A resonance filter is a specific second-order digital system designed to attenuate all frequencies except at and
around a given center frequency. The transfer function of a resonance filter is:
H(z)=(1 - r)(1 - rz-2) / (1 - 2rcos(ωcT)z-1+r2z-2)
where ωc = 2πfc, where fc is the center frequency (in Hz), and r is a parameter that controls the resonance
Show that, for any given resonance frequency ωc and any value of the parameter r, the amplitude response
of the filter equals unity at ω = ωc, i.e. |H(ωc)| = 1.
From the book Digital signal processing: applications and concepts by bernard mulgrew:
H(ω) = F[output] / F[input] = Y(ω) / X(ω) just as H(z) = Y(z)/X(z)
The Attempt at a Solution
Not entirely sure where to begin with this one. In most examples in the book going to H(ω) is done from the H(s) domain. Would I need to go change the transfer function from the z to s domain?
Also is it possible to consider it with the T = 0 inside the cosine to let the cosine value equal 1? making the equation easier to solve.
The steps given in the book to calculate output response of a digital sequence
(i) take the z-transform of the input sequence
(ii) multiply by the transfer function
(iii) take the inverse z-transform
Would I need to do these steps before converting to ω or suffice to do so after?
So far what I think could or should be included in the method is
it is not a proper fraction because highest numerator power equals highest denominator power, so divide both sides by z
use partial fractions to separate the functions (I think this is more so for difference eqn than first question)
The entire question is in the attached pdf and any hints about how to begin and if it is possible to get the answer working in z domain, making the cosine = 0 etc or if I'm on the right track at all would be much appreciated.