The document discusses equalization techniques used to mitigate inter-symbol interference (ISI) in digital communication systems. Equalization aims to remove ISI and noise effects from the channel. It is located at the receiver and uses techniques like linear equalizers, decision feedback equalization, and maximum likelihood sequence estimation to estimate the channel response and minimize the error between transmitted and received symbols while balancing noise. As the wireless channel changes over time, adaptive equalization is used where the equalizer periodically trains and tracks the changing channel response.

1. EQUALIZATION
Zena Mohammed

2. Inter Symbol Interference
delay spread causes inter symbol interference (ISI),
which in turn produces an irreducible error floor in
most digital modulation techniques.
ISI arises when the data transmitted through the
channel is dispersive, in which each received pulse is
affected somewhat by adjacent pulses and due to
which interference occurs in the transmitted signals.

3. Equalization
There are several techniques we can use as
countermeasures to delay spread.
These techniques fall in two broad categories:
o Signal processing.
o Antenna solution.

4. what a mean by Equalization?and what the
goal of it?
Equalization :is the process of remove ISI and noise
effects from the channel.
 It’s located at receiver end of the channel.
The goal of equalization is to mitigate the effects of ISI.
However, this goal must be balanced so that in the
process of removing ISI, the noise power in the
received signal is not enhanced.

5. A simple example, Consider a signal s(t) that is passed
through a channel with frequency response H(f). At the
receiver front end white Gaussian noise n(t) is added to
the signal, so the signal input to the receiver is
W(f) = S(f)H(f)+N(f), where N(f) has power spectral
density N0. If the bandwidth of s(t) is B ,then the noise
power within the signal bandwidth of interest is N0B.
Suppose we wish to equalize the received
signal so as to completely remove the ISI introduced by
the channel. This is easily done by introducing an
analog equalizer in the receiver defined by
𝐻𝑒𝑞(𝑓) = 1/H(f).

6. The receiver signal W(f) after passing through this equalizer
becomes [S(f)H(f) + N(f)] 𝐻𝑒𝑞(f) = S(f) + N0(f),
where N0(f) is colored Gaussian noise with power spectral
density N0/|H(f)|2. Thus, all ISI has been removed from the
transmitted signal S(f).
For an equalizer to mitigate the ISI introduced by the
channel, it must have an estimate of the channel impulse or
frequency response. Since the wireless channel varies over
time, the equalizer must learn the frequency response of
the channel (training) and then update its estimate of the
frequency response as the channel changes (tracking).
The process of equalizer training and tracking is often
referred to as adaptive equalization, since the equalizer
adapts to the changing channel.

7. • Linear1
• Nonlinear2
11.1Equalizer Types

8. Equalizer types(cont.)
 The linear techniques are generally the simplest to implement and
to understand conceptually. However, linear equalization
techniques typically suffer from noise enhancement on frequency-
selective fading channels, and are therefore not used in most
wireless applications.
 Among nonlinear equalization techniques, decision-feedback
equalization (DFE) is the most common, since it is fairly
simple to implement and does not suffer from noise
enhancement.
 symbol-by-symbol SBS equalizers remove ISI from each symbol
and then detect each symbol individually.
 sequence estimators equalizers detect sequences of symbols, so
the effect of ISI is part of the estimation process. Maximum
likelihood sequence estimation (MLSE) is the optimal form of
sequence detection, but is highly complex.

