9.1 - Digital communication systems

From the official description of 9.. videos:

Welcome to Week 8 of Digital Signal Processing.

This week's module is about digital communication systems and this is where it all comes together; from complex-valued signals, to spectral analysis, to stochastic processing, sampling and interpolation: everything plays a role in the design and implementation of a digital modem. Digital communications is an extremely vast and fascinating topic and it is arguably the pinnacle achievement of DSP in the sense that it's the domain where the most extraordinary quantitative progress has been made thanks to the digital paradigm. The fact that MOOCs such as this one are available to such an incredibly vast audience is just one of the tangible results of digital communication systems. It is only fitting, therefore, to devote the last module of our class to this subject.

We will start with the basics of data modulation and demodulation and we will progress to describing how your ADSL box works by way of its direct predecessor, the voiceband modem that spearheaded the Internet revolution by allowing for the first time the delivery of substantial data rates in the home.

Hi, and welcome to module nine of digital
signal processing.

This is the last module in our class, and
this is really where it all comes

together.
In this module we will review the

principles behind the success of digital
communication systems.

And we will look at different
communication systems starting from the

voice band modems that were popular a few
years ago and that you can still hear

when you use a fax machine, to the most
recent incarnations like the ADSL box

that you have in your home and that
you're probably using to watch this

video.
Digital communication systems need no

introduction.
The amount of information that we consume

and that we produce every day is
staggering by an historical standard.

And what is even more amazing is that we
can access this wealth of information

from basically anyway via a small device,
like the smartphoen that you have in your

pocket.
There is actually a joke about that and

suppose that someone from the
Renaissance, like Leonardo, was

teleported to today.
And you'd have to explain to them what

your smartphone does.
Well you have to say this is a small

device that allows me to access
everything that has been done, written

about, or were said by mankind since the
beginning of history.

And I use it mainly to look at pictures
of cats.

But jokes aside the truth remains that
communications systems, digital

communications systems.
Are really the pinnacle achievment of

digital signal processing.
So in this module we'll start from the

basic principles in module nine one and
we'll see the kind of signals that we

have to design in order to be able to
transmit them over a physical channel.

Now a physical channel whether it's a
wireless channel, whether it's a piece of

wire or an optical fiber will always
impose two fundamental constraints on the

kind of signal that can transit over the
channel.

The first one is a bandwidth constraint,
which means that we will only have a

certain range of frequencies over which
we can send information.

And the second constraint is a power
constraint.

It limits the amount of power that we can
inject onto the channel.

So in module 9.2, we will tackle the
banther constraint, in detail.

And in module 9.3, we will look at the
power constraint.

And we will see in the end how these two
constraints limit the maximum amount of

information that we can send over a
channel.

In Module 9.4, we will look at the
modulation and demodulation techniques

that are specially designed to transmit
data over the telephone channel.

And in Module 9.5, we will examine the
several signal processes and tricks that

are put in place to implement a receiver,
which turns out to be much more

complicated than the transmitter, because
the receiver has to undo all the nasty

things that happen to the signal.
When it travels over the channel,

including distortion and noise and so on.
As a matter of fact, module 9.5 is like a

teaser that will probably whet your
appetite for more advanced signal

processing techniques that you will be
able to study in more advanced classes.

And finally in module 9.6, we will study
the ADSL protocol.

Now it turns out that ADSL is just one
big DFT.

And so, the fact that we can implement it
efficiently with the FFT algorithm, is

really the reason behind the
extraordinary commercial success, of the

ADSL setup box.
You will see that everything that we've

studied so far really find it's place in
the design of a sophisticated digital

processing system.
So we hope you have enjoyed this initial

ride into the world of digital signal
processing and hopefully we'll see each

other again in more advanced classes in
the future.

Thank you.
Hi and welcome to module 9.1 of Digital

Signal Processing.
In this module we will start to look at

digital communication systems.
In particular, we will look at the many

incarnations that a signal will undergo
from its source to its destination.

This incarnations will travel through a
variety.

of different analog channel.
And each channel will have a different

set of constraints that the signal will
have to submit itself to.

And in this module we'll start to look
how to design signals that fulfill the

channel constraints.
If you remember in the beginning of this

class we gave you a little overview of
the major improvements and through put

for channels that we implicitly use every
day.

