MIMO 2 If Rxx = p0 I and Ree = σv I then frequency selective channelsVijaya Krishna 1 1 A, Shashank V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) d σvMotivationSignal modelBlock For the case of symbol by symbol detection,processing N−1Filterbank 1 1framework IF (H) = log2 N p0 ∗ −1 J∗Mutual k =0 Jd (I + 2 H H) σv d k ,kinformationSimulationsConclusion
  • 40. Mutual Informationwith filterbank equalization Observation for MIMO frequency 1 1 selective IF (H) = log2 p0 ∗ −1 J∗ | channels N |Jd (I + 2 H H) dVijaya Krishna σv A, Shashank V 1 1 IB (H) = log2 −1Motivation N(P + LH − 1) p0 ∗ I+ 2H H σv P PSignal modelBlockprocessingFilterbank Remarkframework The MI for filterbank equalization depends on theMutual pinformation determinant of N by N submatrix of (I + σ0 H∗ H)−1 . So we 2 vSimulations can choose the delay so as to maximize MIConclusion
  • 41. Mutual Informationwith filterbank equalization Observation for MIMO frequency 1 1 selective IF (H) = log2 p0 ∗ −1 J∗ | channels N |Jd (I + 2 H H) dVijaya Krishna σv A, Shashank V 1 1 IB (H) = log2 −1Motivation N(P + LH − 1) p0 ∗ I+ 2H H σv P PSignal modelBlockprocessingFilterbank Remarkframework The MI for filterbank equalization depends on theMutual pinformation determinant of N by N submatrix of (I + σ0 H∗ H)−1 . So we 2 vSimulations can choose the delay so as to maximize MIConclusion
  • 42. Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion p0 ∗ −1 Choose submatrix of (I + 2 H H) σv with lowest determinant
  • 43. Simulations Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank 4×3 Rayleigh fading channels of length LH = 8 V Block processing case: no of inputs symbols per blockMotivation P = 20Signal modelBlock Filterbank case: Length of equalizer LF = 21processingFilterbankframeworkMutualinformationSimulationsConclusion
  • 44. Simulations Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion Figure: Comparison between block processing and Filterbank equalizers
  • 45. Simulations Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion Figure: MI with variation in delay. SNR=15 dB
  • 46. Simulations Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivationSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion Figure: MI for different LF ’s. SNR=15 dB
  • 47. Conclusion Mutual Informationwith filterbank equalization for MIMO frequency selective Filterbank equalization achieves significantly higher channels information rate when compared to block processingVijaya Krishna A, Shashank V We have the flexibility of choosing the delay so as toMotivation maximize MISignal modelBlock Disadvantage of this scheme: Processing complexity,processingFilterbank similar to BPframeworkMutual Future: Mutual information using zero forcing equalizersinformationSimulationsConclusion
  • 48. Conclusion Mutual Informationwith filterbank equalization for MIMO frequency selective Filterbank equalization achieves significantly higher channels information rate when compared to block processingVijaya Krishna A, Shashank V We have the flexibility of choosing the delay so as toMotivation maximize MISignal modelBlock Disadvantage of this scheme: Processing complexity,processingFilterbank similar to BPframeworkMutual Future: Mutual information using zero forcing equalizersinformationSimulationsConclusion
  • 49. Conclusion Mutual Informationwith filterbank equalization for MIMO frequency selective Filterbank equalization achieves significantly higher channels information rate when compared to block processingVijaya Krishna A, Shashank V We have the flexibility of choosing the delay so as toMotivation maximize MISignal modelBlock Disadvantage of this scheme: Processing complexity,processingFilterbank similar to BPframeworkMutual Future: Mutual information using zero forcing equalizersinformationSimulationsConclusion
  • 50. Conclusion Mutual Informationwith filterbank equalization for MIMO frequency selective Filterbank equalization achieves significantly higher channels information rate when compared to block processingVijaya Krishna A, Shashank V We have the flexibility of choosing the delay so as toMotivation maximize MISignal modelBlock Disadvantage of this scheme: Processing complexity,processingFilterbank similar to BPframeworkMutual Future: Mutual information using zero forcing equalizersinformationSimulationsConclusion
