Question

A digital filter is given by the following difference equationy[n] = x[n] − x[n − 2] −1/4y[n − 2].(a) Find the transfer function of the filter.(b) Find the poles and zeros of the filter and sketch them in the z-plane.(c) Is the filter stable? Justify your answer based on the pole-zero plot.(d) Determine the filter type (i.e. HP, LP, BP or BS) based on the pole-zeroplot.

Answer 1

Answer:

$y(z) = x(z) - x(z) {z}^{ - 2} - \frac{1}{4} y(z) {z}^{ - 2} \\ y(z) + \frac{1}{4} y(z) {z}^{ - 2} = x(z) - x(z) {z}^{ - 2} \\ y(z) (1 + \frac{1}{4}{z}^{ - 2}) = x(z)(1 - {z}^{ - 2}) \\ h(z) = \frac{y(z)}{x(z)} = \frac{(1 + \frac{1}{4}{z}^{ - 2})}{(1 - {z}^{ - 2})}$

The rest is straightforward...