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visit MIT OpenCourseWare at ocw.mit.edu. DENNIS FREEMAN: Hello. Welcome to the first lecture in
the last topic of this class. So we'll spend this lecture
and the next lecture talking about modulation. Modulation, like sampling,
is an excellent illustration of the power of thinking
about signals in terms of their Fourier transforms. We'll see, just like
we saw in sampling, that a problem that
was potentially very complicated to
understand, sampling, was very easy when you thought
about it in the Fourier domain. And precisely the
same thing happens with modulation, which is
precisely the reason we're talking about it now. So I'll talk about modulation in
the context of a communication system. That's just for convenience. In fact, modulation is used
in lots of other places. In fact, next time,
in the next lecture, I'll show you an
application of modulation from my research group,
where we used modulation to improve the resolution
of an optical microscope-- nothing whatever to do
with communications. But it's still-- the
optical microscope-- the enhancement to
resolution that we achieve is directly based
on the principles that I'll start
to describe today. So I'll talk about
modulation today in the context of
communications. Probably the most
convenient, easiest to think about communication
system to all of us, being humans, is speech. We use speech for
communication all the time. It's very easy to think
about how speech works as a communication medium. Somebody talks,
somebody listens. One of the first ways
of thinking about it as a technological feat
was the telephone-- the idea being that you convert
the sound that I'm emitting when I'm speaking
by a microphone into an electrical
representation that gets shot down a wire. Then at the other end, you
take the electrical signal that's coming down the wire
and turn it back into sound. That's the principle
of a telephone-- works very well. Modulation comes
up when we start to think about how would
we generalize this notion for wireless communication. In particular, if we do
cell phone communication, cell phones transmit the
signal that is picked up from the microphone. That signal gets converted
into electromagnetic signals. That's the basis by which
cell phones communicate with the cell tower. The towers may communicate
with other towers via lot of different
kinds of technologies. I'm ignoring those for now. But the communication between
the phone and the tower is via electromagnetic waves. And there's an
interesting thing that happens when you try to recode
the signal that would have been perfectly happy running
down the copper wire, when you try to recode that signal
into an electromagnetic wave. And that has to do
with basic physics of electromagnetic waves,
which I'm sure you all know. And so I'll just remind
you of one simple idea. So for efficient
transmission from an electrical representation
to an electromagnetic wave, first off, that transduction
is mediated by something we call an antenna. Antennas will take
an electrical signal and convert it into an
electromagnetic wave, and vice versa. But the efficiency with
which an antenna works has to do, among other
things, with its size. It's very difficult-- by which
I mean it takes lots of power-- to transform a signal from
an electrical representation to an E&M representation
if the antenna is smaller-- is significantly smaller-- than
the wavelength of interest. So it's not that
you can't do it. It's that it takes
lots of power. So if you don't want
to burn a lot of power doing that
transformation, then you need to make an
antenna that's roughly the size of the wavelength
that you're trying to transmit as an electromagnetic wave. So if we were thinking
about this kind of a scheme for the
communication of voice-- we've talked about
voice many times-- Telephone-quality
voice is usually defined to be frequencies
between about 200 hertz and about 3 kilohertz. If I had a signal
that was composed of those kinds of
frequencies, how big should I make the antenna
to get efficient coupling between the electrical
representation in the cell phone and
the electromagnetic wave that the cell phone
wants to launch to get to the cell tower? So look at your neighbor. That involves turning your head. [LAUGHTER] Say hello and figure out how
long should the antenna be. [INTERPOSING VOICES] DENNIS FREEMAN: Does
everybody have an answer? Raise your hand if you
don't have an answer. How do you like that
for a switch of-- Raise your hand if you
