So all these cartoon drawing, hopefully, is intuitive. I want this, this to be
intuitive to you. This is, critical. The rest is just math, right? The rest is,
there is a formula, whether you memorize the formula or not doesn't make a
difference. If you don't know that concept in these cartoons, then you're in trouble,
okay? Good? Fat window, skinny window, you get to decide and, for the application you
have, what you get to do is just say, hey, I want to build a Gabor transform. How
should I, engineer it? Well, you'd better ask what kind of application you're,
you're looking at. Do I really need good time resolution? Do I need good frequency
resolution? Which one's more important to me? That would tell you how you should
dry, design your, your Gabor window and how fat it is, okay? Cuz you get to pick
it. It's not like there's a formula. Here's what, how you should pick your
Gabor window to be. There is nothing like that. It's all kind of manipulation that
you get to do, okay? Questions on that or does everybody feel good about it. Now.
Yes. I mean, you can like count the pixels cross and then just the Window open.
Depends on how you use the, the Window. Depends on how you use Windows? Yeah, you
don't say that constant that window here. You just like change your window every
time. Okay. Oh, well so what you do is you slide the Window across and by the way if
you guys, have you guys all gone to the web page? Have you looked at that little
picture on the right? That's just a spectrogram and you know what the
difference in the four pictures are? That width so as I run different widths
through. If you, so if you, we, we're going to do that example in class, but as
you run the different widths through what you're really doing is saying, Hey, let me
analyze this, and let me change the width and see kind of what I need to get. You
don't know ahead of time what you should pick, but since you have freedom, and
MATLAB was easy, you just do it and see what you get, okay? So, and you can maybe
use information from both, righ t? Your signal is there and you can process
it in different ways, okay? a couple comments, first. Just like before you
transform. When we actually apply this Gabor transform it's not on an infinite
domain. It's not an integral where you integrate over all time and all space.
Now, like Fourier transform you're going to say I'm going to discretize my time and
I'm going to discretize my space. If you have measurements of a signal, presumably
you've got a clock. You don't have the signal with infinite intestinal precision.
What you have is okay I'm going to sample my signal every second, for twenty
minutes. That one second sets your discritization. Right? You always have a
discritize. So it's always a finite number of frequencies. Finite number of time
points. Okay. That's comment number one. Number two, I wanted to at least introduce
you to a couple other kind of important applications of this or not applications
but a couple other for short fort short, a couple of other short alright ready short
time Fourier transform methods, okay? That's what I meant to say it took me a
little while. one of them and by the way the reason I introduced these two in
particular is because they are particularly important for radar sonar
technologies, okay? So all we're doing here is very broad-brush strokes. This
stuff was forefront like, research back in the 50s and 60s when radar technology was
being developed, right? And so, I mean, Gabor, the, the power of what Gabor did
was that they realized early on that this fourier transform, although cool is so
limited cause you lose all the time information. Which means you can't
localize where's this plane coming from, I know there's a plane, right? If you pick
up a frequency omega not and go hey there's a plane out there somewhere. The
whole point is to say oh, and its over there about five miles. If you can't make
that statement, then it's stupid right? sweet, there's a point somewhere, k. So,
they work on it very early on, and a couple of the more important ones, one of
them is called the Zac transform also known as the Weil-Brezin, okay? and here
it is. Someone's got an iPhone. Oh, negative infinity, infinity so here's the
discreet version of it you take this function, right? And, you see what you do
is you don't necessarily build a, a filter when instead you do, is you basically
slide your function across the frequencies, okay? So this is very
important for periodic signals or quasi-periodic signals. This transform
here. I don't want to, want to say anything else about it except that I just
want you to know that transform and the other one I want you to have heard about
was is this one here the Vigner. Excuse me. Yes. . good question I don't think I
even specify here what it was. it looks like we won't have time to talk about
that. I don't know what it is. I, I didn't write it down. But typically what it is, I
think it's going to be it's a sequence going to be over the domain time where you
discretize A of N. I think what it does is, this A of N is you take your domain
from zero to time capital T. You chop it up into a certain number of points and
then what you do is you recenter this across. What's that? Square root, yeah,
okay? The other one that's important is the Vigner-Ville. and let me show you this
one. This is probably the most important in terms of radar applications cuz this is
kind of what people use a lot and this is and the vigular transform is actually
probably, one of the more famous ones when you start looking at time frequency
representations of signals. And here it is. Oh, I just won 25 cents. There's a
typo in there, you know it's right there. That couch should be up here, not down
there. I'm going to go give myself 25 cents as soon as I get back to the office.
And you didn't even catch it did you? He's already got $1.25 from me, but at least I
saved myself $1.50. So, alright, this is the transform. Again, generically, you can
pick different Gs But this Vigner-Ville basically, notice what it does. It takes
this function. Offsets it tell it over two one way. Takes this th ing, tell it over
two the other way. And then you slide, you slide it this, so in one version of this
thing, you're doing this, the other version you're doing this, dance moves I
know, you could go to the Microsoft store, have you ever gone down there, I got kids
that go down there, I think, I, anyway we won't talk about dancing.