LECT_17. Equalization (1)

8/19/2019 LECT_17. Equalization (1)

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Graduate Program

Department of Electrical and Computer Engineering

Chapter 8: Communication Through Band-limited Channels


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Signals Through Band-Limited Channels

• So far, we saw the design of modulator and demodulator

filters for band-limited channels• Assumption: Channel characteristic is known a priori

• However, in most communication systems, the frequencyresponse of the channel is

• Either not sufficiently known• Or it varies with time

• E.g., telephone channels, wireless channels, …

Digital Comm. – Ch. 8: Communication Through Bandlimited Channels 2Sem. I, 2012/13


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Signals Through Band-Limited Channels

• We now consider the problem of receiver design taking:

• The presence of channel distortion and• Under the condition that the channel is AWGN and its

characteristics are not known a priori

• The channel distortion results in inter-symbol interference



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Signals Through Band-Limited Channels …

• The strategy employed to solve the ISI problem is to design

a receiver that• Compensates for the distortion introduced by the channel

• Or reduce the ISI in the received signal – Equalization

• Equalization methods

• Based on maximum likelihood (ML) sequence detectioncriterion that is optimum from error probability point of view

• Based on the use of linear filters with adjustable coefficients

• Decision-feedback equalization: Uses previously detectedsymbols to suppress the ISI in the symbol being detected



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Digital Signals over Channels with Distortion

• Consider the following transmitted signal

• Passed through the channel, the equivalent lowpass of thereceived is given by

• where we define


∑+∞

−∞=

−=n

t nl nT)(t g I (t)S

z(t)nT)h(t I (t)r n

nl +−= ∑+∞

−∞=

)()()()(and )()()( t gt ct gt xt ct gt hr t t

∗∗=∗=

Gt (f) C(f) Gr (f)

z(t)

sl(t) r l(t)


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Digital Signals Over Channels with Distortion …

• Note that the optimum demodulator can be realized as:

• A filter matched to h(t)• Followed by a sampler operating at the symbol rate 1/T and

• A subsequent processing algorithm for estimating the informationsequence {In} from the sample values

• Let us next see an optimum Maximum-Likelihood receiver



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Optimum Maximum-likelihood Receiver

• The received signal may be expanded in the series

• Where• { f k (t)} is a complete set of orthonormal functions

• {r k } are the observable random variables obtained by projectingr l(t) onto the set { f k (t)}


(t) f r (t)r k

N

1k

k

N

l ∑=∞→

= lim


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Optimum Maximum-likelihood Receiver …

• It can be shown that

• Where• hkn is the value obtained by projecting h(t-nT) onto the set { f k (t) }

• zk is the value obtained by projecting zk (t) onto f k (t)

• The sequence of {zk} is Gaussian with zero-mean andcovariance


k kn

n

nk zh I r +=∑

km0m

*

k δ N } z E{z

2

1=


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• The joint probability density function of the random variable

r N ≡ {r 1 , r 2 ,…….r N } conditional on the transmitted sequence I P ≡ { I 1 ,I 2 ,…….I P} where P ≤ N is

• As the number N becomes large, the logarithm of p(r N |Ip)will be proportional to the metrics PM(Ip), defined as follows


−−

= ∑ ∑

=

2 N

1k

kn

n

nk

0

N

0

p N h I r N 2

1exp

πN 2

1) I p(r

∑∑ ∫∫ ∑ ∫

∫ ∑∞

∞−

∗∗∞

∞−

∞

∞−

∗∗

∞

∞−

−−+

−+−=

−−−=

n m

mn

n

ln

2

l

2

n

nl p

dt mT)h(t nT)(t h I I dt nT)(t h(t)r I Re2dt (t)r

dt nT)h(t I (t)r )PM(I


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• The maximum likelihood estimate of the symbols I 1 ,I 2 ,…..I P

are those that maximize this quantity• However, since the integral of the square of the magnitude

of r l(t) is common to all metrics it may be discarded

• The other integral involving r l(t) gives rise to the variables

• These variables can be generated by passing r(t) througha filter matched to h(t) and sampling the output at the

symbol rate 1/T


dt nT)(t h(t)r (nT) y y ln −=≡ ∗∞

∞−∫


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• The samples { yn} form a set of sufficient statistics for the

computation of PM( I p) or, equivalently, of the correlationmetrics

• Where by definition x(t) is the response of the matchedfilter to h(t ) and is given by

• x(t) represents the output of a filter having an impulseresponse h*(-t) and an excitation h(t)

• Hence, x(nt) represents the autocorrelation of h(t) sampledperiodically at the rate of 1/T


