0:50

So from the equation, the concept of the inner

Â product expresses this idea that the signal,

Â x of n when we project it on the complex sinusoid

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

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

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

Â [MUSIC]

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

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

Â Under result is this spectrum express in units,

Â so we see the magnitude and the phase spectrum.

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

Â and in the phase we are identifying the location of these

Â sinusoids with respect to time zero.

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

Â we'll understand this concept a little bit better.

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

Â 2:29

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

Â signal X in our DFT equation by these complex sinusoid.

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

Â we can sum the exponents, and we obtain a single

Â complex sinusoid with a more complex exponent.

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

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

Â And by basically inspecting this equation we can see that when

Â k is not equal to k sub 0, the denominator

Â is not 0, and the numerator is 0.

Â Therefore, all the output signal X of k

Â 3:51

So this is the DFT of a complex sinusoid, so

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

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

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

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

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

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

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

Â 5:03

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

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

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

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

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

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

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

Â We can sum the exponents, except that the phase term

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

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

Â 8:22

And if we pluck this real sinusoids in the DFT and

Â then we express it as the sum of two complex sinusoids

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

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

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

Â as the sum of two DFTs of each sinusoid separately.

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

Â one is of a frequency of negative frequency and

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

Â with the given amplitude, each one.

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

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

Â two frequency locations.

Â For the frequency location of k sub zero.

Â And for the frequency location of minus k sub zero.

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

Â 10:59

So this is the equation of the inverse DFT

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

Â And then we do a similar operation,

Â like the DFT, we multiply by complex exponentials.

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

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

Â So were basically multiplying the spectrum by a complex exponential and

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

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

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

Â are not conjugated, so we have a positive exponent.

Â And there is this normalization factor,

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

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

Â we are regenerating the sinusoid,

Â we are recomputing the sinusoids that we identified.

Â So, let's put an example.

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

Â which there was one positive value at k = 1.

Â So we started from a sequence of four samples and

Â we obtained a positive value as k = 1.

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

Â Therefore, we multiply each of these spectral samples

Â by the samples of four sinusoids or complex sinusoids.

Â Of different frequencies,

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

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

Â 4 J minus 4 and minus 4 J, so

Â this is the inverse transform of this spectrum.

Â And let's show an example.

Â So for real signals, we do not need the complete

Â spectrum in order to recover the original signal.

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

Â and typically we use the positive of the spectrum.

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

Â of course we have a phase spectrum,

Â then we can do the inverse of that.

Â And we can compute it using these equations.

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

Â the positive part will be the magnitude multiplied

Â by the complex exponential tool, the phase.

Â And the negative part is going to be the magnitude again

Â multiplied by the negative part of the phase.

Â Okay, and then if we do the inverse DFT,

Â we apply that equation into these whole sequence,

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

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

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

Â The spectrum had 32 positive samples and

Â 32 negative samples, and the inverse for

Â your transfer has this 64 samples of a real sinusoid.

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

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

Â So again, you can find a lot of information about

Â the Discrete Fourier transform in Wikipedia and

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

Â the standard credits that we have in every class.

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

Â And in the second part,

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

Â We have also explained the members DFT.

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

Â You should have no problem with the rest.

Â So, see you next class.

Â Thank you.

Â