On Linear Frequency Domain Turbo-Equalization Of …trsys.faculty.jacobs-university.de/turbo/presentations/...1/21 Context Volterra channel model Frequency domain turbo-equalizer SummaryResults

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Context Volterra channel model Frequency domain turbo-equalizer Summary Results and conclusions

On Linear Frequency Domain Turbo-EqualizationOf Non Linear Volterra Channels

Bouchra Benammar 1

Nathalie Thomas1, Charly Poulliat 1, Marie-Laure Boucheret 1

and Mathieu Dervin 2

1 University of Toulouse (ENSEEIHT/IRIT) 2 Thales Alenia Space

August 21, 2014

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Table of Contents

1 ContextState of The Art

2 Volterra channel modelTime domain modelVectorial notationsFrequency domain model

3 Frequency domain turbo-equalizerLinear frequency domain MMSEReduced complexity equalizerSoft Demapper

4 SummaryFrequency domain equalizer structureImplementation comparisonComplexity comparison

5 Results and conclusions

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Context

Fixed satellite communications

Higher order modulations for better spectral efficiency (16 and32)

Nearly saturated amplifiers

Non linear distortions

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Context





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Context





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Context





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State of The Art

State of The Art-1

Different models have been proposed to describe satellite amplifiersdistortions:

Time domain Volterra nonlinear channel: Benedetto et al. 1.

Linear model plus additive noise and warping: Burnet et al. 2

1S. Benedetto and Ezio Biglieri, “Nonlinear Equalization of Digital SatelliteChannels,” IEEE Journal on Selected Areas in Communications, 1983.

2C.E. Burnet and W.G. Cowley, “Performance analysis of turbo equalization fornonlinear channels,” in ISIT 2005.

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State of The Art

State of The Art-1

Different models have been proposed to describe satellite amplifiersdistortions:

Time domain Volterra nonlinear channel: Benedetto et al. 1.

Linear model plus additive noise and warping: Burnet et al. 2

1S. Benedetto and Ezio Biglieri, “Nonlinear Equalization of Digital SatelliteChannels,” IEEE Journal on Selected Areas in Communications, 1983.

2C.E. Burnet and W.G. Cowley, “Performance analysis of turbo equalization fornonlinear channels,” in ISIT 2005.

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State of The Art

State of The Art-2

Several equalizers for the Volterra model have been proposed:

Time domain equalizers :

Non iterative equalizers: Benedetto et al. 3(Non linear),Falconer 4(adaptive).

Iterative equalizers:

Optimal MAP equalizer: Su et al. 5

Linear equalizers: Benammar 6

Factor graph equalizer: Colavolpe et al 7.

3Benedetto, S.; Biglieri, Ezio, ”Nonlinear Equalization of Digital SatelliteChannels,” Selected Areas in IEEE Journal on Communications, 1983

4Falconer, D.D., ”Adaptive equalization of channel nonlinearities in QAM datatransmission systems,” The Bell System Technical Journal, 1978

5Su, Yu-T; Mu-Chung Chiu; Yen-Chih Chen, ”Turbo equalization of nonlinearTDMA satellite signals,” GLOBECOM 2002

6B. Benammar, N. Thomas, C. Poulliat, ML. Boucheret, and M. Dervin, “Onlinear mmse based turbo-equalization of nonlinear volterra channels,” in ICASSP, 2013

7Colavolpe, G.; Piemontese, A, ”Novel SISO Detection Algorithms for NonlinearSatellite Channels,” GLOBECOM 2011

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State of The Art

State of The Art-2



Non iterative equalizers: Benedetto et al. 3(Non linear),Falconer 4(adaptive).Iterative equalizers:









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State of The Art

State of The Art-2












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State of The Art

State of The Art-2












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State of The Art

State of The Art-2












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State of The Art

State Of The Art-3

Non iterative frequency domain equalizers:

Block LMS adaptive equalizer: Sungbin. 8.Yangwang 9

8Sungbin Im, ”Adaptive equalization of nonlinear digital satellite channels using afrequency-domain Volterra filter,” MILCOM ’96

9F. Yangwang; J. Licheng; P. Jin, ”Volterra filter equalization: a frequency domainapproach,” WCCC-ICSP 2000

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System description

Figure: System model description

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Frequency domain Turbo Linear equalizer

Problem formulation

Objective: Derive a frequency domain turbo-equalizer for theVolterra model

Equalizer criterion: Linear Minimum Mean Square Error.