don't have an answer. You all have answers? OK, so how big should
the antenna be-- 1, 2, 3, 4, or 5? OK, very good. So the answer is really big. So you think about the
relationship between wavelength and distance, so you can think
about the relationship is given by speed. The speed of a wave
is the wave length-- how long it goes per cycle-- times the number of
cycles per second, f. So you can solve that
expression to find out the wavelength in terms
of the speed, which for an electromagnetic wave is
the speed of light, 3 times 10 to the eighth meters per second. The hardest one to launch
is the lowest frequency. That'll take the
biggest antenna. So the lowest frequency
in telephony-- in telephone-quality
speech-- is 200 hertz. So we get about
1,500 kilometers-- kind of big, kind of useless. If you were thinking
about trying to make cellular communication
and your antenna actually had to be 1,500 kilometers,
that just isn't going to work. So what do we do? We obviously don't do this. So the answer is that
you would need an antenna hundreds of miles in length. So what frequency
should you be using if you wanted to build a phone
that kind of fit in your hand-- 10 centimeters or so? What would be the frequency-- what would be the interesting
range of frequencies that you would want to
use for such a device? And the answer is-- Of course. So the answer is, you
go to the bottom one, and it's bigger
than a gigahertz. And it's just running
the same expression in the other direction. So you think about, the
frequency that you would like would be the speed of light
divided by the wavelength. If you wanted the wavelength
to be 10 centimeters, then you would end
up with a frequency on the order of gigahertz. And it shouldn't come
as a surprise to you then that that's what we use
in cellular communication. So a modern cell phone
uses 2.1 gigahertz. So the point is that
when we're thinking about how we would like to use
the electromagnetic spectrum for a communications
task, that spectrum is not necessarily well-matched
to the communications problem of interest. You might think that
that cellular example is an exception. In fact, that's the rule. If you try to have a signal
of interest transmitted over a medium or stored
on some sort of a medium, it is generally the case that
there is a matching problem, that the characteristics
of the medium don't match the
characteristics of the message. And so part of
communications engineering is trying to come up
with a coding scheme that matches the characteristics
of the message to the characteristics
of the medium. And so what I'm
going to do today is talk about some matching
schemes based on modulation. So we just saw that if we wanted
to do cellular communication of voice, voice might have
a spectrum represented by this magnitude
function, X, where the bandwidth is on the
order of a few kilohertz. But we might want to
transmit a signal that has the same information. However, we would
like the frequencies to be up around 2 gigahertz. Which of these coding
schemes, Y as a function of X, achieves this transformation? Take the stuff that you have
in a low frequency range and shift it to a
high frequency range. What would you do? The obvious answer is to
stare with a blank face in it. It'll definitely
come to you, right? You don't want to
talk to somebody else. That would give it away. [INTERPOSING VOICES] DENNIS FREEMAN: So what's the
relationship between X of t and Y of t? Is it relation 1, 2, 3,
4, or none of the above? OK, it's about 80% correct. So how do I think about it? So let's see, I
want to figure out a relationship between the
Fourier transform of Y and X. If I wanted to figure out
the Fourier transform of Y, I would integrate Y of t e
to the minus j omega t dt. That looks kind of right. And Y would be X of the t-- let's try the first one-- X of t e to the j omega c t
e to the minus j omega t dt. Well, that's almost the Fourier
transform of X. All I've done is shifted omega. So in fact, that's the same
as X of j omega minus omega c. And that's what I want to do. So the idea is that,
if you multiply by this complex
exponential, the effect of that multiplication in
time is to shift frequencies. Can somebody say that to me-- say the same transformation but
in slightly different words? Guess that was kind of big. Yes. AUDIENCE: Couldn't you
[INAUDIBLE] multiplication by the exponential in time. DENNIS FREEMAN: [INAUDIBLE]
by the exponential in time-- AUDIENCE: Yeah equal to-- DENNIS FREEMAN:
--should correspond to-- AUDIENCE: Which gives you
a frequency [INAUDIBLE]. DENNIS FREEMAN: So
