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

MP3 compressed music sounds pretty good.

However it has been compressed highly from its original format by a factor of ten,

20, or more, and therefore, some information has been lost, but

the human hearing system has been fooled not to notice this loss.

So compression, which will be discussed later in the class, does introduce some

noise, some approximation error, and the trick is to shape the noise so

that it's not picked out by the human hearing system.

We're going to start by simply listening to music and two versions of adding noise.

One which is a not so clever way to put noise and

the second one is a clever way which is using MP3.

Both noise levels are of the same magnitude so

they have the same energy of noise that is being added, so the single to noise ratio

which is the ratio between the useful signal and the noises in both cases 13 dB.

So let's start and play the original noiseless music.

[MUSIC]

Now we play the original music plus what I call Noise number one,

which is simply white noise.

And has a certain dB signal to noise ratio.

[MUSIC]

I guess everybody would agree this is pretty bad.

It's pretty obvious that this is a noisy signal.

Now we're going to add exactly the same amount of noise, or

SNR is also 13 dB, however we shape the noise in a clever fashion.

[MUSIC]

And it's probably pretty difficult to distinguish the original from this

version, which is the MP3 version.

So we have the same amount of noise.

There is an impressive difference.

Lets understand this secret.

And the secret, as you may guess, is found in the Fourier domain so

we plot the DFT spectrum of the music again on the scale.

So that's the top graph and you see that there is some spectral decay and

it varies the particular shape, which depends on this particular piece of music.

Then the first noise, which is certain dB signal to noise ratio noise,

has an, essentially, flat spectrum across all the spectrum.

Whereas the second noise, it's a clever noise, as you can see,

it has a shape that resembles the shape of the music.

And so when the music has all the Fourier energy, we can add some noise,

whereas the music has very energy, we put very little noise.

So that's called noise shaping So

we summarize this in this graph where the blue curve is the spectrum of the music,

the red curve is a spectrum of the noise exactly adapted to the music.

And the compression algorithm used in MP3s of course very complex.

But one of the key elements is to shape the errors and

actively as the Fourier spectrum changes, because, of course,

music evolves over time so the Fourier spectrum does not always look like this.

It changes all the time.

So you have to constantly adapt where you put the noise

as the Fourier spectrum of the music changes.

And so MP3 magically minimizes the loss of quality

by shaping the compression errors in a clever manner, and

this is done in something that is called the short time Fourier transform domain.

Which we don't explain now in detail, but

the idea is essentially a DFT that is local, so with small pieces of music,

you take a discrete Fourier transform You look at the spectrum of the music and

you put some noise exactly where it's not going to be heard.

This is a magic metronome,

this is what has made MP3 such a successful music compression algorithm.

By the way, this is called perceptual compression, and

this is an art, because you have to understand not only the mathematics,

but all the perceptual effects of the human hearing system.

And the people that derived the MP3 compression standard,

that's a lot of people are involved in this.

Have actually brought together all the qualities from mathematical analysis to

human perceptual knowledge to derive a very powerful algorithm.

By the way MP3 is not the only audio compression algorithm out there.

There are other ones.

But all the successful ones will have noise shaping

in the same style as what we have just seen for MP3.

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