W2_L4_P5-Other short-time transforms

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.