[Martin Vetterli] 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.
(Digital Signal Processing - Module 1: Introduction) 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
on the website here indicated on the slide. ( http://www.spdcomm.org )
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 of a physical phenomenon.
For example, 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 numbers,
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, on a LP recording,
the grey level on paper for a black and white photograph,
some flickering colors on a 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 about more sophisticated things, functions, derivative, and integrals?
(Raphael's "The School of Athens") The question of discrete versus continuous,
or analog versus discrete,
goes probably back to the earliest time of science, for example, to 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 shoot an narrow, will it ever arrive at 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,
then a quarter that's D, another eighth that's E etc. will you ever get there?
And of course, we know you get there
(mathematical formula written in the video) because a of 1 over 2 to the n is equal to 1,
a beautiful formula that we'll see several times reappearing in this class.
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 René 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 a physical phenomenon.
For example, Galileo was able to describe the trajectory of a bullet,
using infinitesimal 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 its average.
As is well known, this will be the integral of the function, divided by the length of the interval,
and it is shown here in red dots.
What would be the equivalent in discrete time symbol processing?
We have a set of samples between say, 0 and capital N-1.
The average is simply 1 over n, the sum of the individual 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 then formalized in 1948.
If the function x(t) is sufficiently slow, then there is a simple interpolation formula for x(t):
it's the sum of the samples xn, interpolatated with the function that is called sinc function.
It looks a little but complicated now,
but it's something we're going to study in great detail because it's one of the fundamental formulas
linking 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 sinc 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 discrete time, discrete amplitude.
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,
it has 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(t) here.
It's a sinusoid, and investigate discrete time first.
We see this with x[n] and discrete amplitude.
We see this with these levels of the amplitudes which are also discretized.
And so this signal looks very different from the original continuous time signal x(t).
It has discrete values on the time axis
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 we 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.
(TX --> channel -->RX) Data transmission has also gone from analog to digital.
So let's look at the very simple model here, you've on the left side a transmitter,
(new diagram, described in the audio comment) you have a channel, on the right side you have a receiver.
What happens to analog signals when they are sent over a channel?
So x(t) goes through the channel, its first multiplied by 1/G
because there is path loss and then there is noise added,
indicated here with the σ(t).
The output here is x̄(t).
Let's start with some analog signal x(t).
Multiply it by 1/G,
and add some noise.
(New diagram, described in audio comment) How do we recover a good reproduction of x(t)?
Well, we can compensate for the path loss, so we multiply by G, to get x̄1(t).
But the problem with that x̄1(t) is x of t (that's the good news) +Gσ(t).
So the noise has been amplified.
Let's see this in action.
We start with x(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 transatlantic 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.
(New diagram, described in the audio comment) 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 N times.
Each time the paths loss would be compensated, but the noise would be amplified by a factor of N.
Let us see this in action:
so start with x(t),
path loss by G,
added noise,
amplification by G, with the amplification of the noise and the signal.
For the second segment we have the path loss again, so x̄1 is divided by G.
And added noise, then we amplify to get x̄2(t)
which now has twice an amount of noise, 2Gσ(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.
(New diagram, described in audio comment) 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 take values +1 and -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 signum operations.
So x1hat, is signum of x of t, plus 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.
It goes through the channel, so it loses amplitude by a factor of G,
and there is some noise added.
We multiply by G, so we recover x(t) plus G times the noise of σt.
Then we apply the thresholding operation.
And true enough, we recover a +5 -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
had 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 the 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 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.