generally, a multiplication in time corresponds to-- AUDIENCE: [INAUDIBLE] DENNIS FREEMAN: --convolution. So equivalently, instead of
saying it shifted in frequency, you could say-- AUDIENCE: [INAUDIBLE] DENNIS FREEMAN:
--it got convolved-- AUDIENCE: With delta. DENNIS FREEMAN: --with
the delta function. So a different way of
saying the same thing would be that you
think about convolving with the delta
function in frequency-- same thing. Now, we don't actually do this. When we're doing the
cell phone thing, we don't actually multiply
it by e to the j omega c t. Anyone know why? This is kind of the simplest
way that you could imagine. I've taken frequency
content centered near 0 and turning it into frequency
content centered near omega c. But that's not
what we really do. Why don't we really do that? AUDIENCE: [INAUDIBLE] DENNIS FREEMAN: Exactly. We don't really do it, because
the signals aren't real. So how do you know the
signal is not real? AUDIENCE: Because
magnitudes [INAUDIBLE]. DENNIS FREEMAN: If the
signal had been real, the Fourier transform
would have been-- If the signal had
been real the Fourier transform would have been-- X of j omega. If the signal had been
real, the Fourier transform would have been-- AUDIENCE: Symmetry. DENNIS FREEMAN: --some
kind of symmetry. How do I see symmetry
in that expression? AUDIENCE: Conjugate. AUDIENCE: Conjugate. DENNIS FREEMAN:
Conjugate symmetry. How do I see conjugate
symmetry in that expression? Well, this is cos omega
t plus j sine omega t. So if this is a real signal,
then it gives rise to something here that is symmetric. The cosine terms are all
symmetric about the origin. And it has an
imaginary part that is antisymmetric
about the origin, because you add together
a bunch of cosines. You can't get anything that's
anything other than symmetric about the origin. You add together
a bunch of sines. You can't get anything
except something that is antisymmetric about the origin. So you know this
thing has to be-- so a real signal would have
had conjugate symmetry. The real part would
have been symmetric. The imaginary part
would be antisymmetric. And you can see that this
is not conjugate symmetric. Everybody knows what
I'm talking about? AUDIENCE: [INAUDIBLE] DENNIS FREEMAN:
Conjugate symmetry would mean that the
real part of the signal is symmetric about the origin,
which means that if this is supposed to
represent a real signal, then there should have been a
reflection over here to make it symmetric about the origin. AUDIENCE: Just was
it symmetric before? DENNIS FREEMAN:
It was symmetric, but I shifted it by
a complex number. I shifted it by cos omega
c t plus j sine omega c t. So by that multiplication, I
generated a complex number. AUDIENCE: [INAUDIBLE]. DENNIS FREEMAN: So this signal
was complex valued and not conjugate symmetric. So the point is
trying to get you to remember the kinds of things
that we're supposed to know about Fourier transforms. So by shifting with a
complex exponential, we wreck the realness
of the original signal. The real original signal would
have been conjugate symmetric in frequency. But the wrecking of it
gave rise to a signal that was no longer conjugate
symmetric in frequency. So we don't really
modulate this way. But we do the obvious extension. What we would do is
modulate with a cosine wave. So now, if instead
of multiplying by a complex exponential, I
multiply by the cosine omega c t, by Euler's expression,
I can think about that as being the sum
of two components-- one at plus omega c and
one at minus omega c. And now, when I do
the convolution, I get a signal that is
conjugate symmetric. So when I convolve
this with this one, this one gives me a
copy of this here. And when I convolve
this with this one, this one gives me a copy
of this one down here. So now, the resulting signal,
which I know by construction-- if this was real and this
was real, then that's real-- I know by construction that
this signal must have been real. But I can also see
it in the transform, because I can see
now that there's a symmetry that is
consistent with it being conjugate symmetric. Yes. Somebody had a question? So that's what we
mean by modulation. This is modulation. Modulation just is a fancy
word that means multiply. So what we're going to
do is multiply the signal by a carrier. The carrier is going to
be a signal that carries the message through the medium. The carrier is chosen
so that it goes efficiently through the medium. And then the carrier carries
the message through the medium. So we think about
this as modulation. And we want to be familiar
