Use of equalizers in communication systems for Intersymbol Interference

In communication systems with high data rates, Intersymbol Interference (ISI) is a major problem (ISI).

I have read in books that one way to combat this problem is to use time domain equalizers (ZF, MMSE, DFE) at the receivers or frequency domain equalizers (ZF, MMSE).

I am just here to ask whether my understanding is correct.

1. Are we using the equalizers to compensate for the channel impulse response (in case time domain) or channel transfer function (frequency domain equalizers) and therefore we are combating ISI?
2. If yes, why is it that by compensating for the channel we are getting rid of ISI?
3. In order to obtain this time domain filter or transfer function we should have estimated the channel before hand?
4. Is the output of the equalizers are estimated symbols of what has been transmitted?

I would appreciate helping me out in this confusion of ideas maybe with an example.

A (linear) equalizer tries to compensate for the (linear) filtering effect of the channel. This filtering introduced by the channel creates intersymbol interference (ISI), so the goal of the equalizer is to reduce (or, ideally, eliminate) ISI, such that the symbol error rate is reduced to an acceptable level. Linear filtering creates ISI by smearing out the symbols over more than one symbol period, making neighboring symbols interfere with each other.

Time-domain and frequency-domain equalizers are just specific implementations of equalizers, their basic function is the same. Furthermore, compensating for the impulse response of the channel is the same as compensating for its transfer function because impulse response and transfer function are just two different descriptions of the same phenomenon: linear distortion.

Unknown channels can be estimated, but usually adaptive equalizers are used, which can adapt to unknown and/or time-varying channel characteristics. Usually, a training sequence of known symbols is used for an initial adaptation of the equalizer. There are also methods that do not require a training sequence; these methods are referred to as blind equalization (e.g. constant modulus algorithm).

The output of the equalizer is indeed an estimate of the transmitted signal; the final decision is done by the slicer following the equalizer.

• thank you very much. So in summary an equalizer is there to counter act the linear effect of the channel (whether in frequency or time domain). I have one further clarification, what do you mean by we use the sequence of known symbols for initial adaptation? Do we usually send the training sequence on periodic basis to update the equalizer? Thanks alot for clear explanation. Commented May 25, 2015 at 16:47
• Also, if one uses pulse shaping at the transmitter, is there a need for an equalizer at the receiver? Should the transfer function of the equalizer in this case take into account that we have used a pulse shaping at the transmitter? And is there a need for Matched filter? Sorry for asking too many questions. I am trying to understand the basic components of digital system @Matt L. Commented May 25, 2015 at 16:54
• @Tyrone: For a time-varying channel, the training sequence is usually repeated periodically. Even with pulse shaping, if the channel distorts the signal, or if it is time-varying, you need an equalizer. Often the matched filter is integrated into the equalizer. Commented May 25, 2015 at 17:25
• So does the pulse shaping change the equalizer transfer function at the receiver? The channel can be thought of as Heffective = H(f)P(f) if we used pulse shaping, do we need to find an equalizer to combact Heffective or just H? Also do you mean that the equalizer and matched filter are the same block and have same function? Thanks again. Commented May 25, 2015 at 17:30
• @Tyrone: For different transmit pulse shapes, the corresponding equalizer will be different. The equalizer computes an optimal filter which depends on the channel as well as on the transmit pulse. A matched filter is not the same as an equalizer. What I meant is that the function of the matched filter is often integrated into the equalizer. You can have a matched filter followed by an equalizer, but you can also have an equalizer which integrates the function of the matched filter. Commented May 25, 2015 at 17:59