# Noisy Signal auto/cross-correlation

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?

• 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. – Dilip Sarwate Jun 5 '14 at 13:13
• 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 $y(t)$. – Dilip Sarwate Jun 6 '14 at 11:10