with going back and forth between time and frequency. You can also think about the
result of modulation in time. So if this were
my message, which is intended to be represented
as a low frequency, and this is my carrier,
which is intended to be represented as a
higher frequency, then when I modulate it, I
get a modulated signal, by which I mean-- this is called
amplitude modulation-- the amplitude of the carrier
is modulated by the message. That's all we mean. So you can see that this
transformation, which has the property of
moving the information from a low frequency
that's hard to transmit to a high frequency that
is easy to transmit, that has the effect of doing
a very particular pattern to the time-domain waveform. Now, it would be completely
useless as a communication scheme if it weren't
easy to invert. So imagine that I
have this signal. And what I'd like
to do is recover x. What should I do to recover
x, the original message? So the idea is, I have
an original message available in my cell phone. It gets modulated so
that it can be launched into electromagnetic waves. The electromagnetic waves go to
a receiver thousands of miles away. And now, the idea is to
reconstruct my original signal x. What would I do-- what kind of a system would
I use to recover x from y? AUDIENCE: [INAUDIBLE]. DENNIS FREEMAN: I'm sorry. AUDIENCE: Couldn't you
divide out the cosine? DENNIS FREEMAN: You could
divide out the cosine. So you could take x of t cosine
omega c t times something-- what do I want to
say-- a of t designed so that this times this is 1. That's kind of ugly. Anybody see anything
ugly about that? Yeah. AUDIENCE: [INAUDIBLE] shift
it back the other way? DENNIS FREEMAN: You could also
shift it back the other way. That's right. So before we do that,
why is this ugly? AUDIENCE: The zeros. DENNIS FREEMAN: The zeros. So if we wanted to take a signal
that looks like a cosine wave and multiply it by some
signal that generates 1, that's not too hard to do here. You would do that with
something like this. But it becomes very
hard to do here. So you would end up making some
signal that does some awful-- So it would
periodically be a mass. But you can do what you said. An alternative would
be to multiply it by another cosine, which
in the frequency domain is easy to think about. It would just shift it back-- so convolve with
a pair of impulses to move something that was at
DC out to some high frequency, convolve again to take the
thing that was at high frequency and bring it back to DC. And you can think about that
in either frequency or time. It's easy to think
about it in time. If you think about it
in time, here we've got the product of two cosines. But the product of two
cosines is just 1/2 plus half the cosine of double. Well, that's good. Why is that good? Well, if you multiply
x of t by this, this is a super high frequency. If omega c was a high frequency,
2 omega c is even higher. So what you could do is remove
x times 1/2 cos 2 omega c t with a low pass
filter, since omega c is such a high frequency. And that would just leave you
with half the message, which would be easy then
to reconstruct, because what you would do is
just put it through a low pass filter and then multiply by 2. You can similarly think about
the same thing in frequency. If I took y and convolved
it-- if I multiply in time by another cosine
wave, that second cosine wave is a pair of impulses--
one at minus omega c and one at omega c. And now, when I
convolve the y signal, this one shifts these
two up, and this one shifts these two down. And two of them land
on top of each other. But each of these was
only of height 1/2. So by Euler's expression,
cosine of something was 1/2 e to the whatever plus
1/2 e to the whatever. So I got 1/2's on each
of those amplitudes. So the result is then that
I have to multiply the low frequency part by a factor
of two to undo the 1/2's. So this kind of a scheme
is especially nice, because you can scramble
together multiple messages and still get them separated
at the destination. If you imagine having three
similar transmitters that use their own omega c-- so the first one uses
omega 1, omega 2, omega 3-- so if each one of
the transmitters had their own frequency, and if
the frequencies were far enough apart, and if the frequencies
were all big compared to the message frequency, then
you could combine them all and select out the
one of interest by tuning the receiver, by
choosing the demodulation frequency. So now, if the receiver
chose omega c equals omega 1, you would decode message 1. If omega c were omega 2,
