Linear Equalizer

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Communications DSP

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Digital Transmission Through Bandlimted Channels

CONTENTS

Characterization of Bandlimited Channels

Characterization of Intersymbol Interference

Signal design for bandlimited Channels

Linear Equalizers

Adaptive Linear Equalizers

Non-Linear Equalizers

Textbook : J. Proakis and M. Salehi: Contemporary communication systems

using MATLAB, 1st

Edit ion. Brooks/Cole, Thomson Learning.

2000. (2nd

Edition available on 2004).



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1. Intersymbol Interference

Many communication channels, including telephone channels and some

radio channels, may be characterized as bandlimited linear fi lters with

frequency response

)()()( f je f A f C θ =

(2-1)

where )( f A and )( f θ are the amplitude and phase responses, respectively.

A channel is non-distort ing or ideal within the bandwidth W if,

c A f A =)( , and f f c ⋅= τ θ )( , W f ∈ , (2-2)

where c A and cτ are constants (i.e. constant amplitude and linear-phase

responses).



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If )( f A is not constant, the distortion is called amplitude distort ion, and

if cτ is not constant, there will be phase distortion.

A measure of the phase linearity or phase distortion of the system is the

envelope delay or group delay

df

f d f

)(

2

1)(

θ

π τ ⋅−= .

(2-3)

Due to the amplitude and phase distortion caused by non-ideal channel, a

succession of pulses transmitted through the channel at a rate

comparable to the bandwidth W are smeared.

Individual pulses might not be distinguishable at the receiver and we

have intersymbol interference (ISI).



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Note the received pulsefor a non-ideal channel

does not have zero

crossings at T ± , T 2± , and

so on.

It is possible to

compensate for the non-

ideal frequency response

characteristic of the

channel by use of a filter

or equalizer at thereceiver.



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2. CHARACTERIZATION OF BANDLIMITED CHANNELS

Telephone Channels

Usable band of the channel :

300Hz to 3200Hz.

Impulse response duration of

an average channel is ~ 10 ms

= L.

If the transmitted symbol rates

is Rs=2500 pulses or symbols

per second, the intersymbol

interference might extend over

20 to 30 symbols

( 310102500 −××=× L Rs ).



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Time-dispersive wireless channels:

e.g. short-wave ionospheric propagation

(HF), tropospheric scatter, and mobile

cellular radio.

Time dispersion, and hence ISI, is

the result of multiple propagation

paths with different path delays.

The number of paths and the

relative time delays can vary with

time. For this reason, they are

called time-variant multipath

channels. The channels can be

characterized by the scattering

function: a 2D representation of

the average received signal power

as a function of relative time delay

and Doppler frequency spread.



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The total time duration (multipath spread) of the channel response is

approximately sμ 7.0 on the average. If transmission occurs at a rate of 710

symbols/sec over the channel, the multipath spread of s7.0 will result in

intersymbol interference that spans about 7 symbols ( 76 10107.0 ⋅× − ).

Exercise: GO THROUGH ILLUSTRATIVE PROBLEM 6.5 (ON MULTIPATH

CHANNEL SIMULATION) AND THE M-FILE.



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2.3 Eye Diagram

The amount of ISI and noise can

be viewed in an oscilloscope:

Display the received signal )(t y on

the vertical input with the

horizontal sweep rate set to 1/T

(the symbol rate).

ISI causes the eye to close:

1) reducing the margin for addit ive

noise (higher detection errors).

2) distorting the position of the

zero-crossings and causes the

system more sensitive to

synchronization errors.



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3 Signal Design for Bandlimited Channels

Necessary and sufficient condition

for a signal )(t x to have zero ISI is

⎩⎨⎧

≠

==

01

00)(

n

nnT x T

T

m f X

m

=+⇔ ∑∞

−∞=

)(

(3-1)

where 1/T is the symbol rate.

One commonly used signals has a

raised-cosine-frequency response

characteristics.

