We're going to talk about a real communication system today, a so called
modulated communications systems. The idea is of course, instead of using
base band signalling we're now going to shift the message signal up to a higher
band of frequencies. This allows it to get through wire line
and wireless channels more easily. We'll talk about the transmitter and
receiver. Turns out we already understand how these
work. Once I show you how they function, you
know how to think about them because we'lve already talked about it in previous
videos. And finally we'll compute the SNR that
results when you use modulated communication.
So, what does the transmitter look like for modulated communication?
It's very straightforward. So here's our message.
And we get this somewhat curious thing in which we add one.
And the assumption here is, is that the amplitude of the message has been scales
so it's less than or equal to 1, which makes this greater than equal to 0.
Now the reason for this has to do with your sum receivers that require that this
be a positive quantity, but not all, but in order to satisfy everybody most
amplitude modulation systems do this, adding this offset and then scaling the
message accordingly. The transmitter provides a gain, usually a
as a big number. And multiplies by a cosine at what's
called the carrier frequency. And so the idea is that the cosine here is
carrying the message. That's the idea and so fc defines the
characteristics of the transmitters so There's how much, essentially, power is
radiated is determined by a, and what frequency, range is used is determined by
fc. So, it's easier to understand what's going
on here by looking in the frequency domain.
So here's my typical message spectrum, assume it has this rooftop look, and as we
know, from previous videos when we talked about the Fourier transform, we've already
figured out the spectrum of such signals like this.
So, you get a line spectrum corresponding to the fact that, that periodic par,
portion comes out by itself. Added to this, when you multiply a, low
pass message by a cosine, shifts it up and down in frequency.
So you get these two replicas of the message itself.
Now, I want to point out that the message bandwidth here is W.
Don't forget, bandwidth is defined to be the part of the positive frequency axis
devoted to the message, where it has power.
Well, for the modulated message, looks like it has half the bandwidth of the
original message, and this is always true for simple amplitude modulation is that
the transmission bandwidth doubles and that's just a fact of life.
And this kind of transmitter is really easy to deal with.
This fact that it doubles the transmission bandwidth won't be that much of a factor.
So, let's look at more detail, well, some of the consequqnces of using this kind of
scheme. Suppose we have two different modulation
communication schemes going on at the tsame time.
Because one of them is using a carrier frequency 1, the other one is using
carrier frequency 2. As we know from the wireless situation, if
both signals come in my receiving antenna, so I get 'em both.
Well, as long as these two spectra do not overlap, I can band pass filter.
And remove the second unwanted signal. I don't have to even know it's there.
This is called frequency division multiplexing, where multiplexing means
using more than once. And what we're doing Is using separating
signals in frequency. So all radio stations are all transmiting
on their own carrier frequencies at the same time.
So if you look in the time domain you'll see this mess.
All cellphone signals, all radio signals, all those signals are all occurring at the
same time, but in different frequency bands.
In, which means they're, they're being separated in frequency, which means simple
band pass filtering will allow you to focus on the message that you want.
And there is a little problem here. Suppose the second carrier is not
well-designed, and suppose it overlaps in frequency with somebody else.
Well now, neither, neither of these signals Can be demodulated without error.
This overlap here would cause interference, and it's something you can't
get rid of by linear filtering at all. So this is bad.
So it turns out you have to have some mechanism for regulating.
These carrier frequencies and these bandwidths.
So that there's no overlap among all the various uses that were used in wireless
communication and that's where the government comes in.
In the United States the regulation for what each frequency band can be used for
is very tightly regulated. And this is a rather beautiful depiction
of what can occur in every frequency band. And so let me, let's go through this a
little bit. This is a logarithmic frequency scale,
first of all. Frequency increases going down the page.
It starts at 3 kHz and this first one goes up to 300 kHz.
The rest of these are one decade, 300 kHz, 2-3 MHz, 3 MHz, 30 MHz, etc.