you'd decode message 2. And that's because the medium
works approximately linearly. So if you launch multiple
waves into the air-- I don't want to get too much
into electromagnetic theory-- so the presence of the antennas
distort it from linearity. But once the antennas
are all there, then it's perfectly
linear system. And the thing that gets into
the air as a result of a sum is the sum of the
individual parts. So the idea then is
illustrated here. If I had three different
messages represented by different style houses, and
each one of the messages was at a different frequency--
omega 1, omega 2, omega 3-- by tuning the omega
c of the receiver, if I put omega c at omega 1,
the convolution of this one would suck this one up to here. And by this one would
lower that one down there. You get overlap of the
lowest frequency pair. And so if you built
a low pass filter of exactly the right width,
you would decode message 1. Where, if you just changed the
frequency of the demodulator-- if you make the demodulation
frequency now be omega 2-- now, the effect of shifting
that different amount means that the low pass filter
recovers message 2, rather the message 1. So that's the idea. So that's the idea that we
use in commercial AM radio. And that was, in fact,
a revolutionary idea that enabled people to think
about for the first time a communication system
that did a lot of things that were very different from
previous communication systems. In particular, it went
at the speed of light. Even more importantly,
or at least as important, is the fact that it
was a broadcast system. So broadcast was
an idea that was championed by David Sarnoff. Sarnoff was a visionary. He was the person who was
very excited about the idea of broadcast, which is a
little ironic, because he got his start with Marconi. Anybody ever hear of Marconi? Good, good, you're supposed
to have heard of Marconi. So Sarnoff got his start-- this is Sarnoff;
this is Marconi-- Sarnoff got his
start with Marconi. Marconi made his mint
with wireless telegraphy. Anybody ever hear of telegraphy? Of course not. So telegraphy, that's telegraph. Shake your heads yes. It's ancient, I realize. So Marconi made his fortune
with wireless telegraphy. Telegraphy was what we
call point to point. The idea in telegraphy
was precisely the same as the idea of the
US post office, except it was at
the speed of light, or not quite the speed of light. So the idea in the post office
is you take a sheet of paper and do something to it that
makes it magically appear at somebody else's place. So point A communicated
to point B. Telegraphy was
precisely the same. You take your sheet of paper
to the telegraph office. And somebody who's very
skilled with their hands, or specifically
with their finger, would do something that
caused that piece of paper to be regenerated
hundreds of miles away. Then that piece of
paper got delivered. So message went from
point A to point B. So telegraphy came
long before Marconi. And it was a revolution in
how you do communications. But it was point to
point-- one person sent one message to one person. Sarnoff got his start
in newspaper business. He was a Russian immigrant,
impoverished, and had a newspaper route as a kid. And he was ambitious. Newspapers are broadcast. The idea in point to point and
broadcast are very different. In broadcast, you're
allowed to spend a fortune on the transmitter-- the printing press--
but not on the thing that the individuals get. The individuals get newsprint. So the paper and ink
have to be cheap. The printing press doesn't. So that was
Sarnoff's background. So he was interested as a kid
in broadcast, in newspaper. But then he got his reputation
working for Marconi in wireless telegraphy. Marconi, the inventor of radio,
thought of a way of doing point-to-point telegraphy
wirelessly via radio-- radio telegraphy. And he sold it to ships. And Sarnoff made his
reputation, because he was the guy operating
the radio telegraphy system at that Marconi
company when the Titanic sank. Everybody know about
the Titanic, right? Big ship sank. So Sarnoff was known as an
amazing telegraphy operator. And he stayed at the station
for 72 consecutive hours getting emergency messages
from the Titanic, telling everybody
everything that he could, trying to tell them the
situation, whose family was in good shape, whose family
was not in good shape. It had an enormous impact-- big enough that
Congress made a law saying that every ship had
to have wireless telegraphy. Now, that made Marconi
extremely rich. And it indirectly made
