Mutual Informationwith filterbank equalization  for MIMO  frequency   selective         Mutual Information with filterbank  ...
Outline    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                   1  ...
Motivation    Mutual Informationwith filterbank equalization  for MIMO  frequency                     MIMO systems: Higher ...
Motivation    Mutual Informationwith filterbank equalization  for MIMO  frequency                     MIMO systems: Higher ...
Motivation    Mutual Informationwith filterbank equalization  for MIMO  frequency                     MIMO systems: Higher ...
Motivation    Mutual Informationwith filterbank equalization  for MIMO  frequency                     MIMO systems: Higher ...
Mutual Informationwith filterbank equalization    Filterbank equalizers:  for MIMO  frequency   selective                 I...
Mutual Informationwith filterbank equalization    Filterbank equalizers:  for MIMO  frequency   selective                 I...
Mutual Informationwith filterbank equalization    Filterbank equalizers:  for MIMO  frequency   selective                 I...
Mutual Informationwith filterbank equalization  for MIMO  frequency      Mutual information: Acheivable data rate   selecti...
Mutual Informationwith filterbank equalization  for MIMO  frequency      Mutual information: Acheivable data rate   selecti...
Mutual Informationwith filterbank equalization  for MIMO  frequency      Mutual information: Acheivable data rate   selecti...
Signal model    Mutual Informationwith filterbank equalization                     Consider M×N frequency selective LH tap ...
Signal model    Mutual Informationwith filterbank equalization                     Consider M×N frequency selective LH tap ...
Signal model    Mutual Informationwith filterbank equalization                     Consider M×N frequency selective LH tap ...
Signal model    Mutual Informationwith filterbank equalization                     Consider M×N frequency selective LH tap ...
Block processing    Mutual Information         Block processing: Zero padding schemewith filterbank equalization           ...
Block processing    Mutual Information         Block processing: Zero padding schemewith filterbank equalization           ...
Block processing    Mutual Information         Block processing: Zero padding schemewith filterbank equalization           ...
Mutual Informationwith filterbank equalization    For flat fading channel with channel matrix H, mutual  for MIMO  frequency...
Mutual Informationwith filterbank equalization    For flat fading channel with channel matrix H, mutual  for MIMO  frequency...
Mutual Informationwith filterbank equalization    For flat fading channel with channel matrix H, mutual  for MIMO  frequency...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                 Optionally,Vijay...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                 Optionally,Vijay...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                 Optionally,Vijay...
Mutual Informationwith filterbank equalization  for MIMO                 Symbol by symbol detection:  frequency   selective...
Mutual Informationwith filterbank equalization  for MIMO                 Symbol by symbol detection:  frequency   selective...
Mutual Informationwith filterbank equalization  for MIMO                 Symbol by symbol detection:  frequency   selective...
Filterbank framework    Mutual Informationwith filterbank equalization                      y(z) = H(z)x(z) + v(z)  for MIM...
Filterbank framework    Mutual Informationwith filterbank equalization                                     z −d X(z) = F(z)...
Filterbank framework    Mutual Informationwith filterbank equalization                                     z −d X(z) = F(z)...
Filterbank framework    Mutual Informationwith filterbank equalization                                     z −d X(z) = F(z)...
Mutual information    Mutual Informationwith filterbank       Idea is that error vector is orthogonal to the estimate equal...
Mutual information    Mutual Informationwith filterbank       Idea is that error vector is orthogonal to the estimate equal...
Proof    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective                                 I(...
Proof    Mutual Informationwith filterbank                                2                     If Rxx = p0 I and Ree = σv ...
Proof    Mutual Informationwith filterbank                                2                     If Rxx = p0 I and Ree = σv ...
Mutual Informationwith filterbank equalization  for MIMO                                    2                     If Rxx = ...
Mutual Informationwith filterbank equalization  for MIMO                                    2                     If Rxx = ...
Mutual Informationwith filterbank equalization    Observation  for MIMO  frequency                            1            ...
Mutual Informationwith filterbank equalization    Observation  for MIMO  frequency                            1            ...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, Shashank       ...
Simulations    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, ...
Simulations    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, ...
Simulations    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, ...
Simulations    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, ...
Conclusion    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective        Filterbank equalizatio...
Conclusion    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective        Filterbank equalizatio...
Conclusion    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective        Filterbank equalizatio...
Conclusion    Mutual Informationwith filterbank equalization  for MIMO  frequency   selective        Filterbank equalizatio...
References    Mutual Informationwith filterbank equalization  for MIMO  frequency         Vijaya Krishna. A, A filterbank pr...
References    Mutual Informationwith filterbank equalization  for MIMO  frequency         P. P. Vaidyanathan, Multirate sys...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channelsVijaya Krishna A, Shashank       ...
Mutual information    Mutual Informationwith filterbank equalization  for MIMO  frequency                      I(X ; Y ) = ...
Mutual information    Mutual Informationwith filterbank equalization  for MIMO  frequency                      I(X ; Y ) = ...
Mutual information    Mutual Informationwith filterbank equalization  for MIMO  frequency                      I(X ; Y ) = ...
Mutual information    Mutual Informationwith filterbank equalization  for MIMO  frequency                      I(X ; Y ) = ...
Mutual information    Mutual Informationwith filterbank                                2                     If Rxx = p0 I ...
Mutual information    Mutual Informationwith filterbank                                2                     If Rxx = p0 I ...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                 ObservationVijay...
Mutual Informationwith filterbank equalization  for MIMO  frequency   selective   channels                 ObservationVijay...
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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

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NCC 2011 presentation

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 53. Mutual Informationwith filterbank equalization for MIMO frequency selective channelsVijaya Krishna A, Shashank VMotivation THANK YOUSignal modelBlockprocessingFilterbankframeworkMutualinformationSimulationsConclusion
  54. 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. 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. 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. 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. 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. 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. 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. 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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