Analytically show that the magnitude squared frequency response equals unity at all frequencies. i.e |h(w)|sq?

Question

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

H(z) = a+(z to the power -1) / 1 +(a*z to the power -1)

I have not got a clue how to work out this question any help most welcome.

Answer 1

|H(e^(jωT))| will give the gain. So let's set z=jωT

(a+e^(-jωT))/(1+ae^(-jωT))=

(a+cos(-ωT)+jsin(-ωT))/(1+acos(-ωT)+jasin(-ωT))

The magnitude squared is

((a+cos(ωT))^2+sin^2(ωT))/(1+acos(ωT)^2+(asin(ωT))^2)

expanding
(a^2+2acos(ωT)+cos^2(ωT)+sin^2(ωT))/(1+2acos(ωT)+a^2cos^2(ωT)+a^2sin^2(ωT))

intermediate step
a^2cos^2(ωT)+a^2sin^2(ωT))=a^2

simplifying
(a^2+2acos(ωT)+1)/(a^2+2acos(ωT)+1)

this is magnitude 1