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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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 2 - Advanced Fourier Analysis

- Paolo PrandoniLecturer

School of Computer and Communication Science - Martin VetterliProfessor

School of Computer and Communication Sciences

There are three broad categories of signals,

depending on where the spectral energy actually is mostly concentrated.

The easiest one and

most natural one is lowpass signals, sometimes called baseband signals.

Then we have highpass signals, where the frequency

content is mostly around high frequencies, and in between, you have bandpass signals.

Now there will be a difference between discrete time and continuous time signals,

as we shall see, but these basic categories are present in both cases.

So let's look at the lowpass spectrum.

As you can see, the energy is mostly concentrated around the origin,

around 0, and there is no energy outside.

This is now highpass signal, the energy is around pi or minus pi, and

there is no energy around the origin.

Finally, the bandpass signal, in this case,

it's concentrated around pi over 2, at minus pi over 2, and pi over 2.

Since this is an example of a real spectrum,

it has this symmetry that we have seen in the properties of the DTFT.

Consider now sinusoidal modulation.

This is obtained by taking a signal,

x(n) and multiplying it by a cosine of omega C times n.

What will this produce on the spectrum when we know x(n) and

this DTFT, X of e to the j omega?

So it is DTFT of x(n) multiplied by a cosine of omega c times n.

So that's a DTFT of using Euler's formula as usual of

x(n) multiplied by both e to the j omega c n, and e to the -j omega c n.

And this simply creates a double spectrum.

Namely is equal to 1/2 X(e to the j),

omega shifted to omega c and shifted to -omega c.

So usually we take x(n) as a baseband, and omega c is called the carrier frequency.

Now to get an intuition for this formula, think of the following case.

Think of x(n) being the constant.

So, we simply have the DTFT of cosine omega c n, which, of course,

has these two peaks at omega c, and

-omega c, as we know from the DTFT of a cosine function.

So, this gives intuition.

So, if x(n) is a very, very narrow band, lowpass signal,

it looks a little bit like a constant, and then through modulation,

it will be moved to these two peaks at omega c and -omega c.

Let's do this pictorially.

So, we start with a spectrum here, it's a triangle or a spectrum around the origin.

We move it to omega c, multiplied by 1/2.

That's the first green spectrum, then we move it to -omega c,

it's a blue spectrum, also multiplied by 1/2, and this is the result.

It's a red spectrum now after modulation.

So the central peak has been moved into two half as big peaks at -omega c and

omega c.

We know that the spectrum is 2pi periodic, so

let's show a few periods here for -4pi to +4pi.

Shift it to omega c, green spectrum, shift it to -omega c,

blue spectrum, and the resulting red spectrum.

Now one has to be careful if the modulation frequency grows

beyond a certain point, and we're going to demonstrate this again pictorially.

So here, omega c is very close to pi as a maximum frequency,

close to -pi as a blue spectrum.

And we see now is that we have a funny looking spectrum around +/-pi and +/-3pi.

This is not exactly what we had expected.

So if we blow it up,

we can see that we don't have the triangle of spectrum anymore.

We have a piece of the triangle and something funny around -pi and pi.

Let us look at some applications of what we have just learned about

signal modulation.

So for example, voice and music are typically lowpass signals.

They don't have infinitely high frequencies because anyways,

it wouldn't be heard by the human hearing system.

Radio channels, on the other hand,

are bandpass signals because we need to modulate them high up,

otherwise there is too much interference or too much loss in transmission.

Modulation is a process of bringing a baseband signal, for example,

a voice signal into the transmission band for radio transmission.

And demodulation is the inverse or the dual of modulation,

and it will bring back the signal from a bandpass down to the basement.

So, let us look at this demodulation process.

It is simply done by multiplying the receive signal by the same carrier again.

So, we have y(n) is x(n) times cosine of omega c n.

Its spectrum, we have seen before, Y(e to the j omega) is

a combination of the two spectras shifted to omega c and -omega c.

The DTFT of y(n) multiplied by 2 cosine of (omega c n),

while it's going to be a combination of Y shifted to omega c and

to -omega c, then we replace the formula we just had before, so we have four terms.

One shifted by 2 omega c, another one by -2 omega c, and

two terms that are actually at the origin, and

so we have indeed X(e to the j omega) + 1/2 and

two modulated versions at 2 omega c and -2 omega c.

Let's do this pictorial.

So the DTFT of x(n) is shown here.

So it's a triangle spectrum around the origin, then its

modulated version has two peaks at -omega c and +omega c.

Then y(n) multiplied by cosine omega c n has two shifted version,

one to the right by omega c, it's the green one,

one to the left by omega c, it's a blue one.

And the total is the sum of these two which has a peak around the origin,

which is of height 2, and the two other peaks, which are around pi.

We have now the picture of the DTFT of the demodulated version.

It looks like the original spectrum around the origin, but it has these two peaks

closer to -pi and pi, which were not present in the original signal.

So we have the baseband, but we have these two spurious high-frequency components,

and we will have to learn how to actually get rid of them.

And this will be the topic of the next module.

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