W01_L03_P01 - Averaging over white-noise in signals

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.