W01_L03_P03 - Moving signals and averaging in frequency

Alright. So, one other problem associated
with this, which is you might say,

something like, well, okay, this is all
fine and good. In fact, if it's really

noise, then even if I go to the time
domain, I could just take my time domain

and just kinda like average my time domain
and I should get the little bump out that

I'm expecting in the time domain. Okay. So
I'm gonna show you one example where that

actually does not work. Cuz what happens
if it's moving in a time domain? That's

what we wanna address. Okay? Everybody
good with that? Kinda cool? Simple

filtering, simple making use of the fact
that you know something about noise.

Alright. So, let's do something a little
more complicated and let's come back up

actually to the board real quick here and
to motivate it, cuz I, I just you know,

it's hard to get motivated. That's helping
out. Alright. If I could get a [INAUDIBLE]

Oh, there's my eraser. Alright. So here is
the deal. If you really do have this

airplane flying around up here, it's
moving, and in some sense, you kinda go

like you know, presumably it's moving
pretty fast. Okay? So in 30 seconds, this

thing can go a pretty long ways if you've
got a plane going you know, close to Mach

one or Mach two or Mach three.
Okay? Normally, we think of that going

super fast and you're saying, well, I have
a signal now that as I keep taking

information out of this, the information
actually is translating along. Okay? So,

if you are thinking about saying, well, I
could just send a time domain, take the

signal, and just average it. Well, the
signal's here and now it's moving that way

like my, Mach two.
Okay? So you can't just average this out,

because this will just average out to zero
as well. The signal is gone. If you look

in the time domain, you average a moving
signal, it's gone. However, the frequency

signature doesn't change. So if I bounce
stuff off of here, it would still come

back, as, let's say we're making up, that
doesn't change, so my frequency signature

doesn't change. Now, if you are really
clever engineer and you could actually

find a way to make your plane shift its
frequency and start getting frequent,

signatures, frequency signature all over
the place, the filter wouldn't be able to

pick it up very well. Right? Cuz you'd
shift it you know, oh, wait, wait a

minute, I was, I was filtering over here
but now you've moved your you know, if you

can, if you can do something like that.
Okay? Through frequency conversion then

you could screw up a detector, but what
we're gonna do is very simple, which is if

you do this, no is coming back. This thing
is moving and if you try to average in the

time domain, you get zero. You get nothing
out of it. However, you average in the

frequency domain, you can recover
everything. Okay? Again, like your dog

problem. Nathaniel, how's it going back
there? Do you go by Nate? Nathaniel?

Nathan? Nate, all right. Okay, so there's
no confusion, because I go by Nate Dogg,

cuz it's my rapper name. So as long as you
stay away from that one, we're all good.

Okay. So the dog problem Nate, right, is
there's this moving marble in the

intestine of your dog that you gotta blow
up. So you gotta know where it is

accurately or else it'll blow up other
parts of the dog. By the way, it's kind of

interesting. anybody in biomedical
engineer in here? Right. So is there

somebody in my class that had this, couple
years back that they were looking at, like

trying to you know, shoot kidney stones.
And, I was amazed when they're talking on

the side in a thesis, how many times they
miss? Like, they shoot these big things to

crush the kidney stone, but you know,
people move, fluid stuff moves, and then

it just misses. Well, that does damage to
other parts of your body. Anyway, just

want to let you know that. So sit really
still if you gotta get a shoot kidney

stone shot. Okay? And so that's what
you're doing with your dog. The particle

itself is moving and the signature is
constant in frequency. So you gotta have

to figure out where that frequency sits
and go after that. Okay? Alright. So

that's also here, it's moving, that's
okay. If this is moving super fast, all

you got to key on is the frequency
signature because that plane can't shift

the, can't shift the frequency that you're
sending out. Yeah. Would effects due to

the Doppler Effect ever be important in a
problem like this or? Yeah, in fact you

know, what I'm talking about is such basic
radar stuff, right? But, actually, you can

do all kinds of data processing making use
of that. Yeah, in fact, you would, you

would definitely make use of that. That
would give you sort of, for instance, how

fast is that thing moving, much more
accurately and quickly. It's giving you

some kind of information. Okay. So let's,
let's program up, then a signal and I'll

show you sort of what it might look like
in time and frequency before you

pre-process. And, you get to learn a
little bit of fancy MATLAB. Okay. So,

let's come back to here and what I wanna
do is, I wanna plot a signal that's moving

in time and then I wanna plot what its
frequency spectrum looks like as well.

Okay. So let's come back up to here and
I'm gonna kill it from there. Okay. So we

still have all our stuff here. We have our
k, our t, and now what I'm gonna do in

fact, I think I made this a little bit
bigger. I'm gonna do this to be 60. So,

here's what I'm gonna do. I'm gonna define
a thing called slice. It's a vector, goes

from zero, steps from 0.5 to ten. This is
gonna be like my time slices times zero, I

take a reading. A time and a half, I take
a reading. could be so for instance that I

take or, or half a minute, I take a
reading. Whatever this happens to be in

units and I just keep taking readings, at
every 0.5 all the way to ten. So I have a

total of 21 readings I'm gonna take the
data, 'kay, which isn't a whole lot, but

fine just for example. And what I wanna do
is define some new variables, call them T

and S, which is gonna be what's called a
meshgrid(t,slice).

Yes? You have already, already defined
capital C on line three. Is that gonna

conflict it all with. Oh, I just overwrote
it. That's okay. Yeah, but. Tha t's okay.

