1) Consider the following filtering scheme where, y[n]=P{x[n−2],x[n−1],x[n],x[n+1],x[n+2]} .

The function P(·) performs a local quadratic polynomial regression: p[k]=a0 +a1k+a2k2 for(−2≤k≤2)

(1)

to the input and returns p[0] = a0.

1. Find the solution for y[0] for an arbitrary input x[n]. (HINT: a0, a1, a2 are a solution
2. of a least-squares problem)
3. Find the solution for y[1] for an arbitrary input x[n].
4. What can you say about the properties of this filtering scheme?
5. (a) Is it linear? Is it shift-invariant? is it stable? (b) Does it have a frequency response?
6. (c) Do you really need to perform a polynomial regression for every n?