EEVblog Electronics Community Forum
Electronics => Beginners => Topic started by: TheUnnamedNewbie on August 05, 2016, 09:34:43 am
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I'm having some difficulties understanding how they go from the transfer function (in the frequency domain) to the amplitude response diagram. I understand going from the transfer function to the pole-zero plot, but the course book then mentions (translated from dutch):
"The behaviour of digital or discrete filters in the neighbourhood of poles and zeros is different (.. from that of continuous filters): ln the vicinity of a pole, a local peak will occur in the spectrum, where in the vicinity of a zero, a trench (I don't know what the correct English term for this is) will occur."
That all makes sense, but then as examples, they have these plots:
(http://teksyndicate.eu/pics/.nero/Dave/DSC_5251[1].JPG)
And this one:
(http://teksyndicate.eu/pics/.nero/Dave/DSC_5252[1].JPG)
Can someone explain to me how you go from the pole zero plot to the frequency response? Why is the first always 0 except around the two poles, and the second always A high value except at the frequency where the zero's and the poles are? Why is there no negative peak in the first plot around 0 and ws/2?
(Dutch custom book for the course by Prof. P. Wambacq at KUL - I hope he doesn't mind me posting images from his book on this forum)
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Conceptually, as the frequency goes from 0 to +pi (or -pi), you are tracing the unit circle around the origin (e^(jw) = e^(j*0) -> e^(j*pi)), and the magnitude approaches infinity the closer it approaches a pole, and approaches zero the closer it approaches a zero.
This is why a pole on the unit circle means that the system is unstable; the transfer function approaches infinity at the frequency where the pole occurs.
The first plot is tied to zero by the zeros at w = 0 and w = pi, and gets pulled higher around w = +/- pi/2 by the poles.
For the second plot, you can imagine that the further away from the pole/zero pairs you are, the more they cancel out, so the transfer function is constant. However, as you approach the pole/zero pairs, the zero dominates (it's on the unit circle), so the transfer function is pulled to zero.
There isn't a negative peak in the first plot because the zeros make the transfer function zero, not negative infinity, or some other value.
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That makes a lot of sense. Now that I've also spent more time studying the chapter, I think I'm starting to get a better understanding of what is going on. It was just the wording in the beginning of the chapter that put me off (from that I first thought I should see a flat line, with a bump in the neighborhood of the poles and a negative bump for zeros). I also go thrown off by the "constant" levels which, if I now understand it correctly, are determined by a factor that you cannot derive from the pole-zero plot. So I couldn't quite understand why the first transfer function was zero except around poles, and the second was one.
Thanks for the help!