Hi all, let me explain what that title is on about.

My z-transform function is:
\( \frac{1}{2j} \left(\frac{z}{z-exp(jw_0T)} - \frac{z}{z-exp(-jw_0T)}\right)\) (1)

Now, the question asks for the phase, where \(w_s = \frac{2\pi}{T}\)
for \(w_s = 3w_0\)

So I make the answer to be:
\(w_0 = \frac{2\pi}{3T}\) Just by combining the two equations together and then substituting this into (1) gives:

\( \frac{1}{2j} \left(\frac{z}{z-exp(\frac{j.2\pi}{3})} - \frac{z}{z-exp(\frac{-j.2\pi}{3})}\right)\)

So \( \phi = \pm e(\frac{j.2\pi}{3})\) ?

The answer that has been provided has

\( \phi = \pm e(\frac{j.\pi}{3})\)

 

I would think you would need to make the substitution

\(z=exp(j\omega T\)

in the transfer function.

This would give the frequency response traversing the unit circle as the angular frequency ω varies.

To find the phase limits one presumably evaluates the function at ω=0 and ωs.

Interestingly at w=w0 your function leads to a case of division by zero.