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 ?²_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 ?²_w is very large (large noise power) and when ?²_w is very small (small noise power). Does y[n] make sense for these extreme cases?

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of In] y[n] = ° (x[n] – m[n]) + m;[n]. o[n] %3D 1 o{n] =E (x[k] - m,[r))*. k=n-1 n+1 m.[n] =E *[k). 2 x[k], k-n-1 of[n] – o, o(n] > 0. of In] = { 0, otherwise,