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Alright, welcome to the lecture today. As always, a few short test questions. And you

noticed a little sheet of paper that's attached to your short test. That's the evaluation

numbers that most of you probably know. I'll say a few words about that later, but please

take your time now to read through the questions and note your answers.

Alright, thank you.

Alright, thank you.

Alright, thank you.

Alright, thank you.

Alright, thank you.

Okay, so even also for the late comers, welcome to the lecture. And let's go through the few

short test questions. So last week we completed the section on filtering and we spoke about

a few filter concepts. So filters in the frequency domain, for example. And we learned the concept

of the notch filter. So for the notch filter, you are building a band stop filter. So a

certain frequency band is stopped from going through the filter. And the question here

is how do you define the zeros of a notch filter in the set plane. So what's the connection

of set plane, zero positions.

So in the set plane, what frequency corresponds to this position of set equals plus one?

Zero hertz, that's correct. And if you look at the point just below one, if you come to

the zero, to the point of set equals one from this direction, so what point would this represent?

Two pi, that's correct. And what frequency? No, fs. Because here you have fs divided by

two, so half the sampling frequency. And as you might remember from signals and systems,

if you heard it, or if not, then you just have to accept that everything that happens

from zero to fs half in real valued signals is mirrored at this axis. So everything happening

below the real axis is just a mirror version of the upper part of the real axis. And that

means interesting for us is just this part. And that also rings a connection with the

Nyquist theorem, so you can reproduce just everything up to half the sampling frequency.

And also in digital filtering, only what happens until half the sampling frequency is interesting.

Everything else is just repetition. So that means that the interesting part of this filter

is from zero hertz to fs half. Let's assume fs to be a thousand hertz. So if I ask you

in an oral exam to design a notch filter, that for this signal that sampled at a thousand

hertz designs a notch at 50 hertz, where would you place the zeros? Anybody volunteers?

Yeah, can you give me the formula that we had in the lecture for that? For the angles?

It's a simple formula, so when I ask formulas in an oral exam or in an exam, I just ask

a simple formula. So it's fs, fo divided by fs times 2 pi. It's very simple. You can

even derive that yourselves if you want. And it's, by the way, the formula that you just

gave. So fo divided by fs half times pi, which is the same concept. So the zeros in the case

where we have the notch at 50 hertz and the sampling frequency of the thousand hertz would

be somewhere here, so at one tenth of the angle. Now is this correct, this image? Disregarding

the exact angle placement? Yes, thank you. So this answer gives you a 1.3, so almost

everything correct, and this answer gives you the 1.0. Okay, so the next question I'll

not do at the blackboard, so the schematic of a comb filter I have here because I'm lazy,

and you can all draw that yourselves. So we have notches, repeating notches, at 60, 180,

300, and 420 hertz, and I typically not ask you to draw a phase response of the filter,

which is the lower part of the plot, because it will not be exact anyway, and also schematics

don't make sense. So what we typically discuss, also in experiments, is for example, is this

filter a linear phase filter, and what's the advantages, and what does linear phase mean,

and so on. And by the way, this is also for your notes, this is the same plot that we

had in last lecture. And now we need a little bit more space, so 3. So what's the Wiener-Hopf

equation, the optimal filter. This is, no, just warming your hands. We talked about it,