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.