Down here, at very low frequencies, you probably can't read that, it says not
allocated. You can do anything you want up to about
10 kilohertz carrier frequency. And like I said for wireless it doesn't
propagate very well at all, and no one's going to really care if you build your own
transmitter, frequency range. It won't go very far so it doesn't
interfere with anybody else. But above that it's very tightly
regulated. The client's frequency here is 300
gigahertz, that's very, very high. And everything in between, you can see is
modulated. The AM radio band is here.
The FM band is somewhere here, and it may look like the FM band is narrower then the
AM band, but that's only because this is a logarithmic scale.
This is going from 300 kilohertz to three megahertz.
This scale goes from 50 megahertz up to five, I'm sorry, 30 megahertz, up to 300
megahertz. So this bandwidth is actually much wider
than it appears. And things become busier, if you will,
more things are allocated because it's the bandwidth that matters in a logarithmic
scale. Things look much more compressed than they
would be in the linear scale. The we point out that in some frequencies
bands you can use them for more than one thing as long as you don't interfere with
something else. But we there is so much demand for carrier
frequencies that some of them are, the government regulates them to be multi-use,
we have to do that. And actually, some are reserved for radio
astronomy. It's kind of interesting, I think that's
one of the bands reserved for radioastronomy, interesting.
Okay, well, how are we going to receive this thing, how are we going to receive a
amplitude modulated message. So, this is what we get what the, what
comes out of the channel and out of the, in the frequency domain.
We get the broadband noise, white noise and there's our, our message signals.
And pretty clearly, we want to get rid of all this.
Stuff that out-of-band noise and we do that by band pass filtering.
So this is a, stands for band pass filter as a center frequency equal to the carrier
frequency and has a badwidth of 2w because that's frequency range.
So once you go through that band pass filter, this is what you get.
You're only left with in-band noise and the message that we want.
Now I point out that the message that we want is not at the right frequency.
The original message started down here. How do we get the message down in
frequency, back to where it originally started?
We put it up here originally so we can get through the channel, this is the
transmitter matching the channels characteristics, antennas with much, much
better at high frequencies than at base band.
So we did that we got it through now we have to move it back.
How do we do that? Well, little surprise here.
So, let's look at this little mathematical property.
Suppose we take x, and multiply it again by a cosine.
So the original transmitted message was, a times 1 plus n times a cosine, and
multiplying it again by cosine gives me cosine squared.
What's cosine squared of theta? Well, cosine squared of theta is one half
times one plus cosine of 2 theta. So we wind up with something at twice the
carrier frequency that we started with. Let's expand this.
So there's the first part and you get the same thing times this cosine.
So, believe it or not, remodulating by the cosine gives us back a signal that's now
in it's original frequency band. We also though, have a signal they're
sitting up at twice the carrier frequency. I think I know how I'm going to get rid of
that. I'm still, I'm simply going to filter it
out to get rid of that, and when I do that I'm left with my original message.
So this is the where we started coming out of the front end, and after multiplication
by the cosine and low pass filtering, we wind up with our message down where we
want it. We're going to low pass it to get rid of
these things. And we're just going to have to live with
the noise that wound up in the same band and that's where we'll have to calculate a
signal to noise ratio. So, here's the demodulator in all of it's
glory. Consists of what we call the front-end.
Which just is a simple band pass filter to get rid of Extraneous communications,
interference we don't care about and to get rid of out of hand noise.
And as we know that gives us a spectrum that looks like this.
Then after we multiply by a cosign and low pass filter we're left now with our.
Demodulated message consisting of the original message, that's what we want,
plus noise. So we need to calculate some signal to
noise ratios. Well, for all kinds of reasons, I want to
compute the signals and rates ratio here and here.
So the front end output is going to be of interest because it's a little bit simpler
to compute. And also it has it, it's interesting to
compare these two s, s and r's. So, for the front end the message part is
easy. So this is just x of t, times the
amplitude that we used and this is the attenuation introduced by the channel and
this is the power in this. And it turns out that power, that one half
comes from the cosine basically because it, the power in cosine is half.
