So let’s see how that looks like.

Again, we have seen these equation before.

So, if we start from a signal x, that is real sinewave and

then we take the DFT of the windowed version of this sinewave,

we see that of course, the sine wave can be expressed as the sum

of two complex sinewaves that are then multiplied by the window we use.

And being the sum of two exponential sinewaves,

we can split that into two summatory, so separate DFTs.

So we'll have the DFT of the negative frequency and

the DFT of the positive frequency.

Each one, again, multiplied by a window.

And these complex exponentials can be grouped together.

And basically this is the DFT of a shifted version of the transform of the window.

Okay, so basically, at the end we see that

the first summatory is the DFT of the function W.

So it's W and the frequency index is shifted, so

we have shifted the window and is a scale by the amplitude

of the cosine, by half of the amplitude of the cosine.

And the other element is the same window, but

shifted by the positive frequency, and

also scaled by the same amplitude.

So if we start from a sinewave and we want to show it that the plot

of one single spectrum of this window sinusoid, we can see it like this.

So this is the positive spectrum.

So we don't see the two windows, we only see the positive one.

So we are seeing the positive frequencies and

so the contribution of the positive exponential.

And we see the shape of the window that we use,

but of course, centered at the frequency of the sinusoid, which is 440 hertz,

which of course we can listen to the sinewave.

[SOUND] And this is its spectrum.

So a peak centered at 440 hertz.

And the phase that during the main lobe is flat

that corresponds to the phase of that sinewave at location zero.