Sarnoff extremely rich too. Sarnoff then got very
interested in extending the idea of radio,
which then was point to point to broadcast. So the idea was to somehow-- he called it a radio music box. Somehow, it was supposed
to be like a newspaper but at the speed of light. So the idea was to somehow
make mass consumption of radio. At the time, radio was
per ship, point to point. So you have the land-based
station talking to ship A, or the land-based
station talking to ship B, or ship
A talking to ship B, but it was all point to point. Sarnoff's idea was, let's
make a newspaper out of this. The key to doing that was
making a cheap receiver. It's like newsprint. The printer can cost a mint. The transmitter for
radio broadcast music is allowed to cost a
mint, but the receivers are not allowed to cost a mint. That's like the newsprint. So the trick was to make
an inexpensive receiver. The problem with making an
expensive receiver is that the scheme that we
just talked about, where you decode the signal by
multiplying by cos omega c t-- omega c chosen to be the
frequency that you want to listen to-- the problem with
that-- that's called synchronous demodulation--
the problem with that is that you've got to
be exactly synchronized. If you want to listen to
the message at omega 2, omega c must equal omega
2, not omega 2 plus 3. So if you want to listen
to a particular frequency, to a particular message, you
had to have the frequency chosen to match the carrier
of the message you wanted to listen to. Today, that's not so hard. The way we would make
frequencies today is with crystal. Crystals are great because
the frequencies are determined by the distances in
crystal lattices, which are determined by nanomechanical
processes very precisely. And in fact, we can make
crystals with no problem with frequency resolutions
of 10 to the minus seventh, even 10 to the minus eighth. So the errors are very small
compared to the frequencies that we're trying to generate. Even that wouldn't
be good enough. So back then-- so Sarnoff was
working back in the 1800s-- back then, they
couldn't possibly do 10 to the minus eighth precision. They were doing
something more like 10 to the minus second precision. They didn't have a
technology based on crystals. So it would have been
impossible to match even to within a factor
of 10 to the minus third. But even if they
could have matched to within 10 to the minus
eighth, which we could today, even that wouldn't
work, because not only does the frequency have to match,
the phase has to match. So if you're multiplying
by cos omega c t and you want omega
c to be omega 2, it better actually be
the cosine of omega 2 and not the sine of omega 2. Sine and cos differ by a phase. So the question is, what's the
effect of phase when you're trying to demodulate a signal? So look at your neighbor. What would happen if
you tried to demodulate by precisely the right frequency
but you were slipped by phase? [INTERPOSING VOICES] AUDIENCE: [INAUDIBLE] over here
and be much smaller at the end. [INTERPOSING VOICES] AUDIENCE: And then it's
to the point [INAUDIBLE]. [INAUDIBLE] [INTERPOSING VOICES] AUDIENCE: And then
if you [INAUDIBLE] [INTERPOSING VOICES] DENNIS FREEMAN: So
what would happen if there is a shift
between this carrier that was modulating the signal
and the carrier that is demodulating the signal? What would happen
if there's a phase shift of phi between those two? What's the effect
of phi not being 0? Ideally, phi would be 0. Ideally, phi would be 0. What would happen
if phi were not 0? You did all that talking,
and you don't have-- Yes. AUDIENCE: So the low pass
filter of the two will not work. It would be-- It will be lots more
than what you want it to be, like 1/2 the original. DENNIS FREEMAN: So
that's exactly right. So the effect of phi-- if you think about-- so now, you have cosine of some
omega c t and cos omega c t plus phi. So you have cos A cos
B. so that gives you the cos of the difference
and the cos of the sum. The cos of the
difference was supposed to be-- the difference
was supposed to be 0. The cos of the
difference would be the cos of zero,
which would be 1, and that's where
the 1/2 came from. But now, the difference
isn't 0 anymore. So instead of
getting 1/2 cos of 0, we get 1/2 cos of
phi, which means that if phi is a constant, you
just get the wrong amplitude. But if phi is a slowly varying
signal, which it would be, even if you had a frequency
reliability of one part in 10 to the minus seventh-- one way of thinking about
that would be that there's a slowly varying phase-- the effect of the
