0:00:00.000,0:00:09.112
Alright. So, one other problem associated[br]with this, which is you might say,

0:00:09.112,0:00:13.526
something like, well, okay, this is all[br]fine and good. In fact, if it's really

0:00:13.526,0:00:18.060
noise, then even if I go to the time[br]domain, I could just take my time domain

0:00:18.060,0:00:23.131
and just kinda like average my time domain[br]and I should get the little bump out that

0:00:23.131,0:00:28.270
I'm expecting in the time domain. Okay. So[br]I'm gonna show you one example where that

0:00:28.270,0:00:33.079
actually does not work. Cuz what happens[br]if it's moving in a time domain? That's

0:00:33.079,0:00:37.172
what we wanna address. Okay? Everybody[br]good with that? Kinda cool? Simple

0:00:37.172,0:00:42.300
filtering, simple making use of the fact[br]that you know something about noise.

0:00:43.300,0:00:53.006
Alright. So, let's do something a little[br]more complicated and let's come back up

0:00:53.006,0:00:58.386
actually to the board real quick here and[br]to motivate it, cuz I, I just you know,

0:00:58.386,0:01:04.227
it's hard to get motivated. That's helping[br]out. Alright. If I could get a [INAUDIBLE]

0:01:04.227,0:01:12.570
Oh, there's my eraser. Alright. So here is[br]the deal. If you really do have this

0:01:12.570,0:01:22.371
airplane flying around up here, it's[br]moving, and in some sense, you kinda go

0:01:22.371,0:01:27.903
like you know, presumably it's moving[br]pretty fast. Okay? So in 30 seconds, this

0:01:27.903,0:01:33.503
thing can go a pretty long ways if you've[br]got a plane going you know, close to Mach

0:01:33.503,0:01:38.023
one or Mach two or Mach three.[br]Okay? Normally, we think of that going

0:01:38.023,0:01:43.150
super fast and you're saying, well, I have[br]a signal now that as I keep taking

0:01:43.150,0:01:48.412
information out of this, the information[br]actually is translating along. Okay? So,

0:01:48.412,0:01:53.607
if you are thinking about saying, well, I[br]could just send a time domain, take the

0:01:53.607,0:01:59.064
signal, and just average it. Well, the[br]signal's here and now it's moving that way

0:01:59.064,0:02:03.243
like my, Mach two.[br]Okay? So you can't just average this out,

0:02:03.243,0:02:09.181
because this will just average out to zero[br]as well. The signal is gone. If you look

0:02:09.181,0:02:17.342
in the time domain, you average a moving[br]signal, it's gone. However, the frequency

0:02:17.342,0:02:23.878
signature doesn't change. So if I bounce[br]stuff off of here, it would still come

0:02:23.878,0:02:29.028
back, as, let's say we're making up, that[br]doesn't change, so my frequency signature

0:02:29.028,0:02:34.460
doesn't change. Now, if you are really[br]clever engineer and you could actually

0:02:34.920,0:02:40.941
find a way to make your plane shift its[br]frequency and start getting frequent,

0:02:40.941,0:02:45.756
signatures, frequency signature all over[br]the place, the filter wouldn't be able to

0:02:45.756,0:02:49.454
pick it up very well. Right? Cuz you'd[br]shift it you know, oh, wait, wait a

0:02:49.454,0:02:53.395
minute, I was, I was filtering over here[br]but now you've moved your you know, if you

0:02:53.395,0:02:57.141
can, if you can do something like that.[br]Okay? Through frequency conversion then

0:02:57.141,0:03:01.374
you could screw up a detector, but what[br]we're gonna do is very simple, which is if

0:03:01.374,0:03:05.753
you do this, no is coming back. This thing[br]is moving and if you try to average in the

0:03:05.753,0:03:09.549
time domain, you get zero. You get nothing[br]out of it. However, you average in the

0:03:09.549,0:03:14.288
frequency domain, you can recover[br]everything. Okay? Again, like your dog

0:03:14.288,0:03:21.582
problem. Nathaniel, how's it going back[br]there? Do you go by Nate? Nathaniel?

0:03:21.582,0:03:27.146
Nathan? Nate, all right. Okay, so there's[br]no confusion, because I go by Nate Dogg,

0:03:27.146,0:03:31.956
cuz it's my rapper name. So as long as you[br]stay away from that one, we're all good.

