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Let's get started.
We want to explore continuous-time.

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This is the time we live in the physical
reality of the world.

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And this we will contrast to discrete
time.

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The time that lives inside a computer.

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Module 6: Interpolation and Sampling.

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So the outline of the module is the
following.

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First we look at continuous-time signals.

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Mainly signals that live in the real world
that we inhabit every day.

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Then, we look at how to go from a
sequence, the elements

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we have seen so far, into continuous time
signals, and that's onto interpolation.

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Then we look at sampling.

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The cornerstone result here

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of this module.

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How to go from a continuous time signal to
a sequence.

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And under which conditions.

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This is a faithful representation of the
continuous time signal.

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This will involve to look at aliasing

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a nasty phenomenon that we need to
understand.

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Then we'll look at interpolation and
sampling in practice.

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Finally we look at the most important
application

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which is discrete time processing of
continuous time signals.

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This is when we take a signal from the
real word, we process

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it in a computer and we render the signal
in the real word.

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Module 6.1, The Continuous Time Paradigm.
First, we look

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at some models of the world and compare
digital with analog views of the world.

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Then we study continuous time signals in
some detail.

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We also introduce a continuous

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time fully transform, which is the last
form

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before we transform, we have not
encountered yet.

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And therefore, this is a time to meet this
very interesting mathematical object.

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We have seen these pictures before, these
are the two views of the world.

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On the left, you see the analog world, the
one we live in, and is shown by a picture

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of the Earth taken by the Apollo mission.
On the right side, you have a digital view

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of the world.
This is taken by digital camera.

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Then very strongly pixelized and the
values are actually also discrete.

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So this is a view that you have of the
world inside a computer.

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In short, this is a contrast between and
analog continuous view versus a discrete

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and digital view of the world.
We can show this in a block diagram form.

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We live in an analog world, so input and
output are

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from and to the analog world, but inside

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we process on the computer, which is
discrete time.

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The input is x of t.

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The output is y of t.

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But inside the box, we process sequences,
x of n, y of n.

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Examples of such boxes are for example the
processing we do for MP3 music

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or what we do on a digital camera.
Other examples are the following.

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You have

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a digital world.
For example, you synthesize an image.

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And you process it and then show it on a
computer screen.

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This is a typical example in computer
graphics or in video games.

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So the input to the box is a sequence.

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It is processed into an output, which is a
continuous time or continuous space image.

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Sometimes the box is used to take a
decision.

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So the input is analog, from the analog
world.

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The output is discrete, for example, to
decide

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something, should we take an action or
not.

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The processing, again, inside the box is
done on

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sequences, x[n] from the input, y[n] to
the output.

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This is typical, for example, in
monitoring applications.

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Is a certain level of temperature reached,
and therefore in control systems.

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Let's summarize.

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The digital world view is the one of
arithmetic, of combinatorics, what is

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usually referred to as computer science,
but

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also the world of discrete signals
processing.

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The analog view is the view of calculus.
Integrals and so on.

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Distribution, system theory.
Continuous time analog electronics.

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So you can see these are two sides of a
coin.

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Because we always have to deal with one
and

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the other when we want to process signals
in the real world.

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To make it slightly more mathematical, the

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digital view is you have a countable
index.

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So x-n, where n is an integer.

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So we have sequences.

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And we like to have sequences that are
square summable.

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So xn is an l2 of z.

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We have a frequency that is limited
between minus pi and pi.

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And we have a Fourier transform called the
discrete time Fourier transform.

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Which maps from l2 of z to capital l2

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of minus pi to pi.

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The analog world view is that we have real
time in second.

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T is measured in second.

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We have functions on the real line, x of
t.

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We like them square summable, that's what
we call capital L, two of r.

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The frequency is given in radians per
second

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and is unlimited unlike in the discrete
time case.

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And we have a fully transform that maps l2
of r functions into

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l2 of r functions, as we will see shortly.

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So how do we bridge the gap between these
two worlds?

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So here's an example where we have, for
example,

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computer synthesis generates a sequence
xn, we have a

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sound card on the computer, the sound card
has

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a system clock, so every Ts second it will
generate.

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An output based on xn.
This output will be driving a loudspeaker.

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In the other direction, we are speaking
into a microphone like I do right now.

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There is a sound card.

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It has a system clock, again with a
sampling interval Ts.

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And we get samples xn based on what was
measured by the microphone.

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So we can see that there is a very
intimate

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relationship between going from continuous
time, xt, to xn, the sequence.

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That's called sampling or from

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xn, the sequence, to x of t, that's called
interpolation.

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And we have to be able to understand very
clearly.

