Our next topic is wavelets.

But before I talk about wavelet,

I would like to go back to

Fourier transform for just a second and remind you

that using Fourier transform allows us to

transition between the time and

frequency domains and if you remember,

if we have a signal composed of

mode frequencies and we

look at the signal in the time domain,

we see a picture like this one

that chooses the strength of

the signal as a function of

time and if we apply Fourier transform,

we end up in the frequency domain where we can

isolate the individuals frequencies in the signal.

Okay. You already know this,

but there is a catch,

the catch is that Fourier transform works really well,

but it works really well for

signals generated by stationary processes.

Let me show you what I mean.

Look at this signal, this is a signal

generated by a stationary process

which means that this signal,

simply put, never changes, right?

If I look at the signal now,

it looks like this.

If I go do something else

come back to the signal in one hour,

take another look in another moment in time,

the signal looks exactly the same.

If I look at the signal in

one day or one week or after one year,

it looks exactly the same,

the signal never changes.

Another way to put this is to say that

the signal contains all its frequencies always,

it doesn't matter at what moment in time you look at

the signal all the frequencies

that make up the signal are always there.

All right. The problem is

that in real life that's not always the case.

You might end up with a signal that

looks like this and if you look at the signal,

say in at this moment in

time you see something like this.

If you look at the signal at this moment in time,

you see something completely different and

then again something more similar to

the first observation and

then something again completely different.

The problem is that Fourier transform doesn't

work very well in this type of situations

and quite often these bursts in the signal

like this guy here and this guy

here are exactly what we are looking for.

This could be some kind of a sensor reading,

this could be some kind of an anomaly.

We want to be able to detect those things

and to handle them in a more intelligent way.

So let me show you what happens if we have

a very simple signal generated by a stationary process,

you see here I have

a frequency of one, amplitude of three,

over a period of 10,

the phase shift is zero

and you know that's a very simple,

very basic signal this guy here.

Then I apply Fourier transform and that's where I see,

exactly what I expect to see over this period,

I have 10 repetitions of the signal,

so you know a frequency of 10 and I can

apply the inverse Fourier transform

and reconstruct the signal.

So far so good.

Now, imagine that this signal I add a second component.

I add another signal with a frequency of four,

amplitude of 20, right?

But I'm not simply adding

this in parallel to the first signal,

I'm adding this component at a specific time at T2.

So then what happens is my signal will look like this.

At time T2, I have this burst in

the signal right and if I try to apply Fourier transform,

I get something like this.

Okay, I still have my main frequency in the signal,

I can identify it,

it's the dominant frequency.

But then, this short burst kind of factorizes into

all these small frequencies here

which if I add them back together,

if I apply the inverse Fourier transform,

I will get the same thing,

I will reconstruct the signal,

but if I look at

this image down here in the second part of the plot,

if I look here only,

I have no idea what happens in the signal.

What frequencies, I see the frequencies in the signal,

but I I can't identify that

this was actually a short burst in the signal right.

If I look at the top of the plot,

if I look here,

I can identify the burst it's here,

but I have no idea at one specific point,

what frequencies are part

of the signal in this specific moment in time.

So I am kind of

confined either to the time domain

or the frequency domain,

but I never get the complete picture.

That's the problem with signal generated by

non-stationary process and this

is where wavelets come in.