0:03:31.956,0:03:36.012
Okay. So the dog problem Nate, right, is[br]there's this moving marble in the

0:03:36.012,0:03:40.358
intestine of your dog that you gotta blow[br]up. So you gotta know where it is

0:03:40.358,0:03:44.531
accurately or else it'll blow up other[br]parts of the dog. By the way, it's kind of

0:03:44.879,0:03:49.051
interesting. anybody in biomedical[br]engineer in here? Right. So is there

0:03:49.051,0:03:54.163
somebody in my class that had this, couple[br]years back that they were looking at, like

0:03:54.163,0:03:59.455
trying to you know, shoot kidney stones.[br]And, I was amazed when they're talking on

0:03:59.455,0:04:04.262
the side in a thesis, how many times they[br]miss? Like, they shoot these big things to

0:04:04.262,0:04:08.717
crush the kidney stone, but you know,[br]people move, fluid stuff moves, and then

0:04:08.717,0:04:13.706
it just misses. Well, that does damage to[br]other parts of your body. Anyway, just

0:04:13.706,0:04:19.197
want to let you know that. So sit really[br]still if you gotta get a shoot kidney

0:04:19.197,0:04:24.218
stone shot. Okay? And so that's what[br]you're doing with your dog. The particle

0:04:24.218,0:04:29.441
itself is moving and the signature is[br]constant in frequency. So you gotta have

0:04:29.441,0:04:34.317
to figure out where that frequency sits[br]and go after that. Okay? Alright. So

0:04:34.317,0:04:38.684
that's also here, it's moving, that's[br]okay. If this is moving super fast, all

0:04:38.684,0:04:43.230
you got to key on is the frequency[br]signature because that plane can't shift

0:04:43.230,0:04:47.918
the, can't shift the frequency that you're[br]sending out. Yeah. Would effects due to

0:04:47.918,0:04:52.052
the Doppler Effect ever be important in a[br]problem like this or? Yeah, in fact you

0:04:52.052,0:04:56.291
know, what I'm talking about is such basic[br]radar stuff, right? But, actually, you can

0:04:56.291,0:05:00.372
do all kinds of data processing making use[br]of that. Yeah, in fact, you would, you

0:05:00.372,0:05:04.506
would definitely make use of that. That[br]would give you sort of, for instance, how

0:05:04.506,0:05:08.431
fast is that thing moving, much more[br]accurately and quickly. It's giving you

0:05:08.431,0:05:14.828
some kind of information. Okay. So let's,[br]let's program up, then a signal and I'll

0:05:14.828,0:05:20.785
show you sort of what it might look like[br]in time and frequency before you

0:05:21.429,0:05:30.101
pre-process. And, you get to learn a[br]little bit of fancy MATLAB. Okay. So,

0:05:29.810,0:05:37.579
let's come back to here and what I wanna[br]do is, I wanna plot a signal that's moving

0:05:37.579,0:05:45.056
in time and then I wanna plot what its[br]frequency spectrum looks like as well.

0:05:45.056,0:05:52.835
Okay. So let's come back up to here and[br]I'm gonna kill it from there. Okay. So we

0:05:52.835,0:05:59.209
still have all our stuff here. We have our[br]k, our t, and now what I'm gonna do in

0:05:59.209,0:06:06.118
fact, I think I made this a little bit[br]bigger. I'm gonna do this to be 60. So,

0:06:06.118,0:06:14.802
here's what I'm gonna do. I'm gonna define[br]a thing called slice. It's a vector, goes

0:06:14.802,0:06:20.334
from zero, steps from 0.5 to ten. This is[br]gonna be like my time slices times zero, I

0:06:20.334,0:06:26.229
take a reading. A time and a half, I take[br]a reading. could be so for instance that I

0:06:26.229,0:06:31.340
take or, or half a minute, I take a[br]reading. Whatever this happens to be in

0:06:31.340,0:06:36.881
units and I just keep taking readings, at[br]every 0.5 all the way to ten. So I have a

0:06:36.881,0:06:41.876
total of 21 readings I'm gonna take the[br]data, 'kay, which isn't a whole lot, but

0:06:41.876,0:06:47.349
fine just for example. And what I wanna do[br]is define some new variables, call them T

0:06:47.349,0:06:54.493
and S, which is gonna be what's called a[br]meshgrid(t,slice).

0:06:54.497,0:06:59.965
Yes? You have already, already defined[br]capital C on line three. Is that gonna

0:06:59.965,0:07:04.149
conflict it all with. Oh, I just overwrote[br]it. That's okay. Yeah, but. Tha t's okay.

