Turbo Equal1 Ver3

7/31/2019 Turbo Equal1 Ver3

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Lecture Project

ECE 243BDepartment of Electrical & Computer Engg,

University of California, Santa Barbara

June 7, 2007

Anindya Sarkar and Sandeep Ponnuru

An Introduction to TurboEqualization


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Encoder

Interleaver

Mapper

Channel

ak

bk

ck

xk

yk

Binary data bits

Binary code bits

Scrambled version of bk

Channel symbol

Samples fromreceiver filter output

Systemconfigurationfor a digital

transmitter aspart of a

communicationlink


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Role of the various parts

Encoder dataprotection throughredundancy

Interleaver to avoid

long error bursts

Mapper maps the

interleaved coded bitsinto channel symbolssuitable for modulation

00 01 10 11

-3A -A A 3A


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Receivers Job:

Rx should estimate the data that was transmitted,

using info of how the channel has corrupted the dataalong with added redundancy to protect the data

Nullifying the ISI effects on Tx data is equalization(detection)

Recovering the data bits from equalized symbolstream using ECC is decoding


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Receiver A Optimal Detector

Optimal detector

yk

ak^

Main problem with MAP approach is that computationof APP (a-posteriori prob.) is computationallydemanding

P(ak = a|y) =X

a:ak=a

P(a|y) =X

a:ak=a

p(y|a)P(a)

p(y)

As y depends on the entire sequence

a=(a1,a2,..,aK) of data bits ak, MAP approach isinfeasible for large block lengths K because of theexponentially growing number 2K of terms p(y|a)


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Decoder

Deinterleaver

Demapper

Equalizer/Detector

yk

s(xk)xk^

s(ck)

s(bk)

ak

bk

^

^

ck^

Receiver B

One time equalization anddecoding using hard or soft

decisions (receiver B)

1) First process the

observations to best estimatethe transmitted channelsymbols

2) After estimating channelsymbols, they are de-mapped into their associatedcode bits, de-interleavedand decoded using anoptimal decoder


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Equalizer/Detector

Demapper Mapper

Deinterleaver Interleaver

Decoder

+

+-

-

yk

s(ck)

s(bk)

^ak

s(bk)

s(ck)

Receiver C

Once the error controldecoding processes thesoft information, it can

generate its own soft infoindicating the relativelikelihood for each Tx bit

The soft info from thedecoder can then beproperly interleavedand accounted forduring equalization,creating a feedbackloop between theequalizer and decoder

This process is beliefpropagation or messagepassing


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Separate Equalization and Decoding

Standard approach to reduce the computational complexity is tosplit the detection problem into 2 subpartsequalization and

decoding.

We consider 2 classes of equalization: trellis based and linear

filtering based methods (also consider soft decoding)

Trellis based method - MAP symbol detector sets symbolestimate xk to that x from {1,-1} which maximizes the APP P(xk= x| y)

^


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Z-1 Z-1xk

h0 h1 h2nk

yk

Tapped delay linecircuit of channelmodel (2 taps):we assume itsimpulse responseis known

yk = nk + vk, vk =L

X=0 h`xk`, k = 1, 2, , NFor a L-tap line, given a binary input alphabet {1,-1}, the channel can be in

one of 2L states, ri, i=1,2,..,2L, for the 2L possible delay elements

NB: given x v can be found as we assume channel is known


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r0 r0 r0r0

r1 r1 r1 r1

r2 r2 r2 r2

xij/vij

r3 r3 r3 r3

k-1 k+1 k+2k

Trellis representation of the channel - r0 (00),r1 (01),r2 (10) and r3

(11) are 4 possible states. The aim is to compute P(Xk = x|y),given received sequence y, so as to find out the a posteriori mostlikely sequence

x01/v01

x21/v21

x20/v20

NB: given xij, vij can be found as we assume channel is known

Li E li i


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Linear Equalization

For a L-tap channel and a m-ary alphabet, MAPdetection for channel requires mL states in

trellishence linear equalization

Linear filtering based methods perform onlysimple operations on the received symbols

Given yk, and assuming knowledge ofchannel H, estimate xk can be obtained

through a linear/affine transform


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using channel model: yk = Hxk + nk

best affine estimate: xk = fTk

|{z}to estimateyk + bk

|{z}to estimatezero forcing case with noise: xk = xk + f

Tk nk


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MSE = E(|xk xk|2)

