A commonly used numerical operation called the first backward difference is defined as 

y[n] = ∆(x[n]) = x[n] – x[n – 1],

where x[n] is the input and y[n] is the output of the first-backward-difference system.

(a) Show that this system is linear and time invariant.

(b) Find the impulse response of the system.

(c) Find and sketch the frequency response (magnitude and phase).

(d) Show that if

x[n] = f[n] * g[n],

 then

 ∆(x[n]) =∆(f [n])* g[n] = f[n] * ∆(g[n]),

(e) Find the impulse response of a system that could be cascaded with the first-difference system to recover the input; I e., find h_i[n], where

h_i[n] * ∆(x[n] = x[n].