9. 11.2Folded Spectrum and ISI-Free Transmission
Input symbol
H(t)=p(t)*c(t)
the transmitted signal is thus given by d(t) ∗p(t) ∗c(t)
for d(t) = 𝑘 𝑑 𝑘 𝛿(𝑡 − 𝑘𝑇) the train of information symbols.
Let f(t) denote the combined baseband impulse response of the transmitter,
channel, and matched filter : f(t) = p(t)∗c(t)∗g(−t)
where 𝑛 𝑔(𝑡)= n(t) ∗g(−t) is the equivalent baseband noise at the equalizer input.
Resulting signal
y(t) = d(t)∗f(t) + 𝑛 𝑔(t)
= 𝑑 𝑘f(t − kT) +𝑛 𝑔(𝑡)

10. If we sample y(t) every T seconds we obtain 𝑦𝑛= y(nT) as
desired data bit ISI sampled baseband noise.
We now show that the condition for ISI-free transmission, 𝑓𝐾= 𝛿 𝑘 𝑓0, is satisfied if
and only if:
the folded spectrum
𝐹 (f) = 𝑓0 implies that the folded spectrum is flat.
We now show that 𝑓𝑘= 𝛿 𝑘 𝑓0 implies a flat folded spectrum. If 𝑓𝑘= 𝛿 𝑘 𝑓0
So 𝑓𝑘 is the Fourier transform of F(f). Therefore, if 𝑓𝑘= 𝛿 𝑘 𝑓0, F(f) =𝑓0.

11. 11.3 Linear Equalizers
A linear equalizer minimizes the error between the received symbol and
the transmitted symbol without enhancing the noise. Although linear
equalizer performs better, but its performance is not enough for channels
with severe ISI. An obvious choice for channels with severe ISI is a non-
linear equalizer.

12. 11.3.1Zero Forcing (ZF) Equalizers

13. 11.3.2Minimum Mean Square Error (MMSE)
Equalizer
In MMSE equalization the goal of the equalizer design is to minimize
the average mean square error (MSE) between the transmitted
symbol 𝑑 𝑘 and its estimate 𝑑 𝑘at the output of the equalizer, i.e we
want to find the {𝑤𝑖}s to minimize E[𝑑 𝑘− 𝑑 𝑘]. Since we are dealing
with linear equalizers, the equalizer output 𝑑 𝑘̂ is a linear
combination of the input samples 𝑦 𝑘:

14. There are three interesting things to notice about this result:
First of all, the ideal infinite length MMSE equalizer cancels out the noise
whitening filter.
Second, this infinite length equalizer is identical to the ZF filter except for the
noise term N0, so in the absence of noise the two equalizers are equivalent.
Finally, this ideal equalizer design clearly shows a balance between inverting
the channel and noise enhancement:

15. Maximum-likelihood sequence estimation (MLSE) avoids the problem of noise
enhancement since it doesn’t use an equalizing filter: instead it estimates the
sequence of transmitted symbols . Using a Gram-Schmidt orthonormalization
procedure we can express w(t) on a time interval [0, LT] as
11.4Maximum Likelihood Sequence
Estimation

16. 11.5Decision-Feedback Equalization
• The DFE consists of a feedforward filter with the received
sequence as input (similar to the linear equalizer) followed by a
feedback filter with the previously detected sequence as input.
the DFE determines the ISI contribution from the detected symbols {𝑑 𝑛} by passing
them through the feedback filter. The resulting ISI is then subtracted from the
incoming symbols. Since the feedback filter D(z) in Figure below sits in a feedback
loop, it must be strictly causal, or else the system is unstable. The feedback filter of
the DFE does not suffer from noise enhancement because it estimates the channel
frequency response rather than its inverse. For channels with deep spectral nulls,
DFEs generally perform much better than linear equalizers.

17. 11.6Equalizer Training and Tracking
• in wireless channels c(t) = c(𝜏,t) will change over time, the system
must periodically estimate the channel c(t) and update the
equalizer coefficients accordingly. This process is called equalizer
training.
• Note that the bit decisions 𝑑 𝑘 output from the equalizer are
typically passed through a threshold detector to round the
decision to the nearest bit value2. The resulting roundoff error
can be used to adjust the equalizer coefficients during data
transmission. This is called equalizer tracking.
• Tracking is based on the premise that if the roundoff error is
nonzero then the equalizer is not perfectly trained, and the
roundoff error can be used to adjust the channel estimate
inherent in the equalizer.

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