The sampled waveform )(t xδ can be written as



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⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+>

+≤<

−⎥⎦

⎤⎢⎣

⎡⎟

⎠

⎞⎜⎝

⎛ −−+

−≤≤

=

T f

T f

T T f

T T

f T

f X rc

2

1,0

2

1

2

1,

2

1cos1

2

2

10,

)(

α

α α α

α

π

α

(3-2)

where 10 ≤≤α is called the roll-off factor , range and 1/T is the symbol

rate.

When 0=α , )( f X rc reduces to an ideal “ brick wall” physical

nonrealizable response with bandwidth occupancy 1/(2T), called the

Nyquist frequency.

For 0>α , the bandwidth occupied by )( f X rc beyond the Nyquist

frequency is called the excess bandwidth, usually expressed as a

percentage of the Nyquist frequency. For 2/1=α , the excess

bandwidth is 50%, and when 1=

α , the excess bandwidth is 100%.



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The signal pulse )(t xrc having the raised-cosine spectrum is

222 /41

)/cos(

)/(

)/sin()(

T t

T t

T t

T t t xrc

α

πα

π

π

−= (3-3)

In absence of channel distortion, there is no ISI from adjacent symbols.

In presence of channel distortion, a channel equalizer is needed to

minimize its effect on system performance.

Exercise: GO THROUGH ILLUSTRATIVE PROBLEM 6.7 (DESIGN OF

TRANSMIT AND RECEIVE FILTER).

This method is called frequency sampling. Suppose the fi lter length is 2N+1.

The desired analog frequency response is )(Ωd H . The desired frequency

response of the discrete-time fil ter is then )()(~

)(~

Ω== Ωd

T j

d

j

d H e H e H sω . [For

designing the pulse shaping fi lter, to avoid aliasing, we chooseT T s

41= .]



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Sample )(~ ω j

d e H at 2N+1 equally spaced points over the unit circle:

},...,0,...,:)12/(2{ N N k N k k −=+= π ω , we get a set of points ][)( k H e H k jd =

ω . If the

DT-FT of the filter pass through these points then,

][][)( k H enhe H N

N n

jn j k k == ∑−=

− ω ω

Taking the inverse, one gets ∑∑−=

+

−=

== N

N k

N kn j N

N k

jk ek H ek H nh n )12/(2][][][ π ω .

Other methods for designing the transmit and receive fil ters (Nyquist f ilters)

include semidefinite programming (SDP) and eigenfi lter methods.

Normally, )( f X RC will be designed first. It is then factored by a method

called spectral factorization to obtain )( f GT and )( f G R .



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4 Linear Equalizers

The most common type of channel equalizer used in practice to reduce ISI is

a linear FIR fi lter with adjustable coefficients }{ k c .

The ISI is usually negligible

beyond a certain number

of symbols.

The number of terms in the

ISI term is thus fini te.

The linear fi lter is therefore

usually implemented as

finite-duration impulse

response (FIR) fil ter , with

adjustable tap coefficients.



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4.1 Symbol- and Fractionally-spaced Linear Equalizers

The time delay τ between adjacent tap may be selected as large as T, and

the equalizer is called a symbol-spaced equalizer . The input to the

equalizer is )(kT y . However, the excessive bandwidth of the signal causes

those components above the Nyquist frequency 1/(2T) to aliase with those

below. The equalizer only compensates for the aliased channel-distorted

signal.

If τ is shorten to T/2, i.e. the received signal is sampled at 1/(2T) Hz, then

even for 100% excess bandwidth, there will not be any aliasing in the

received signal. The channel equalizer is said to have fractionally spacedtaps, and it is called a fractionally spaced equalizer . However, higher

operating speed is needed.



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For zero ISI, the product of the various transfer functions should equal to

a Nyquist channel, say )( f X rc , the raised cosine spectrum

)()()()()( f X f G f G f C f G rc E RT = . (4-1)

The receive pulse shaping filter )( f G R are usually matched to transmit

pulse shaping f ilter )( f GT with

)()()( f X f G f G rc RT = , (4-2)

i.e. it forms a Nyquist channel with no ISI in absence of channel distortion.

The frequency response of the equalizer should be

)( f G R )(

)(

1

)(

1)(

f j

E ce f C f C f G θ −

== , (4-3)

In this case, the equalizer is said to be the inverse channel filter to the

channel response. It might not be stable and realizable.



