So this is the way we will pack the signal in

what we call the FFT buffer before calling the FFT.

And if we compute the FFT of that and

then compute the spectrum in dB and the phase with unwrap.

Unwrapping the phase, we see this visualization in which we see

the symmetry of the magnitude spectrum and we see it quite nicely.

We see it quite smoothly.

And the phase, we see the odd symmetry of the phase and

we see a very smooth phase visualization because of two reasons.

Because we did the zero-phase windowing and because we did the unwrapping.

So because of the zero-phase windowing,

basically we are getting rid of the shifting distortion.

That would occur if we had not centered all the samples around zero.

And of course, the unwrapping allows us to see this very smooth visualization.

Okay, so this is the last part of what I wanted to talk about.

So we have seen the DFT, we have seen the different properties,

so now we can put it together.

Doing the analysis and synthesis of the DFT in

what we call the analysis/synthesis type of operation.

So we can start from a signal,

compute the FFT represented correctly in the magnitude and phase.

And since there are symmetries, there is need to only show half of it,

the positive side.

So this is the positive side of the magnitude spectrum and

the positive side of the phase spectrum.

So the spectrum was twice as long.

And then we can do the inverse Fourier transform from these and

reconstruct the original signal and it should be exactly the same.

So if we do things right, there are input signal,

the output signal should have exactly the same values.