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?