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.