MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Mutual Information with filterbankequalization for MIMO frequency selective

channels

Vijaya Krishna A Shashank V

Department of ECEP E S Institute of Technology, Bangalore

NCC 2011

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Outline

1 Motivation2 Signal model3 Block processing4 Filterbank framework5 Mutual information with filterbank equalization6 Conclusion

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Motivation

MIMO systems: Higher rate, more reliability

Frequency selectivity: Equalization required at receiver

Typically, block processing used:Zero padding or cyclic prefixing: Convert frequencyselective fading to flat fadingRedundancy of the order of channel length required

Lower data ratesAdditional processing required: coding, etc

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Motivation

MIMO systems: Higher rate, more reliability

Frequency selectivity: Equalization required at receiver

Typically, block processing used:Zero padding or cyclic prefixing: Convert frequencyselective fading to flat fadingRedundancy of the order of channel length required

Lower data ratesAdditional processing required: coding, etc

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Motivation

MIMO systems: Higher rate, more reliability

Frequency selectivity: Equalization required at receiver

Typically, block processing used:Zero padding or cyclic prefixing: Convert frequencyselective fading to flat fadingRedundancy of the order of channel length required

Lower data ratesAdditional processing required: coding, etc

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Motivation

MIMO systems: Higher rate, more reliability

Frequency selectivity: Equalization required at receiver

Typically, block processing used:Zero padding or cyclic prefixing: Convert frequencyselective fading to flat fadingRedundancy of the order of channel length required

Lower data ratesAdditional processing required: coding, etc

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank equalizers:Instead of converting to flat fading, view the channel asFIR filter

Equalization: Inverse filtering

By adding no/minimal redundancy, we can find FIRinverse filters

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank equalizers:Instead of converting to flat fading, view the channel asFIR filter

Equalization: Inverse filtering

By adding no/minimal redundancy, we can find FIRinverse filters

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank equalizers:Instead of converting to flat fading, view the channel asFIR filter

Equalization: Inverse filtering

By adding no/minimal redundancy, we can find FIRinverse filters

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information: Acheivable data rate

I(X ;Y ) = H(X )− H(X |Y )

Aim: Quantify data rate: Mutual information for Filterbank case

Our Contribution:1 Derivation of expression for MI with filterbank

equalization for the MMSE criterion2 MI expression for the case of symbol by symbol

detection

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information: Acheivable data rate

I(X ;Y ) = H(X )− H(X |Y )

Aim: Quantify data rate: Mutual information for Filterbank case

Our Contribution:1 Derivation of expression for MI with filterbank

equalization for the MMSE criterion2 MI expression for the case of symbol by symbol

detection

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information: Acheivable data rate

I(X ;Y ) = H(X )− H(X |Y )

Aim: Quantify data rate: Mutual information for Filterbank case

Our Contribution:1 Derivation of expression for MI with filterbank

equalization for the MMSE criterion2 MI expression for the case of symbol by symbol

detection

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Signal model

Consider M×N frequency selective LH tap MIMOchannelSignal model:

y [n] =LH−1∑k=0

H(k) x(n − k) + v(n)

Y(ejω) = H(ejω)X(ejω) + V(ejω)

Mutual information of channel:

I(H)M= I(X ;Y ) =

12πN

ˆ π

−πlog∣∣∣∣IN +

p0

σ2v

H∗(ejω)H(ejω)

∣∣∣∣dωDifficult to evaluate

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Signal model

Consider M×N frequency selective LH tap MIMOchannelSignal model:

y [n] =LH−1∑k=0

H(k) x(n − k) + v(n)

Y(ejω) = H(ejω)X(ejω) + V(ejω)

Mutual information of channel:

I(H)M= I(X ;Y ) =

12πN

ˆ π

−πlog∣∣∣∣IN +

p0

σ2v

H∗(ejω)H(ejω)

∣∣∣∣dωDifficult to evaluate

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Signal model

Consider M×N frequency selective LH tap MIMOchannelSignal model:

y [n] =LH−1∑k=0

H(k) x(n − k) + v(n)

Y(ejω) = H(ejω)X(ejω) + V(ejω)

Mutual information of channel:

I(H)M= I(X ;Y ) =

12πN

ˆ π

−πlog∣∣∣∣IN +

p0

σ2v

H∗(ejω)H(ejω)

∣∣∣∣dωDifficult to evaluate

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Signal model

Consider M×N frequency selective LH tap MIMOchannelSignal model:

y [n] =LH−1∑k=0

H(k) x(n − k) + v(n)

Y(ejω) = H(ejω)X(ejω) + V(ejω)

Mutual information of channel:

I(H)M= I(X ;Y ) =

12πN

ˆ π

−πlog∣∣∣∣IN +

p0

σ2v

H∗(ejω)H(ejω)

∣∣∣∣dωDifficult to evaluate

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Block processing

Block processing: Zero padding schemey(n) = HP x(n) + v(n)

HP =

H(0) . . . H(LH − 1) 0 . . . 0

0. . . . . . . . .

