A causal LTI system has the system function (a) Write the difference equati...

A causal LTI system has the system function

(a) Write the difference equation that is satisfied by the input x[n] and output y[n] of this system.

(b) Plot the pole–zero diagram and indicate the ROC for the system function.

(c) Make a carefully labeled sketch of |H(e^jω)| . Use the pole–zero locations to explain why the frequency response looks as it does.

(d) State whether the following are true or false about the system:

(i) The system is stable.

(ii) The impulse response approaches a nonzero constant for large n.

(iii) Because the system function has a pole at angle π/3, the magnitude of the frequency response has a peak at approximately ω = π/3.

(iv) The system is a minimum-phase system.

(v) The system has a causal and stable inverse.