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Presented By :
• Communication system model
• Need for equalization (ISI)
• equalizer design
• Zero Forcing Equalizer
• Minimum Mean Square
)(*)(*)()( tgtctgth RT
A digital communication system requires transmit and receive filter
A digital communication system requires transmit and receive filter.
Transmit Filter shapes transmitted signal to meet spectral requirements.
Receive filter accomplishes two roles:
Recover symbol sent and limit noise effect
)(*)()( tctgth T
)()( tThtg TR Where constitute for
Inter Symbol Interference
• When a system is designed according to matched filter
criterion, there is no inter-symbol interference(ISI).
• When sampling at t with period T we get back same signal.
• At these h(t)0
when h(t) = ∞, bandwidth and c(t) is limited. This will
always lead to ISI with,
Tail of other symbols response overlaps
Which can lead to lower performance with
• Therefore in channels ISI cannot be avoided so:
• and are designed as matched pairs.
• These effects of 𝑐(𝑡) is countered by means of an
Tx filter Channel
Rx. filter DetectorEqualizer
Regardless of ISI, the role of transmit ﬁlter is important to ﬁt
the transmitted spectrum into the appropriate spectral
This requires the usage of longer duration waveforms
and spectral occupancy reduction
• Equalization is all about compensating other effects
that distorts the signal
• Equalizers are basically used to remove the ill effect
generated in signal through channel by using
• For example if we have a properly shaped transmit
pulse as a sinc function which has no ISI but when
transmitted it would be combined with additive
noise which resembles ISI.
• The natural way to compensate the ISI effect is by using
inverse filter if we know the channel impulse response.
𝐻(𝑒 𝑗𝑤) = 𝐻(𝑒−𝑗𝑤)
• That means convolution of the channel response and the
equalizer response must be equal to 1
Methods to meet requirement
• Two main techniques are employed to meet filter
• Automatic Synthesis and adaptation.
• In automatic Synthesis typically equalizer just
compares the received signal to original stored
signal to determine error and coefficient of an
inverse filter.(ZFE and LMS)
• In adaptation the equalizer attempts to minimize
an error signal based on the difference between
the output of the equalizer
Zero Forcing Equalizer
• In we have a transfer function of equalizer h 𝑓 , the simplest
way to remove ISI is to setting a transfer function inverse of
h 𝑓 = 1/𝑐(𝑓).
• This is known as zero forcing equalizer.
• Basically it applies the inverse frequency response of the
channel to the received signal.
• Let’s say a ZF equalizer has tap coefficients W which are
chosen to minimize the peak distortion of the equalized
channel, defined as,
𝐷 𝑝 =
|𝑞 𝑛 − 𝑞 𝑛|
• Where 𝑞 𝑛=( 𝑞0 … . . 𝑞 𝑋) X= 𝑁 + 𝐿 − 1, is the desired channel and
the delay d is a positive integer 𝑑 = 𝑑1 + 𝑑2.
• Condition if , 𝐷 𝑝 < 1, then 𝐷 𝑝 is minimized by N taps
values which cause 𝑞 𝑗 = 𝑞 𝑗 for 𝑑1−𝑑2 ≤ 𝑑1 + 𝑑2.
• Therefore if the initial distortion is greater than
unity the ZF is not guaranteed to minimize the peak
• For the case when 𝑞0 = 𝑒 𝑑
the equalized channel is
𝑞 = 𝑞0, . , 𝑞 𝑑1−1, . 0. . , 0,1,0. . . 0 𝑞 𝑑1+𝑁, … , . 𝑞 𝑁+𝐿−1
In this case the equalizer forces zeroes into the
equalized channel and, hence, the name “zero
Equalizer Tap Solution
• For a known channel impulse response, the tap
gains of the ZF equalizer can be found by the direct
solution of the set of linear equations. Therefore
for that we for matrix
• 𝑃 = 𝑝 𝑑1 , … … 𝑝 𝑑 , … . , 𝑝 𝑁 + 𝑑1 − 1
and the vector,
𝑞= 𝑞 𝑑1 … . . 𝑞 𝑀
Then the vector of the optimal tap gains, 𝑤 𝑜𝑝,
= 𝑞 𝑇 → 𝑤𝑜𝑝 = 𝑃−1 𝑇
Drawbacks in ZFE
• ZFE strategy suffers from the noise enhancing issue
at high frequencies.
• In which it relies on perfect estimation of ℎ(𝑛).
• The noise has not been taken into account at all.
• Since ZFE is generally an inverse filter it applies high
gain at upper frequency which increases noise.
• The training signal is basically an impulse which is
inherently a low signal, which corresponds to low
• MMSE tries to reach a trade-off between noise and
Minimum Mean Square Error
• A FIR filter can equalize the worst case ISI only when
the peak distortion is small1.