...

0. . . . . . . . .

......

. . . . . . . . ....

0 . . . H(0) · · · H(LH − 1)

M(P+LH -1) by NP Block Toeplitz matrixP: no of input symbols per blockx(n) = [xT (Pn), xT (Pn − 1), ....., xT (P(n − 1)− 1)]T

Results of flat fading channels can be used for blockprocessing

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Block processing

Block processing: Zero padding schemey(n) = HP x(n) + v(n)

HP =

H(0) . . . H(LH − 1) 0 . . . 0

0. . . . . . . . .

...

0. . . . . . . . .

......

. . . . . . . . ....

0 . . . H(0) · · · H(LH − 1)

M(P+LH -1) by NP Block Toeplitz matrixP: no of input symbols per blockx(n) = [xT (Pn), xT (Pn − 1), ....., xT (P(n − 1)− 1)]T

Results of flat fading channels can be used for blockprocessing

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Block processing

Block processing: Zero padding schemey(n) = HP x(n) + v(n)

HP =

H(0) . . . H(LH − 1) 0 . . . 0

0. . . . . . . . .

...

0. . . . . . . . .

......

. . . . . . . . ....

0 . . . H(0) · · · H(LH − 1)

M(P+LH -1) by NP Block Toeplitz matrixP: no of input symbols per blockx(n) = [xT (Pn), xT (Pn − 1), ....., xT (P(n − 1)− 1)]T

Results of flat fading channels can be used for blockprocessing

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

For flat fading channel with channel matrix H, mutualinformation:

I(H) = 1N

log2

∣∣∣∣I + P0

σ2vH∗H

∣∣∣∣Mutual information with zero padding:

IB(H) =1

N(P + LH − 1)log2

∣∣∣∣I + P0

σ2vH∗PHP

∣∣∣∣lim

P→∞IB(HP) = I(H)

Can be realized using joint ML detection at receiver

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

For flat fading channel with channel matrix H, mutualinformation:

I(H) = 1N

log2

∣∣∣∣I + P0

σ2vH∗H

∣∣∣∣Mutual information with zero padding:

IB(H) =1

N(P + LH − 1)log2

∣∣∣∣I + P0

σ2vH∗PHP

∣∣∣∣lim

P→∞IB(HP) = I(H)

Can be realized using joint ML detection at receiver

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

For flat fading channel with channel matrix H, mutualinformation:

I(H) = 1N

log2

∣∣∣∣I + P0

σ2vH∗H

∣∣∣∣Mutual information with zero padding:

IB(H) =1

N(P + LH − 1)log2

∣∣∣∣I + P0

σ2vH∗PHP

∣∣∣∣lim

P→∞IB(HP) = I(H)

Can be realized using joint ML detection at receiver

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Optionally,1. Successive interference cancellation (MMSE-SIC)

2. Eigenmode precoding

May not be feasible. Suboptimal MMSE with symbol bysymbol detection used.

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Optionally,1. Successive interference cancellation (MMSE-SIC)

2. Eigenmode precoding

May not be feasible. Suboptimal MMSE with symbol bysymbol detection used.

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Optionally,1. Successive interference cancellation (MMSE-SIC)

2. Eigenmode precoding

May not be feasible. Suboptimal MMSE with symbol bysymbol detection used.

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Symbol by symbol detection:

For the k th symbol, rate is

IBk ,MMSE =

1N(P + LH − 1)

log2

1[I + p0

σ2vH∗PHP

]−1

k ,k

Total rate is

IBMMSE =

1N(P + LH − 1)

MP−1∑k=0

IBk ,MMSE

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Symbol by symbol detection:

For the k th symbol, rate is

IBk ,MMSE =

1N(P + LH − 1)

log2

1[I + p0

σ2vH∗PHP

]−1

k ,k

Total rate is

IBMMSE =

1N(P + LH − 1)

MP−1∑k=0

IBk ,MMSE

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Symbol by symbol detection:

For the k th symbol, rate is

IBk ,MMSE =

1N(P + LH − 1)

log2

1[I + p0

σ2vH∗PHP

]−1

k ,k

Total rate is

IBMMSE =

1N(P + LH − 1)

MP−1∑k=0

IBk ,MMSE

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank framework

y(z) = H(z)x(z) + v(z)

y(n) = Hx(n) + v(n)

H =

H(0) . . . H(LH − 1) 0 . . . 0

0. . . . . . . . .

...

0. . . . . . . . .

......

. . . . . . . . ....