slowly varying phase would be to make the
amplitude vary with time. So we call that fading. And it would be a
very distracting thing to have happen. So this kind of a
technology would even be difficult today, when we can
match frequencies very well. It was completely out of the
question back in the 1800s, when they couldn't match
frequencies that well. So the trick was to not
only send the message but also send the carrier. So ideally, when we first
talked about modulation, there is no C path. All you do is you
take the signal, and you modulate
it by a carrier, and you send that
out on the antenna. Now, instead, add on a
little bit of the carrier. That way what's in the air is
the carrier and the message. So now when you
receive it, somehow if you can receive
the carrier, you can use the carrier to tell
you information about not only the frequency but also
the phase of the carrier. And you can use that then
to demodulate the message. That's the idea. Notice that adding in a
little bit C of the carrier is precisely the same as adding
a constant to the message before you modulate--
mathematically identical. And that gives an easy way
of thinking about the effect of this carrier. If you think about
the message added to C and if C is big enough, you can
make the message positive only. You remember in the
previous illustration, every time the message went
through 0, which might happen here, the modulating
message went through 0 also, which means that sign
changes were affected by a 180-degree phase
shifts of the carrier, which was kind of a subtle thing. So now, the message
appears entirely as the positive
envelope of the carrier. Well, that's nice, because
that makes it very easy to decode in a way that
has no dependence whatever on the carrier frequency. If the carrier frequency
is big enough-- if the frequency of the
carrier is sufficiently larger than the maximum
frequency of the message, there's a trivial way to decode
such a system with a nonlinear circuit of this type. What's intended here is that
you take the message that's received from the antenna,
reconstruct y, which is intended to be the
output message, such that if z, the signal
on the antenna, exceeds the current value of
the message, the diode comes on. And that makes the blue
line, the decoded signal, rapidly go back up
to the red line-- the thing that's
coming off the antenna. But if the antenna signal
shrinks below the blue line, let the blue line
discharge, because there is an RC decay constant. So there's a fast attack
through the diode, so that the blue quickly goes
to the peak value of the red and slowly decays back toward 0. The result is that if the
difference in frequency between the carrier frequency
and the message frequencies is sufficiently large,
you can effectively separate the blue from the red
with a very simple circuit. And that's the way they do it-- or that's the way they
did it in the early 1900s. But there's still a
problem with that. The problem is
that the messages-- audio of the type that
I'm speaking, speech-- is characterized by an
enormous peak-to-average ratio. The strongest pressures
that are generated by speech are enormously more powerful
than the average pressure that's generated in speech. You can see that in this
diagram by these peaks. There are several things
that generate peaks. Peaks are generated at about 60
or 70 hertz by my vocal chords. But they're also generated
by my lips in plosives. When I do plosive,
there's a sudden jump in the instantaneous frequency
that's not there on average. And for normal
speech, that ratio can be as high as 35 to 1. 35 to 1, not a big deal. The problem is that power goes
like the square of voltage. So 35 to 1 becomes 1,000 to 1. It takes 1,000 times more
energy to code the peaks than it does to
code the average. And the problem with that is
that in this coding scheme that we talked about,
you have to add a constant that is big enough
so that the signal never goes negative. So the constant that you
add has to go in proportion to the peak value. So you end up transmitting
almost all of your power By the ratio of
1,000 to 1, that's the amount of power
that gets used to transmit the carrier
compared to the message. Well, that's a terrible scheme,
if what we were trying to do is point to point. Imagine your cell
phone, if you had to transmit enough power to, in
the worst case, do the peaks, you would on average
be transmitting power at 1,000 times the rate that
you would necessarily have to. So that's OK for broadcast. So for example, WBZ