And if you work it all out you easily decide that this is the power in the
message. The power in the noise is just as easy, in
fact easier, to calculate because it lives in a frequency band that's centered around
the carrier frequency. And so it's been with this 2W and you
multiply by 2 times naught over 2, 2 to the positive, negative frequency, 2 out of
2 is the specter of light, and we wind up with a signal to noise ratio coming out of
the front end filter that's given by this expression, which should look pretty
familiar compared to the base band case. We clearly the transmitters amplitude gain
is to compensate for the attenuation introduced by the channel.
And the bigger the noise is, the smaller the SNR, and the less happier we're going
to be. And the smaller the attenuation the less
happy we're going to be the SNR may not be good enough.
Well now let's calculate the signal to noise ratio in the N hat.
So, to do that we simply look at the message part.
And on a previous slide I showed you. And after you multiply by the cosign which
you get for the message part and it's just going to be this power which I think is
pretty easily seen to be that we square all the constants up front in the power m.
I should point out that the power in n is not very big and since we've done consider
it, because remember we imposed the condition that the amplitude of the
message that we never got bigger then one in absolute value.
So this power is not a big number. Any gain that's provided in, in amplitude
modulation schemes is provided by the transmitter through.
Now the noise is the part where we have some fun.
So after you do the moving up, moving down of the spectrum and, and wide up down
here. The message part is given by the Voice
power spectrum in R tilda, the part of R tilde that's related to the white noise, I
call that N tilde. It, after multiplying by the cosine that
power spectrum got shifted up and down in frequency, and they both wound up here.
There are other parts that wound up in. Frequency ranges we won't care about
because they're going to be filtered out. And the four again comes from the power
considerations in the cosine. So, all we have to do is add up these
components here. So, we get a two for positive and negative
frequency. Each of those had spectral height N naught
over 2. The bandwidth here is W and the 2 comes
from the fact that powers add, that means that power spectra add.
So, they're both a constant power spectrum, so you just get a factor of 2
and over 4 is, Is because we're already in the power domain here, the formatted power
spectrum. So, we get the four, put it all together,
the noise power is in the D over 2. So that gives us a signal to noise ratio
that's perceived and a received message that's given by this.
It turns out you can show that there is going to be no other receiver that gives
you a bigger signal from noise ratio than that quantity.
So this is what we call, is called optimal.
You can't do better than this. And so you cannot improve the signal to
noise ratio over this quantity. And it depends on channel and transmitter
characteristics in an obvious way. Now we point out that the snr here is half
of what it is here. So there's a gain in this receiver part,
by a factor of two over the SNR that's here.
And it's very interesting story about how that comes about.
It's not at all obvious, and have to leave that for another day.
So, now we have a characterization of virtually all modulated communication
systems, at least in terms of amplitude modulation.
So, the transmitter multiples the message by echo sine.
The receiver as a front end filter followed by another.
Essentially modulation scheme followed by a low pass.
These are very easy to build and construct.
So, because we amplitude modulate, we can choose these carrier frequencies and now
send many, many, many signals all at the same time but in different frequency
bands. It's just the advantage of thinking about
signals in the frequency domain. In communication systems, it's very clear.
Frequency domain is very, very important to understand.
We also, you move it up in frequency, so it gets through these wireless channels
more easily. Antennas work much, much more efficiently
when you move them up. In the way that we've seen.
You now have the muliplexing situation. You can do this multiplexing and the
signal to noise ratio we wind up with is fundamental.
Very important to remember this. This is a guildeline for how well
assisting you may be concerned about works in terms of.
The amplitued provided by the transmitter and how much the attenuation is in the
channel. And don't forget, we've assumed that alpha
is some constant divided by the distance between the transmitter and receiver.
So the further away you are from the transmitter, the smaller your SNR is going
to be by the square of the distance.