sketching magnitude of frequency response of H(z)

Question

I am trying to plot the magnitude of H(z), I got it to factore, and would like to sketch its plot of magnitude. But I am have trouble evaluating the function. $$ H(z)= \frac{(1-2z^{-1})(1+0.5z^{-1})(1+0.9z^{-1})}{(1-z^{-1})(1-0.7jz^{-1})(1+0.7jz^{-1})} $$

I know we replace z with $e^{jw}$ and try to evaluate it at various w such as pi, pi/2 etc. But that is really complicated and when I tried it I couldn't get the right answer as the text book. Is there an easier way to evaluating magnitude and phase by hand?

Thanks

Answer 1

Since you have the transfer function factored into poles and zeros, it's easiest to draw the pole-zero diagram to get an idea of the overall shape of $H(z)$. The first thing you notice is the pole at $z=1$, i.e. at DC. So your filter is unstable and $|H(e^{j\omega})|$ goes to infinity at DC. Furthermore there's a complex conjugate pole pair on the imaginary axis, i.e. at half the Nyquist frequency. So you can expect a small bump around that frequency. Finally, I would calculate the value of $H(z)$ at Nyquist, i.e. at $z=-1$:

$$H(-1)=\frac{3\cdot 0.5\cdot 0.1}{2\cdot 1.49}\approx 0.05$$

A more detailed sketch than that is indeed difficult, but that's probably not the idea of the exercise.