0 . . . H(0) · · · H(LH − 1)

MLF by N(LF +LH -1) block Toeplitz matrixLF : Length of FIR filter used for equalization

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank framework

z−d X(z) = F(z)Y(z)

x(n − d) = FHx(n) + Fv(n)

MMSE inverse: FMMSE = RxxJdH∗(HRx xH∗ + Rv v )−1

Jd = [0N×Nd IN×N 0N×N(LH +LF−d−2)]

Jd =

0 · · · 0 1 · · · 0 0 · · · 0...

. . ....

. . . . . . . . . . . . . . ....

0 · · · 0 0 · · · 1 0 · · · 0

If Rxx = I and Rv v = σ2

v I

FMMSE = JdH∗(HH∗ + σ2v I)−1

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank framework

z−d X(z) = F(z)Y(z)

x(n − d) = FHx(n) + Fv(n)

MMSE inverse: FMMSE = RxxJdH∗(HRx xH∗ + Rv v )−1

Jd = [0N×Nd IN×N 0N×N(LH +LF−d−2)]

Jd =

0 · · · 0 1 · · · 0 0 · · · 0...

. . ....

. . . . . . . . . . . . . . ....

0 · · · 0 0 · · · 1 0 · · · 0

If Rxx = I and Rv v = σ2

v I

FMMSE = JdH∗(HH∗ + σ2v I)−1

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Filterbank framework

z−d X(z) = F(z)Y(z)

x(n − d) = FHx(n) + Fv(n)

MMSE inverse: FMMSE = RxxJdH∗(HRx xH∗ + Rv v )−1

Jd = [0N×Nd IN×N 0N×N(LH +LF−d−2)]

Jd =

0 · · · 0 1 · · · 0 0 · · · 0...

. . ....

. . . . . . . . . . . . . . ....

0 · · · 0 0 · · · 1 0 · · · 0

If Rxx = I and Rv v = σ2

v I

FMMSE = JdH∗(HH∗ + σ2v I)−1

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

Idea is that error vector is orthogonal to the estimateand

X = AxyY + X⊥Y = X + E

X = X|Y = AxyY = RxyR−1yy Y

IF (H) = log2|Rxx ||Ree|

Theorem

IF (H) =1N

log2|Rxx |

|Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx |

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

Idea is that error vector is orthogonal to the estimateand

X = AxyY + X⊥Y = X + E

X = X|Y = AxyY = RxyR−1yy Y

IF (H) = log2|Rxx ||Ree|

Theorem

IF (H) =1N

log2|Rxx |

|Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx |

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Proof

I(X ;Y ) = h(X )− h(X |Y )

For the MMSE equalizer, h(X |Y ) = h(E), the entropy ofthe error vector

IF (H) = h(X )− h(E) =1N

log2|Rxx ||Ree|

Ree = E{xx∗} − E{xy∗}E{y y∗}E{yx∗}

Ree = Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Proof

If Rxx = p0I and Ree = σ2v I then

Ree = p0I − p0JdH∗(p0HH∗ + σ2v I)−1HJdp0

Using matrix inversion lemma,Ree = p0Jd(I +

p0σ2

vH∗H)−1J∗d

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Proof

If Rxx = p0I and Ree = σ2v I then

Ree = p0I − p0JdH∗(p0HH∗ + σ2v I)−1HJdp0

Using matrix inversion lemma,Ree = p0Jd(I +

p0σ2

vH∗H)−1J∗d

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

If Rxx = p0I and Ree = σ2v I then

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

If Rxx = p0I and Ree = σ2v I then

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Observation

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

IB(H) =1

N(P + LH − 1)log2

1∣∣∣∣(I + p0σ2

vH∗PHP

)−1∣∣∣∣

RemarkThe MI for filterbank equalization depends on thedeterminant of N by N submatrix of (I + p0

σ2vH∗H)−1. So we

can choose the delay so as to maximize MI

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Observation

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

IB(H) =1

N(P + LH − 1)log2

1∣∣∣∣(I + p0σ2

vH∗PHP

)−1∣∣∣∣

RemarkThe MI for filterbank equalization depends on thedeterminant of N by N submatrix of (I + p0

σ2vH∗H)−1. So we

can choose the delay so as to maximize MI

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion Choose submatrix of (I + p0σ2

vH∗H)−1 with lowest

determinant

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Simulations

4×3 Rayleigh fading channels of length LH = 8Block processing case: no of inputs symbols per blockP = 20Filterbank case: Length of equalizer LF = 21

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Simulations

Figure: Comparison between block processing and Filterbankequalizers

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Simulations

Figure: MI with variation in delay. SNR=15 dB

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Simulations

Figure: MI for different LF ’s. SNR=15 dB

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Conclusion

Filterbank equalization achieves significantly higherinformation rate when compared to block processing

We have the flexibility of choosing the delay so as tomaximize MI

Disadvantage of this scheme: Processing complexity,similar to BP

Future: Mutual information using zero forcing equalizers

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Conclusion

Filterbank equalization achieves significantly higherinformation rate when compared to block processing