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When we can go from one to the other, when
these two

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are tightly related or they are not
face-full images of each other.

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So let's study continuous time for a
while.

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This is a new object.

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Because we have looked only at sequences
so far.

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So we have a real-time variable t.

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We have a signal x of t.

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Which is typically a complex function of
that real variable t.

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As announced, we like them to be of finite
energy,

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so the square integral of x of t is
finite.

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And on L2 of R, we can very naturally

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define an inner product between.

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Two functions, X of T, Y of T as an
integral of X star of T, Y of T DT.

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And here X star of course is a conjugate
of X of T.

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So complex conjugation.

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The energy is defined as inner product of
X with itself.

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We have just spent a module studying
discrete time filtering.

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So naturally, there are also continuous
time filters.

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These are called analog-linear timing
variant filters.

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They are given by a box, h.

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The notation is x star h.
Use the convolution.

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This can be written as an interproduct or
conveniently as

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an integral from minus infinity to
infinity of x tau.

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H of t minus tau, d tau.

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Please note very importantly the t minus
tau,

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so that corresponds to a time reversal
when

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you compute this integral.

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We promised a Fourier transform for
continuous time signals.

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Remember, in discrete time, there was this

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maximum angular frequency of plus minus
pi.

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In continuous times, there will be no such

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maximum frequency, it can go off to
infinity.

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So it's a concept of frequency is the
same.

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We take an inner product.

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Between the function xt and e to the minus
j omega t dt.

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Now this is a capital omega to be sure

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we understand this is a continuous time
Fourier transform.

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So, capital X of g, capital omega is such
a Fourier transform.

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We can invert this Fourier transform by
renormalizing with 1 over 2 pi,

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taking the integral of the Fourier
transform with respect to e to the j.

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Capital omega t.

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We will not study in great detail

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the conditions when it exists, under what
conditions

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it will converge, et cetera, but let us
assume for the sake of

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practicality that this transform, for the
object

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of interest for us, does actually exist.

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So this real world frequency's very
important because it

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does interact with what we have an
increasion for.

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Capital omega is expressed in radians per
second, we can also

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talk about hertz, that would be capital
omega divided by 2

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pi, so hertz is 1 over seconds.
And the period is one over the frequency

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in hertz, or 2pi over capital omega.
Let's look at an example.

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This is a good old Gaussian bell-shaped
curve.

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So of x of t is, is exponential of
t-square normalized by sigma square.

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And it has the usual look.

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It's continuous time for a transform
happens to

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be also a bell-shaped curve, simply
rescaled appropriately.

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This is rather an oddity.

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Usually for a transform of a function
doesn't look

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like itself, as we will see in the sequel.

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We have seen the convolution theorem for
discrete time convolution.

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The very same theorem holds for continuous
time as well.

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So if we have x of t convolved with H
leading to y of t, it's fully transformed,

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is simply the product of the two fully
transformed capital X and capital H.

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We introduce a new concept, namely the
concept of bandlimited functions.

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Omega N bandlimitedness means that the
Fourier

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transform of a function that is
bandlimited

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is strictly 0 outside an interval from
minus capital omega N to capital omega N.

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This is a very simple function.

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It's a rect function between minus omega n
and

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omega n, centered around zero, and of
height, capital G.

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So this was a Fourier transform of these
band

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limited functions, a prototype of a band
limited function.

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What is the time domain function?

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Capital phi is a rect function.

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So phi of t is going to be

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the inverse continuous time Fourier
transform written here.

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We had calculated this in module.

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5.5.
So let's not do it again.

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Please look it up if you don't remember
it.

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And the time delay function is

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up to a scaling.
One of these sync functions.

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We will normalize the sync function, so
that the area

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is equal to 2 pi in the fully transformed
domain.

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Then the inverse continuous time Fourier
transform will

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have a maximum of one at the origin.

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So lets fix this, G is equal to pi over
capital omega

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00:11:01,870 --> 00:11:07,310
N so total bandwidth is 2 capital omega N,
and we define Ts

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as 2 pi divided by the bandwidth, or
subsequently pi over

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capital omega N and that will be a very
important sampling interval for this.

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Sinc function.

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So now we have this prototypical
bandlimited function.

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On the one hand, capital of phi is the
rect function.

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And phi of t is the sinc function, t over
capital Ts.

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00:11:35,260 --> 00:11:38,060
And we see it now in normalized fashion so
it's

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a box function from minus capital omega to
capital omega.

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00:11:41,800 --> 00:11:43,950
The height is pi over capital omega.

200
00:11:45,740 --> 00:11:49,880
Its time domain function phi of t has a
maximum as the origin

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equal to one.

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And it has zero crossings at multiples of
Ts.