We have seen how to calculate spectra of
signals, of sequences.
Now, once, you have the spectrum and it is
not the wide band spectrum, but it is
mostly around the 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
low-pass signal.
If it has the energy concentrated
somewhere else, we call it a band-pass
signal.
And if it's energy is around minus pi or
plus pi frequency, that is called a
high-pass signal.
Now, we're going to see a modulation
theorem for the fluid transform, which
allows to transform a signal.
Let's say from low-pass into a band-pass
signal or to deimmolate the signal which
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 particular string to another
string or to a reference sinusoid, I use
in some of the modulation theorem, may be
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 low-pass, high-pass and
band-pass 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
mainly 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
low-pass 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 the low-pass spectrum.
As you can see, the energy is mostly
concentrated around the origin, around 0
and there is no energy outside.
This is now high-pass signal.
The energy is around pi or minus pi and
there is no energy around the origin.
Finally, the band-pass 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 of n, and its DTFT capital
X of e to the j omega?
So it is the DTFT of xn multiplied by a
cosine of omega c times n.
So that's the DTFT of using Euler's
formula, as usual of x of n multiplied by
both e to the j omega cm and e to the
minus j omega cn.
And, this simply creates a double
spectrum, namely is equal to 1 half,
capital x of e to the j omega shifted to
omega c and shifted to minus omega c.
So usually, we take xn as a baseband and
omega c is called the carrier frequency.
Now, to get the intuition for this
formula, think of the following case.
Think of x of n being as a constant, so we
simply have the DTFT of cosine velocity n,
which of course has these two peaks at
omega c and minus omega c as we know from
the DTFT of a cosine function.
So this gives an equation so if xn is a
very, very narrow band low-pass 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
minus 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 multiply it by 1
half.
That's the first green spectrum, then we
move it to minus omega c.
It's a blue spectrum also multiplied by 1
alpha, and this is the result, the red
spectrum now after modulation.
So the central peak has been moved into
two alphas big peaks at alpha omega c.
We know that spectrum is 2 pi periodic, so
let's show a few periods here for minus 4
pi to plus 4 pi shifted to omega c, green
specturm shifted to minus omega c, 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, omega c is very close to pi, the
maximum frequency, close to minus pi, the
blue spectrum.
And we see now that we have a funny
looking spectrum around plus or minus pi
and plus or minus 3 pi.
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 minus 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 low-pass 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 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 yn is xn times cosine of omega
cn.
It's spectrum, we have seen before y equal
to g omega is a combination of the two
spectra shifted to omega c and minus
omega, omega c.
The DTFT of yn multiplied by two cosine of
omega cn.
Well, it's going to be the combination of
capital Y shifted to omega c and to omega
c and minus omega c.
Then, we replace the formula we just had
before so we have four terms.
One shifted by 2 omega c and also one by
minus 2 omega c, and two terms that are
actually at zero origin.
And so, we have indeed capital Xe to the j
omega on plus 1 half and two modulated
versions at 2 omega c and minus to omega
c.
Let's do this pictorially.
So the DTFT of x n is shown here.
So it's a spectrum, triangle spectrum
around the origin.
Then it's modulated version has two peaks
at minus omega c and plus omega c.
Then yn multiplied by cos omega cn has two
shifted version, one to the right by omega
c, it's the green one, one to the left by
omega c is the blue one.
And totally, the sum of these two which
has a peak around the origin, which is of
height 2 and a 2 also peak, 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 minus 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.
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 reference sinusoid at some frequency
omega 0.
You have a tunable sinusoid of frequency
omega.
And we would like to make omega, omega 0
as close as possible, actually equal and
this only by listening to it.
And what we are going to do here 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 omega is equal to omega note
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 omega close to omega
north.
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 xn, which is a sum of cos
omega 0 n plus cos of omega n, in terms of
a sum and a difference of these two
frequencies, which finally can be written
approximately as two times the cos of the
difference, delta of omega times n and the
cosine of the base frequency omega naught.
From this formula we see there are two
components, there are the errors signals,
the cosine of delta omega N and there is a
modulation signal, The cosign at omega 0.
When omega is close to omega 0, 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 omega 0 is 2 pi times 0.2.
Omega is 2 pi times 0.22, its a difference
which is actually half of the difference
between the 2 is 2 pi times 0.01.
We see now interestingly, we have the
carrier frequency, which is red curve
modulated by the difference by cosine of
delta of omega.
So we see this, the beeping in blue
overlaid to the red curve which is
modulation.
We can change the frequency omega to 0.21
times 2 pi.
The difference now is 0.005, the beating
is slower.
We pick omega is equal to 2 pi times
0.205, the beating is even slower.
And here, we take an example where omega
is very close to omega 0 and the beating
is extremely slow.
And we almost see only the modulating
frequency omega 0 and the very slow
variation due to the beating by cosine of
delta of omega.
It's time to see a video demonstration how
to tune a guitar, using this very simple
principle.
You've 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 and 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 short 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
their 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 or something else.