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

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

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mostly around the certain frequency, we
can specify types of signals.

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

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low-pass signal.
If it has the energy concentrated

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somewhere else, we call it a band-pass
signal.

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And if it's energy is around minus pi or
plus pi frequency, that is called a

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high-pass signal.
Now, we're going to see a modulation

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theorem for the fluid transform, which
allows to transform a signal.

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Let's say from low-pass into a band-pass
signal or to deimmolate the signal which

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from a high frequency into a low
frequency.

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

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

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is actually tuning a guitar.
So tuning particular string to another

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string or to a reference sinusoid, I use
in some of the modulation theorem, may be

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

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

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Module 4.8, Sinusoidal Modulation.
We are going to look at different types of

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signals, namely low-pass, high-pass and
band-pass signals.

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

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signal, for example lowpass signal, to
become a bandpass signal.

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

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mainly tuning of a guitar.
There are three broad categories of

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signals depending on where the spectral
energy actually is mostly concentrated.

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The easiest one and most natural one is
low-pass signals, sometimes called

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baseband signals.
Then we have high-pass signals, where the

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frequency content is mostly around high
frequencies, and in between, you have

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band-pass signals.
Now, there will be a difference between

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discrete time and continuous time signals
as we shall see, but these basic

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categories are present in both cases.
So let's look at the low-pass spectrum.

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As you can see, the energy is mostly
concentrated around the origin, around 0

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and there is no energy outside.
This is now high-pass signal.

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The energy is around pi or minus pi and
there is no energy around the origin.

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Finally, the band-pass signal.
In this case, it's concentrated around pi

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over 2 at minus pi over 2 and pi over 2.
Since this is an example of a real

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

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consider now sinusoidal modulation.
This is obtained by taking a signal x n

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and multiplying it by a cosine of omega c
times n.

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What will this produce on the spectrum
when we know x of n, and its DTFT capital

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X of e to the j omega?
So it is the DTFT of xn multiplied by a

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cosine of omega c times n.
So that's the DTFT of using Euler's

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formula, as usual of x of n multiplied by
both e to the j omega cm and e to the

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minus j omega cn.
And, this simply creates a double

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spectrum, namely is equal to 1 half,
capital x of e to the j omega shifted to

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omega c and shifted to minus omega c.
So usually, we take xn as a baseband and

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omega c is called the carrier frequency.
Now, to get the intuition for this

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formula, think of the following case.
Think of x of n being as a constant, so we

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simply have the DTFT of cosine velocity n,
which of course has these two peaks at

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omega c and minus omega c as we know from
the DTFT of a cosine function.

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So this gives an equation so if xn is a
very, very narrow band low-pass signal.

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It looks a little bit like a constant.
And then, through modulation, it will be

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moved to these two peaks at omega c and
minus omega c.

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Let's do this pictorially.
So we start with a spectrum here.

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It's a triangle or a spectrum around the
origin.

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We move it to omega c multiply it by 1
half.

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

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

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spectrum now after modulation.
So the central peak has been moved into

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two alphas big peaks at alpha omega c.
We know that spectrum is 2 pi periodic, so

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let's show a few periods here for minus 4
pi to plus 4 pi shifted to omega c, green

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specturm shifted to minus omega c, blue
spectrum, and the resulting red spectrum.

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Now, I want us to be careful if the
modulation frequency grows beyond a

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

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So here, omega c is very close to pi, the
maximum frequency, close to minus pi, the

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blue spectrum.
And we see now that we have a funny

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looking spectrum around plus or minus pi
and plus or minus 3 pi.

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

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don't have the triangle spectrum anymore,
we have a piece of the triangles and

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something funny around minus pi and pi.
Let us look at some applications of what

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we have just learned about signal
modulation.

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So, for example, voice and music are
typically low-pass signals.

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They don't have infinitely high
frequencies, because anyways, it wouldn't

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be heard by the human hearing system.
Radio channels, on the other hand, are

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

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Otherwise, there is too much interference
or too much loss in transmission.

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

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signal into the transmission band for
radio transmission.

