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.