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

The first man made signal from outer space.

Sputnik was the first artificial Earth satellite launched by

the Soviet Union on October 4th, 1957.

Its main spherical body was surrounded by four external radio antennas,

which transmitted a signal.

This signal is thus the first man made signal sent from outer space.

Although, Sputnik was in orbit for three months, its signal only lasted for

22 days, at which point it ran out of power.

Amateur radio operators could detect the signal all over the world.

So, just imagine you're one of these radio operators back in 1957.

All excited to participate in this historic moment.

And let us listen to the received signal.

[SOUND].

The transmitted signal was just a sequence of beeps,

which is what we can in fact hear.

We also perceive a large amount of background noise,

which is due both to the type of equipment of the day.

But also, because of the background noise that is natural in such recordings.

Let us have a look at the signal.

This plot shows only two seconds of the audio recording.

We clearly see the beep,

as well as the noise contained in the signal between the beeps.

In class, we also studied another tool,

namely the Fourier transform, which allows representing a signal in terms of

the different frequency components it contains.

It is just a change of point of view, and

the operation can be inverted to recover exactly the original signal.

We are plotting the magnitude of the Fourier Transform of

the signal transmitted by Sputnik.

There is a large component at omega is equal to 0,

the origin, which corresponds to the constant component of the signal.

This is the so-called DC component in electronics.

We also observe two small peaks,

which correspond to the frequency of the transmitted beeps.

Fourier presentations are defined on the interval minus pi to plus pi, and

are two pi periodic.

However, it might not be clear how the concept of discrete frequency relates to

the more intuitive one of continuous frequency.

The latter are easy to understand from our daily life and

expressed in one over seconds, or hertz.

The continuous frequency f and

the discrete frequency omega are related by the following formula.

Where fs is a sampling rate of the signal that is

the frequency at which we measure the signal.

This link and the explanation of this formula will be made clearer when we

study the chapter on sampling and interpolation.

We see that the two peaks appear now at plus minus 1653 hertz.

Let us further study what happens when moving from time to frequency domain.

In the time domain let us simply fire our signal and

model as a product between a train of square pulses and a sinusoid.

So, the square pulse here turns on and off,

a sinusoid here, by simply multiplying the two signals.

Now let us study what happens in the frequency domain.

The train of pulses is a periodic signal whose DFT consist of deltas

placed at the multiples of the fundamental frequency at which the pulse oscillates.

So, the FT of the sinusoid consists of two

deltas placed at the positive and negative digital frequency of the sinusoid.

Moreover, as seen in the lecture when we studied the properties of

the Fourier transform,

the product in time domain correspond to a convolution in frequency domain.

Thus, we obtain a DFT that looks like the figure at the bottom.

So, we convulve this spectrum with this one, and this replicates the little

spectrum here at the two locations of the xerox given by the sinusoid.

How does this compare with our original signal?

First, notice that the frequency of the sinusoid is

much higher than the frequency of the train of pulses.

So, the axles of pulses are going to be very small and

masked by the overall noise.

So, two main peaks at 16 63 hertz correspond to the derack of the sinusoid.

The transmitted signal did not contain any particle or

information, so what makes it such an important scientific milestone?

Its first impact lies in the fact that it demonstrated the possibility of

satellite communications.

It would take another 10 years to become reality, whereas it is ubiquitous today.

The successful launch of Sputnik also arose at the time of

intense tension between the Soviet Union and the USA, called the Cold War.

With the successful launch of Sputnik, the Soviet Union clearly demonstrated it's

scientific advance, which led to a crisis in the US called the Sputnik Crisis.

So, US government started intensely funding research and education programs in

engineering and sciences to catch up with the Soviet Union.

This program culminated ten years later with the Apollo program and

the US being the first country to land on the moon.

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