For instance, the transatlantic cables
that allow telephoning from Europe to the

Unites States have seen an improvement
That went from five bits per second in

1866 with the first cable to 60 terabytes
per second last year.

Similarly something you use every day at
home, your modem that allows you to

connect to the internet, has increased
its data rate from 1,200 bits per second

in the 50s to 24 megabits per second with
the current incarnation of ADSL.

Now what are the reasons behind this
incredible success?

Well, the first one clearly is the power
of the DSP paradigm.

The fact that DSP works with integers
means that, for instance signals are very

easy to regenerate.
We have seen an example in the

introduction, and we will see it again in
a second.

Also digital filters allow us to
implement very precise phase control, and

we will see how important phase is in the
detection of a transmitted signal.

And finally, we can seamlessly integrate
adaptive algorithms into a DSP system.

Adaptive algorithms are algorithmic
procedures that adapt their behavior.

As a function of the received signal.
These are very hard things to do in

analog hardware, but very easy to do in
digital hardware.

As a reminder of what happens when we use
digital signals for communication, think

of the problem of transmitting a string
of binary digits over an analog channel.

To do that, we build a very simple
signal, an analog signal, where we

associate the values plus 5 volts to the
symbol 0.

And minus 5 volts to symbol one.
Now the signal is analog, but it encodes

binary information, namely it encodes a
string of integers.

When we transmit this over wire, two
things happen.

The signal gets attenuated and noise gets
added to the signal.

So what we'll receive at the other end of
the channel is The original signal

attenuated by effect of G, summed to some
random noise that corrupts the original

signal.
Now, if we want to regenerate the signal,

the first thing we do is, undo the
attenuation.

So we multiply the received signal by, a
gain factor, that is the reciprocal of

the attenuation.
So we multiply the signal by g, we obtain

a signal that has, once again the
amplitude of the original signal but in

so doing we also amplified noise.
And so, we have very unclean levels here,

which could cause all sorts of problems.
But since we know that signal is bi level

all we need to do is threshold.
This signal, and when we see that it's

positive, we set it plus 5.
And when we see that it's negative, we

set it minus 5.
This is easily accomplished in digital

domain by taking the sign of the signal
before undoing the attenuation factor.

And this is the signal that we get at the
other end of the transmission channel.

And we can repeat this procedure as many
times as we need and that explains why we

can send so much information over very,
very long cables that go all the way

under the ocean.
The second success factor for digital

communications today comes from the
algorithmic nature of DSP techniques.

We have seen an example in image coding,
in JPEG, where signal processing

techniques such as the discreet cosign
transform could be matched seamlessly to

information theory techniques that
involve the compression of bit streams.

And this interplay between these two
techniques from different domains.

Creates such powerful compression
algorithms.

Other everyday examples can be found in
CDs or DVDs.

Where you have encoding of acoustic or
video information matched to powerful

error correcting codes.
So that DVDs or CDs that are scratched or

dusty still play.
And in communications systems.

Techniques such as trellis coded
modulation and Viterbi decoding are used

to exploit all the capacity of an analog
communication channel.

The third success factor for digital
communications is related to hardware

advancements.
We can have today miniaturized devices

that we can keep in our pocket, we can
have general purpose platforms used to

develop advanced communication systems,
so we don't need to develop specific

hardware for each different task.
And communication devices have become

very power efficient, so that we can have
Large data centers, or central offices

that process an enormous number of
communication channels in peril.

So let's have a look at what happens when
you place a call from your mobile phone

to someone that has their phone at home.
The information is first sent over the

air to the closest base station where it
is now converted to a different format

and sent over copper wires to a switch.
The switch is designed to find the

routing pattern that will send the
information to the final destination.

The switch will send information over
what is going to most likely an optic

fiber channel to the global telephone
network.

The telephone network will route your
information to the central office that is

closest to the person you'll calling.
The central office will then send the

same information in yet a different
format over a coax cable to the switch

that is closest to the telephone that is
being called and finally from the closest

switch to the phone in the house.
There is what is called the last smile

which is a longish piece of copper wire.
So, you see at every change of channel

many many things can happen.
The signal can be converted to digital

again and then back to analog.
The modulation schemes and the signal

formats that we will have to use on this
different stretches of the channel will

have to adopt to the physical
characteristics of the medium.