To minimize MSE, linear model used is as follows

xk = E(xk) + fTk (ykE(yk))

fk = Cov(yk,yk)1Cov(yk, xk)

D di


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Decoding

Decoding

Encoder is convolutional code

Trellis representation of convolutional code

Forward backward algorithm

Observation y

Prior L(a)

Forward/

Backward

A posteriori L(b)

Observation =0 & Prior from equalizer

P f f R i B


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Performance for Receiver B

BER for Receiver B with hard (dash) and soft (solid)


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Drawbacks of model

Separate equalization- decoding

K information bits N coded bits

Equalizer calculates P(xk/y)

Not all trellis paths are valid

EqualizerDemapper +Deinterleaver Decoder

y xk bk ak

Equalizers trellis


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Equalizer s trellis

At each trellis state we need to know if the branch leading to it is in valid:

Out of the 2N possible paths, identifying the 2K valid paths requires a look up of all

possible sequences a computationally prohibitive task for large N and K

INVALID

PATH

VALID PATH

Turbo equalizer


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Turbo equalizer

Equalizer

DeinterleaverInterleaver

Observation y

Lext(ck/y)

Lprior(ck/y)

L(bk/p)

+

_ +

Lext(bk/p)

L(ck/y)

L prior(bk/y)-

L(ak)

DecoderThis acts as prior P

extrinsic information


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Equalizers integration

MAP equalizer

Just like convolutional decoder

Exchange bit LLRs

MMSE

Use Prior LLRs (P(Xk=xk)) to get better estimates of xk The extrinsic info for xk should not contain any prior info

about xk

Avoid limit cycles by interleaving

Underlying code


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Underlying code

The inner code is channel code!

Recursive inner codes performed well

Add pre-coder to make the channel code recursive

Pre-coder memory < Channel memory

(1+D2, 1+D+D2) Interleaver+Mapper

Z-1 Z-1

h0 h1 h2

nk

ykXk~

Xk

Outer code Inner code


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Pre-coder Design

Consider a simple precoder 1/(1+D)

Input word with weight 1, Output word very large

weight Pre-coder transforms low weight outer-code

words to high weight Two types

Lower multiplicity of error events - 1/(1+Dk) (k=1,2)

Higher weights of error events - 1/ (1 + D2 + D3 )

Pre coder Design


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Pre-coder Design

Plot of BER showing two types of precoders

Precoder A: 5,3,11

(1/(1+DK))

Precoder B: 15, 17

(1/ (1 + D2 + D3 ))

Without precoder:

1

performs better at

higher SNR due tohigher weight ofo/p codewords


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Performance

Performance of turbo equalization with Accumulating Precoder for

various iterations

Without

precoder

With

Precoder

Best resultsafter

precoding -20 iterations

Shannon Limit

L t S


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Lecture Summary

ISI channel as a rate 1 convolutional code BCJR like decoding

MMSE type decoding for big constellations

Separate equalization and decoding

far off Shannon Limits

Answer: Turbo equalization Simplified decoder for channel code (e.g. MMSE)

efficient precoder design

Main References:


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Main References:1) Turbo equalization: principles and new results

Tuchler, M. Koetter, R. Singer, A.C.Inst. for Commun. Eng., Tech. Univ. of Munich;This paper appears in: Communications, IEEE Transactions onPublication Date: May 2002

Volume: 50, Issue: 5On page(s): 754-767

2) The effect of a precoder on serially concatenated coding systems with anISI channelInkyu LeeWireless Syst. Res. Lab., Murray Hill, NJ;This paper appears in: Communications, IEEE Transactions onPublication Date: Jul 2001

Volume: 49, Issue: 7On page(s): 1168-1175
http://ieeexplore.ieee.org/search/searchresult.jsp?disp=cit&queryText=%28%20koetter%20%20r.%3CIN%3Eau%29&valnm=+Koetter%2C+R.&reqloc%20=others&history=yeshttp://ieeexplore.ieee.org/search/searchresult.jsp?disp=cit&queryText=%28%20singer%20%20a.%20c.%3CIN%3Eau%29&valnm=+Singer%2C+A.C.&reqloc%20=others&history=yeshttp://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=21711&isYear=2002http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=20235&isYear=2001http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=20235&isYear=2001http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=21711&isYear=2002http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26http://ieeexplore.ieee.org/search/searchresult.jsp?disp=cit&queryText=%28%20singer%20%20a.%20c.%3CIN%3Eau%29&valnm=+Singer%2C+A.C.&reqloc%20=others&history=yeshttp://ieeexplore.ieee.org/search/searchresult.jsp?disp=cit&queryText=%28%20koetter%20%20r.%3CIN%3Eau%29&valnm=+Koetter%2C+R.&reqloc%20=others&history=yes


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THANK YOU

QUESTIONS??


