# Pole zero plot, normalizing frequency response plot?

I'm asked to plot the frequency response (amplitude) given a specific pole-zero diagram.

$$H(z) = H_0 \frac{\prod\limits_{m=1}^{M} (z - q_m)}{\prod\limits_{m=1}^{M} (z - p_m)}$$

$$H(e^{i\omega}) = H_0 \frac{\prod\limits_{m=1}^{M}(e^{i\omega} - q_m)}{\prod\limits_{m=1}^{M}(e^{i\omega} - p_m)}$$

If I understood it correctly, the amplitude at frequency $\omega$ is (the magnitude of the distance from $e^{i\omega}$ to all the zeroes) divided by (the magnitude of the distance from $e^{i\omega}$ to all the poles), i.e:

$$\Big|H(e^{i\omega})\Big| = \Big|H_0\Big| \frac{\prod\limits_{m=1}^{M}|e^{i\omega} - q_m|}{\prod\limits_{m=1}^{M}|e^{i\omega} - p_m|}$$ where $q_m$ are the zeroes and $p_m$ are the poles and $H_0$ is the constant gain factor.

The problem is that in the graph where I need to draw the frequency response, the frequency and amplitude range $0\to1$ like so:

After I get a value by calculating the poles and zeroes I almost always get a value above $1$. What do I need to do with the value to fit it into the graph? How do I normalize(?) the value?

For the frequency axis, you're going to have to normalize by $\pi$ to plot digital frequencies in $[0,\pi]$.
For the magnitude axis, normalize by $\mathrm{max}\left(|H\left(e^{i\omega}\right)|\right)$.