0:07:04.149,0:07:09.609
Yeah, I mean, if I wanted to use this[br]again at some point, this is the kind of

0:07:10.034,0:07:15.423
programming mistake that's so rookie, I[br]would never make it. I just did it in

0:07:15.423,0:07:21.375
class to show you what kind of mistakes[br]you can make. Yeah, rookie move. We'll

0:07:21.375,0:07:29.513
stay with it though, just to show that no[br]matter how far in life you go. you know, I

0:07:29.513,0:07:35.069
have a couple of mantras about teaching[br]and you've heard them I think, in my class

0:07:35.069,0:07:39.377
before it knows, there's two things you[br]never do, ever. I mean this is the best

0:07:39.377,0:07:44.309
advice I have to give ever in my whole[br]life. first you don't do, you don't spell

0:07:44.309,0:07:48.617
in public and you don't do algebra in[br]public, cuz you will eventually do

0:07:48.617,0:07:52.869
something so stupid, like wow, like I[br]can't spell dog or can't add two to ten to

0:07:52.869,0:07:57.461
get cuz you're up there you know, and[br]you're nervous and everybody sees it, so

0:07:57.461,0:08:01.940
just don't do if you don't have to. I[br]would love to say don't program in public.

0:08:03.500,0:08:09.810
But unfortunately for this class, I cannot[br]get around that one. Alright. So there you

0:08:09.810,0:08:16.273
go. now what does the meshgrid do by the[br]way? What I'm thinking about right now, is

0:08:16.273,0:08:23.945
I'm taking time slices of data, so what t[br]is, right? Little t, here. T is, goes from

0:08:23.945,0:08:32.606
you know, -30 to 30 and I have a signal on[br]that. So, I have this domain where, okay,

0:08:32.606,0:08:37.780
so I have, for every, every little burst[br]of time, I take examples of length 60.

0:08:38.040,0:08:43.189
Okay? And I divide that 60 by 512, and[br]then, 0.5 units later is a different

0:08:43.189,0:08:50.884
units. I take another sample, another[br]measurement, again, with 512 points in it

0:08:50.884,0:08:56.260
and another in 512 points. Okay? So I'm[br]gonna collect this and what meshgrid does,

0:08:56.260,0:09:01.360
it gives us a sense of direction. It makes[br]a two-dimensional grid where in

0:09:01.636,0:09:06.667
one-dimension, it's time, in the other[br]dimension, it's this slice variable. Okay?

0:09:06.667,0:09:11.698
So t and slice now become capital T and[br]capital S, which are now matrices

0:09:11.698,0:09:17.212
conveying information about you know, ones[br]in one direction. One is in an orthogonal

0:09:17.212,0:09:24.624
direction to it. Okay? I wanna use those[br]to define our signal and we also need this

0:09:24.624,0:09:35.382
in the frequency domain, K, S. We're gonna[br]do the same thing here with the frequency

0:09:35.382,0:09:39.274
components .[br]In one direction, it's the wave numbers,

0:09:39.274,0:09:43.924
but then, it's wave number per sample in,[br]in the other direction, it's number

0:09:43.924,0:09:49.361
samples. Okay? Now, let's define a[br]function. Here it is, u=sech Now, I use

0:09:49.361,0:09:56.979
these variables cuz now they know about[br]their direction and I'll just show you

0:09:56.979,0:10:04.297
what I've got here, and so you can, it's a[br]simple function. I'm gonna, I'm gonna

0:10:04.297,0:10:09.086
leave this zero for right now. I'm gonna[br]tell you why in a moment cuz I wanna shift

0:10:09.086,0:10:13.425
the center frequency around. This is a[br]center frequency. If I leave it at zero,

0:10:13.425,0:10:18.158
it means that in the frequency domain I'm[br]centered at k equals zero, but we can make

0:10:18.158,0:10:22.271
this anything we want. I can make it[br]centered at k equals whatever else. I

0:10:22.271,0:10:26.722
want, in fact, in your dog problem, I'm[br]telling you right now if you put a filter

0:10:26.722,0:10:32.035
around k equals zero, your dog will die.[br]Tim, don't let your dog die. Okay.

0:10:32.035,0:10:37.287
Alright? So, alright, so don't put a[br]filter near zero, cuz it won't save

0:10:37.287,0:10:42.996
anybody. Alright. So that's gonna be my[br]function and so what I'm gonna do is

0:10:42.996,0:10:50.189
subplot this. I'm gonna say, okay, I'm[br]gonna use what's called the waterfall

0:10:50.189,0:10:59.509
command. And I like waterfall, because[br]it's a black and white picture, and it's

0:10:59.509,0:11:04.455
nicer to plot in color, however, I want to[br]set a view angle on it. The problem with

0:11:04.455,0:11:09.140
plotting in color is if you go to a[br]journal, it costs you a lot of money.