broadcast radio, WBZ uses a 50-kilowatt
transmitter. 50 kilowatts is
the amount of power that would otherwise be
sufficient to generate 500 100-watt light bulbs. That's a fair amount of power. Imagine the heat that comes
off 500 100-watt light bulbs. That's how much power
is being radiated by the antenna for WBZ. That power is not necessary to
transmit the average message. It's necessary to
transmit the peak message. You can imagine how
long your cell phone battery would last if you were
transmitting 50 kilowatts. That doesn't work. So that's how the
broadcast radio takes advantage of broadcast. It makes no sense to
use this coding scheme for a point-to-point system. It's fine if what
you're trying to do is have one
transmitter, WBZ, that services a million listeners. That's fine. The problem with this
scheme for decoding is it still doesn't
separate different channels. And the way to fix that was
developed by Edwin Armstrong. So Sarnoff who was
kind of the visionary. He had the idea for
broadcast radio. He's the entrepreneurial type,
who thought of how to do this. Armstrong was the
technical genius. He knew how to do it. So Armstrong's idea here,
which we call superheterodyne, was let's make the signal
look like it always comes from omega i regardless
of what channel it comes from. So omega i, the
intermediate frequency, will always take whatever
frequency you're interested in and turn it into omega
i and will do that by just modulating. And the cleverness
had to do with a lot of technical details. He worked out a scheme where
this modulation was very easy. The cutoff-- the sidebands
on the bandpass filters didn't have to be
very steep, which made them easy to implement. The sharp band pass filters
were all at that one intermediate frequency. So he had to generate one
very sharp band pass filter, but that same sharp bandpass
filter could then be used for all the different channels. So the idea then was use
a coarse tunable filter to map the frequency
of interest to omega i, put that through a very
sharp filter, of which there is exactly one in each
receiver, and then use this decoding scheme to
demodulate the carrier, wi. That's how they did it. We would never do it that way. That's part of the
theme of the course. We are interested
in schemes that let us map continuous
time to discrete time, that sort of thing. So one way we might do it is
implement a radio digitally. So the idea would be, what
if you took the antenna, put it through a sampler,
turned the radio signal, which contains a gazillion
number of bands-- for commercial radio, there's
100 channels in the frequency band 500 to 600 kilohertz-- but just take the whole
signal off of the antenna, turn it into a bunch
of bits, run that through some digital
logic that does, by magic, picks out the one
that you're interested in, generates a new
stream of bits, yd, from which you can do
bandlimited reconstruction. So this is the
last two lectures-- do you do this, how you do this. Now, all we do is we put a
particular algorithm in there. And we've got a radio. That's the idea. So the key to being
able to do that is whether or not you
can build that sampler. So what would be the required
sampling time in order to make a digital radio? And since I'm
running out of time, I'll just tell you that the
important thing it's sampling-- it's what we did last time--
the answer is you need T, so that the sampling
frequency is at least twice the maximum frequency
of the thing that you're trying to code. So the biggest frequency
here is 1,600 kilohertz. You need to sample that with
omega sampling more than twice that frequency-- so bigger than 2
pi 1,600 kilohertz. And if you work that out,
that leaves sampling time of about 1/3 of a microsecond. And the point is that that's
easy to do these days. That's the kind of part that
you get from DigiKey for $2. So that's easy. So the only thing
that you need to do is worry about, well, then
how much computation is there? And that also turns
out to be easy. The principal thing you need to
do is make a bandpass filter. The question is, how would
you make a bandpass filter? And here are three
possible systems for making a bandpass filter. Should I take my digitized
antenna signal modulate low pass modulate, or modulate
low pass modulate, modulate low pass modulate, cosine,
sine, cosine, sine, or put it through a filter
that looks just like that unit sample response,
except that it's multiplied by cos omega c t n? Some number of those work. And I'll leave it for you to
figure out which of those work. Good to see you. Have a good day.
