The z-Transform - Synthesis Lectures on Signal Processing - page 68

For the direct evaluation, we substitute exp(j*pi/4) for z in the z -transform and make the following

call:

ansZD = 1./(1 - 0.9*exp(-j*pi/4))

which yields ansZD = 0.6768 -1.1846i . To estimate the frequency response using time domain correlation,
we obtain the impulse response (for the first 150 points, which should be adequate since it decays quickly),
and correlate with the first 150 samples of the complex exponential power series of exp(-j*pi/4)

ansTD = sum((0.9.ˆ(0:1:149)).*(exp(-j*pi/4). ˆ(0:1:149)))

which produces the identical answer to six decimal places.
A VI that allows you to drag a pole or pair of complex conjugate poles in the z -plane while
observing the magnitude and phase responses of the z -transform, as well as the real and imaginary parts
of the equivalent impulse response, is

DemoDragP olesZxf ormV I

an example of which is shown in Fig. 2.21.

Figure 2.21: A VI that allows the user to drag a pole or pair of complex conjugate poles in the z -plane
and observe the effect on impulse response and z -transform of an IIR formed from the pole(s).