∑∑∑ −∗∗ −

=

n m

mnmn

n

nn p x I I y I Re2)CM(I

dt nT)h(t (t)h x(nT) xn +=≡ ∫∞

∞−

∗


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• Substituting for r l(t)

• Where

• Note that yk is corrupted by ISI• We can reasonably assume that the ISI effects cover a

length L ≤ |n|

• The output of the modulator can be considered as the

output of a finite state machine• I.e., the channel output with ISI may be represented by a trellis

diagram

• The ML estimator of the information is the most probable paththrough the trellis give the received sequence yn


k n

nk nk v x I y +=

∑ −

∫ −= ∗

dt kT)(t h z(t)vk


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• It can be shown that the correlation matrix can be

computed recursively in the Viterbi algorithm according tothe relation


−−+= ∑

=−

∗−−

L

1m

mnmn0nn1n1nnn I x2 I x y2 I Re)(I CM )(I CM


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Discrete-time Model for a Channel with ISI

• We note that:

• Transmitter sends discrete-lime symbols at a rate of 1/T• Sampled output of the matched filter at the receiver is also a

discrete-time signal with samples occurring at a rate of 1/T

• An equivalent discrete-time transversal filter having tap gaincoefficients { xk } can be used to represent the cascade of• Transmitter filter with impulse response g(t)

• Channel with impulse response c(t)

• Matched filter at the receiver with impulse response h*(-t) and

• The sampler

• Input of the filter is the sequence of information symbols{ I k } and its output is the discrete-time sequence { yk }



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Discrete-time Model for a Channel with ISI …


Equivalent discrete-time model of channel with ISI


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Discrete-time Model for a Channel with ISI

• The output of the receiver filter is given by

• The received signal sample is obtained as

• The optimum receiver is matched to h(t), that is gr (t) = h*(-t)


(t)g z(t)nT) x(t I t)(g(t)r y(t) r n

nr l∗

∞

−∞=

∗ ∗+−=−∗= ∑

∑

∞

−∞= −

+==n

k k nnk v x I KT y y )(

∫∫∞

∞−

∗∞

∞−

∗ +=+= dt nT t ht zvdt nT t ht h x k n )()( ;)()(


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• The autocorrelation of the noise samples at the correlator

output is given by

• which is the correlated noise sequence at the receiver filter

output• This complicates the detection process by introducing

distortion!

• To solve the problem we use a whitening filter


≤−

= −

otherwise0

L jk x N )v E(v

2

1 k j0 j

*

k

)(

X(z) H(z) I n

vn

yn vm


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• Let X(z) denote the two sided Z transform of the sampled

autocorrelation function { xk }

• Since xk =x-k *, it follows that X(z) =X*(1/z*) and the 2L roots

of X(z) have symmetry such that if ρ is a root, 1/ ρ* is also aroot

• Hence, X(z) can be factored and expressed us

• Where F(z) is a polynomial of degree L and roots ρ1, ….., ρL and F*(z-1) also has degree L and roots 1/ρ1*, ….., 1/ρL*


∑−=

−= L

Lk

k

k z x X(z)

)(zF F(z) X(z) 1−∗=


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• We can now choose a whitening filter as H(z)=1/F*(z-1)

having all roots inside the unit circle resulting in aphysically realizable & stable recursive discrete-time filter

• The passage of the sequence { yk } through the whiteningfilter results in an output {vk }

• {η k } is a zero-mean white Gaussian noise sequence withvariance σ2n

• { f k } is a set of tap coefficients of an equivalent discrete-timetransversal filters with transfer functions F(z)


k

L

0n

nk nk η I f v += ∑= −


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• The cascade of g(t), c(t), h*(-t), the sampler, and the

whitening filter 1/F(z-1

) is represented as an equivalentdiscrete-time transversal filter with {f n} as its tap coefficients

• The additive noise sequence {ηk} is white Gaussian noisesequence having zero mean and variance No



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• Example: Consider that the

• Transmitter signal pulse g(t) has duration T and unit energy• The received signal pulse is h(t) = g(t) + ag(t-T)

• Determine the equivalent discrete-time white noise filter model

• The sampled autocorrelation function is given by

• The Z transform of xk is


=

=+−=

=

∗

)1(

)0(1)1(2

k a

k ak a

x k

1) z1)(a(az

az)a(1 za

z x X(z)

1

12

1

1k

k k

++=

+++=

=

∗−

−∗

−=

−

∑


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• Assuming that |a| < 1, we can choose F(z)= az + 1

• The two tap coefficients of the transversal filter are f 0 = 1 and f 1=a


Equivalent discrete-time model of ISIchannel with AWGN


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Assignments!

• Read the working principle of

• Linear filters with adjustable coefficients• Decision-feedback equalization


LECT_17. Equalization (1)

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