Let x[n] and y[n] respectively denote the input and output of a system. The input–output...

Let x[n] and y[n] respectively denote the input and output of a system. The input–output relation of a system sometimes used for the purpose of noise reduction in images is given by

Where

and σ²_w is a known constant proportional to the noise power.

(a) Is the system linear?

(b) Is the system shift invariant?

(c) Is the system stable?

(d) Is the system causal?

(e) For a fixed x[n], determine y[n] when σ²_w is very large (large noise power) and when σ²_w is very small (small noise power). Does y[n] make sense for these extreme cases?