Let's get started.
We want to explore continuous-time.
This is the time we live in the physical
reality of the world.
And this we will contrast to discrete
time.
The time that lives inside a computer.
Module 6: Interpolation and Sampling.
So the outline of the module is the
following.
First we look at continuous-time signals.
Mainly signals that live in the real world
that we inhabit every day.
Then, we look at how to go from a
sequence, the elements
we have seen so far, into continuous time
signals, and that's onto interpolation.
Then we look at sampling.
The cornerstone result here
of this module.
How to go from a continuous time signal to
a sequence.
And under which conditions.
This is a faithful representation of the
continuous time signal.
This will involve to look at aliasing
a nasty phenomenon that we need to
understand.
Then we'll look at interpolation and
sampling in practice.
Finally we look at the most important
application
which is discrete time processing of
continuous time signals.
This is when we take a signal from the
real word, we process
it in a computer and we render the signal
in the real word.
Module 6.1, The Continuous Time Paradigm.
First, we look
at some models of the world and compare
digital with analog views of the world.
Then we study continuous time signals in
some detail.
We also introduce a continuous
time fully transform, which is the last
form
before we transform, we have not
encountered yet.
And therefore, this is a time to meet this
very interesting mathematical object.
We have seen these pictures before, these
are the two views of the world.
On the left, you see the analog world, the
one we live in, and is shown by a picture
of the Earth taken by the Apollo mission.
On the right side, you have a digital view
of the world.
This is taken by digital camera.
Then very strongly pixelized and the
values are actually also discrete.
So this is a view that you have of the
world inside a computer.
In short, this is a contrast between and
analog continuous view versus a discrete
and digital view of the world.
We can show this in a block diagram form.
We live in an analog world, so input and
output are
from and to the analog world, but inside
we process on the computer, which is
discrete time.
The input is x of t.
The output is y of t.
But inside the box, we process sequences,
x of n, y of n.
Examples of such boxes are for example the
processing we do for MP3 music
or what we do on a digital camera.
Other examples are the following.
You have
a digital world.
For example, you synthesize an image.
And you process it and then show it on a
computer screen.
This is a typical example in computer
graphics or in video games.
So the input to the box is a sequence.
It is processed into an output, which is a
continuous time or continuous space image.
Sometimes the box is used to take a
decision.
So the input is analog, from the analog
world.
The output is discrete, for example, to
decide
something, should we take an action or
not.
The processing, again, inside the box is
done on
sequences, x[n] from the input, y[n] to
the output.
This is typical, for example, in
monitoring applications.
Is a certain level of temperature reached,
and therefore in control systems.
Let's summarize.
The digital world view is the one of
arithmetic, of combinatorics, what is
usually referred to as computer science,
but
also the world of discrete signals
processing.
The analog view is the view of calculus.
Integrals and so on.
Distribution, system theory.
Continuous time analog electronics.
So you can see these are two sides of a
coin.
Because we always have to deal with one
and
the other when we want to process signals
in the real world.
To make it slightly more mathematical, the
digital view is you have a countable
index.
So x-n, where n is an integer.
So we have sequences.
And we like to have sequences that are
square summable.
So xn is an l2 of z.
We have a frequency that is limited
between minus pi and pi.
And we have a Fourier transform called the
discrete time Fourier transform.
Which maps from l2 of z to capital l2
of minus pi to pi.
The analog world view is that we have real
time in second.
T is measured in second.
We have functions on the real line, x of
t.
We like them square summable, that's what
we call capital L, two of r.
The frequency is given in radians per
second
and is unlimited unlike in the discrete
time case.
And we have a fully transform that maps l2
of r functions into
l2 of r functions, as we will see shortly.
So how do we bridge the gap between these
two worlds?
So here's an example where we have, for
example,
computer synthesis generates a sequence
xn, we have a
sound card on the computer, the sound card
has
a system clock, so every Ts second it will
generate.
An output based on xn.
This output will be driving a loudspeaker.
In the other direction, we are speaking
into a microphone like I do right now.
There is a sound card.
It has a system clock, again with a
sampling interval Ts.
And we get samples xn based on what was
measured by the microphone.
So we can see that there is a very
intimate
relationship between going from continuous
time, xt, to xn, the sequence.
That's called sampling or from
xn, the sequence, to x of t, that's called
interpolation.
