Welcome to module 2 of Digital Signal
Processing.
In this module, we are going to see the
basics of Digital Signal Processing and
we'll start by considering discrete time
signals and
operators of discrete time signals in
module 2.1.
In module 2.2, we are going to consider
complex exponentials.
These are the most elementary and
fundamental discrete time signals.
Finally,
in module 2.3, we are going to take
building blocks, elementary ones from
signal processing to see
some elementary operations on signals,
like moving averages
and recursive filters, and finally build a
simple synthesizer.
Let's now get started and see what
discrete time signals really are.
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Module 2.1, Discrete-time signals.
The Overview is the following.
We will definie discrete-time signals to
know what we are
talking about here as a main component in
the class.
We are going to look at different types of
signals, give examples.
And then go onto operators, which are
elementary blocks,
building blocks for more complex systems
that will be
used in the discrete-time signal
processing.
And finally finish with two concepts which
will come also over and over in
the class which is the energy of a signal
and the power of a signal.
What is the earliest discreet time signals
we can think of?
Well, probably the earliest one that has
been recorded are the floods of the Nile.
So the Nile of course is extremely
important in the agriculture of Egypt and
so from very early on, people recorded how
high the Nile would come in
any given year, and this representation
has been caught here on this beautiful
historical record that is about 4 and a
half thousand
years old, and these are representation of
the flood data.
Unfortunately, we don't have digital
version of this data,
so we can really do signal processing on
this.
Maybe somebody wants to pick up a
research project and actually, do this
transformation.
In the meantime, we have access to more
recent data.
So this is an example of the, flood in the
Nile in cubic meters per second.
Yearly measurements over the last 100
years or so.
you can see the representation is these
lollipop diagrams.
So as a lollipop is, at any given year, we
represent one data point.
So we have here,
almost a bit more than 100.
It's a very busy signal, but overall, even
if you just look at it approximately, you
can sort of see there is a trend, okay?
And the trend is not a good one, means
there is less and less water in the Nile.
The second example of, signal is daily
temperature.
It's probably your first scientific
experiment.
Every day, at 8 am, you look at the
temperature outside.
You do this every day for a year, for a
couple of years and so on.
And you record something that could also
be
a lollipop diagram, like in the previous
slide.
Except there are so many points here, that
we decided to actually join them.
So it
looks almost as a continuous time function
even though
it's one measurement every day, or a few
thousand days.
Now, when you look at this, you, of
course, immediately see that there are
seasons happening.
So the temperature here in Centigrades,
you know, goes up and down
on a, with a yearly variation, and this we
can actually show.
We can fit the sinusoid, here.
It's a blue curve
that has been added.
And this is simply the trends that these
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seasonal changing, daily temperature.
Now, this blue sinusoid is actually one of
the
things we're going to do here in the
class.
We're going to take a data set like the
original data set and
going to fit a sinusoid instead of our
algorithmic methods to actually do this.
The next signal is, so called solar
activity.
So as you probably know, the sun has
different intensity due to
solar spots, and this has been measured
over several hundred years now, and
we again see this this curve here, which
has a lot of variation.
It's not seasonal variation.
It's a different variation and
we shall also analyze this variation later
on in the class.
Now all the signals so far had to do with
physics.
Now we can see a man made signal
that's a world population, from the
beginning of current
times until now, and you see that it's a
slowly evolving curve up to the industrial
revolution here.
This is the industrial revolution, and
then for various reasons which
are not the topics of this class, there
was a population explosion.
By now we are close to 8 billion, and
it's not completely clear what's going to
happen next.
Another man-made signal is, the value of
the stock market.
Here, it's the Dow Jones and again it's
something that looks a
little bit like the world population in
the beginning, so very small.
Then there is an exponential increase,
then there is the internet bubble,
and the financial crisis, etc.
Okay, so it's a good question mark.
What is going to happen next?