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And demodulation is the inverse or the
dual of modulation and it will bring back

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the signal from a bandpass down to the
baseband.

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So let us look at this demodulation
process.

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

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So we have yn is xn times cosine of omega
cn.

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It's spectrum, we have seen before y equal
to g omega is a combination of the two

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spectra shifted to omega c and minus
omega, omega c.

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The DTFT of yn multiplied by two cosine of
omega cn.

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Well, it's going to be the combination of
capital Y shifted to omega c and to omega

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c and minus omega c.
Then, we replace the formula we just had

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before so we have four terms.
One shifted by 2 omega c and also one by

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minus 2 omega c, and two terms that are
actually at zero origin.

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And so, we have indeed capital Xe to the j
omega on plus 1 half and two modulated

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versions at 2 omega c and minus to omega
c.

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Let's do this pictorially.
So the DTFT of x n is shown here.

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So it's a spectrum, triangle spectrum
around the origin.

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Then it's modulated version has two peaks
at minus omega c and plus omega c.

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Then yn multiplied by cos omega cn has two
shifted version, one to the right by omega

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

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

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height 2 and a 2 also peak, which are
around pi.

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

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

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

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

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components, and we will have to learn how
to actually get rid of them, and this will

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be the topic of the next module.
Finally,we're going to see a real

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

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The abstraction of the problem is that you
have reference sinusoid at some frequency

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omega 0.
You have a tunable sinusoid of frequency

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omega.
And we would like to make omega, omega 0

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as close as possible, actually equal and
this only by listening to it.

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And what we are going to do here is a
beating between these two frequencies when

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they are close enough.
And then by tuning, we can bring this

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beating to essentially frequency zero at
what point omega is equal to omega note

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

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So how are we going to go about this?
Well, first we bring omega close to omega

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

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of musical ear.
When these two frequencies are close, we

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play both sinusoids together, then we have
to remember trigonometry.

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And we write xn, which is a sum of cos
omega 0 n plus cos of omega n, in terms of

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a sum and a difference of these two
frequencies, which finally can be written

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approximately as two times the cos of the
difference, delta of omega times n and the

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cosine of the base frequency omega naught.
From this formula we see there are two

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components, there are the errors signals,
the cosine of delta omega N and there is a

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modulation signal, The cosign at omega 0.
When omega is close to omega 0, the error

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signal is very low frequency, so we cannot
really hear it, because it's such a low

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frequency.
So the modulation will bring it up to the

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hearing range and we are going to actually
hear it as an oscillation of the carrier

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frequency.
And we're going to see this pictorially in

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just a moment.
Let's look at the pictorial demonstration

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here.
So we start omega 0 is 2 pi times 0.2.

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Omega is 2 pi times 0.22, its a difference
which is actually half of the difference

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between the 2 is 2 pi times 0.01.
We see now interestingly, we have the

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carrier frequency, which is red curve
modulated by the difference by cosine of

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delta of omega.
So we see this, the beeping in blue

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overlaid to the red curve which is
modulation.

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We can change the frequency omega to 0.21
times 2 pi.

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The difference now is 0.005, the beating
is slower.

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We pick omega is equal to 2 pi times
0.205, the beating is even slower.

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And here, we take an example where omega
is very close to omega 0 and the beating

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is extremely slow.
And we almost see only the modulating

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frequency omega 0 and the very slow
variation due to the beating by cosine of

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delta of omega.
It's time to see a video demonstration how

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to tune a guitar, using this very simple
principle.

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You've probably have seen musicians doing
it on stage, now you understand the math

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behind it.
Okay, after all these maths, let's try to

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do something useful like tuning an
electric bass.

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

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

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

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to make equal.
As long as they are not, there will be a

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beating that we can adjust, we are going
to hear this in just a moment.

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

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The short harmonic is at 164.8.
For the A string, the first harmonic is at

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110 and the second harmonic is at 165.
And we're going to use these two harmonics

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to do the tuning and you can hear when
their out of tune.

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

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slows until it's actually zero beating.
And then the two frequencies are similar

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and then we can start playing something
like or something else.