Hi all,

I am stuck on a particular question with regards to obtaining a difference equation from a given transfer function. I know how to do this in most cases, but I can't do this one:

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The transfer function of a certain digital filter, running at a sample rate of fs = 10kHz is given by:

Where Wc and r are constants that parameterise the filter coefficients: Wc is an angular frequency and r is a real number.

**(a) what is the difference equation of this filter?**
(b) Is the impulse response of this filter finite or infinite?

(c) What is the range of (real) values of r for which the filter is stable?

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(Btw, Wc is "Omega small subscript c")

Here's my answer (which is drastically wrong btw):

H(z) = Y(z)/X(z)

Y(z)(1-2rcos(WcT)z^-1 + r^2z^-2) = X(z)(1-2rcos(WcT)z^-1 + r^2z^-2)

y(n) - (1-2rcos(WcT)y(n-1) + r^2y(n-2) = x(n) - rx(n) + r^2x(n-2)-x(n-2)

y(n) = x(n) - rx(n) + r^2x(n-2) - x(n-2) + 2rcos(WcT)y(n-1) + r^2y(n-2)

It's important I obtain the answer to this correctly otherwise I can't go on to do parts (b) and (c) properly.

Appreciate if anyone knows the answer to this.