I have a following signal given:
$$S(n) = [-1, 0, 2, -1]$$
I have to find the Z-transform of the signal and find amplitude-frequency characteristic and amplitude-phase characteristic of the signal.
So far I've managed to get to the point that the Z transform of the S(n) signal will be:
$$Z(z) = 2z^{-2} - z^{-3} - 1$$
However, I have no idea where to go from this. How do I find the characteristics?
Suppose the Z-transform of your time sequence $s(n)$ is denoted as $S(z)$, you already got $S(z)=-1+2z^{-2}-z^{-3}$. The magnitude-frequency characteristics can be obtained as
$|S(e^{j\omega})|=|-1+2e^{-j2\omega}-3e^{-j3\omega}|$.
If you use MATLAB, you can use the function of 'freqz.m' to do this. For example,
$>> [h, f] = freqz(s,1,n,fs);$
$>> plot(f, abs(h));grid$
where, $s=[-1\ \ 0\ \ 2\ \ -1]$ is your time sequence, $fs$ is the sampling frequency, $n$ is the number of points (default 512) to calculate. Then the magnitude is $|h|$.