0:11:09.420,0:11:16.942
Okay. So there is like my plane moving[br]around, whatever you want to call it. This

0:11:16.942,0:11:21.536
thing is moving in time, right? So I have[br]this signal that's moving around and I

0:11:21.536,0:11:25.892
made a very simple movement. I made it[br]move like a sign wave in the slice

0:11:25.892,0:11:30.644
direction. So, it's just doing this. Okay?[br]All right. Now if I add noise to this and

0:11:30.644,0:11:37.385
I. No it's time domain, here. We're going[br]to look at the frequency in a moment. So,

0:11:37.385,0:11:42.870
if I add noise to this you know, we could[br]bury this whole thing, we could hide it.

0:11:42.870,0:11:47.876
And he said, well, yeah but I could just[br]average it to zero. If you average that,

0:11:47.876,0:11:52.675
this stuff all gets washed out. You get no[br]signal out. You have to do it in a

0:11:52.675,0:11:58.023
frequency domain. Okay? Alright. So, let's[br]take a look at the frequency domain of

0:11:58.023,0:12:03.440
this thing. So I have the signal moving[br]around in time, but if I just take this

0:12:03.783,0:12:08.788
Fourier transform then I can plot this[br]thing there. So, okay, so let's go ahead

0:12:08.788,0:12:24.340
over here. That's the first plot, and, so[br]I gotta do is that gonna go through this

0:12:24.340,0:12:29.633
and one slice at a time for every slice[br]take Fourier transform. Okay? So I say,

0:12:29.633,0:12:36.860
okay, fine. So what I'll do is I'll go[br]over here and I'm gonna say I will go grab

0:12:36.860,0:12:48.146
the first, first, let's say row of this u.[br]I'm going to ftt it. I just Fourier

0:12:48.146,0:12:55.406
transformed that first row and what I'm[br]going to do with this Fourier transform is

0:12:55.406,0:13:03.808
I'm going to plot it. So let's call this[br]UT j, okay? And, once I go through all of

0:13:03.808,0:13:12.110
this. Oh, come on. And actually, I don't[br]care about fft j or I, what I want to do

0:13:12.110,0:13:17.272
is maybe plot also, if fftshift it right[br]now. Okay? Cuz I just want to plot what

0:13:17.272,0:13:24.936
this looks like. There is my thing there.[br]And, might as well take the absolute value

0:13:24.936,0:13:33.536
while we're at it, for a moment, we're[br]gonna just this thing, absolute value. So

0:13:33.536,0:13:37.869
what I did here is I went to each row,[br]Fourier transformed it, then I have to

0:13:37.869,0:13:41.939
fft(shift) it with, with the absolute[br]value. And then what we can do is say,

0:13:41.939,0:13:52.091
okay, how about we look in subplot(2) and[br]I'm gonna do the waterfall again. But now,

0:13:52.091,0:14:04.380
I need the fftshift. Actually, did I do[br]this yeah, of, of K versus S versus UT.

0:14:05.100,0:14:14.198
Okay? So that's gonna be this Fourier[br]Transform. Oh, I should set the same view

0:14:14.198,0:14:30.817
angle, by the way. Sorry. Okay. So here's,[br]here is, here is perfect signal, which

0:14:30.817,0:14:36.684
doesn't exist in reality, right? Which is[br]I have this thing moving around and it's

0:14:36.684,0:14:45.524
moving in time. Tell me what it's doing in[br]frequency domain. Nothing. Awesome, right?

0:14:45.524,0:14:49.559
Because it's very important you know[br]what's happening in time and the whole

0:14:49.559,0:14:53.650
section at the beginning is kind of[br]understanding this idea between time

0:14:53.650,0:14:57.965
frequency dynamics. The time domain can be[br]doing all kinds of stuff, only that

0:14:57.965,0:15:02.392
frequency fixed. Okay? Remember, when we[br]do the noise reduction, your key thing is

0:15:02.392,0:15:07.043
to say, well, if I, if this thing is fixed[br]in frequency, all I got to do is build a

0:15:07.043,0:15:11.437
little filter out there in freque ncy,[br]capture that out. Okay? Cuz this thing is

0:15:11.437,0:15:16.225
not moving around. So I can be moving all[br]over the place, flying around my jet,

0:15:16.225,0:15:21.137
you're still giving off a fixed frequency[br]and this is what you'd lock in on, if

0:15:21.137,0:15:22.879
you're gonna do a detection.