And we have to be able to understand very
clearly.
When we can go from one to the other, when
these two
are tightly related or they are not
face-full images of each other.
So let's study continuous time for a
while.
This is a new object.
Because we have looked only at sequences
so far.
So we have a real-time variable t.
We have a signal x of t.
Which is typically a complex function of
that real variable t.
As announced, we like them to be of finite
energy,
so the square integral of x of t is
finite.
And on L2 of R, we can very naturally
define an inner product between.
Two functions, X of T, Y of T as an
integral of X star of T, Y of T DT.
And here X star of course is a conjugate
of X of T.
So complex conjugation.
The energy is defined as inner product of
X with itself.
We have just spent a module studying
discrete time filtering.
So naturally, there are also continuous
time filters.
These are called analog-linear timing
variant filters.
They are given by a box, h.
The notation is x star h.
Use the convolution.
This can be written as an interproduct or
conveniently as
an integral from minus infinity to
infinity of x tau.
H of t minus tau, d tau.
Please note very importantly the t minus
tau,
so that corresponds to a time reversal
when
you compute this integral.
We promised a Fourier transform for
continuous time signals.
Remember, in discrete time, there was this
maximum angular frequency of plus minus
pi.
In continuous times, there will be no such
maximum frequency, it can go off to
infinity.
So it's a concept of frequency is the
same.
We take an inner product.
Between the function xt and e to the minus
j omega t dt.
Now this is a capital omega to be sure
we understand this is a continuous time
Fourier transform.
So, capital X of g, capital omega is such
a Fourier transform.
We can invert this Fourier transform by
renormalizing with 1 over 2 pi,
taking the integral of the Fourier
transform with respect to e to the j.
Capital omega t.
We will not study in great detail
the conditions when it exists, under what
conditions
it will converge, et cetera, but let us
assume for the sake of
practicality that this transform, for the
object
of interest for us, does actually exist.
So this real world frequency's very
important because it
does interact with what we have an
increasion for.
Capital omega is expressed in radians per
second, we can also
talk about hertz, that would be capital
omega divided by 2
pi, so hertz is 1 over seconds.
And the period is one over the frequency
in hertz, or 2pi over capital omega.
Let's look at an example.
This is a good old Gaussian bell-shaped
curve.
So of x of t is, is exponential of
t-square normalized by sigma square.
And it has the usual look.
It's continuous time for a transform
happens to
be also a bell-shaped curve, simply
rescaled appropriately.
This is rather an oddity.
Usually for a transform of a function
doesn't look
like itself, as we will see in the sequel.
We have seen the convolution theorem for
discrete time convolution.
The very same theorem holds for continuous
time as well.
So if we have x of t convolved with H
leading to y of t, it's fully transformed,
is simply the product of the two fully
transformed capital X and capital H.
We introduce a new concept, namely the
concept of bandlimited functions.
Omega N bandlimitedness means that the
Fourier
transform of a function that is
bandlimited
is strictly 0 outside an interval from
minus capital omega N to capital omega N.
This is a very simple function.
It's a rect function between minus omega n
and
omega n, centered around zero, and of
height, capital G.
So this was a Fourier transform of these
band
limited functions, a prototype of a band
limited function.
What is the time domain function?
Capital phi is a rect function.
So phi of t is going to be
the inverse continuous time Fourier
transform written here.
We had calculated this in module.
5.5.
So let's not do it again.
Please look it up if you don't remember
it.
And the time delay function is
up to a scaling.
One of these sync functions.
We will normalize the sync function, so
that the area
is equal to 2 pi in the fully transformed
domain.
Then the inverse continuous time Fourier
transform will
have a maximum of one at the origin.
So lets fix this, G is equal to pi over
capital omega
N so total bandwidth is 2 capital omega N,
and we define Ts
as 2 pi divided by the bandwidth, or
subsequently pi over
capital omega N and that will be a very
important sampling interval for this.
Sinc function.
So now we have this prototypical
bandlimited function.
On the one hand, capital of phi is the
rect function.
And phi of t is the sinc function, t over
capital Ts.
And we see it now in normalized fashion so
it's
a box function from minus capital omega to
capital omega.
The height is pi over capital omega.
Its time domain function phi of t has a
maximum as the origin
equal to one.
And it has zero crossings at multiples of
Ts.