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4.2 Zero-forcing equalizers (ZF equalizers)

The impulse response of the FIR equalizer is

∑−=

−=K

K n

n E nt ct g )()( τ δ , (4-4)

and the corresponding frequency response is

∑−=

−=K

K n

fn j

n E ec f G τ π 2)( , (4-5)

where }{ nc are the 2K+1 equalizer coefficients, and K is chosen sufficiently

large so that the equalizer spans the length of the ISI.

Zero-forcing assumes the received pulse shape (or channel) is known and finds an equalizer to minimize the ISI at t ime instants, nT, n=1,…..



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Let )()()()( f X f G f C f G RT = and )(t x the time waveform corresponding to

)( f X . The equalized output signal pulse q(t) is

∑−=

−==K

K n

n E nt xct qt gt x )()()(*)( τ , (4-6)

The samples of )(t q taken at times mT t = , are given by

∑−=

−=K

K n

n nmT xcmT q )()( τ , K m ±±= ,....,1,0 . (4-7)

For zero ISI, )(mT q should ideally be zero except at m=0.

Since there are 2K+1 equalizer coefficients, we can control only 2K+1

sampled values of )(t q . We therefore impose the zero-forcing

condit ions to determine the ZF-equalizer coefficient }{ nc

⎩⎨⎧

±±±=

==

K m

mmT q

,...,2,1,0

0,1)( . (4-8)



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(4.7) is a system of linear equation in unknown }{ nc which can be written

as

q Xc = , (4-9)

where X is an )12()12( +×+ K K matrix with elements )( τ nmT x − , c is the

)12( +K coefficient vector, and q is the )12( +K column vector with one

nonzero element.

The equalizer coefficient is given by

q X c1−= , (4-10)

FIR zero-forcing equalizer does not completely eliminate the ISI

because it has a finite length. As K is increased, the residual ISI can be

reduced, and as ∞→K , ISI is completely eliminated under noise free

condition.



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A drawback of zero-forcing equalizer is that it ignores the presence of

additive noise. It might lead to significant noise enhancement. In a

frequency range where )( f C is small, the channel equalizer

)(/1)( f C f G E = compensates by placing large gain in that frequency

range. The noise in that frequency range is greatly enhanced.

Example: Suppose that the received output is equal to

2)/2(1

1

)( T t t x +=

,(4-11)

where T /1 is the symbol rate. The pulse is sampled at the rate T /2 and is

equalized by a zero-forcing equalizer. Determine the coefficients of a

five-tap zero-forcing equalizer.



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Solution

Since the pulse is sampled at the rate T /2 , it is a fractionally-spaced equalizer

with 2/T =τ . Also, as the equalizer is of 5 taps, 22/)15( =−=K .

The zero-forcing condition in (4.8) becomes

∑−=

−=K

K n

n

nT mT xcmT q )

2()( , 2,....,1,0 ±±=m . (4-12)

From (4.7), the matrix X and vector q are given by

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

5/110/117/126/137/1

12/15/110/117/1

5/12/112/15/1

17/110/15/12/11

37/126/117/110/15/1

X , and

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

0

0

1

0

0

q . (4-13)



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Solving for l inear equation q Xc = yields

⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢⎢

⎣

⎡

−

−

−

==

⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢⎢

⎣

⎡

= −

−

−

2.2

9.4

3

9.4

2.2

1

2

1

0

1

2

q X c

c

c

c

c

c

opt . (4-14)



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4.3 Minimum Mean-Square Error (MMSE) Equalizers

Let )(t z be the noise-corrupted output of the FIR equalizer

∑−=

−=K

K n

n nt yct z )()( τ , (4-14)

where )(t y is the input to the equalizer given by (2.3).

The equalizer is sampled at t imes mT t = , and we have

∑−=

−=K

K n

n nmT ycmT z )()( τ . (4-15)

The desired response at the output of the equalizer at mT t = is the

transmitted symbol ma (assume to be known during training of the

equalizer).

The error is defined as the difference between ma and )(mT z .



