# Drawing the modulus from a Transfer function

As you may guess from my other questions i am a student so pardon me any ignorance and guide me to the truth. Thanks

In this exercise from an exam i was given that transfer function with poles and zeros. I was asked to draw the modulus and the "UHD imagive below the Transfer function" image was my answer.

I know that when you encounter a zero, you "go down" and when you encounter a pole you "go up" in the graph. When you encounter both "aligned", you get a cero as they cancel each other.

The answer from the teacher was that the maximum, which is 1, is at pi/8 (and -pi/8), not at pi/4, because at pi/4 (and -pi/4) he draw the minimum.

Why is that? and what i did got wrong or missed in my understanding?

Thanks for answer.

EDIT: Here i add another example from a recent exam. I would like if someone could do a brief/short explanation in how to properly draw it and what should i know or notice.

The question is the same, but i lack proper understanding of it.

• You might find this answer useful in developing your understanding about the way poles and zeroes work together in a frequency response. – A_A Jan 23 '18 at 16:38
• So, "my maximum" that i set at pi/4 couldn't be because the function will go into the infinity? – WhiteGlove Jan 23 '18 at 16:43

## 1 Answer

Your sketch is not bad at all. I don't know why your teacher drew a "minimum" (do you mean a zero?) at $$\pi/4$$, because there shouldn't be any. The zeros are clearly at $$\pm\pi/2$$ and at $$\pi$$, and the main peak should indeed be close to $$\pi/4$$. By the way, I'm missing a pole. There are $$4$$ zeros and only $$3$$ poles. The fourth one should probably be at the origin. Maybe your teacher didn't give this exercise as much thought as he should have ...

EDIT:

Concerning your second example, it's always the same: a pole gives a bump in the magnitude response, the closer the pole to the unit circle, the more pronounced (sharp) is the bump. A zero on the unit circle gives an exact zero in the magnitude response, a zero away from the circle gives a dip. The closer the zero to the circle, the more pronounced (deeper) is the dip. Note that the exact maxima and minima in the magnitude response caused by poles and zeros close to the circle are often not exactly at the frequencies that correspond to the angles of the poles or zeros, but often they are close. This is all about a basic sketch, so no need to try to be overly precise.

• i can post another similar exercise, the exam is tomorrow and i'm having quite a problem with it. – WhiteGlove Jan 24 '18 at 18:55
• @WhiteGlove: As I said in my answer, you basically got it right. – Matt L. Jan 24 '18 at 18:57
• I will upload another, i wonder if you could help me out and understand it well with some more detail than just my crappy graph. I'm really tired of this subject... – WhiteGlove Jan 24 '18 at 19:17