# sketching magnitude of frequency response of H(z)

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

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