Every analog channel.
Has two unescapable limits that we have

to reckon with.
The first is a bandwith constraint.

The signals that we can send over an
analog channel will have to be limited to

a certain frequency band, and the second
limit is the fact that we cannot use

arbitrary power over that band.
There will be limits on the power of the

signal we can send.
The maximum amount of informatin we will

be able to send with the channel given
this contraints is called a capicity of

the channel.
We will see a remarkable result of

information theory later on that exactly
quantifies the capcity of the channel

given it's signal to noise ratio and it's
bandwidth.

As communication system engineers we are
given the specifications of a chennel.

And we want to design a system that sends
as much information over this channel.

And as reliably as possible give this
unescapeable capacity constraint.

Amount of information and reliability are
concepts that are still a little fuzzy

for the time being.
They will become clearer later on but we

can certainly look at the intuition
behind this problem.

For instance, if we look at the
relationship between bandwidth and

capacity, we can do this very simple
thought experiment.

Suppose we are going to transmit
information encoded as a sequence of

digital samples over a continuous time
channel.

So, what we do we take the samples we
interpolate the samples with a certain

sampling period Ts now if we make Ts very
small it means that we can send more

samples per second.
But if we make Ts small we know that the

bandwidth will grow as the reciprocal of
Ts you remember the formula for

interpolate signal.
In the sampling theorem, it says that the

analog spectrum will be zero outside of a
band that goes from omega n to minus

omega n.
And omega n is Pi over Ts.

If we make ts small the bandwidth will
grow with 1 over Ts.

So we see, that capacity, and the amount
of information that we can send per

second, are related in some way.
Similarly, the relationship between the

power constraint and capacity, can be
appreciated, because we can never do away

with noise.
So, at the receiver, when we send the

sequence of integers for instance, we
will have to guess What has been set

after it has been corrupted by noise.
So suppose we have a channel that

introduces a noise variance of 1 and
suppose we are transmitting the integer

between 1 and 10.
If the variance is 1 lots of transmitted

integers will have and error that will
send them very close to the next integer

in line.
So suppose I'm sending the integers

between 1 and 10.
And so I'm sending say one but because of

the noise the one will be 1.75 for
instance.

So I'm not really sure if what was sent
was one or was two.

And then the strategies say okay.
Let's transmit only odd numbers.

So instead of everything I will not just
be at 0, we'll transmit 1 and then I will

not transmit 2 but I will transmit 3.
So I'm increasing the gap between

possible symbols and so the noise that
before Had probably me misguessing the

transmission of 1, will still be small
enough to bring me back to the original

signal.
Now it is rather intuitive that, all

other things being equal.
A signal with a wider range will have a

larger power.
So, if I want to keep the power constant,

I will still have to send symbols between
zero and 10, but now there are only half

as many odd integers between zero and 10
that there are integers, and so the

amount of information that I can send per
unit of time.

will be halved.
Let's now look at some common

communication channels and see what their
power and bandwidth constraints are.

Maybe the simplest communication channel
that we're still familiar with, is the AM

radio channel.
AM stands for amplitude modulation, and

indeed the radio transmitter is very
simple.

We take an analog signal, it can be voice
or music, we do a low-pass filtering

operation to limit its bandwidth, And
then we do a very, very simple sinusoidal

modulation with the cosine of a given
carrier.

The result in modulated signal, is simply
put to an antenna, and it will be

propogated in the radial spectrum.
The radial spectrum is a very scarce

resource.
There's only one radial spectrum,

everybody has to share it.
Therefore, every frequency band in the

spectrum, is strictly regulated by law.
In the case of AM, the band is from 530

kilohertz to 1.7 megahertz.
This is divided into 8 kilohertz wide

channels.
And each radio station gets allocated a

specific channel.
The power is limited by law for a variety

of reasons.
The first is that the propagation

patterns for AM waves is very different
during the day, and during the night.

In particular at night time, AM radio
waves travel much further than during the

day.
So, they can create all source of

interferences in distant places if the
power is not limited.

Also you don't want radio stations to use
too much power because it wouldn't be

healthy for people live in the vicinity
of the transmitter and on the channel

where all are familiar with is the
telephone channel.

The telephone network is more properly
called the switched telephone network

because instead of taking the
combinatorial approach and having each

phone connected to every other phone in
the world.

