EQUALIZATION

Zena Mohammed

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.

EqualizationThere are several techniques we can use as countermeasures to delay spread.

These techniques fall in two broad categories: o Signal processing.o Antenna solution.

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.

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 isW(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 receivedsignal 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 = 1/H(f).

The receiver signal W(f) after passing through this equalizer becomes [S(f)H(f) + N(f)] (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.

• Linear1

• Nonlinear2

11.1Equalizer Types

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.

11.2Folded Spectrum and ISI-Free Transmission

Input symbolH(t)=p(t)*c(t)

the transmitted signal is thus given by d(t) p(t) c(t) for d(t) = 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(t) g(−t) is the equivalent baseband noise at the equalizer input.

Resulting signaly(t) = d(t)f(t) + (t)

=f(t − kT) +

If we sample y(t) every T seconds we obtain = y(nT) as

desired data bit ISI sampled baseband noise.

We now show that the condition for ISI-free transmission, = , is satisfied if and only if:

the folded spectrum

(f) = implies that the folded spectrum is flat.We now show that = implies a flat folded spectrum. If =

So is the Fourier transform of F(f). Therefore, if = , F(f) =.

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.

11.3.1Zero Forcing (ZF) Equalizers

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 and its estimate at the output of the equalizer, i.e we want to find the {}s to minimize E[]. Since we are dealing with linear equalizers, the equalizer output is a linear combination of the input samples :

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:

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

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 {} 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.

11.6Equalizer Training and Tracking

• in wireless channels c(t) = c(,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 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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