  • 51. References Mutual Informationwith filterbank equalization for MIMO frequency Vijaya Krishna. A, A filterbank precoding framework for selective channels MIMO frequency selective channels, PhD thesis, IndianVijaya Krishna Institute of Science, 2006. A, Shashank V G. D. Forney Jr., “Shannon meets Wiener II: On MMSEMotivation estimation in successive decoding schemes,” In Proc.Signal model Allerton Conf., Sep. 2004.Blockprocessing (http://arxiv.org/abs/cs/0409011)Filterbankframework X. Zhang and S.-Y. Kung, “Capacity analysis for parallelMutual and sequential MIMO equalizers,” IEEE Trans on Signalinformation Processing, vol. 51, pp. 2989- 3002, Nov. 2003.SimulationsConclusion
  • 52. References Mutual Informationwith 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.Blockprocessing A. Scaglione, S. Barbarossa, and G. B, Giannakis,Filterbankframework “Filterbank transceivers optimizing information rate inMutual block transmissions over dispersive channels,” IEEEinformation Trans. Info. Theory, Vol. 45, pp. 1019-1032, Apr. 1999.SimulationsConclusion
  • 53. Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivation THANK YOUSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion
  • 54. Mutual information Mutual Informationwith filterbank equalization for MIMO frequency I(X ; Y ) = h(X ) − h(X |Y ) selective channelsVijaya Krishna For the MMSE equalizer, h(X |Y ) = h(E), the entropy of A, Shashank V the error vectorMotivationSignal model 1 |Rxx | IF (H) = h(X ) − h(E) = log2Block N |Ree |processing Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ }Filterbankframework ˜ ˜˜ ˜MutualinformationSimulations Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯Conclusion
  • 55. Mutual information Mutual Informationwith filterbank equalization for MIMO frequency I(X ; Y ) = h(X ) − h(X |Y ) selective channelsVijaya Krishna For the MMSE equalizer, h(X |Y ) = h(E), the entropy of A, Shashank V the error vectorMotivationSignal model 1 |Rxx | IF (H) = h(X ) − h(E) = log2Block N |Ree |processing Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ }Filterbankframework ˜ ˜˜ ˜MutualinformationSimulations Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯Conclusion
  • 56. Mutual information Mutual Informationwith filterbank equalization for MIMO frequency I(X ; Y ) = h(X ) − h(X |Y ) selective channelsVijaya Krishna For the MMSE equalizer, h(X |Y ) = h(E), the entropy of A, Shashank V the error vectorMotivationSignal model 1 |Rxx | IF (H) = h(X ) − h(E) = log2Block N |Ree |processing Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ }Filterbankframework ˜ ˜˜ ˜MutualinformationSimulations Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯Conclusion
  • 57. Mutual information Mutual Informationwith filterbank equalization for MIMO frequency I(X ; Y ) = h(X ) − h(X |Y ) selective channelsVijaya Krishna For the MMSE equalizer, h(X |Y ) = h(E), the entropy of A, Shashank V the error vectorMotivationSignal model 1 |Rxx | IF (H) = h(X ) − h(E) = log2Block N |Ree |processing Ree = E{xx ∗ } − E{x y ∗ }E{y y ∗ }E{y x ∗ }Filterbankframework ˜ ˜˜ ˜MutualinformationSimulations Ree = Rxx − Rxx Jd H∗ (HRx x H∗ + Rv v )−1 HJd Rxx ¯¯ ¯¯Conclusion
  • 58. Mutual information Mutual Informationwith 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 channelsVijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d vMotivation 1 1Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dBlock σvprocessingFilterbankframework For the case of symbol by symbol detection,Mutual N−1information 1 1Simulations IF (H) = log2 N p0 ∗ −1 J∗Conclusion k =0 Jd (I + 2 H H) σv d k ,k
  • 59. Mutual information Mutual Informationwith 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 channelsVijaya Krishna A, Shashank Using matrix inversion lemma, p V Ree = p0 Jd (I + σ0 H∗ H)−1 J∗ 2 d vMotivation 1 1Signal model IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dBlock σvprocessingFilterbankframework For the case of symbol by symbol detection,Mutual N−1information 1 1Simulations IF (H) = log2 N p0 ∗ −1 J∗Conclusion k =0 Jd (I + 2 H H) σv d k ,k
  • 60. Mutual Informationwith filterbank equalization for MIMO frequency selective channels ObservationVijaya Krishna A, Shashank 1 1 V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dMotivation σvSignal model 1 1Block IB (H) = log2 N(P + LH − 1) p0 ∗ I+ 2H Hprocessing σv P PFilterbankframeworkMutualinformationSimulationsConclusion
  • 61. Mutual Informationwith filterbank equalization for MIMO frequency selective channels ObservationVijaya Krishna A, Shashank 1 1 V IF (H) = log2 p0 ∗ −1 J∗ | N |Jd (I + 2 H H) dMotivation σvSignal model 1 1Block IB (H) = log2 N(P + LH − 1) p0 ∗ I+ 2H Hprocessing σv P PFilterbankframeworkMutualinformationSimulationsConclusion

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