We have the flexibility of choosing the delay so as tomaximize MI

Disadvantage of this scheme: Processing complexity,similar to BP

Future: Mutual information using zero forcing equalizers

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Conclusion

Filterbank equalization achieves significantly higherinformation rate when compared to block processing

We have the flexibility of choosing the delay so as tomaximize MI

Disadvantage of this scheme: Processing complexity,similar to BP

Future: Mutual information using zero forcing equalizers

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Conclusion

Filterbank equalization achieves significantly higherinformation rate when compared to block processing

We have the flexibility of choosing the delay so as tomaximize MI

Disadvantage of this scheme: Processing complexity,similar to BP

Future: Mutual information using zero forcing equalizers

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

References

Vijaya Krishna. A, A filterbank precoding framework forMIMO frequency selective channels, PhD thesis, IndianInstitute of Science, 2006.

G. D. Forney Jr., “Shannon meets Wiener II: On MMSEestimation in successive decoding schemes,” In Proc.Allerton Conf., Sep. 2004.(http://arxiv.org/abs/cs/0409011)

X. Zhang and S.-Y. Kung, “Capacity analysis for paralleland sequential MIMO equalizers,” IEEE Trans on SignalProcessing, vol. 51, pp. 2989- 3002, Nov. 2003.

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

References

P. P. Vaidyanathan, Multirate systems and filter banks,Englewood Cliffs, NJ: Prentice-Hall, 1993.

Vijaya Krishna. A, K. V. S. Hari, ”Filterbank precoding forFIR equalization in high rate MIMO communications,”IEEE Trans. Signal Processing, vol. 54, No. 5, pp.1645-1652, May 2006.

A. Scaglione, S. Barbarossa, and G. B, Giannakis,“Filterbank transceivers optimizing information rate inblock transmissions over dispersive channels,” IEEETrans. Info. Theory, Vol. 45, pp. 1019-1032, Apr. 1999.

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

THANK YOU

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

I(X ;Y ) = h(X )− h(X |Y )

For the MMSE equalizer, h(X |Y ) = h(E), the entropy ofthe error vector

IF (H) = h(X )− h(E) =1N

log2|Rxx ||Ree|

Ree = E{xx∗} − E{xy∗}E{y y∗}E{yx∗}

Ree = Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

I(X ;Y ) = h(X )− h(X |Y )

For the MMSE equalizer, h(X |Y ) = h(E), the entropy ofthe error vector

IF (H) = h(X )− h(E) =1N

log2|Rxx ||Ree|

Ree = E{xx∗} − E{xy∗}E{y y∗}E{yx∗}

Ree = Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

I(X ;Y ) = h(X )− h(X |Y )

For the MMSE equalizer, h(X |Y ) = h(E), the entropy ofthe error vector

IF (H) = h(X )− h(E) =1N

log2|Rxx ||Ree|

Ree = E{xx∗} − E{xy∗}E{y y∗}E{yx∗}

Ree = Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

I(X ;Y ) = h(X )− h(X |Y )

For the MMSE equalizer, h(X |Y ) = h(E), the entropy ofthe error vector

IF (H) = h(X )− h(E) =1N

log2|Rxx ||Ree|

Ree = E{xx∗} − E{xy∗}E{y y∗}E{yx∗}

Ree = Rxx − RxxJdH∗(HRx xH∗ + Rv v )−1HJdRxx

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

If Rxx = p0I and Ree = σ2v I then

Ree = p0I − p0JdH∗(p0HH∗ + σ2v I)−1HJdp0

Using matrix inversion lemma,Ree = p0Jd(I +

p0σ2

vH∗H)−1J∗d

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Mutual information

If Rxx = p0I and Ree = σ2v I then

Ree = p0I − p0JdH∗(p0HH∗ + σ2v I)−1HJdp0

Using matrix inversion lemma,Ree = p0Jd(I +

p0σ2

vH∗H)−1J∗d

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

For the case of symbol by symbol detection,

IF (H) =1N

N−1∑k=0

log21[

Jd(I +p0σ2

vH∗H)−1J∗d

]k ,k

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Observation

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

IB(H) =1

N(P + LH − 1)log2

1∣∣∣I + p0σ2

vH∗PHP

∣∣∣

MutualInformation

with filterbankequalization

for MIMOfrequencyselectivechannels

Vijaya KrishnaA, Shashank

V

Motivation

Signal model

Blockprocessing

Filterbankframework

Mutualinformation

Simulations

Conclusion

Observation

IF (H) =1N

log21

|Jd(I +p0σ2

vH∗H)−1J∗d |

IB(H) =1

N(P + LH − 1)log2

1∣∣∣I + p0σ2

vH∗PHP

∣∣∣