We could claim that thanks to signal
processing,
we can predict the future of the stock
market.
But I will be honest, this is a very hard
problem.
If we knew how to do this, we probably
wouldn't be, teaching free online course.
We would be do, doing something else for a
living.
But anyway, it's a typical discrete time
signal.
You can measure the Dow Jones every day
and record this as we did here.
Again, it would be a lollipop a stick
diagram.
But here, we have joined the points
because there
are so many points over this period of
time.
So let's be a little bit more formal.
We saw several examples.
All of these examples where a number per
day, a number per second, a number per
year.
in the channel case this number can be a
complex number, so
for us a digi, discrete-time signal is a
sequence of complex numbers.
We will look at mostly one dimensional
signals and the dimension typically is
time.
The notation, that's very important to
note is that x is the name of the signal,
then we have a square bracket and finally
we have the index m which is an
integer.
We typically have two sided
sequences, so they start from minus
infinity, to,
let's say x minus 1, x at 0, x at 1, etc
and goes of to
plus infinity.
And the index is dimension-less.
So it could be seconds, years, micro
seconds or whatever,
but we just index it as a dimensionless
integer index.
The analysis of a signal means we take
periodic measurements.
This is a topic that we'll study in detail
under the name of sampling.
So for example, in the case of the Nile,
we took a sample every year.
In the case of the stock market, every
day.
And synthesis is when we generate a
sequence of samples.
We will see this when we talk about the
Karplus-Strong Algorithm later in
this module, where we indeed generate a
musical signal by generating discrete time
samples.
So among the formal signals, or elementary
signals, we will be use in this class.
We have for example the delta signal.
So delta signal denoted by delta of n
here, is very simple.
It's the simplest possible signal.
It's 0 everywhere except at the origin,
where it's equal to 1.
Okay.
Now do such signals exist in reality or is
it a pure mathematical abstraction?
It is an abstraction.
Most signals will not be that simple, but
let
me show you one signal which comes close
to that.
Let us look at natural device called the
clappers that
is used in Hollywood studios to
synchronize audio and video.
Why do they need this?
Well, sometimes the audio is recorded on
one machine, the video,
on another machine, and later, you have to
splice them together.
And so the way it is done is that you use
this clapper.
You write down what scene is shot by what
director in what movie.
And what you do is that you have the
clapper that is open.
You put it in front of the camera then you
close it and that gives a click sound.
And we see the click sound here at the
bottom.
And it''s not exactly a d rack.
It's more complicated.
But it is really an impulse.
An impulse of sound when you listen to it
it sounds like a d rack.
When you look at it in detail it has
oscillations
because it's a piece of wood that is going
to vibrate.
But to a first approximation, this is as
close as it gets to a d rack.
The next simple signal is a so called unit
step.
It is sequence that is 0 from minus
infinity till the origin.
At the origin it flips to 1 and is 1
everywhere after that.
Okay.
Does such a signal exist in reality?
A physical implementation would be a
switch
like this blade switch that we see here.
We call this the Frankenstein switch, if
you have seen the movie.
Dr. Frankenstein, at some point, gives
life to the monster.
So, he flips a switch, a blade switch and
then some current flows into the monster
that comes alive.
So, that's a beautiful picture of one of
these blade switches.
The third, signal that is very typical is
a so called exponential decay sequence.
So it's a combination of two things.
Okay, it has, a unit step,
so the signal is 0 before the origin and
then it is positive.
And that's at point we apply an
exponential here
with a root a that is smaller than 1.
Taken to the nth power.
And so it nicely comes down as an
exponential.
Okay, if we did not apply the unit step,
then of course the exponential would grow
up here exponentially,
and it would be unbounded.
So that would be, not signal you would
encounter in practice.
So we see here a signal that has been
specifically engineered that
it looks actually like a signal that you
would encounter in real life.
And real life is coffee.
