The frequency response of a quadratic factor used as a filter
Here's where we make an important substitution. Until now, our z has been a mystery variable with no obvious relationship to the behaviour of a sinewave. Let's introduce the expression that actually defines the z-transform. The key relationships between z , z^-1 and the frequency f of a sinewave input can be expressed in either an exponential or a trig format:
where fs is the sample rate. You can make a mental connection to the effect of a small time delay (equal to the sampling interval) on the phase of a sinewave of frequency f. z is a complex variable that 'rotates round' the complex plane between purely real and purely imaginary, as the frequency f affects the argument of the trig functions.
You'll often encounter the exponential form  in filter books, but the complex trig form of  somehow seems more relevant to the engineer's habit of stuffing a sinewave into something and seeing what happens. So, let's substitute  into Part 1's equation  and see if we can develop an expression for the frequency response.
The first line of the boxed portion of  shows the real part, and the second line shows the imaginary part. Now, for a quadratic section to give a null in the frequency response, there must be a frequency fn for which Q(fn)=0. This means that both the real and the imaginary parts of equation  must be identically zero at that frequency. It should be easy to see that for the imaginary term in  to vanish for a frequency fn, we simply need to have
and if this is the case, we can substitute it back into the real part of  and set that to equal zero, from which we get
Bingo! This corroborates our earlier observation, and what's more, it gives us a direct expression for the quadratic factor that's needed in order to locate a frequency response null at any fn that we want. Figure 2 shows the frequency response of quadratic factors (three-tap filters) built using equation , for several notch frequencies.
Equations  and  give the real and imaginary parts, and we can see that there are some special cases. For zero frequency (z term = -2) and half the sample rate (z term = +2), the factor Q(z) is the product of two equal real factors, either (z-1)*(z-1) or (z+1)*(z+1). For a frequency of Fs/4, the roots are purely imaginary, and Q(z)=(z+j)*(z-j).
Figure 2: Factors of the form of equation  for various values of null frequency