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I am reading about channel equalization and I am somewhat confused on where in the signal chain channel equalization is normally applied and if it's before or after a correlator detector or matched filter.

My confusion is that if equalization occurs before matched filtering/correlator detection, then a training sequence used to estimate the channel will need to be based upon the output of the ADC, not of the matched filter. So if a pseudo-random sequence is used as a pilot tone it won't consist of symbols but of some other modulation type.

If the equalization is based upon the symbol rate output of the matched filter how are fractional delays handled?

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    $\begingroup$ This really does depend on what you're doing. Often, you can model an equalizer as an LTI system (e.g. a filter); and then, you can use the commutation property of convolution to prove it doesn't matter where you do what. The interesting question is where you get your channel state information from; but that's not necessarily the place where you apply your equalization! $\endgroup$ – Marcus Müller Nov 7 '18 at 16:47
  • $\begingroup$ I see, but in terms of estimating the channel(for example using a training sequence) does the pilot sequence consist of a sequence of symbols or something else? I have a read a few papers on this but no one actually talks about where in the signal chain this is applied. $\endgroup$ – FourierFlux Nov 7 '18 at 16:51
  • $\begingroup$ It consists of something of known properties. Often it's easiest to use symbols, because, hey, you already have a transmitter for these, but sometimes the symbols might have disadvantegous properties (e.g. not DC-free enough, no possibility to make white in spectrum), so that you use something else. $\endgroup$ – Marcus Müller Nov 7 '18 at 16:59
  • $\begingroup$ As in everything you do in engineering, it depends; you take a problem and you solve it. The solution to the problem (here: estimating a channel) depends on your problem's specifics. $\endgroup$ – Marcus Müller Nov 7 '18 at 17:00
  • $\begingroup$ Ok so if you're using symbols you apply your LS criteria using the output of the correlator? Again it's not clear at all where this is actually applied. $\endgroup$ – FourierFlux Nov 7 '18 at 17:03
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Generally, any matched filtering would be done before equalization, since there is no practical advantage that I am aware of to applying a matched filter afterwards. Most textbooks that I've read present an architecture in which a matched filter is applied first, the signal is sampled at 1 sample/symbol, then an equalizer is applied at 1 sample/symbol. But it does not have to be this way: you could apply a matched filter, but not downsample, then apply a fractionally-spaced equalizer and sample the output at 1 sample/symbol; or your could skip the matched filter entirely and apply an equalizer directly, in which case the equalizer would adapt to include both the matched filter and equalizer response.

Regarding fractional delays: if the equalizer is symbol-spaced, then we must perform symbol timing recovery prior to downsampling so that we sample the matched filter output at the correct times. We need to perform symbol timing recovery anyway to correct any rate offsets, but if we are downsampling to 1 sample/symbol, we must also get the symbol clock phase correct. If the equalizer is fractionally-spaced, then it can adaptively refine the sample time, although it can't correct any substantial rate offsets. This is an advantage of using a fractionally-spaced equalizer; the primary disadvantage is the higher input sampling rate.

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  • $\begingroup$ Thank you for the clarification. In the case the delay time spacing is less than the symbol time, does this cause a problem for symbol spaced equalizers? I' $\endgroup$ – FourierFlux Nov 7 '18 at 17:38
  • $\begingroup$ I'm not sure I understand exactly what you are asking. But if you use a symbol-spaced equalizer, the sample rate must be exactly equal to the symbol rate and the sample times (which you could look at as the delay) must also be accurate. If not, this will degrade the performance of the symbol-spaced equalizer and the link as a whole. $\endgroup$ – Ill-Conditioned Matrix Nov 7 '18 at 17:55
  • $\begingroup$ Sorry, I meant to say if the delayed copy was delayed a fraction of the symbol time if this would cause an issue. Also it seems like least squares channel estimation is normally based upon a pilot sequence of symbols and thus would need to be after a correlator/matched filter(otherwise you're just sampling individual points). $\endgroup$ – FourierFlux Nov 7 '18 at 18:06
  • $\begingroup$ A fractional delay that wasn't compensated for would indeed cause an issue. This is one advantage of fractionally-spaced equalization. As for channel estimation, yes, it is normally based on a pilot sequence. While you would probably put it after a matched filter, I don't see why you have to. The matched filter is just a known linear, time-invariant system in your channel. $\endgroup$ – Ill-Conditioned Matrix Nov 7 '18 at 18:12
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    $\begingroup$ In principle you could put the equalizer before the matched filter, but I don't know why you would. I don't follow the thinking about "sampling the signal at the symbol rate and losing a ton of information." Both a symbol-rate channel estimate (pilot symbols in, 1 sample/symbol coming out) and a multi-rate channel estimate (pilot symbols in, multiple samples/symbol coming out; essentially, the effective pulse shape of the channel) are valid channel estimates and useful in different contexts. $\endgroup$ – Ill-Conditioned Matrix Nov 7 '18 at 18:34

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