So, before we start talking. A couple things that we want, you want to
understand. So, we talked about how to do filtering. So, you have this center
frequency, right? And you have this noisy signal. And let's say, in the frequency
domain, there's my version of noise, okay? You get to all this junk, right? Any
ideas, you say is there anything there or not? And then, you would put like some
kind of filter cuz you know what frequencies you're going to look at. And
then, you basically multiply these two together, kill everything off. Okay, so
that's one way to do things. Now, couple things to notice about this. First, I know
where to look. Now in a radar problem generically, you don't know where to look.
But in your DOG problem, I'm not letting, I'm not telling you what frequency to look
for. You've got to find it, okay? There's one other piece of information we haven't
used here, okay? The other piece of information we haven't used here is, is
something about the noise itself. What did we say this noise was? We say this noise
was white noise. Now, the nice thing about detectors is you can keep often times keep
resampling things, right? So, for instance, if I'm in the radar problem,
right? Okay, that's my radar on top of a mountain, and there's your airplane coming
in. Alright. That's a really bad airplane. But anyway, there's my airplane coming in,
or missile, whatever happens to be and I'm trying to get this thing. I can keep, I'm
continually taking continually taking information in on this. So, I'm sampling
not just once, not just twice, but I have a continual sample string, okay? So, I
haven't made use of that at all in this filtering problem. So, I want to try to
think about how to make use of that. The fact is, when you have white noise, here's
the specifics about white noise. White noise is random. And, by the way, normally
when it's randomly distributed you say, well, okay. So, how did I do, do it in my
computer? I said, mean zero, unit variance. Okay? So, when I add two pieces
of noise together, they shou ld be incoherent. If I have three, four, five,
six, none of it should be coherent. In fact, I add a bunch of noise together and
I take its average and its average is zero. Awesome. So I did one sample. What
if I took a bunch? Then, the idea would be that if I took a bunch of noise, then if I
add all of the noise together, it should be zero. And by the way, if it's not
noise, it won't go to zero. So, I'm going to use that to your, to my advantage. But
really, that's like to your advantage, too. Anything that's to my advantage right
now is to your advantage. Okay. The only thing that's to my advantage and not to
your advantage is my, maybe the homework. Cuz then I already wrote it. You got to do
it. So, okay. So, so we're going to try to work on that and see how that can work for
us, okay? So, what I want to do is write aother code today building upon what we
did last time with the aim of making, kind of exploring this idea of what noise does,
okay? So, I'm going to turn the lights down in front. Now, you guys in the back
there, the TV screens, what do you got there on the TV screens? Chow, can you see
my can, computer? Alright, great. Okay. Alright. So, here we go. We're going to
start of with a little bit of code we had last time. So, here it is. Let's just go
through it again. You'll be writing a lot of code like this in your homework, first
homework for instance. And of course, hopefully everybody likes these commands,
clear all, close all, clc. This just starts us off on a fresh slate for the
code. And then here, I define a time domain. So, I'm going to sample my signal
for 30 units. Whatever that happens to be. Let's say, call it seconds. I'm going to
sample for 30 seconds. And what I want to do with that 30 seconds and see what's,
what's in that signal. So, I get to have a certain number of points I'm going to
sample with. So here, there's 512 points and 30 seconds. So, I'd sample at that
rate, okay? And now, I can define my time interval as well as my frequency
components in, in that sampling. So here, thi s T2 and T. So, remember the lin space
is a linear space that goes from -T over two to T over two, so it's negative
fifteen to fifteen. And I break it up into 512 plus one points. So, 512 points plus
the last point is the same as the first, periodic. Using, remember I'm using
Fourier modes, Fourier coefficients. Those are all sines and cosines of two pi
periodic, okay? Whenever you use Fourier components, assumptions, periodic. So, I
got periodic and I throw away the last point by just taking the first one through
n here. So, T is what I really want. And then, I define my wave numbers or
frequency components, with two pi over T, is there because the FFT thinks you're
working on a two pi periodic domain. So, this is a re-scaling. And you go from zero
to n / two -one, and to n over, -n / two -one.