• The MMSE gives the filter coefficients to keep a MSE
between the output of the equalizer and the desired
• The MMSE equalizer requires a training sequences (d(t)).
• 𝑦(𝑡) and 𝑣(𝑡) are signals affected by noise
1. Peak distortion: magnitude of the diﬀerence between the output of the channel and the desired signal
The aim is to minimize:
𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡
)(td )( fGT
)( fC )( fGR
MMSE Block Diagram
MSE vs. equalizer coefficients
quadratic multi-dimensional function of equalizer
MMSE aim: find minimum value directly (Wiener solution), or use an
algorithm that recursively changes the equalizer coefficients in the correct
direction (towards the minimum value of 𝜀 )!
Illustration of case for two real-valued equalizer
coefficients (or one complex-valued coefficient)
𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡
• MMSE criterion can be read as
𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡
Where 𝑣(𝑡) = 𝑦(𝑡) ∗ ℎ 𝑒𝑞(𝑡) = 𝑛=0
𝑏 𝑛 𝑦(𝑡 − 𝑛𝑇)
• The minimization parameters are accordingly,
= 0 = 2E[(v t − d t )
• Where it has been taken into account the linearity of the E[.] and
, and the fact that expected value is taken over noise
distribution, independently of their filter coefficient.
• This leads to the next orthogonality condition.
• Therefore the error sequence between the output of
the equalizer and the desired signal and the received
data sequence should be statistically orthogonal.
𝐸 𝑣 𝑡 − 𝑑 𝑡 𝑦(𝑡 − 𝑚𝑇) = 0 = 𝑅 𝑦𝑣 𝑚𝑇 − 𝑅 𝑦𝑑 𝑚𝑇 = 0
• 𝑚 = 0, … . . 𝑁
• 𝑅 𝑦𝑣 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑣 𝑡 + 𝑚𝑇
• 𝑅 𝑦𝑑 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑑 𝑡 + 𝑚𝑇
• All random process involved are considered widesense
stationary and jointly wss
• A process is wss when its mean and covariance do not vary
• Two random Process A and B are jointly wss when they are
wss and their cross correlation only depends on the time
• 𝑅 𝐴𝐵 𝑡, 𝑡′
= 𝑅 𝐴𝐵(𝑡 − 𝑡′)
• 𝑅 𝑦𝑣 𝑚𝑇 can be written as
• 𝑅 𝑦𝑣 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑣 𝑡 + 𝑚𝑇
• = 𝐸 𝑦 𝑡 𝑛=0
𝑏 𝑛 𝑦 𝑡 + 𝑚 − 𝑛 𝑇 = 𝑛=0
𝑏 𝑛 𝑅 𝑦( 𝑚 − 𝑛 𝑇)
• Where 𝑅 𝑦(𝑡′) is the autocorrelation of 𝑦 𝑡 .
• This set of equations can be written in vector matrix form as,
• 𝑅 𝑦 𝑏 𝑀𝑀𝑆𝐸 = 𝑅 𝑦𝑑
• They are known as Wiener-Hopf equations.
• The filter coefficients can be finally calculated as
• 𝑏 𝑀𝑀𝑆𝐸 = 𝑅 𝑦
• Note the similarities with the process with ZFE
• We calculate a matrix depending on the channel
response, and a vector depending on the expected
• The matrix is inverted and we get a solution.
• Note that the matrices and vectors have statistical
• The minimum mean square error we arrive at is
𝜀 𝑚𝑖𝑛 = 𝐸 𝑑 𝑡 2 − 𝑅 𝑦𝑑
= 𝑅 𝑦𝑑
−1 𝑅 𝑦
• Equalization is a process implemented at receiver that is
mandatory for almost any modern digital
• There is a variety of equalization strategies and possible
• Adaptive, blind, ML and so on.
• It is not a closed a field and it is subject to ongoing
research and improvements.
• There is not an all-powerful equalization technique, it all
depends on the kind of channel, hardware availability,
target performance, tolerable delay, and so on.
• Equalizers are not left alone: FEC systems can also
contribute to compensation of residual ISI effects.
• Ziemer, R.E., and Peterson, R.L., Introduction to Digital Communication, Macmillan, 1992.Simon Haykin -
Adaptive Filter Theory.
• David Smalley, Equalizer Design, Atlanta Regional technology Center, Texas instrument
• Lovrich, A. and Simar, R., “Implementation of FIR/IIR Filters with the TMS32010/TMS32020”, Digital
Signal Processing Applications with the TMS320 Family, Volume 1, Texas Instruments, 1989.
• Proakis, J.G., “Adaptive Equalization for TDMA Digital Mobile Radio”, IEEE Transactions on Vehicular
Technology, Volume 40, No. 2, May 1991.
• S.Kay – Statistical Signal Processing – Estimation Theory
• John G.Proakis – Digital Communications.
Apr. 15, 2021
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This presentation is about Equalizer and its design principles, with application point of view.