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The mean-square error (MSE) between the actual output sample )(mT z and

the desired values ma is

])([2

mamT z E MSE −=⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= ∑

−=

2

)( m

K

K n

n anmT yc E τ

[ ]∑ ∑−= −=

−−=K

K n

K

K k

k n k mT ynmT y E cc )()(* τ τ

[ ]∑−=

−⋅−K

K k

mk k mT ya E c )(2 * τ )|(| 2ma E + .

∑ ∑−= −=

−=K

K n

K

K n

yk n k n Rcc )( ∑−=

−K

K k

ayk k Rc )(2 )|(| 2ma E + .

(4-16)

where )]()([)( * τ τ k mT ynmT y E k n R y −−=− , )]([)( * τ k mT ya E k R may −⋅= ,

The expectation [ ]⋅ E is taken with respect to the random information

sequence }{ ma and the additive noise.



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The minimum MSE solution is obtained by differentiating (4.15) with

respect to the equalizer coefficients }{ nc .

The condition for minimum MSE is

∑−=

=−K

K nay yn

k Rk n Rc )()( , K k ±±±= ,...,2,1,0 , (4-17)

which is a system of linear equation with )12( +K equations in )12( +K

unknown }{ nc .

In practice, the autocorrelation sequence )(n R y and the cross-correlation

sequence )(n Ray are unknown a priori. They have to be estimated using

the time-average estimates

∑=

−=K

k

y kT ynkT yK

n R1

* )()(1

)(ˆ τ , ∑=

−=K

k

k ay ankT yK

n R1

** )(1

)(ˆ τ (4-18)



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The symbols k a are assumed known, which are transmitted to the

receivers during the so-called training mode.

In contrast to the zero-forcing solution, these equations depend on the

statistical properties (the autocorrelation) of the noise as well as the ISI

through the autocorrelation )(n R y .



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5 Adaptive Linear Equalizers

On channels whose frequency response characteristics are unknown

but time-invariant (does not change with time, such as telephone lines),

we may measure the channel characteristics by sending knownsymbols (training symbols) to the receiver and adjust the parameters of

the equalizer. Once adjusted, the parameters remain fixed during the

transmission of data. Such equalizers are called preset equalizers.

Adaptive equalizers update their parameters by sending periodically

training symbols to the receivers during the transmission of data, so

they are capable of tracking a slowly time-varying channel response.



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Both the zero-forcing and MMSE equalizers require the solution of a systemof linear equation of the form

d Bc = (5-1)

where B is a )12()12( +×+ K K matrix, c is a column vector representing the

)12( +K equalizer coefficients, and d is an )12( +K column vector.

The solution is

d B c1−=opt (5-2)

Solving (5.2) directly will require very high arithmetic complexity:

))12(( 3+K O . In practical implementation, it is solved using iterative

methods such as the Least mean squares (LMS) or recursive least squares

(RLS) algori thms.

C i i DSP



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5.1 LMS algorithm

The LMS algori thm is based on the method of steepest descent:

1. One begins with an arbitrarily chosen coefficient vector say 0 c .

2. Each tap coefficient is changed in the direction opposite to its

corresponding gradient component in the gradient vector g , which is the

derivative of the MSE with respect to the )12( +K filter coefficients:

k k k g c c ⋅Δ−=

+1 (5-3)

where Δ is the step-size parameter.

C i ti DSP



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In the LMS algorithm, g is estimated continuously. A commonly used

method is to approximate the MSE by the instantaneous error2*2 )()( k

T

k k k ae y c−= . Thus, the estimated gradient is

*2 22)(ˆk k k k k k yeeee

k k −=∇⋅=∇= c c g (5-4)

The LMS or stochastic gradient algorithm is given by

*1 k k k k e y c c ⋅Δ+=+ ,

*k

T

k k k ae y c−= .(5-5)

Step-size selection:

One commonly used step-size parameter Δ in order to ensure

convergence and good tracking capabilities in slowly varying channels is

RPK )12(5

1

+=Δ (5-6)

RP denotes the received signal-plus-

noise power, which can be estimated

from the received signal.