Yeah, I mean, if I wanted to use this
again at some point, this is the kind of

programming mistake that's so rookie, I
would never make it. I just did it in

class to show you what kind of mistakes
you can make. Yeah, rookie move. We'll

stay with it though, just to show that no
matter how far in life you go. you know, I

have a couple of mantras about teaching
and you've heard them I think, in my class

before it knows, there's two things you
never do, ever. I mean this is the best

advice I have to give ever in my whole
life. first you don't do, you don't spell

in public and you don't do algebra in
public, cuz you will eventually do

something so stupid, like wow, like I
can't spell dog or can't add two to ten to

get cuz you're up there you know, and
you're nervous and everybody sees it, so

just don't do if you don't have to. I
would love to say don't program in public.

But unfortunately for this class, I cannot
get around that one. Alright. So there you

go. now what does the meshgrid do by the
way? What I'm thinking about right now, is

I'm taking time slices of data, so what t
is, right? Little t, here. T is, goes from

you know, -30 to 30 and I have a signal on
that. So, I have this domain where, okay,

so I have, for every, every little burst
of time, I take examples of length 60.

Okay? And I divide that 60 by 512, and
then, 0.5 units later is a different

units. I take another sample, another
measurement, again, with 512 points in it

and another in 512 points. Okay? So I'm
gonna collect this and what meshgrid does,

it gives us a sense of direction. It makes
a two-dimensional grid where in

one-dimension, it's time, in the other
dimension, it's this slice variable. Okay?

So t and slice now become capital T and
capital S, which are now matrices

conveying information about you know, ones
in one direction. One is in an orthogonal

direction to it. Okay? I wanna use those
to define our signal and we also need this

in the frequency domain, K, S. We're gonna
do the same thing here with the frequency

components .
In one direction, it's the wave numbers,

but then, it's wave number per sample in,
in the other direction, it's number

samples. Okay? Now, let's define a
function. Here it is, u=sech Now, I use

these variables cuz now they know about
their direction and I'll just show you

what I've got here, and so you can, it's a
simple function. I'm gonna, I'm gonna

leave this zero for right now. I'm gonna
tell you why in a moment cuz I wanna shift

the center frequency around. This is a
center frequency. If I leave it at zero,

it means that in the frequency domain I'm
centered at k equals zero, but we can make

this anything we want. I can make it
centered at k equals whatever else. I

want, in fact, in your dog problem, I'm
telling you right now if you put a filter

around k equals zero, your dog will die.
Tim, don't let your dog die. Okay.

Alright? So, alright, so don't put a
filter near zero, cuz it won't save

anybody. Alright. So that's gonna be my
function and so what I'm gonna do is

subplot this. I'm gonna say, okay, I'm
gonna use what's called the waterfall

command. And I like waterfall, because
it's a black and white picture, and it's

nicer to plot in color, however, I want to
set a view angle on it. The problem with

plotting in color is if you go to a
journal, it costs you a lot of money.

Okay. So there is like my plane moving
around, whatever you want to call it. This

thing is moving in time, right? So I have
this signal that's moving around and I

made a very simple movement. I made it
move like a sign wave in the slice

direction. So, it's just doing this. Okay?
All right. Now if I add noise to this and

I. No it's time domain, here. We're going
to look at the frequency in a moment. So,

if I add noise to this you know, we could
bury this whole thing, we could hide it.

And he said, well, yeah but I could just
average it to zero. If you average that,

this stuff all gets washed out. You get no
signal out. You have to do it in a

frequency domain. Okay? Alright. So, let's
take a look at the frequency domain of

this thing. So I have the signal moving
around in time, but if I just take this

Fourier transform then I can plot this
thing there. So, okay, so let's go ahead

over here. That's the first plot, and, so
I gotta do is that gonna go through this

and one slice at a time for every slice
take Fourier transform. Okay? So I say,

okay, fine. So what I'll do is I'll go
over here and I'm gonna say I will go grab

the first, first, let's say row of this u.
I'm going to ftt it. I just Fourier

transformed that first row and what I'm
going to do with this Fourier transform is

I'm going to plot it. So let's call this
UT j, okay? And, once I go through all of

this. Oh, come on. And actually, I don't
care about fft j or I, what I want to do

is maybe plot also, if fftshift it right
now. Okay? Cuz I just want to plot what

this looks like. There is my thing there.
And, might as well take the absolute value

while we're at it, for a moment, we're
gonna just this thing, absolute value. So

what I did here is I went to each row,
Fourier transformed it, then I have to

fft(shift) it with, with the absolute
value. And then what we can do is say,

okay, how about we look in subplot(2) and
I'm gonna do the waterfall again. But now,

I need the fftshift. Actually, did I do
this yeah, of, of K versus S versus UT.

Okay? So that's gonna be this Fourier
Transform. Oh, I should set the same view

angle, by the way. Sorry. Okay. So here's,
here is, here is perfect signal, which

doesn't exist in reality, right? Which is
I have this thing moving around and it's

moving in time. Tell me what it's doing in
frequency domain. Nothing. Awesome, right?

Because it's very important you know
what's happening in time and the whole

section at the beginning is kind of
understanding this idea between time

frequency dynamics. The time domain can be
doing all kinds of stuff, only that

frequency fixed. Okay? Remember, when we
do the noise reduction, your key thing is

to say, well, if I, if this thing is fixed
in frequency, all I got to do is build a

little filter out there in freque ncy,
capture that out. Okay? Cuz this thing is

not moving around. So I can be moving all
over the place, flying around my jet,

you're still giving off a fixed frequency
and this is what you'd lock in on, if

you're gonna do a detection.