So the way coffee will lose its
temperature is governed by partial
differential equation, that we see here
goes actually back to Newton, one of the
greatest scientist of all time.
So, the law of cooling says that the rate
of cooling is proportional to
the difference of temperature between the
cup of coffee and the environment, okay.
And there is a constant c here that
says how fast actually the temperature
will decay.
So negative sign means the temperature
will go down when the coffee is warm and
the environment is cold.
And the solutions of the differential
equation is
given here and it has, indeed, an
exponential term.
There are some practical issues.
you have only convection at work and there
has to be sufficient conductivity,
and then you will see a decay, that looks
a little bit like this.
Okay?
And here, let's say the ambient
temperature is 0.
We start with
the coffee at 100 degrees, boiling water.
And it will start to decay like this.
And the speed of decay will depend
how much convection there is, how much
conductivity.
Okay.
So this was all to show you a little bit a
sense
of you have physics at work and you see
signals in real life.
A discreet time version of these signals
will be typically sequences we'll be
interested
in in this class.
The next character in the play is a
sinusoid.
So here is the sinusoid indicated as sine
of frequency omega naught
times n, the index, plus a phase, theta.
Okay.
In the drawing, we see theta is equal to
0.
actually, it's not equal to zero.
I'm sorry.
theta is equal to pi over 2.
So, at the origin it's maximum and then we
have the sinusoidal variation.
Now sinusoidal signals are everywhere.
The heartbeat is an example.
The whistle on
the whistle used on a train that we
will also analyze later is, has sinusoidal
components.
Waves that are showing up on the shore are
another example of sinusoidal components.
And, of course, musical instruments,
really leave off, strings
that have sinusoidal models, something
we'll also study in detail.
There are really four classes of signals.
And, each one of them is a natural example
of a signal and is used in different
cases.
The first one, the simplest one is, the
finite length signal, then the infinite
length sequence, periodic signals, and
finite support sequences.
Let's look at each of them in a little bit
more detail.
So finite length signal can be either
written in the sequence notations.
So the sequence notation we have seen
before x of square bracket n.
But n is only going from 0 to n minus 1.
So it's a length's capital n signal.
The vector notation puts this into a
vector, and in this class vectors are
column vectors, so we have the entry of
the vectors at x0, x1 up
to xn minus 1, a transpose to get a column
vector.
Finite length signals or vectors are very
useful, for
example, in numerical packages like Matlab
and so on.
Infinite length signals.
We again have the sequence notations that
we had seen before.
And the index and n now is from the
integer
set, meaning from minus infinity to plus
infinity.
This is very good for abstraction, for
theorems, for analysis, using
Furey analysis, for example, z transforms,
that we will see later.
But, of course, in real life, and
fortunately,
nothing lives from minus infinity to plus
infinity.
That depends a little bit on your beliefs,
but
if you believe in big bang, then it
started at
some point.
And if you are not certain about
reincarnation, it will stop at some point.
So the sequence notation is really a
mathematical abstraction.
Periodic signals we told earlier about
sinusoids.
So periodic sequences do exist also in
real life.
They are denoted by x tilde, okay.
So the notation here tilde means there is
a periodicity and it means that x
n is equal to x n plus k times the period.
So, capital N, is the period,
k is an integer, it means how many periods
we shift and n is the usual index for
time.
So, one of these periods of lengths n has
all the information needed.
The rest is just a repetition.
Okay.
And very often, periodic signals are a
bridge between finite and infinite length
signals because
you start with a, with a finite-length
signal and maybe you let the period go
to infinity, or you take a periodic signal
and you
let the N, the length of the period, go to
infinity.
A finite-support sequence is defined by x
over bar, given here and
it's a sequence x n that is only different
from 0 for n between 0 and N minus 1.
It is 0 otherwise, and it has the same
information as a finite-lengths signal of
lengths N, and it
is another possible bridge between finite
and infinite length signals.
