W01_L03_P02 - Convergence to signal through repeated sampling

There. So now what I'm going to show you
is this. I'm going to sample this over and

over again. Okay so in other words suppose
I have my detector send out a read a

signal. I read a signal this. I keep
reading 30 second slices of time. Okay and

presumably I keep getting different noise
every time, right, cause white noise. So

what I'm going to do is ask what happens
to these if I go through and I start

sampling over and over again. Okay. I can
do anything I want. I am master of my

code. I can make noise a million. And by
the way, that might just be one

realization. I run it again. Okay, so
there's a little bit of a spike there,

fine, we'll go 30. I want to try to clean
that up. As far as you can see in that

blue there is no evidence of anything
right remember you got to launch a missile

or do something cool like that or save
your dog. So you got to figure this out,

don't want any dead dogs in this class,
cuz I'll be held accountable, probably.

Alright, here's what we're going to do,
we're going to do a little loop, for J

equals 1-30. So what I going to do in this
loop is the following. I'm going to take

my. Deal here. Let's, let's actually this
just needs to be defined once so let's

bring this outside Lou so I have some
noise which is 30. Yeah let me take away

these sub-plotting routines for a moment.
Okay. Here's what I want to do. In this

loop we've gone this loop 30 times and
each time we do it I'm going to generate a

new signal. Sorry I have the same signal,
but I'm going to generate a new signal

because I'm going to add noise to it. So
it's the same signal that I keep reading,

right? So in a radar problem if there's a
plane out there, I keep picking up that

signal from the plane but it's buried, but
it's buried inside noise which keeps

changing in time. Okay so every time I do
this I have this issue of t that stays the

same but now the noise changes every time.
So every time I do this I get different

noise on it, different noise on it. So
each one of'em just looks like just what

we had before nothing there. So what I'm
going to do is I'm going to just keep

doing this. And I'm going to ask if I do
this and I go through this loop I want to

actually just calculate the average of
what my spectrum looks like. I'm going to

add if I take 30 measurements. I'm going
to add them together and divide by 30.

What happens to the noise? What I know is,
is white noise, when it's uncorrelated.

All that noise should have mean. Sweet.
Everything just going to just fair

anybody. One. Tough crowd. Tough crowd.
Alright. I'm going to show ya a magic

trick, right? Okay, all right laptops
usually don't come out of your sleeves

[LAUGH] iPhones can. but. Okay. One
believer in the class. Let's see how many

of the others I can make believe in this.
Alright, so here we go. . What's that?

Inverse. Inverse voice transform. So let's
go through this and say, okay, here's what

I'm going to do. I've got to keep track of
all this, right? So what I'm going to do

is the following. I'm going to make a
vector, called average, And I'm going to

start off with just a bunch of 0s in it.
And what I'm going to do with this vector

is after I've done computing the signal
I'll say well my average is what it was

before plus this UTN. We'll just keep
stacking the information in it. And at the

very end, after done this 30 times, we go
through this loop 30 times, keep stacking

my average information there. At the end
I'm just going to take this signal, and

divide by the total number of times I did
this, right. So I could say, alright well,

how about we do this? Average is equal to
absolute value of. So first I'm going to

shift it. I took 30 slices so just divide
by 30 we'll make this a little more

general in a minute actually let's do the
following. so we took 30 realizations is

a, I don't like to make variable names
that long. Cuz I misspelled them that

long. so what I'm going to do is I'm going
to take, I'm going to basically, remember

av is basically I've added it all
together, by the way you don't add the

absolute value, okay? So I'm like you take
the, , absolute value of each, an add them

together you take the raw signal,
otherwise you've, the minus and plus's

that are there don't cancel, it's all
positive, and it all adds up and it'll

look like no it's worse if you take more
data, right? It should always be better

you sample 30 times, you got 30 things now
to make a decision with, right? Okay, so I

add them all together. I'm going to ask
that you shift it, take steps of five,

divided by the realization and I want to
plot that, okay? So, let's go ahead and

plot it. All right, so here's why I don't
have a plot in here first. The original

signal. The absolute value at oop. So this
is my perfect signal, we crop that in red.

So remember what I did is I, before I
polluted this thing with noise, I had

this, I know what the answer's supposed to
be, right, and I've added noise to it. So

this is what the answer's supposed to be.
That red line is going to come out, okay?

