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I am trying to understand the basics of the cross/auto-correlation if the noisy signal $y(t)$ which is received from the channel at the receiver. Here at receiver we want to estimate the noise in the received signal and then get a clear signal of $y(t)$.

My questions are below:

  1. How autocorrelation or cross-correlation is performed for this $y(t)$ signal? And what does tell us from the correlation function?
  2. What does the Fourier Transform of the correlation function say about the noisy signal?

Thanks in advance!

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  • $\begingroup$ Let $y(t)=x(t)+n(t)$ be the received signal. Do you want to recover the entire signal $x(t)$ with as much of the noise $n(t)$ removed as possible (e.g. $x(t)$ is an audio signal that you wish to listen to and so fidelity, i.e. recovering the waveshape as much as possible, is important), or do you merely wish to detect whether a known signal $x(t)$ is present in $y(t)$? That is, decide whether $y(t)$ is just $n(t)$ or is $x(t)+n(t)$? (This situation arises in radar where $y(t)$ might, or might not, have an echo of known transmitted waveform $x(t)$ reflected off a target. $\endgroup$ – Dilip Sarwate Jun 5 '14 at 13:13
  • $\begingroup$ Thanks Dilip, So we want to get the signal x(t) from the noisy signal y(t)=x(t)+n(t). To perform this activity we need to cross-correlate the y(t) with n(t), right? Or we first want to estimate the noise in y(t) signal, and for this the cross-correlation is between the known x(t) and noisy signal y(t)=x(t)+n(t)? $\endgroup$ – tuner Jun 6 '14 at 7:01
  • $\begingroup$ I would also appreciate if you could also explain me the followings: 1-If we cross-correlate the y(t) with n(t), what does the correlation function say? And how can we evaluate the fourier transform of this cross-correlation function? 2- If we cross-correlate the known x(t) with y(t), what does the correlation function say? And how can we evaluate the fourier transform of this cross-correlation function? Thanks in advance! $\endgroup$ – tuner Jun 6 '14 at 7:02
  • $\begingroup$ If you know $x(t)$ and it is, say, an audio signal, then there is no need to recover it from its noisy version $y(t)$; just play your stored copy of $x(t)$ and enjoy! Cross-correlation is used when you are trying to detect whether $x(t)$ is present in $y(t)$ or not; the result of the cross-correlation is so grossly distorted from what $x(t)$ is, even in the absence of any noise, that it cannot be said to be any kind of recovery of $x(t)$ from the noisy $y(t)$. $\endgroup$ – Dilip Sarwate Jun 6 '14 at 11:10

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