0:50

So from the equation, the concept of the inner

product expresses this idea that the signal,

x of n when we project it on the complex sinusoid

it's e to the minus j to pi k n over N.

We are basically measuring the amount of the sinusoids in the signal.

If we show an example, this violin sound, so we are taking a fragment of this sound.

[MUSIC]

Okay, so we are taking capital N samples of this violin sound and

we are projecting it into these complex sinusoids that we are generating.

Under result is this spectrum express in units,

so we see the magnitude and the phase spectrum.

In the magnitude we see the amount of each of the sinusoids present in the signal,

and in the phase we are identifying the location of these

sinusoids with respect to time zero.

So, if we plan how to compute the DFT of one single complex sinusoid,

we'll understand this concept a little bit better.

So let's start from an input signal, x sub 1, which is

2:29

And what we going to do is, we're going to substitute our input

signal X in our DFT equation by these complex sinusoid.

So therefore, we have a product of two complex sinusoids,

we can sum the exponents, and we obtain a single

complex sinusoid with a more complex exponent.

And this, in fact, is the sum of a geometric series, and

therefore, it has a closed form that can be expressed by this equation.

And by basically inspecting this equation we can see that when

k is not equal to k sub 0, the denominator

is not 0, and the numerator is 0.

Therefore, all the output signal X of k

3:51

So this is the DFT of a complex sinusoid, so

on top we see this complex sinusoid that k is equal to 7,

so basically it means that it has 7 periods in the length of capital N.

And in this case, we have defined N as 64, so there's 7 periods in these 64 samples,

and of course, we see the cosine and the sine, and

when we compute, the DFT, again we see the magnitude and phase.

The phase so let's not talk about that right now,

let's just focus on the magnitude and here we see clearly, the value

5:03

Let's start with the signal X sub 2 in which is a complex exponential, but.

The frequency is not one of the frequencies of the DFT sinusoids.

So the frequency is expressed by f sub 0 and it has an initial phase.

And It has the same duration, so it has a duration of N, but

it doesn't have lets say, a fix number of periods in that duration.

So, anyways, so lets put this sinusoid into the DFT equation,

and we again, get the product of two complex exponentials.

We can sum the exponents, except that the phase term

of the sinusoid can be pulled outside, because it does not depend on N.

And also, being a geometric series, we can have a closed form.

8:22

And if we pluck this real sinusoids in the DFT and

then we express it as the sum of two complex sinusoids

we basically can do the same operation that we did in the previous case,

being the DFT linear function, and we'll talk about that.

We basically can express the DFT of this sum of two complex sinusoids

as the sum of two DFTs of each sinusoid separately.

Therefore, the result is basically two DFTs that we basically have seen,

one is of a frequency of negative frequency and

the other is the DFT of a frequency of positive frequency, and

with the given amplitude, each one.

So what the result, basically we go through the logic that we did before

is that it's going to have an amplitude, a sub zero over two for

two frequency locations.

For the frequency location of k sub zero.

And for the frequency location of minus k sub zero.

And it will have 0 for the rest of k and let's see, plot for it.

10:59

So this is the equation of the inverse DFT

in which our input signal now is the spectrum, is X of K.

And then we do a similar operation,

like the DFT, we multiply by complex exponentials.

But in this case, it's not a negative exponential,

it's a positive exponential because were not taking the conjugate.

So were basically multiplying the spectrum by a complex exponential and

then we are summing over this result of over N sample.

And then there is a normalization factor that we include, which is 1 over n.

So the main differences with the DFT is that the complex exponential

are not conjugated, so we have a positive exponent.

And there is this normalization factor,

apart from that, is basically the same, but conceptually is very different.

Basically, what we're doing here, it's kind of a synthesis,

we are regenerating the sinusoid,

we are recomputing the sinusoids that we identified.

So, let's put an example.

If we start from spectrum, like one we saw before in

which there was one positive value at k = 1.

So we started from a sequence of four samples and

we obtained a positive value as k = 1.

So this is a spectrum of a sequence and now if we apply this Inverse DFT function.

Therefore, we multiply each of these spectral samples

by the samples of four sinusoids or complex sinusoids.

Of different frequencies,

we will see that the result is basically the signal we started with.

So this is a complex signal, the result that has for

4 J minus 4 and minus 4 J, so

this is the inverse transform of this spectrum.

And let's show an example.

So for real signals, we do not need the complete

spectrum in order to recover the original signal.

We saw that it was symmetric so it's enough to have half of the spectrum,

and typically we use the positive of the spectrum.

So if we have for example in these figure we have a given magnitude spectrum and

of course we have a phase spectrum,

then we can do the inverse of that.

And we can compute it using these equations.

So we first have to generate the negative part of the spectrum so

the positive part will be the magnitude multiplied

by the complex exponential tool, the phase.

And the negative part is going to be the magnitude again

multiplied by the negative part of the phase.

Okay, and then if we do the inverse DFT,

we apply that equation into these whole sequence,

these whole spectrum X [k] we will get back a real sinusoid.

Okay, so this is a sinusoid that has the length of the spectrum

we started from, in this case, it's 64 samples.

The spectrum had 32 positive samples and

32 negative samples, and the inverse for

your transfer has this 64 samples of a real sinusoid.

Okay, so we will come back to these concepts in the next lectures so

do not worry if you still are not understanding completely this concept.

So again, you can find a lot of information about

the Discrete Fourier transform in Wikipedia and

of course on the website of Julius and here you have all

the standard credits that we have in every class.

So in the first part of this lecture we introduced the DFT equation.

And in the second part,

we have seen how the DFT works when the input is a sinusoid.

We have also explained the members DFT.

If you have been able to understand this, you are doing very good.

You should have no problem with the rest.

So, see you next class.

Thank you.