Key assumptions:

Constant amplitude modulations (possibly extended to APSKmodulations)Time invariant No-Apriori MMSE solution.

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Problem formulation



Key assumptions:


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Problem formulation



Key assumptions:


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Problem formulation



Key assumptions:Constant amplitude modulations (possibly extended to APSKmodulations)

Time invariant No-Apriori MMSE solution.

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Problem formulation



Key assumptions:Constant amplitude modulations (possibly extended to APSKmodulations)Time invariant No-Apriori MMSE solution.

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Time domain model

Time domain model

The time domain Volterra model writes as:

z̃n =

M−1∑i=0

hix̃n−i +

M−1∑i=0

M−1∑j=0

M−1∑k=0

hijkx̃n−ix̃n−j x̃∗n−k + wn (1)

Using a cyclic prefix of length M yields the following:

zn =

M−1∑i=0

hix<n−i>N︸︷︷︸1D circular convolution

(2)

+

M−1∑i=0

M−1∑j=0

M−1∑k=0

hijkx<n−i>Nx<n−j>N

x∗<n−k>N︸︷︷︸3D circular convolution

+wn

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Time domain model

Time domain model

The time domain Volterra model writes as:

z̃n =

M−1∑i=0

hix̃n−i +

M−1∑i=0

M−1∑j=0

M−1∑k=0

hijkx̃n−ix̃n−j x̃∗n−k + wn (1)

Using a cyclic prefix of length M yields the following:

zn =

M−1∑i=0

hix<n−i>N︸︷︷︸1D circular convolution

(2)

+

M−1∑i=0

M−1∑j=0

M−1∑k=0

hijkx<n−i>Nx<n−j>N

x∗<n−k>N︸︷︷︸3D circular convolution

+wn

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Time domain model

Vectorial notations

The time domain model writes in the vectorial form as follows:

z = H︸︷︷︸Circulant

x +∑i

∑j

∑k

Hijk︸︷︷︸diagonal

xijk + w (3)

where

H =

h0 0 . . . 0 hM−1 . . . h2 h1

h1 h0 0 . . . 0 hM−1 . . . h2

.... . .

. . ....

......

...

0 . . . 0 hM−1 . . . h2 h1 h0

(4)

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Frequency domain model


Based on the N -block time domain circular model and usingonly 1D-DFT , the frequency domain model writes as:

Zm = Hd(m)Xm +√N

N−1∑p=0

N−1∑q=0

N−1∑r=0

H(3)p,q,rXpXqXrδN (p+ q + r −m)

+ Wm (5)

where

Hd is the N -1D-DFT of 1st order Volterra kernels hi.

H(3)p,q,r are the N -3D-DFT of 3rd order Volterra kernels hi,j,k.

δN (m) = 1 if < m >N= 0 is the modulo-N Kronecker’sdelta.

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Zm = Hd(m)Xm +√N

N−1∑p=0

N−1∑q=0

N−1∑r=0


+ Wm (5)

where




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Zm = Hd(m)Xm +√N

N−1∑p=0

N−1∑q=0

N−1∑r=0


+ Wm (5)

where




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Linear frequency domain MMSE

Frequency domain Turbo Linear MMSE-1

Frequency domain solution writes as:

X̂ , F x̂ = HHd C

−1ZZ,d (Z− E [Z]) + CE [X] (6)

where :

E[Z] = FE[z] and E[X] = FE[x]

The covariance of symbols Z:

CZZ,d = HdHHd +

∑(i,j,k)

|hijk|2IN + σ2wIN

= HdHHd + σ2

w̃IN (7)

The constant C = 1N

∑N−1i=0

|Hd(i)|2σ2w̃+|Hd(i)|2

.

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X̂ , F x̂ = HHd C

−1ZZ,d (Z− E [Z]) + CE [X] (6)

where :

E[Z] = FE[z] and E[X] = FE[x]The covariance of symbols Z:

CZZ,d = HdHHd +

∑(i,j,k)

|hijk|2IN + σ2wIN

= HdHHd + σ2

w̃IN (7)

The constant C = 1N

∑N−1i=0

|Hd(i)|2σ2w̃+|Hd(i)|2

.

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X̂ , F x̂ = HHd C

−1ZZ,d (Z− E [Z]) + CE [X] (6)

where :

E[Z] = FE[z] and E[X] = FE[x]The covariance of symbols Z:

CZZ,d = HdHHd +

∑(i,j,k)

|hijk|2IN + σ2wIN

= HdHHd + σ2

w̃IN (7)

The constant C = 1N

∑N−1i=0

|Hd(i)|2σ2w̃+|Hd(i)|2

.

