>> Welcome to module one of Digital Signal Processing.
In this module we are going to see what signals actually are.
We are going to go through a history, see the earliest example of these quick time
signals. Actually it goes back to Egyptian times.
Then through this history see how digital signals, for example, with the telegraph
signals, became important in communications.
And today, how signals are pervasive in many applications, in every day life
objects. For this we're going to see what the
signal is, what a continuous time analog signal is, what the discrete time
continuous amplitude signal is and how these signals relate to each other and are
used in communication devices. We are not going to have any math in this
first module. It is more illustrative and the
mathematics will come later in this class. This is an introduction to what digital
signal processing is all about. Before getting going, let's give some
background material. There is a textbook called Signal
Processing for Communications by Paolo Prandoni and myself.
You can have a paper version or you can get the free PDF or HTML version, Version
owns website here indicated on the slide. There will be quizzes, there will be
homework sets, and there will be occasional complimentary lectures.
What is actually a signal? We talk about digital signal processing,
so we need to define what the signal is. Typically, it's a description of the
evolution over physical phenomenon. Quite simply, if I speak here, there is
sound pressure waves going through the air that will be picked up by a microphone and
that's a typical signal. When you listen to the speech, there is a
loud speaker creating a sound pressure waves that reaches your ear.
And that's another signal. However, in between is the world of
digital signal processing because after the microphone it gets transformed in a
set of members. It is processed in the computer.
It is being transferred through the internet.
Finally it is decoded to create the sound pressure wave to reach your ears.
Other example are the temperature evolution over time, the magnetic
deviation for example, L P recording , is a grey level on paper for a black and
white photograph,some flickering colors on TV screen.
Here we have a thermometer recording temperature over time.
So you see the evolution And there are discrete ticks and you see how it changes
over time. So what are the characteristics of digital
signals. There are two key ingredients.
First there is discrete time. As we have seen in the previous slide on
the horizontal axis there are discrete Evenly spaced ticks and that corresponds
to discretisation in time. There is also discrete amplitude because
the numbers that are measured will be represented in a computer and cannot have
some infinite precision. So what amount more sophisticated things,
functions, derivative, and integrals. The question of discreet versus
continuous, or analog versus discreet, goes probably back to the earliest time of
science, for example, the school of Athens.
There was a lot of debate between philosophers and mathematicians about the
idea of continuum, or the difference between countable things and uncountable
things. So in this picture, you see green are
famous philosophers like Plato, in red, famous mathematicians like Pythagoras,
somebody that we are going to meet again in this class, and there is a famous
paradox which is called Zeno's paradox. So if you should narrow will it ever
arrive in destination? We know that physics Allows us to verify
this but mathematics have the problem with this and we can see this graphically.
So you want to go from A to B, you cover half of the distance that is C, center
quarters that's D and also eighth that's E etc will you ever get there and of course
we know you gets there because the sum from 1 to infinity of 1 over 2 to the n is
equal to 1, a beautiful formula that we'll see several times reappearing in this.
Unfortunately during the middle ages in Europe, things were a bit lost.
As you can see, people had other worries. In the 17th century things picked up
again. Here we have a physicist and astronomer
Galileo, and the philosopher Rene Descartes, and both contributed to the
advancement of mathematics at that time. Descartes' idea was simple but powerful.
Start with a point, put it into a co-ordinate system Then put more
sophisticated things like lines, and you can use algebra.
This led to the idea of calculus, which allowed to mathematically describe
physical phenomenon. For example Galileo was able to describe
the trajectory of a bullet, using infinite decimal variations in both horizontal and
vertical direction. Calculus itself was formalized by Newton
and Leibniz, and is one of the great advances of mathematics in the 17th and
18th century. It is time to do some very simple
continuous time signal processing. We have a function in blue here, between a
and b, and we would like to compute it's average.
As it is well known, this well be the integral of the function, divided by the
length's of the interval, and it is shown here in red dots.
What would be the equivalent in this discreet time symbol processing.
We have a set of samples between say, 0 and capital N minus 1.
The average is simply 1 over n, the sum Was the antidote terms x[n] between 0 and
N minus 1. Again, it is shown in the red dotted line.
In this case, because the signal is very smooth, the continuous time average and
the discrete time average Are essentially the same.
This was nice and easy but what if the signal is too fast, and we don't know
exactly how to compute either the continuous time operations or an
equivalent operation on samples. Enters Joseph Fourier, one of the greatest
mathematicians of the nineteenth century. And the inventor of Fourier series,
Fourier analysis which are essentially the ground tools of signal processing.
We show simply a picture to give the idea of Fourier analysis.
It is a local Fourier spectrum as you would see for example on an equalizer
table in a disco. And it shows the distribution of power
across frequencies, something we are going to understand in detail in this class.
But to do this quick time processing of continuous time signals we need some
further results. And these were derived by Harry Niquist
and Claude Shannon, two researchers at Bell Labs.
They derived the so-called sampling theorem, first appearing in 1920's and
formalized in 1948. If the function X of T is sufficiently
slow then there is a simple interpolation formula for X of T, it's the sum of the
samples Xn, Interpolating with the function that is called sync function.
It looks a little but complicated now, but it's something we're going to study in
great detail because it's 1 of the fundamental formulas linking this discrete
time and continuous time signal processing.
Let us look at this sampling in action. So we have the blue curve, we take
samples, the red dots from the samples. We use the same interpolation.
We put one blue curve, second one, third one, fourth one, etc.