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ak = argmaxa{0,1} P(ak = a|y)

yk = nk +LX

`=0

h`xk`, k = 1, 2, , N

output vk is given by vk =LX

`=0

h`xk`

xk = fkTyk + bk

\LARGE{\begin{equation*}

P(a_k = a| \bf{y}) = \sum_{\forall \bf{a}: a_k=a} P(\bf{a|y}) = \sum_{\forall \bf{a}: a_k=a} \frac{

p(\bf{y|a})P(\bf{a})}{p(\bf{y})}

\end{equation*}}



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Encoder protect the data by adding proper redundancy. Forward error correction protects the data from random

single bit errors or short bursts

Interleaver to ensure that such errors appear random and toavoid long error bursts, interleaver helps to randomize the order

of the code bits prior to transmission

Mapper permuted code bits are converted into electrical

signal levels that can be modulated either at baseband or ontoa carrier (passband). Mapper maps the interleaved coded bitsinto channel symbols suitable for modulation

Channel Effects:


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Channel Effects:

Traditional methods of data protection used inECC do not work when the channel introduces

additional distortions, in form of ISI

When the channel is dispersive, the receiverwill need to compensate for the channel effectsbefore using a standard decoding algorithm

Methods for channel compensation are calledchannel equalization

Tasks for Receiver


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Tasks for Receiver

Rx should estimate the data that was transmitted,using info of how the channel has corrupted the dataalong with added redundancy to protect the data

When channel introduces ISI, adjacent channelsymbols are smeared together creating dependencies

among Tx channel symbols.

Nullifying the ISI effects on Tx

data is equalization(detection) while recovering the data bits fromequalized symbol stream using ECC is decoding

Why soft information??


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Why soft information??

Equalizer can make hard decisions as to whichsequence of channel symbols were transmitted, andfor these hard decisions to be mapped into theirconstituent binary code bits.

The process of making hard decisions on the channelsymbols destroys information pertaining to how likely

each channel symbol may have been

The extra soft information can be changed to

probabilities that each of the received code bits takeson 0/1, which after deinterleaving, can be exploited bya BER optimal decoding algorithm

Optimal Detection


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Opt a etect o

Optimal Rx achieves the minimum probability of errorfor each data bit ak.

This is achieved by setting ak to that value whichmaximizes the a-posteriori probability

given the observed sequence y, i.e.

An algorithm that maximizes the APP is called amaximum a posteriori (MAP) algorithm

( )k k

P a a

( | )kP a a y=

ak = argmaxa{0,1}

P(ak = a|y)

Problem with MAP approach


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pp

Main problem with MAP approach is that computationof APP is computationally demanding

As y depends on the entire sequence a=(a1,a2,..,aK) of

data bits ak, MAP approach is infeasible for large blocklengths K because of the exponentially growingnumber 2K of terms p(y|a)

P(ak = a|y) =X

a:ak=a

P(a|y) =X

a:ak=a

p(y|a)P(a

p(y)

xTapped delay


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xk

yk

h0h

1

h2

nk

line circuit ofchannel

model

(2 taps)

\LARGE{\begin{equation*}

y_k = n_k + \sum_{\ell=0}^L h_{\ell}x_{k - \ell}, ~ k=1,2,\cdots,N

\end{equation*}}

For a L-tap line, given a binary input alphabet {1,-1}, the channel can be in one of 2L

states, ri, i=1,2,..,2L, for the 2L possible delay elements

MAP symbol detection


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y For a L-tap line, given a binary input alphabet {1,-1},

the channel can be in one of 2L states, ri, i=1,2,..,2L,for the 2L possible delay elements

S = {r1,r2,,r2^L} = set of possible states

Given sk, the state sk+1 can only assume one of 2values corresponding to whether 1/-1 is fed in at timek

possible evolution of states is given by trellis

output vk is given by vk =L

X=0

h`xk`


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p(sk, sk+1,y) = p(y)P(sk, sk+1|y)

p(sk, s

k+1,y

) =p

(sk, y

1, , y

k1)| {z }

k(sk)