Now, let's look at this compared to the
average, which is I added everything up,

divided by the total number of
realizations, and I'm going to plot that

in black. Okay, and by the way, let me
plot a single realization. If, if we go

through this loop, we went through this
loop from J equals one to realizations.

The last time we went through the loop we
had this thing called UTN. So that was

just one example of realization. Let's
also plot that. So, KS versus the absolute

value of, and I, FFT shift of ETN, there
and we'll plot that in yellow. I want it

to be sort of subtle. Okay, how about
cyan? Okay, I want it to be sort of not a

strong, dominant color. Or else, it'll
take up the whole picture. Okay, so first

of all, so let's talk about this picture.
The psy N is one measurement. My last one.

I took 30, that's what the psy an is. The
red is the exact answer and the black.

That's averaged over 30, okay? Actually,
that's not so bad look at that kind of

get's my peak, right? Let's see how well
we can do. So right, so un fact, what,

here's what I'm going to do. We're going
to start looking at realizations. What

happe ns if I have one realization, were I
get the blue? If I have two, how well do I

do? You know and then we'll start stepping
it up to see how well does this converge.

What's the answer if I have a thousand
reali, you know, a million, okay? What's

the answer of a thousa, infinity. Sorry,
that's the answer I was looking for. My

daughters their nine and six. And their
trying to understand this thing called

infinity. They've heard of it like, my
youngest daughter, six, is like what

happens if you, and she always makes up
these toy problems. Like, you know ten

billion 35, is that bigger than infinity?
. Now, all you got to go figure what

happens though if you multiply that plus
twenty . The older one king of gets at me,

be a little pissed and the younger one,
she's, she, she's there, gettin in concept

infinity, and if you go to infinity. That
red line will be black. The black line

will go red to the red line. And you had a
question. No. Yes. Okay. Yes. Why don't

you need to FFT shift to average? Oh, we
did already. I did. You did it when you

add it . Yeah. I . So now, so now let's
look at this thing here. And, in fact, you

know what? This is going to be pretty
sweet. We're going to make a little movie.

Ready? You're clearly not. But anyway,
okay. . Alright, let's, let's try to do

this. Let's put this inside the loop. And
I will plot it inside the loop. So when J

is one, we're going to plot this, hold.
And so then we're going to do this pause

for half a second. And so when it goes
through the first time, you'll see. In

fact, what we're going to do, is take out
the cyan line, that's the individual run.

And all we're going to plot is.
Essentially the average verses the what

were suppose to get ready? Okay let's see.
J, thanks. Gary, any, anybody else, I

think that, hopefully is it. And now,
we're going to go also through and do this

a hundred times. Oh yeah, it's not a good
idea either, right? Hold on, hold on. Oh,

so I, I just, I should call this something
else. How about that? I faked my code out.

Thank you. I have tw o, okay? Are we
ready? I think we're ready to push the go

button. It will, hold on, here it, I got
pull this in so you can see it, oh come

on, come on, no, okay, now I got to kill
it, okay, there it is okay, fine, it's

going through, see that, see what's
happening. Dang. I know what's it look

like its doing to you. Okay it's looking
pretty good to me. Is it looking good to

you? I don't know what realization we're
on. And what you can see, as we take more

and more realizations, right? We take more
and more averages, we keep sampling this

thing out. The noise cancels itself out.
And you basically get the black line is

just folding right on to the red but it's
suppose to. There's a real signal in

there, okay? Now, if there was no signal
in there it would all just go to zero.

Okay, so this is another way to take your
data subtract the noise out. You know

something about noise make use of it. If
there's anything in the problem that

you're not making use of you're probably
not doing the problem right, okay? Every

piece of information you have should be
used. Here we're using information now

about the noise. Okay, after a hundred,
this is what you get. So, this can be like

your dog problem. Your dog problem is the
following, right? You're going to first of

all, I haven't told you what the center
frequency of your dog problem is. Here I

kind of knew. You know, I embedded, I have
the answer, right? It's on my computer. I

said make a little signal right there in
the middle of your domain. I don't do that

with the dog problem. I make it up, and I
don't tell you where it is. So you've got

to figure out where I should even filter,
right? All right, so in this process you

notice if I sample a lot, I clearly have a
spike right there. So even if I didn't

know where to filter I'd say Oh, maybe I
should filter right around here. There

seems to be something there that's not
just noise. We go that?