Now that we have seen elementary signals,
let's talk a little bit about operators.
So if we have a sequence, we can
scale the sequence by multiplying it with
scalar alpha.
Okay.
So the natural thing is you have a
sequence like this.
You multiply it by 2.
It's going to look the same.
It's simply going to have amplitudes twice
as large.
The sum of two signals is obvious.
Take two sequences and the result will be
the sum of the two.
The product of two sequences, same story,
you take two elementary sequences x n and
z n, you multiply them term by term.
And last but not least, very important,
the shift by k.
Or the delay by k.
So k is a positive integer.
So when we write y n is equal to y n minus
k.
Then y n is a delayed version here of the
sequence x mainly by k terms.
Right, and to look at this in detail, it's
sometimes a little bit confusing,
but if you want to think about it very
quickly, is that y of 0.
So the output sequence at 0 is equal to x
of
0 minus k, so x of minus k.
This is in the past so it
is a single x from index minus k.
Let's look specifically at these shifts on
the various types of sequences we have
seen.
So the shift of finite-lengths sequence.
Well, we have on the left side the
original sequence which is the
decaying set here, of samples, and we are
going to
shift it by extending this sequence with
0's left and right.
Okay.
So this, the only thing we did is that we
added 0's
before the 7 as the eight samples and
after the eight samples.
And if we shift it.
Well, we delayed by 1, so it has shifted
right.
And what has happened is that 0 showed up
here and the x7 fell off on the right
side.
We see this here
at the bottom, that we can follow what
happens as we shift further and further.
Okay, shift by 2.
Again it is delayed by 2.
So we have two 0's at the beginning and
two samples that fall off.
And we see the result at the bottom right
side.
And shift by 3, shift by 4 and you see the
idea.
We can also have a model of the world
where the signal
is periodically extended and then the
shift operator will have another effect.
Let's look at this.
So we start with the sequence eight
samples from x0 to x7.
We extend it periodically.
Okay, so here we have a second
period, here we have the previous period.
And now when we shift,
what will happen is that, we have a shift
pile to the right okay?
x tilde n minus one.
So, x7 enters on the left side, and x7
leaves at the right side, okay.
So, the new signal, given here at the
bottom right, has x7 entering
here, and having left on the other side.
Now this of course
is the same as doing a circular shift,
okay.
Let's do a few more.
So x tilde n minus 2, we have now
two new samples entering and two that fell
off there.
And at the bottom right, we see the
picture again.
X tilde n minus 3, same story.
X tilde n minus 4, one more step and you
start to see what is happening here.
We have now seen the shift operator, which
is one of the most fundamental
ones we are going to use over and over
again, and study in great detail.
Let's talk about two conceptual
characteristics of signals.
The first one is something we call the
energy.
The energy is a sum of squares, of the
samples.
Okay.
It is given in the formula here.
So you take a sequence.
You take the absolute
value of the samples.
You square it.
And you sum from minus infinity to plus
infinity.
This may or may not be finite.
If you take a sinsusoid it's, signals that
starts at minus infinity, goes to plus
infinity, has amplitudes that vary, let's
say between minus 1 and 1.
And of course if you square it and you sum
from minus infinity to plus infinity, It
will blow up, okay.
So, a notion that can still be defined
even for such
signals that might have infinite, energy
is a notion of power.
The notion of power is that, you look at
the energy over
a window, the window is of size, 2 n plus
1, right?
Because we sum from minus n here to plus
n, the sum of squares.
But we normalize this by
the length of the window.
So we put the 1 over 2 n plus 1, and
this is the power, because it's like the
instantaneous energy for the sequence.
So for periodic signals as I have all
ready alluded to, the energy will be
infinite.
And the power is simply the energy of
one period divided by the length of the
period.
So this is given down here.
So, this part is the energy.
1 over gives the power, and this typically
will
be actually finite, even when this guy is
infinite.
End of module 2.1
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