A discrete-time LTI system H has input x[n] and output y[n] related by the linear constant coefficient difference equation y[n] − 1 2 y[n − 1] = x[n] + 1 3 x[n − 1]. (a) Find the transfer function H(z). Note: you can find the functional form of H(z), but in this part you do not yet have enough information to find the ROC. In parts (c) and (e) you will be given more information so that you can find the ROC. (b) Give a pole-zero plot for H(z). (c) Now assume that the system frequency response H(e jω) exists. For this assumption, give the ROC of H(z) and find the system impulse response h[n]. (d) Under the assumption of part (c) – that H(e jω) exists – is the system causal? Is it stable? (e) Now assume that the system H is unstable and is not causal. For these assumptions, give the ROC of H(z) and find the impulse response h[n]