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The optimal solution can be approached after a few hundred iterations.

As the equalizer is updated at the symbol rate, it corresponds to afraction of a second.

Initially, the adaptive equalizer is trained by the transmission of a

known pseudorandom sequence }{ ma over the channel. At the

demodulator, the equalizer employs the known sequence to adjust i ts

coefficients.

Upon initial adjustment, the adaptive equalizer switches from a

training mode to a decision-directed mode, in which case the

decisions at the output of the detector are sufficiently reliable so that

the error signal is formed by computing the difference between thedetector output and the equalizer output

k k k zae −= ˆ (5.7) k a is the output of the detector.

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In general, decision errors at the output of the detector occur infrequently.

Such errors have litt le effect on the performance of the tracking algorithm.

EXERCISE: GO THROUGH ILLUSTRATIVE PROBLEM 6.12 AND RUN THE M-

FILE.

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EXAMPLE

The LMS algori thm is used to identify the following channel

]088.0,038.0,126.0,0,25.0,9047.0,25.0,126.0,088.0,063.0,05.0[ −−−= x

with 11)12( =+K .

Smaller stepsize

leads to faster

convergence but

higher errors.

RLS has faster

convergence but a

high complexity.

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6 Nonlinear Equalizers

The linear filter equalizers are very effective on channels, such as wire line

telephone channels, where ISI is not severe.

The severity of the ISI is directly related to the spectral characteristics of the

channel and not necessarily to the time span of the ISI.

There is a spectral nul l in channel B at f=1/2T (more severe ISI). Channel A

does not have a channel null and has a large span of ISI.

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The energy of thetotal response is

normalized to unity

for both channels.

The time span of the ISI in channel A is 5 symbol intervals on each side of

the desired signal component, which has a value of 0.72.

The time span for the ISI in channel B is one symbol interval on each side

of the desired signal component, which has a value of 0.815.

In spite of the shorter ISI, channel B results in more severe ISI. A linear

equalizer will introduce a large gain in its frequency response to

compensate for the channel null in channel B at f=1/2T, noise is enhanced.

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6.1 Decision Feedback Equalizers (DFE)

A DFE is a nonlinear equalizer that employs previous decisions to

eliminate the ISI caused by previously detected symbols on the current

symbol to be detected.

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It consists of two filters. The first fil ter is a feedfoward filter , which is

generally a fractionally-spaced FIR filter with adjustable tap coefficients.The second one is called a feedback filter , which is an FIR filter with

symbol-spaced taps having adjustable coefficients. Its input is the set

of previously detected symbols.

The output of the feedback filter is subtracted from the output of the

feedforward fi lter to form the input to the detector.

∑∑=

−=

−−=21

11

~)( N

n

nmn

N

n

nm abnmT yc z τ (6.1)

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where }{ nc and }{ nb are the adjustable coefficients of the feedforward

and feedback filters, respectively; nma −~ , 2,..,2,1 N n = , are the previously

detected symbols; 1 N and 2 N are the length of the feedforward fil ter

and feedback f ilters, respectively.

The tap coefficients are usually selected to minimize the MSE criterion

using the stochastic gradient (LMS) algorithm or RLS algorithm.

Decision errors from the detector that are fed to the feedback filterhave a small effect on the performance of the DFE. A small loss in

performance of 1 to 2dB is possible at error rates below 210 , but the

decision errors in the feedback fil ters are not catastrophic.

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In a digital communication system that transmit information over a

channel that causes ISI, the optimum detector is a maximum-likelihoodsymbol detector (MLSD) that produces at its output the most probable symbol

sequence }~{ k a for the given received sampled sequence }{ k y .

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That is, the detector finds the sequence }~{ k a that maximizes the

likelihood function.

}){|}({ln})({ k k k a y pa =Λ (6.2)

where }){|}({ k k a y p is the joint probabili ty of the received sequence}{ k y conditioned on }{ k a . The Viterbi algorithm can be used to

implement the MLSD, but its complexity grows exponentially with the

span of the ISI. They are suitable for short channel with severe ISI,

such as mobile channels.

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Linear Equalizer