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

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

The rest is straightforward...