4.8 - Sinusoidal modulation

See "Chapter 4 Fourier Analysis" in "Signal Processing for Communications" by Paolo Prandoni and Martin Vetterli, available from http://www.sp4comm.org/ , and the slides for the entire module 4 of the Digital Signal Processing Coursera course, in https://spark-public.s3.amazonaws.com/dsp/slides/module4-0.pdf .

And if you are registered for this Coursera course, see https://class.coursera.org/dsp-001/wiki/view?page=week3

We have seen how to calculate spectras of signals, of sequences.

Now, once, you have the spectrum and it is not a wide-band spectrum,

but it is mostly around a certain frequency, we can specify types of signals.

For example, if most of the energy of a signal is around the origin,

we call it a lowpass signal.

If it has the energy concentrated somewhere else, we call it a bandpass signal.

And if its energy is around -π or +π frequency, that is called a highpass signal.

Now, we're going to see a modulation theorem for the Fourier Transform,

which allows to transform a signal,

let's say from low-pass into a band-pass signal, or to demodulate a signal

which is from a high frequency into a low frequency.

This is obtained simply by multiplying by
cosine of the adequate frequency.

Once we have the modulation theorem, we can do a very practical application

which is actually tuning a guitar.

So tuning a particular string to another string or to a reference sinusoid,

by using the modulation theorem.

Maybe you've used this trick or you've heard musicians use it on stage.

Now you'll see how a Fourier modulation
theorem is actually behind it.

Module 4.8: Sinusoidal Modulation.

We are going to look at different types of signals,

namely lowpass, highpass and bandpass signals.

From there, we move to sinusoidal modulation which is a way to shift a signal,

for example lowpass signal, to become a bandpass signal.

Using these tools, we are going to look at a very practical application

which is, namely, tuning of a guitar.

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 high-pass signals,

where the frequency content is mostly around high frequencies,

and in between, you have band-pass 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 a 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 π or -π and there is no energy around the origin.

Finally, the band-pass signal.

In this case, it's concentrated around π/2 at -π/2 and π/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 ωc times n.

What will this produce on the spectrum when we know x [n],

and its DTFT, X of e to the jω?

So it is the DTFT of x[n], multiplied by a cosine of ωc times n.

So that's the DTFT of using Euler's formula, as usual of x of n

multiplied by both e to the jωcn and e to the -jωcn.

And this simply creates a double spectrum, namely it's equal to 1/2 X of e to the jω

shifted to ωc and shifted to -ωc.

So usually, we take x[n] as a baseband and ωc 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 ωcn,

which of course has these two peaks at ωc and -ωc,

as we know from the DTFT of a cosine function.

So this gives the 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 ωc and -ωc.

Let's do this pictorially.

So we start with a spectrum: here, t's a triangle or a spectrum around the origin.

We move it to ωc, multiply it by 1/2.

That's the first green spectrum, then we move it to -ωc.

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

the red spectrum now after modulation.

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

We know that spectrum is 2π periodic, so let's show a few periods here from -4π to +4π shifted to ωc

(green spectrum), shifted to -ωc (blue spectrum) and the resulting red spectrum.

Now, I want us to be careful

if the modulation frequency grows beyond a certain point

and we're going to demonstrate this again pictorially.

So here, ωc is very close to π, the maximum frequency,

close to -π, the blue spectrum.

And we see now that we have a funny looking spectrum around + or - π and + or - 3π.

This is not exactly what we had expected, so if we blow it up,

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

we have a piece of the triangles and something funny around -π and π.

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 anyway,

they 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 the 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 baseband.

So let us look at this demodulation process.

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

So we have y[n] is x[n] times cosine of ωcn.

Its spectrum, we have seen before -- Y to the jω --

is a combination of the two spectras shifted to ωc and -ωc.

The DTFT of y[n] multiplied by 2 cosine of ωcn,

well, it's going to be the combination of Y shifted to ωc and to -ωc.

Then, we replace the formula we just had before, so we have four terms.

One shifted by 2ωc, another one, by -2ωc,

and two terms that are actually at the origin.

And so, we have indeed capital X e to the jω on plus 1/2

and two modulated versions at 2ωc and -2ωc.

Let's do this pictorially.

So the DTFT of x[n] is shown here.

So it's a spectr-- triangle spectrum around the origin.

Then its modulated version has two peaks at -ωc and +ωc.

Then y[n] multiplied by cos ωcn has two shifted version,

one to the right by ωc, it's the green one, one to the left by ωc is the 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 2 other peaks, which are around π.

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 -π and π,

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.

Finally,we're going to see a real application,

a really useful application, that is, it is tuning your guitar.

The abstraction of the problem is that you have a reference sinusoid at some frequency ω naught,

You have a tunable sinusoid of frequency ω.

And we would like to make ω and ω0 as close as possible,

actually equal and this, only by listening to it.

And what we are going to hear

is a beating between these two frequencies when they are close enough.

And then by tuning, we can bring this beating to essentially frequency zero,

at what point, ω is equal to ω naught

and we have tuned our guitar string with respect to a reference frequency.

So how are we going to go about this?

Well, first we bring ω close to ω nought

That's sort of easy if you have a minimum of musical ear.

When these two frequencies are close, we play both sinusoids together,

then we have to remember trigonometry,

and we write x[n], which is a sum of cos ω0n plus cos of ωn,

in terms of a sum and a difference of these two
frequencies,

which finally can be written approximately as 2 times the cos of the difference,

Δ of ω times n and the cosine of the base frequency ω naught.

From this formula we see there are two components: there is the error signal

-- the cosine of Δωn -- and there is a modulation signal -- the cosign at ω0.

When ω is close to ω0, the error signal is very low frequency,

so we cannot really hear it, because it's such a low frequency.

So the modulation will bring it up to the hearing range

and we are going to actually hear it as an oscillation of the carrier frequency.

And we're going to see this pictorially in just a moment.

Let's look at the pictorial demonstration here.

So we start ω0 is 2π times 0.2.

ω is 2π times 0.22, the difference,

which is actually half of the difference between the 2 is 2π times 0.01.

We see now, interestingly, we have the carrier frequency, which is red curve

modulated by the difference, by cosine of Δ of ω.

So we see this, the beeping in blue overlaid to the red curve which is the modulation.

We can change the frequency ω to 0.21 times 2πi.

The difference now is 0.005, the beating is slower.

We pick ω is equal to 2π times 0.205, the beating is even slower.

And here, we take an example where ω is very close to ω0

and the beating is extremely slow.

And we almost see only the modulating frequency ω0

and the very slow variation due to the beating by cosine of Δ of ω.

It's time to see a video demonstration how to tune a guitar,

using this very simple principle.

You probably have seen musicians doing it on stage,

now you understand the math behind it.

Okay, after all these maths, let's try to do something useful like tuning an electric bass.

An electric bass has an E string, [music] which is a frequency of 41.2 hertz,

an A string, [music] which is a frequency of 55 hertz.

To use the result we have just seen, we want to find two frequencies

that we want to make equal.

As long as they are not, there will be a beating that we can adjust,

we are going to hear this in just a moment.

So the first harmonic of the E string here is at [music] 83.4 hertz.

The third harmonic is at 164.8.

For the A string, the first harmonic is at 110 and the second harmonic is at 165.

And we're going to use these two harmonics to do the tuning

and you can hear when they're out of tune.

There is a beating and you should -- as you get them closer and closer,

the beating slows until it's actually zero beating.

And then the two frequencies are similar and then we can start playing something like [guitar strumming]

or something else.