. p(sk+1

, yk|sk)| {z }

k(sk,sk+1)

. p(yk+1

, , yN|sk+1)| {z }

k+1(sk+1)

k(s) =XsS

k1(s)k1(s, s)

k(s) =XsS

k+1(s)k(s, s)

k(sk, sk+1) = P(sk+1|sk)p(yk|sk, sk+1)

k(ri, rj) = P(xk = xij).p(yk|vk = vij), (i, j) B

0, (i, j) 6= B

p(yk|vk) = exp((yk vk)2/(22))/

22

P(xk = x|y) =X

(i,j)B:xij=x

P(sk = ri, sk+1 = rj |y)

We can then obtain the conditional LLR L(ck|y) of the code


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k ybit ck, (assuming BPSK)

The code estimates ck are computed from the sign ofL(ck|y)

^

L(ck|y) = lnP(ck = 0|y)

P(ck = 1|y)

= lnP(xk = +1|y)

P(xk = 1|y)

Computational complexity of the trellis basedapproaches is given by the number of trellis states,equal to 2L, which grows exponentially with L, the

number of delay elements or taps

Linear Equalization


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In contrast to MAP, linear filter based methodsperform only simple operations on the receivedsymbols, which are generally applied sequentially to a

subset yk of the observed symbols

Given yk, estimate x

kof x

kcan be obtained as

The vector fk and scalar bk are parameters obtainedthrough optimization

xk = fkTyk + bk

^

Assumptions on the model:


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Assumptions on the model:

Series of transmit pulse shapes modulated withsymbols xk is transmitted over a LTI channel withknown channel impulse response (CIR)

Coherent symbol-spaced Rx with exact knowledge ofsignal phase and symbol timing receives transmittedwaveforms

Received waveforms are passed through the receivefilter which is matched to transmit pulse shape andCIR

Problem Setup


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A branch of the trellis is a 4-tuple (i,j,xij,vij) such thatstate sk+1=rj at time k+1 can be reached from statesk=ri at time k, with input xk=xij and output vk=vij

The set of all index pairs (i,j) corresponding to validbranches is denoted by B

Aim is to efficiently compute the APP P(xk|y)

We initially compute the prob. that the Tx sequencepath in the trellis contained the branch (i,j,xij,vij) at timek, i.e. P(s

k

=ri

,sk+1

=rj

|y)

( ) ( )P( | )


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p(sk, sk+1,y) = p(y)P(sk, sk+1|y)

p(sk, sk+1

,y) = p(sk, y1, , yk

1)| {z }

k(sk)

. p(sk+1

, yk|sk)| {z }k(sk,sk+1)

. p(yk+1

, , yN|sk+1

)| {z }k+1(sk+1)

k(s) =XsS

k1(s)k1(s, s)

k(s) =XsS

k+1(s)k(s, s)

k(sk, sk+1) = P(sk+1|sk)p(yk|sk, sk+1)

k(ri, rj) = P(xk = xij).p(yk|vk = vij), (i, j) B

0, (i, j) 6= B

p(yk|vk) = exp((yk vk)2/(22))/

22

P(xk = x|y) = X(i,j)B:xij=x

P(sk = ri, sk+1 = rj |y)

Equalization Approaches


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Linear model suggests multiplying yk with avector fk that recovers xk perfectly in theabsence of noise

This Zero Forcing Equalizer suffers from

noise enhancement, when the channelimpulse response matrix is ill-conditioned

This can be avoided using linear minimummean square error (MMSE) estimation

Avoiding Limit Cycles


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Soft info an independent piece of information

However, if decoder formulates the soft info about a given bit,based on soft info provided from the equalizer about exactly the

same bit, then this info is not independent of the channelobservations

This would result in a feedback loop of length 2 equalizerinforms the decoder about a given bit while the decoder re-informs the equalizer what it already knows.

Solution Equalizer tells the decoder new info about a bitbased on info gathered from distant parts of the received signal(using interleaver).

Possible Receiver Configurations:


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Possible Receiver Configurations:

Receiver A: optimal detector

Receiver B: separate equalization and

decoding

Receiver C: turbo equalization jointequalization and decoding

Documents

Turbo Equal1 Ver3