1.

A causal discrete-time system is described by a linear constant-

coefficient difference equation

y[n+2]=y[n+1]−0.5y|n]+x[n+1]

(a)

Obtain the z-tranform of the unit impulse response h[n]. Pro-

vide the ROC and the poles and zeros. Is the system BIBO stable?

Does the Fourier transform converge in the ROC?

(b)

Given the initial conditions y [0] = 0 and y[1] = 1, provide

the ROC of Y (2) and obtain the output signal y[n] for a unit step

input x [n] = u[n] using the partial fraction method for the inverse z-

transform. Express y [n] as a real-valued solution using the polar form

of a complex number reje = rcos 0 + jrsin 0. Give the steady-state

solution of y [n] as n → ∞⁰.