# How to design filter for a stochastic process (signal)?

The given stochastic $$y[n]$$ is composed of two signals, $$s[n]$$ which is the desired signal, and $$\epsilon[n]$$ is the noise which is a zero-mean weak-sense stationary signal. The desired signal is a sinusoid function with unknown amplitude and the initial phase is also a stochastic variable distributed according to a rectangular pdf with support $$[0, 2\pi)$$.

$$y[n] = s[n] + \epsilon[n]$$

I try to compute the FFT of $$y[n]$$ to see what the signal looks like in the frequency domain but it doesn't help much. So my thought is to find cross-correlation and then effect spectrum. In order to find the bandstop frequency.

Can someone guide me on how to solve this kind of problem?

• What is the purpose of your filtering? Are you wanting to estimate the initial phase? the unknown amplitude? Is the frequency of the sinusoid known? Or is the purpose to remove the noise? Please dont reply to these questions. Instead, edit your bastion to put in some of this information. Click on the edit button below your post to do this, – Dilip Sarwate Nov 25 '20 at 3:49
• As said by Dilip, it is wiser to think about the purpose before rushin to a tool. Here, the nature of the signal is well defined: sine of deterministic but unknown amplitude, wss noise. For detection and period estimation, one could think about parametric modelling and periodograms. – Laurent Duval Nov 25 '20 at 7:18
• Autocorrect changed question to bastion. Gaaaah!!!! – Dilip Sarwate Nov 25 '20 at 17:49
• i think the OP wants to isolate $s[n]$. problem is, if the amplitude of $s[n]$ is low enough, we cannot tell the difference from the noise floor. autocorrelation of $y[n]$ might show something, but if the OP cannot see any spike in the spectrum, it's not likely that the OP can see a spike in the autocorrelation. – robert bristow-johnson Nov 25 '20 at 21:41
• to the OP: you said "The desired signal is a sinusoid function with unknown amplitude and .. initial phase". If that is true, does this mean you might know the frequency of the sinusoid? if so, you can make a very sharp bandpass filter (BPF) to extract the sinusoid. – robert bristow-johnson Nov 25 '20 at 21:44