Digital filter response question

[jmher0403]

Homework Statement

A digital filter is defined by y_n = x_n - _n-2 - 0.81*y_n-2.

At what frequency, in terms of f_Nyquist, is the filter response zero and maximum?

Homework Equations

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

The Attempt at a Solution

I figured out that H(z) = [z²-1] /[z² +0.81]

and F(w) = [e^2jwt -1 ] / [e^2jwt + 0.81]
= [cos(2wt) + isin(2wt) - 1] / [cos(2wt) + isin(2wt) + 0.81]


Please give me some advice on how to go on about finding the frequencies please...

Hint was given that wt = ∏ f/f_Nyquist
please also explain why this is so...


Thanks a lot in advance :D

 

[jmher0403]

I figured out the frequencies that give zero are 0 and fN

any hints on how to find the frequency that gives maximum response?/?

 

[milesyoung]

You could use a bit of calculus to find minima/maxima of |F(ω)|, you've probably done this plenty of times before with real-valued functions.

F(ω) has a simple property, though, that, by inspection, enables you to find the value of ω that maximizes |F(ω)|. When the magnitude of the numerator attains its maximum value, the magnitude of the denominator attains its minimum value. That means something for the magnitude of the fraction.

I don't know how familiar you are with complex numbers and their interpretation as vectors in the complex plane, but if this means something to you, you might interpret the numerator as the vector sum of 1∠2ωT and 1∠180°, where T is the sampling time of your system (one might confuse an expression with t as a time-domain expression).