What happens is that when you call on
other phone.

Your phone is connected to the central
office, and the central office determines

which parts of the network have to be
connected together so that your call can

be routed to the destination phone.
So, the piece of wire that connects you

to the central office is up to, maybe
say, a couple of kilometers long, and is

called the last mile.
The central office today is a bunch of

digital switches, in the old days was
mechanical rotary switches The network

can be anything from optical fiber to
satellite links to anything else in

between, and here you have the symmetric
part where you get to your destination.

The telephone channel is conventionally
limited from 300 hertz to 3,000 hertz.

These are historical limits that depend
on the kind of hardware that was used In

the old days in central office and in the
network.

Today these limits are historical
artifact but they are kept because anyway

voice communications are perfectly
intelligible within this band And with

the reduced band, you can multiplex.
Namely, you can put together very many

communications on a wider channel.
The power that you can send on a

telephone wire is limited from 0.2 to 0.7
volts, or root mean square.

And this a strictly enforced limit to
make sure that you don't send signals

that can burn the equipment at the
central office.

And the signal to noise ratio is rather
good because the analog part of the

telephone network operates in the bass
band and there's not a lot of

interference in the low frequencies.
So let's how we're going to go about

designing a communications system.
Probably the most important concept here,

is that we're going to adopt the
all-digital paradigm.

What this means is that, we will keep
everything in the digital domain until we

hit the physical channel.
And if we were to describe this as a

block diagram, it would look like this.
We have a binary bit stream, can

represent any sort of views or data.
We have a transmitter that operates

entirely in digital domain that generates
a discreet time signal s of n.

The last element in the transmission
chain.

Is a digital to analog converter
operating at a given frequency, or at the

given period as you prefer, that
transforms this signal into an analog

signal that we can send over the channel.
So remember the channel constraints.

Look a little bit like a filter design
problem.

We have a band width that is specified in
terms of a maximum and minimum frequency.

So we can only operate over this band.
And then we have a power constraint that

restricts the power associated with the
signal that we produce.

So if you want to convert this to our old
digital paradigm the first thing to do is

to convert the specs into discreet time
specs.

So we choose a frequency for the D2A
converted, fs, this will be our niquist

frequency, fs over 2, and with this we
can convert the specs.

Maximum frequency will be pi, and our
minimum and maximum frequency bands will

be omega min and omega max using the
relation.

Omega equal to 2 pi f over fs.
And you can put here, f min or f.

Now, here are some working hypotheses
that are common to most transmission

systems you will ever see.
We start from a bitstream.

And we will convert this bitstream into a
sequence of symbols.

For samples a of n, via something called
a mapper.

What the mapper does is associate group
of bits to a specific symbol.

Just to give you a concrete example
assume we're going to map each group of

bits to its decimal value.
We want to model the sequence of symbols

as a white random sequence and in order
to do so, we have to assume that the

bitstream is a completely random
sequence.

Now, this is not necessarily the case,
for instance, imagine you're digitizing

audio and you have long stretches of
silence.

This will result into a long sequence of
zeros.

And so, what we do is we put a scrambler
in the line.

What a scrambler does.
It transforms a sequence of bits into a

sequence that looks like a random
sequence but this randomization is

completely invariable at a receiver.
So, we start with the sequence of zeroes

for instance.
We put into the scrambler, it's going to

look like a completely random sequence of
zeroes and one but it's done

algorithmically so we can invert this
randomization on the receiver and

retrieve the original bitstream..
With this we can consider the sequence of

symbol a of n as a wide sequence.
And now we need to convert the sequence

into a continuous time signal within the
constraints.

So here's the updated transmission
scheme.

User data goes into a scrambler.
This is a random binary sequence.

The mapper converts groups of bits to
symbols.

And then we have to decide what to do in
here before converting this into an

analog signal.
The first problem is Fulfilling the

bandwidth constraint.
If we assume that the data is randomized

and therefore the symbol sequence is a
wide sequence, we know that the power

spectral density is simply equal to the
variance and so the power of the signal

will be constant over the entire
frequency band but we actually need to

fit it into the small band here as
specified by the bandwidth constraint.

So, how do we do this.
Well in order to do that we need to

introduce a new technique called up
sampling and we will see this in the next

module.