Question

A causal discrete-time system is described by a linear constantcoefficient 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]. Provide the ROC and the poles and zeros. Is the system BIBO stable? Does the Fourier transform converge in the ROC?

Answer 1

The z-transform of the unit impulse response h[n] is (z - 1)/(z^2 - z + 0.5). The ROC is the entire z-plane. The system is BIBO stable and the Fourier transform converges.

Step-by-step explanation:

Step 1: To obtain the z-transform of the unit impulse response h[n], we substitute y[n+2] and x[n+1] with their respective z-transform expressions: Y(z) = z^2Y(z) - zY(z) + 0.5Y(z) + X(z). Simplifying the expression, we get H(z) = Y(z)/X(z) = (z - 1)/(z^2 - z + 0.5).

Step 2: Finding the ROC: The poles of the transfer function H(z) are the solutions to the denominator equation z^2 - z + 0.5 = 0. Solving this quadratic equation, we find that the roots are complex: z1 = 0.5 + j0.5 and z2 = 0.5 - j0.5. The ROC is the region in the complex plane where all poles lie. In this case, the ROC is the entire z-plane (i.e., it includes all values of z).

Step 3: BIBO stability: A system is BIBO stable if and only if all poles of H(z) are within the unit circle in the z-plane. In this case, both poles lie within the unit circle, so the system is BIBO stable.

Step 4: Fourier transform convergence: The Fourier transform of a discrete-time signal converges if the region of convergence (ROC) of the z-transform includes the unit circle. In this case, since the ROC is the entire z-plane, the Fourier transform of the system converges.