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.