When we sum them all together, we get back the original blue curve.
It is magic. This interaction of continuous time and
discrete time processing is summarized in these two pictures.
On the left you have a picture of the analog world.
On the right you have a picture of the discrete or digital world, as you would
see in a Digital camera for example, and this is because the world is analog.
It has continuous time continuous space, and the computer is digital.
It is discreet time discreet temperature. When you look at an image taken with a
digital camera, you may wonder what the resolution is.
And here we have a picture of a bird. This bird happens to have very high visual
acuity, probably much better than mine. Still, if you zoom into the digital
picture, after a while, around the eye here, you see little squares appearing,
showing indeed that the picture is digital Because discrete values over the domain of
the image and it also has actually discrete amplitude which we cannot quite
see here at this level of resolution. As we said the key ingredients are
discrete time and discrete amplitude for digital signals.
So, let us look at x of t here. It's a sinusoid, and investigate discrete
time first. We see this with xn and discrete
amplitude. We see this with these levels of the
amplitudes which are also discrete ties. And so this signal looks very different
from the original continuous time signal x of t.
It has discrete values on the time axes and discrete values on the vertical
amplitude axis. So why do we need digital amplitude?
Well, because storage is digital, because processing is digital, and because
transmission is digital. And you are going to see all of these in
sequence. So data storage, which is of course very
important, used to be purely analog. You had paper.
You had wax cylinders. You had vinyl.
You had compact cassettes, VHS, etcetera. In imagery you had Kodachrome, slides,
Super 8, film etc. Very complicated, a whole biodiversity of
analog storages. In digital, much simpler.
There is only zeros and ones, so all digital storage, to some extent, looks the
same. The storage medium might look very
different, so here we have a collection of storage from the last 25 years.
However, fundamentally there are only 0's and 1's on these storage devices.
So in that sense, they are all compatible with each other.
Processing also moved from analog to digital.
On the left side, you have a few examples of analog processing devices, an analog
watch, an analog amplifier. On the right side you have a piece of
code. Now this piece of code could run on many
different digital computers. It would be compatible with all these
digital platforms. The analog processing devices Are
essentially incompatible with each other. Data transmission has also gone from
analog to digital. So lets look at the very simple model
here, you've on the left side of the transmitter, you have a channel on the
right side you have a receiver. What happens to analog signals when they
are send over a channel. So x of t goes through the channel, its
first multiplied by 1 over G because there is path loss and then there is noise added
indicated here with the sigma of t. The output here is x hat of t.
Let's start with some analog signal x of t.
Multiply it by 1 over g, and add some noise.
How do we recover a good reproduction of x of t?
Well, we can compensate for the path loss, so we multiply by g, to get xhat 1 of t.
But the problem is that x1 hat of t, is x of t.
That's the good news plus g times sigma of t so the noise has been amplified.
Let's see this in action. We start with x of t, we scale by G, we
add some noise, we multiply by G. And indeed now, we have a very noisy
signal. This was the idea behind trans-Atlantic
cables which were laid in the 19th century and were essentially analog devices until
telegraph signals were properly encoded as digital signals.
As can be seen in this picture, this was quite an adventure to lay a cable across
the Atlantic and then to try to transmit analog signals across these very long
distances. For a long channel because the path loss
is so big, you need to put repeaters. So the process we have just seen, would be
repeated capital N times. Each time the paths loss would be
compensated, but the noise will be amplified by a factor of n.
Let us see this in action, so start with x of t, paths loss by g, added noise,
amplification by G with the amplification the amplification of the noise, and the
signal. For the second segment we have the pass
loss again, so X hat 1 is divided by G. And added noise, then we amplify to get x
hat 2 of t, which now has twice an amount of noise, 2 g times signal of t.
So, if we do this n times, you can see that the analog signal, after repeated
amplification. Is mostly noise.
And that becomes problematic to transmit information.
In digital communication, the physics do not change.
We have the same path loss, we have added noise.
However, two things change. One is that we don't send arbitrary
signals but, for example, only signals that[INAUDIBLE].
Take values plus 1 and minus 1, and we do some specific processing to recover these
signals. Specifically at the outward of the
channel, we multiply by g, and then we take the signa operation.
So x1hat, is signa of x of t, plug g times sigma of t.
Let us again look at this in action. We start with the signal x of t that is
easier, plus 5 or minus 5. 5.
It goes through the channel, so it loses amplitude by a factor of g, and their is
some noise added. We multiply by g, so we recover x of t
plus g times the noise of sigma t. Then we apply the threshold operation.
And true enough, we recover a plus 5 minus 5 signal, which is identical to the ones
that was sent on the channel. Thanks to digital processing the
transmission of information has made tremendous progress.
In the mid nineteenth century a transatlantic cable would transmit 8 words
per minute. That's about 5 bits per second.
A hundred years later a coaxial cable with 48 voice channels.
At already 3 megabits per second. In 2005, fiber optic technology allowed 10
terabits per second. A terabit is 10 to the 12 bits per second.
And today, in 2012, we have fiber cables with 60 terabits per second.
On the voice channel, the one that is used for telephony, in 1950s you could send
1200 bits per second. In the 1990's, that was already 56
kilobits per second. Today, with ADSL technology, we are
talking about 24 megabits per second. Please note that the last module in the
class will actually explain how ADSL The works using all the tricks in the box that
we are learning in this class. It is time to conclude this introductory
module. And we conclude with a picture.
If you zoom into this picture you see it's the motto of the class, signal is
strength.