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The frequency domain estimated symbols write individually as:

X̂m =H∗

d(m)

σ2w̃ + |Hd(m)|2Zm +

(C −

H∗d(m)Hd(m)

σ2w̃ + |Hd(m)|2

)E[Xm]

−√NHd(m)∗

σ2w̃ + |Hd(m)|2N−1∑p=0

N−1∑q=0

N−1∑r=0

H(3)p,q,rE

[XpXqXr

]δN (p+ q + r −m) (8)

Triple sum of N elements is computationally prohibitive.

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Reduced complexity equalizer

Reducing the computational complexity

Simplification when using time domain 3rd order terms:

N−1∑p=0

N−1∑q=0

N−1∑r=0

H(3)p,q,rE

[XpXqXr

]∆N (p+ q + r)

= FM−1∑i=0

M−1∑j=0

M−1∑k=0

HijkE[xijk] (9)

where:

∆N (p+ q + r) = [δN (p+q+r−0), . . . , δN (p+q+r−N−1)]T .

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Soft Demapper

Soft Demapper

The estimation residual error is supposed to be Gaussiandistributed following: en = x̂n − κnxn.

κn = Cov(x̂n, xn) = C (10)

The distribution of the residual error:

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

x

PD

F(x

)

Simulated error pdfTheoretical error pdf

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Frequency domain equalizer structure

Structure of the turbo equalizer

Figure: Frequency domain equalizer

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Implementation comparison

Implementation comparison

Time domain MMSE

Compute E[x], E[xijk] and

E[z]

Compute:

aNA = HH(HHH + σ2

w̃IN)−1

For n = 0 : N − 1

x̂n = aNA (zn − E[zn])+E[xn]

where zn = [zn . . . zn−L]T

Frequency domain MMSE

Compute E[x], E[xijk] and

E[z]

Compute :

FE[x] and FE[z].

For n = 0 : N − 1

X̂n = C1(Zn−E[Zn])+E[Xn]

Compute FHX̂ .

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Complexity comparison

Complexity

Assumptions:

All expectations are available.The simplified computation of third order interference is used.All constant terms are previously initialized.

The complexity in terms of the number of real multiplicationsand additions:

Equalizer ] real multiplications ] real adds

Time NA N((8 + 4I)L+ 4M − 4) N((8 + 4I)L+ 4M − 6)

Freq NA6N log2(N) + (4I − 9)N +

36

12N log2(N)+(2I−4)N+

18 + 2I

where I is the number of Volterra 3rd order kernels.

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Complexity

Assumptions:

All expectations are available.

The simplified computation of third order interference is used.All constant terms are previously initialized.



Time NA N((8 + 4I)L+ 4M − 4) N((8 + 4I)L+ 4M − 6)


36

12N log2(N)+(2I−4)N+

18 + 2I


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Complexity

Assumptions:

All expectations are available.The simplified computation of third order interference is used.

All constant terms are previously initialized.



Time NA N((8 + 4I)L+ 4M − 4) N((8 + 4I)L+ 4M − 6)


36

12N log2(N)+(2I−4)N+

18 + 2I


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Complexity

Assumptions:




Time NA N((8 + 4I)L+ 4M − 4) N((8 + 4I)L+ 4M − 6)


36

12N log2(N)+(2I−4)N+

18 + 2I


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Complexity

Assumptions:




Time NA N((8 + 4I)L+ 4M − 4) N((8 + 4I)L+ 4M − 6)


36

12N log2(N)+(2I−4)N+

18 + 2I


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Results

System : 8-PSK modulation with a 1/2 rate convolutionalcode (7,5).

2 3 4 5 6 7 8 9 1010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0

BE

R

Coded ISI freeMMSE−TDE 1st iteration Exact MMSE−TDE 1st iterationMMSE−TDE 4th iterationExact MMSE 4th iterationMMSE−FDE 1st iterationMMSE−FDE 4th iteration

Figure: BER comparison for different turbo equalizers

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Conclusions

Design of a frequency domain turbo-equalizer for Volterra nonlinear channels.

Gain in the computational complexity under some conditions.

Equivalent performance to the time domain equalizer.

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Questions

Thank you for your attention!Questions?

Documents

On Linear Frequency Domain Turbo-Equalization Of …trsys.faculty.jacobs-university.de/turbo/presentations/...1/21 Context Volterra channel model Frequency domain turbo-equalizer SummaryResults