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So I'm trying to design a band pass filter in MATLAB (with a center frequency of 10kHz and a sampling frequency of 44kHz).

I have calculated the transfer function but I'm not sure how to enter this transfer function into MATLAB to generate a plot of the frequency and phase responses of the filter, and the poles and zeros.

Band Pass Filter transfer function calculation. Ωc (the cutoff frequency in radians) was already found to be 5pi/11, using the formula Ωc = 2π fc/fs = 2π*10000/44000 = 5π/11.

It could also be that the transfer function is calculated wrong? I haven't seen the denominator have so many terms before so I'm not sure..

This is my first time posting on SE, so any help is appreciated.

PS: I kinda always thought that the poles of a band pass filter would be pi/2 always, so I guess I was wrong?

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You've designed a second-order band pass filter, so the number of terms in the denominator is just fine (it's a second-order polynomial). Your design is OK, apart from the scaling, i.e., the maximum of the magnitude of the transfer function is not equal to $1$, which might be desirable.

Your transfer function has the form

$$H(z)=\frac{z^2-1}{(z-p)(z-p^*)}=\frac{z^2-1}{z^2-2zr\cos(\omega_p)+r^2}\tag{1}$$

where $p$ is a complex-valued pole:

$$p=re^{j\omega_p}\tag{2}$$

with radius $r$ and pole angle $\omega_p$. Note that in general the pole angle $\omega_p$ and the peak frequency $\omega_0$ (where the magnitude of the frequency response $H(e^{j\omega})$ achieves its maximum) are not equal. However, in your case they are almost identical because $\omega_p$ is relatively close to $\pi/2$. See this answer for more details on this matter.

If you want the frequency response peak value to be equal to $1$, you need to scale your transfer function by the factor $(1-r^2)/2$ (also explained in the answer quoted above).

The pole angle $\omega_p$ depends on the desired center frequency, so it can't always be equal to $\pi/2$ as you seem to have thought (even though in your calculations you chose it according to the desired center frequency).

The Matlab function freqz computes the complex frequency response of a discrete-time filter. For plotting the poles and zeros you can use the function zplane.

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  • $\begingroup$ Thanks a lot for the answer, Matt! I've read over both your responses and it helped me achieve the desired result. But I'm still confused about how to find the gain factor K so the peak gain is 1. By using (1−r^2)/2, I end up getting a peak gain of less than 1. Through trial and error (changing K and looking at the graphs each time), I found the value of K should be about 0.087. Here are all of my calculations for clarity: imgur.com/a/K7Rj6xX Any ideas on what I may be doing wrong? Thanks again! $\endgroup$ – NikhilSaxena Jun 24 '18 at 18:55
  • $\begingroup$ @NikhilSaxena: Your filter coefficients are correct. The problem must be in the way you plot the frequency response. $\endgroup$ – Matt L. Jun 24 '18 at 19:06
  • $\begingroup$ Ah that's good to know. Thank you! I've accepted and upvoted the answer now, but since my reputation is less than 15, the upvote won't be shown publicly :) $\endgroup$ – NikhilSaxena Jun 24 '18 at 19:32

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