# Filtering White Noise to Generate Harmonic Signal

I have studied convolutions and filters a long time ago. Today, I am trying to review the basics using some notes of mine, but I am finding difficult to solve easy problems. Since I don't have solutions for these problems, I believe that it could be helpful for me if you could show me how to solve one of them.

Let's say that I want to define a filter $b_\alpha(L)$ (where $L$ is the lag operator) that applied to a white noise $x(t)$ approximates a $\cos(\alpha t)$ function (where $\alpha$ is given). How should I proceed to estimate this filter?

If you filter white noise with a narrow band filter that passes $e^{j\alpha t}$ and $e^{-j\alpha t}$ you will get a sinusoidal output at mean frequency $\alpha$ with an amplitude and phase that varies at the maximum rate according to the bandwidth of your filter. The narrower the bandwidth, the lower the power (since the power for white noise is a distribution evenly spread over all frequencies), but the closer you will approximate a pure cosine signal at a random initial offset in phase and random initial amplitude (which will then still proceed to vary, although relatively slowly compared to a winder band filter, in amplitude and phase). If you could do an infinitely narrow filter (which you can't), you would also get no signal.