I have checked the theory to calculate the magnitude of frequency response from the pole-zero plot from the previous posts. As far as I understand(and I hope I am correct), the magnitude can be calculated from this formula.

|$H(z)| = \frac{|\prod_{n=0}^{n=\infty} (z-z_n)|}{|\prod_{n=0}^{n=\infty}(z-p_n)|}$

So, for this question, (no 11, fig. a), here

To visualize the frequency plot, I wrote the following Python code,

w = np.linspace(-np.pi,np.pi,100)
z = np.exp(1j*w)
cof1 = (0.9+0.1j)
cof2 = (0.9-0.1j)
eq = z**2 - ((cof1+cof2)*z)+(cof1*cof2)  //to find the equation from the roots
num = (z+1)**3
den = (z-0.8)*eq
y = num/den

The code is not great but it kind of works (I think so). And, I took some approximate values for coefficient of poles. Anyway, I got the following output, here

which is wrong. The correct output is, here

I don't understand, where I went wrong. Why is there a significant gap in the magnitude between my output and the correct answer? I used the same code to calculate some other plots and it worked fine. But in this particular question, it didn't work. I hope my code is not wrong. Could anybody help me with this? I am trying to play around with the poles and zeros to see its relation with the magnitude of the frequency response curve.

  • $\begingroup$ I don't see anything in that figure given in the solution. What's that supposed to be? Your magnitude plot looks fine, it's just a low pass filter. Take a look at these questions for the relation between pole-zero plots and frequency responses: Q1, Q2, Q3. $\endgroup$
    – Matt L.
    Commented Feb 13, 2021 at 19:13
  • $\begingroup$ @MattL. there is a small bump between $-\pi/2$ and $\pi/2$. Though the magnitude is very small. This is the answer sheet provided by the lecturer and I don't understand it. And the answer to the rest of the figures is also similar. I can't seem to figure out the difference. $\endgroup$
    – Rima
    Commented Feb 14, 2021 at 6:11
  • $\begingroup$ I think I got my mistake. I should have used the range between -1 to 1 instead of $\pi$ and calculated in terms of z rather than $e^(j\omega)$ because of which there is a large gap in the magnitude. thanks for the reference. $\endgroup$
    – Rima
    Commented Feb 14, 2021 at 8:54
  • $\begingroup$ I don't think that you made a mistake. The frequency response is obtained by using $z=e^{j\omega}$, and $\omega$ is in the range $[-\pi,\pi]$. I'm quite sure that the problem lies in the solution. I mean, what are those strange lines supposed to be that extend over all the figures? $\endgroup$
    – Matt L.
    Commented Feb 14, 2021 at 11:49
  • $\begingroup$ This question has a few very related exercises. Many more nice matching exercises can be found here. $\endgroup$
    – Matt L.
    Commented Feb 14, 2021 at 12:34


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.