Learn the fundamentals of digital signal processing theory and discover the myriad ways DSP makes everyday life more productive and fun.

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Do curso por École Polytechnique Fédérale de Lausanne

Processamento Digital de Sinais

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Learn the fundamentals of digital signal processing theory and discover the myriad ways DSP makes everyday life more productive and fun.

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Module 3: Part 1 - Basics of Fourier Analysis

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

So we all know about pitch.

What is a basic frequency for example of a singing voice or

a particular instrument playing a note?

So it is about frequency, or cycles per second of the waveform.

However, there is not only the basic frequencies, there are also harmonics.

So if the basic frequencies only got zero, you have two omega zero,

three omega zero, four omega zero, etc, that appear also typically in the sound.

And the shape of these harmonics,

how big they are actually gives the timbre of the instrument.

The characteristic of particular sound.

So let's compare three particular instruments, a cello, a sax, and

the violin, by looking at their Fourier spectra.

And understanding how the harmonics can look very different and

give very different types of sounds.

If you look at the waveform of a particular instrument it is

very hard to see how it's going to sound.

And we know this from for example, the waveform of a speech signal,

it's very hard to understand speech by looking at the waveform of the signal.

So, here the first case we have a saxophone, and

we see a few periods of the basic sound,

that's the sound pressure that is generated by the saxophone playing.

Let us play this instrument.

[MUSIC]

Now let's look at a very different instrument, namely a violin.

Again, we see a few periods of the sound that was recorded from the violin playing.

It looks different, but we don't exactly know what is the difference going to be,

so let's play this instrument.

[MUSIC]

And let's look at the third instrument, a cello.

The waveform looks different from the violin, so let us listen to it.

[MUSIC]

Okay, so now we have heard three different sounds, and

if we were to play them back to you, can you guess what instrument is playing?

And what is the note that the instrument is playing?

That depends on your training, and many of you'd be able to do this very well.

But we're going to see a mathematical method to do this,

which is to take a fully transformed or as a discreet fully transformed in this case,

of a few periods of this signal.

The message is of course, that in time domain

it is not easy to understand the signal, in Fourier domain it will be much easier.

So here we plot the magnitude of the discrete Fourier transformed

coefficients taken from a few periods of these various signals.

And we plot them between 0 and 1.32 kilohertz.

On top, you see the sax.

It has a few peaks.

As you can see at 220, 440, 660, etc.

Then we have the violin.

It has also a peak at 220, but high peak at 880.

And at the bottom, the cello has also a peak at 220, like the sax it's a big one.

Then 440, and then it decays and goes back up at 1320.

So you see the spectra look very different, and

definitely the violin looks very different from the cello and the sax.

To some extent the saxophone and the cello have somewhat of a similar sound.

A warm basic sound, and the violin as you can see from the peak at 880,

has a more high frequency nature to it.

So the note is each time played a total 220Hz.

The other peaks are what we call harmonics, and without using

the Fourier transform, unless you are a trained musician we might be lost.

In sum, by looking at the Fourier spectrum

we can see that the fundamental note is the first peak in the spectrum.

It's not necessarily the biggest peak, but it's the first one that appears.

That's the note that is being played.

The further peaks are the harmonics, and they are all multiples of the basic note.

And the relative size of the harmonics gives the timber, or

the characteristic of an instrument.

And so here we see in a very simple example where the signal is periodic,

that the discrete Fourier transform is very helpful in understanding these

musical sounds.

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