I think it’s a basic question, but I couldn’t get it’s clear answer. If we have data symbols $x$ vector of length $N$ x $1$ ; it’s convoluted with a multipath channel $h$ of length $L$ x $1$; the resulted signal

$y$ = $h * x$ ;

where $*$ is convolution operation.
if we have $x_{(1:4:N)}$ is the pilot symbols which well-known at the receiver, how can we get $h$ based on $y$?

thank you.


1 Answer 1


What's missing to answer this is the spectral occupancy of x, specifically the modulated version that is convolved with the channel h. Since the convolution is a linear operation, and h is a linear channel, we will only be able to determine h for those frequencies that are occupied by x, and with a relative accuracy that will be proportional to the power at each frequency (We will be able to make a better estimate of the frequency response of h for those portions of the spectrum with higher SNR).

If x does not sufficiently occupy all the channels of interest, we will not be able to completely estimate the linear channel.

  • $\begingroup$ Do you mean it's not possible to estimate h in that case ? I think it's similar to single carrier (SC) modulation; how can we estimate the channel in case of SC ? $\endgroup$
    – Fatima_Ali
    Commented Sep 26, 2020 at 14:09
  • 1
    $\begingroup$ @Fatima_Ali it is also not possible to estimate in SC for the same reason--- For example you can not use a single tone to estimate the channel over its entire bandwidth, you can only measure the phase and amplitude change on that tone, at its one frequency--- how could you possibly know how the channel would distort other frequencies from that? For equalization we typically use sounding waveforms that approximate white noise over the bandwidth of interest. $\endgroup$ Commented Sep 26, 2020 at 15:25

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