Consider a discrete-time system represented by the difference equa-tion y[n] = 0.5y[n 1] + x[n] where x[n] is the input and y[n] the output. (a)

Consider a discrete-time system represented by the difference equa-tion y[n] = 0.5y[n ˆ’ 1] + x[n] where x[n] is the input and y[n] the output.

(a) An equivalent representation of the system is given by the difference equation

y[n] = 0.25y[n ˆ’ 2] + 0.5x[n ˆ’ 1] + x[n]

Is it true? Let x[n] = δ[n] and zero initial conditions and solve the two difference equations to verify this. Determine how to obtain the second difference equation from the first.

(b) Using the first initial difference equation show that the output is

What is this expression? determine the impulse response h[n] of the system from this equation? Explain.

(c) If the output is computed using the convolution sum, and the input is x[n] = u[n] ˆ’ u[n ˆ’ 11], find y[n]. Determine the steady-state value of the output, i.e., y[n] as n †’ ˆž.

(d) What is the maximum value achieved by the output y[n]? when is it attained?

ο0 γίn]Σο.5)' x[n -k k=0 'x[n – k] (0.5