W01_L03_P04 - Signal detection through averaging and averaging

Now let me shift the frequency a little
bit, just to see what it looks like. So to

shift that center frequency. In other
words, off zero because we haven't done

anything off zero yet. We'll just go over
here, and say, ten. Alright, now you may

go like, what happened? Alright? Well I'll
tell you what happened. Moment here, this

goes up. So first of all, I just shifted
the center frequency over by ten, still

fixed in frequency. Now the top picture is
interesting. Now I am looking at this, my

center frequency is zero and zero
frequency, what does that mean? Cosine

zero X, n oscillations. Now my center
frequency is cosine ten X, well cosine ten

X. I mean I have this, I have this e to
the I ten X multiplying everything. So you

see this here? Looks like a cosine 10x
sitting on that thing. So I have a bump,

this hyperbolic secant times the cosine
ten X. So it's a wave packet, oscillation.

So if you want to get rid of it, you'd
say, well how about if I just look at the

absolute value of that thing? And if you
do that, it takes away the phase, and it

gives you back that. Okay, all that
oscillation was, was having a group of

carrier frequencies sitting around in
there. It's a frequency component and it

has a phase. You just get rid of them.
Absolute value, stuff like that. It's all

I care about. You may care about the phase
for more important reasons later, but for

right now, it's all we care about. Notice
a couple things, if I had a lot of noise

on here. I'm going to bury all of this so
in fact let's, let's, let's bury all of it

and just put it in a sea of noise and your
going to see that. Basically, if I take

all of this out and this is all noisy then
what do I do. Okay, first thing you gotta

do, if you have this all buried in noise,
figure out where is this, where's that

center frequency. Second thing you do is,
once you know where the center frequency

is, you can build the filter there, and
build the filter so you can go back to

here, okay. Alright, so let's bury the
noise. And to bury it in noise is, is

amazingly eas y to do, right? So you have
your function, and there it is, and all

you got to do bury it in noise is, for
every frequency component you got. Let me

see where this ends. Here, so I'm FFT'ing
the signal right? And all I gotta do is

add, plus noise, times rand N um., this is
one by N, plus I times rand N, one by N.

That's it, so like we did before except I
haven't told you what noise is. So let's

just do noise is equal to like, twenty.
I'll just put a bunch of noise on it and

let's look at this thing. Whoa, tell me if
you see anything in there okay now that

really bug becomes a real shit alright
whatever or shit test right. Roshard

whatever guy you see in there? Keon do you
see anything in there no nothing dog cat

your house you grew up in no nothing
carpet. Okay, anybody else, want to

volunteer anything else, better than
carpet, no, okay. All right, by the way,

this is what you got, to work with, right?
And here's what a my claim is, if I keep

taking samples of this thing, in fact, I
have this thing, I have 21 samples, right,

I haven't made any use of that fact.
Right, I've 21 samples, these are all

white noise, and in there is just this
clear line, that you cannot see. There is

a signal sitting there, waiting for you to
go get it. Okay, now lets talk about, if

you have this, what does this do to here?
So that's what it looks like here what

does it look like in the time domain. And
not so hard to do you gotta inverse before

you transform everything else you call
that UN. all I gotta do I've gotta

interverse 4A transform it. So I have to
go, basically, now if you shoot. I FFT

shifted it all ahead of time right into
the absolute value. So, let's go and do

some of that. Let's take out the absolute
value. Let's take off that FFT shift.

Okay, all that hard work thrown away, fine
we'll, we'll, we'll FFT shift it later.

All we have to do is take the absolute
value here and FFT shift it there instead.

Okay, but U of N now is just the IFFT of
U, J. Okay, O UT sorry yeah. So I take row

by row this noisy signal. I inversef ully
transform it, and now I can say supply 211

and we can just plot water fall TS
absolute value of U and, Oh I need my view

back. Okay, so here this is your problem.
Tell me if there's anything there. So

remember data analysis is about this. So
what, what I like about these examples is

in your dog problem or in this here. You
clearly saw, we already know the answer

ahead of time. Right? We know there's a
signal in there. We know there's this

thing going like this through here. You
can't see it cause I hit it, right? And

the objective, if you are the airplane is
to hide yourself. The objective if you are

there in that radar is to, is, is hide and
go seek, right, and but it's, it's whoever

has got the better, better data analysis
processing skills, right? There are things

here you have to take advantage of. If you
don't take advantage of them, you get no

information, okay? And what's the
information you could take advantage of?

So first of all in this problem. You may
not even know where a filter exist put.

What happens if this airplane bounces off
your omeganaut? Suppose, okay? Let, let,

let's play a what if game. Suppose you
develop really great stealth technology.

This plane here you know how to put a, a
nice coating around your, your airplane.

Design it so that you send omeganaut out.
Nothing comes back. Now, if you're these

guys, you're like, wow, that's plane
visible to me. However what if you're

emitting some other frequency? There's
something else coming off your plane off

the engine it wasn't something I shot out,
shot out there, but I have this big

spectrum of stuff that I'm reading in.
Maybe you're emitting at some other

frequency some information that I could
get. I don't even know what frequency I

should be lookin for. But if I keep taking
measurements what this is going to allow

me to do is come back to this idea here,
is to say, here's what I'm picking up

which just looks like nothing. And what I
want to start doin, slice by slice, is I

just keep taking information in, start
averaging it. Because anything that's real

signal is not going to cancel out. Noise
should cancel out, right? So then if I get

all the noise cancellation happening.
Okay, sorry for the pause. Alright, ready?

So if I get all the noise cancelling out,
I will start to see emerging out of this

frequency component a signature there,
somewhere. And maybe I don't know where it

is ahead of time, and if I see it. Then
you can say well, you didn't put a filter

around here, get rid of everything else
put the filter there, see what it says

over here. See if there's a signal there,
right. So this is the kind of stuff you

would want to do. It's all about who's
better at signal processing actually, at

the end of the day. Alright, so I've given
you this example here which is, sort of

just using very basic time frequency.
Which is I can represent the whole signal

in time or I can represent it in
frequency. Now what we're going to move

onto is much more sophisticated thinking.
Which is to say, wait a minute, somehow.

When I go from time to frequency, if I'm
in the frequency domain, it tells me

nothing about the time domain. If I'm in
the time domain, it tells me nothing about

frequency. I want both pieces of
information at the same time. And this is

where time/frequency analysis comes in
which is to say, I would trade a little

bit of what I know in the time domain for
a little information on the frequency

domain. I'm not going to give up
everything I know in the time domain.

They're not separate objects. They, I'm
going to use both at the same time, okay?

Remember you can not know everything about
time and frequency at the same time.

Impossible, through the idea of the
Fourier transform, the idea of Heisenberg

uncertainty. That's not a possibility for
you. However, you can sacrifice one for

the other to try to get yourself in a
situation where you can know a little bit

about both and maybe make some information
there. Okay? Alright, so Wednesday, we're

going to talk about window Fourier
transforms, which is this idea of how do I

represent both. In the meantime, you have
your dog problem, and we'll just keep

moving forward from there.