These are like my cosine zero, my cosine 1x, and my cosine 2x. And they're
integers. And why do I order them this way? Remember, the FFT shifts things,
okay? Ot does, it does one of these. So, you've got your four components when you
do the FFT does that. Take a knife, cut it in the middle of the main switches it
over. So is kn minus the number two to, n over two. Yeah, so when you say that,
yeah, then I take away that shift. And now it's just from -n over two to n over two,
but with a two pi over two factor in front of it. Okay. Now, I'm going to define a
function, such. And it's Fourier transform ut, okay? So, we have everything we've
got. We got a function, we have this Fourier transform. And now, what we want
to do, and let me bring this up a little bit. Does that, so there is my function
and such. There is its Fourier transform. And I'm going to add some noise. There it
is. So, I take some, this is just going to be some coefficient. It's going to control
how much noise I want to throw on this. And notice the way I put white noise on,
you put it on the frequency domain. White noise is a collection, right? If, white
noise is all colors, right? So, there's also a thing called colored noise which
is, if you have certain frequen cy components that have noise in it, well,
around those frequencies or colors, you would add noise there. But if we do white
noises, all frequency components have noise, okay? So, what I do here is I go to
every frequency component and add a random variable. This is round in. It means zero
unit variants, okay? And I add both a real and imaginary part. Just do my signal. So,
I've, I've modified my signal made it not so nice. Yeah. Could you explain one more
time why you need to add the imaginary noise as well? Yeah. So, if you do not
have this imaginary piece, what you've added only is real components into your
noise. It makes it symmetric. Remember, when think about your frequency
components, it's e to the i. When you do a Fourier transform, it's e to the i
whatever the, the frequency component has to be. Well, if you only add e to the i
with real component, then what happens is, right? You're saying, okay, I'll add e to
the i. But then, e to the -i looks the same. So, what ends up happening then it,
at symmetric noise. You don't want to do that. And if you only add the imaginary
parts, you add a symmetric noise. So, you add both to get, so there's no correlation
between the cosine i omega terms and the cosine -one omega terms, both of them get
different noise, okay? Alright. Good? Alright. Dealing with a lot of this
Hawaiian thing these days, I don't know why I just developed it in last quarter
where I was doing a lot of this. It's a little shocker, you know? So, if you see a
lot of that, I mean, I don't know what that means. Except I think maybe I'm
having a good time in class, yeah? Oh, that's not so good. Alright. we'll see how
long that lasts. But I, kind of like it. Hawaii made me think of the sunshine,
that's a good thing to think about. Okay. So then, I have my new signal, utn. Can
you guys still see my screen back there? Or is that me, there we go. Perfect.
Alright. u of n. u of n is my noisy signal. So let's plot these, just to take
a look. So you know, we'll subplot them. Subplot oh, by the way, see, I also want
to, remember I have this k of s here? That's a shifted version, I want you to
just remember that I've already done that. So, I'll make a subplot of two rows, one
column. And first, we're going to plot the original signal, t versus u. And we'll
plot that in red. And then, we'll plot this noisy signal, t versus absolute value
of un, plot that in black. And then, second subplot we'll plot, first of all,
we'll plot the nice spectrum, which is ks versus the absolute value of ut which is
the transform. Oh, by the way, FFT shift of ut. And plot that in red. And now,
we'll plot the noisy part which is ks versus absolute value of FFT shift utn.
And I think, there we go. Alright. So, we'll just plot what these things look
like. Here they are. Okay. So, the red is my clean signal. The black is my muddled
signal because it got polluted with noise. The red is my nice clean spectrum. And all
that blue is the fact that I added noise at all these components. So, normally what
you're going to read from your detectors, the.