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If in a digital communication system consecutive pulses are sent over a channel, to avoid ISI, a raised cosine filter is used for pulse shaping. If I understand correctly, then this graph taken from wikipedia shows how consecutive pulses are sent:

consecutive pulses

Due to channel induced ISI a zero forcing equalizer is used at the receiver to eliminate such distortions. As I understand it forces the output the equalizer to be 1 at K = 0 which is the peak of the raised cosine pulse and makes everything else 0

Zero forcing

But the way I see it, K isn't always equal to 0 for all pulses. The peak of each pulse is at another sample point. So how does the equalizer work for multiple pulses? or am I missing something?

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  • $\begingroup$ The equalizer works with the estimated symbols out of the matched filter. It does not deal directly with the pulses. I recommend reading "Software Radio Design" by Johnson et a, and "Digital Signal Processing in Modern Communication Systems" by Schwarzinger. $\endgroup$
    – MBaz
    Commented Oct 22, 2021 at 19:22
  • $\begingroup$ @MBaz So basically the modulated symbols undergo pulse shaping, then are sent throught a channel, get contaminated with noise, but are down sampled at the reciever to retrieve the baseband signal (symbols), then the match filter removes noise, and the clean signal undergoes equalization to mitigate channel induced ISI? $\endgroup$
    – Valdi
    Commented Oct 22, 2021 at 22:08
  • $\begingroup$ Yeah, except that the symbols are estimated after the matched filter, not before. $\endgroup$
    – MBaz
    Commented Oct 23, 2021 at 14:51

1 Answer 1

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A train of pulses can be considered as a superposition (sum) of several pulses separated in time by some delay. Considering that the equalizer is a linear time-invariant (LTI) system, it works in the same way when processing a single pulse or multiple pulses. To see this, remember that, for an LTI system, if you delay the input by T seconds, the only change you will see in the output is a delay of the same T seconds. Therefore, any LTI equalizer will operate in the same way for each pulse in the train, meaning that its output will be the superposition of the system response for each individual pulse delayed by the individual time delay associated with each pulse.

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