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DENNIS FREEMAN: So for
the last couple of times,
we've been looking
at Fourier series
as a way of looking at signals
as a way of being composed out
of sinusoids, much the
way we had previously
looked at frequency responses as
a way of thinking about systems
characterized by sinusoids.
And for the past two sessions,
we've looked at Fourier series
not because they
were terribly useful,
but because they
were terribly simple.
Today, I want to do the
much more difficult task,
but much more interesting
task of thinking
about the general
case for thinking
about a sinusoidal decomposition
of an arbitrary signal, one
that is not
necessarily periodic.
So I should say
upfront, what I'm going
to talk about is motivational.
It's not a proof.
Proving Fourier
series convergence
is actually very complicated.
It's something that
mathematicians worked on
for about 100 years.
So I am not going to
try to prove things
in any rigorous
fashion, but I am
going to try to
motivate things so
that you should at least expect
that such a thing should exist.
So the idea, the
motivation is going to be
how can I think about
an aperiodic signal
within a periodic framework
because I already have
worked out all the details.
The details for Fourier series
are relatively simple, well,
at least compared to a Fourier
transform, which is harder.
Fourier series themselves
are not that easy.
But if I believe the
Fourier series idea,
is there a way to
leverage that to think
about aperiodic signals?
And the idea is going to be
let's take an aperiodic signal.
I've tried to choose
something terribly simple.
It's the simplest thing that I
could think of it isn't zero.
So it's one for a while
and zero most of the time.
But I could make
that signal, which
is clearly not periodic, by
thinking about periodically
extending it.
Copy it.
Add it to itself
many times, each time
shifted by a capital T.
This signal is
obviously periodic.
This transformation
is obviously going
to take any signal
regardless and turn it
into something that
is periodic in cap T.
So if I did that, then I
could kind of trivially say,
well, the aperiodic
thing is just
the limit when capital T goes to
infinity of the periodic thing.
OK, that's pretty trivial.
OK, that's obviously true.
The trick is, what if I took a
Fourier series in the middle?
What if I periodically
extended this thing
to get something
that is periodic,
take a Fourier
series of this thing,
and then take the
limit of the series?
So that's the thing I'm going to
do over the next three slides.
So think about a general,
aperiodic signal, periodically
extended so it's now periodic in
cap T, take a Fourier series--
just to motivate the kind
of math that happens,
I've written out the math for
this particularly simple signal
that is one for a while
and zero most of the time.
The Fourier series
coefficient, a sub k
is obviously 1
over t, the period,
integral over the period--
I took the symmetric period
because it's the easiest one--
signal of interest, basis
function integrated over time.
And that's pretty trivially--
those integrals are easy.
That was chosen that way.
And so I get an answer
that looks like that.
The thing I want you
to see about the answer
is that I can think about it
as a function of omega or k.
And that's what
I've plotted here.
In particular, if I multiply
a sub k by capital T,
so as to kill this
1 over t thing,
and if I plot t times a sub k, I
get a relationship 2 sine omega
S over omega, omega
being k 2pi over t.
But for the Fourier series, that
only exists for k and integer.
So that's what's represented
by the blue bars.
But what I want you to
see is just from the math,
the envelope
doesn't depend on t.
OK, that's the trick.
So the idea is I'm plotting
the Fourier coefficients a sub
k as a function of k.
So k equals 0, 1, 2,
3, 4, 5, et cetera.
But I notice that the envelope
can be written strictly
as a function of omega where
there is a simple relationship
between omega and k.
But omega is defined
across the entire axis,
and it's represented by
this light black curve.
That's more apparent if I
think about increasing capital
T. What if I were to keep
the base waveform the same,
but change capital T?
Say I double capital T.
The thing that happens
is the envelope stays
the same, but the spacing
of the k's becomes condensed.
There's more k's in a
given number of hertzes,
in frequency, than
there was before.
And if I double it
again, it doubles again.
The envelope didn't change.
The k's did.
The interesting thing
about that construction
is that it has separated out the
part that depends on capital T
from the part that doesn't
depend on capital T. Capital
T was this arbitrary
thing that I
used to take an aperiodic
signal and turn it
into a periodic signal.
And it has an
effect on the answer
that can be separated from
the other part of the answer.
Some part of the answer
depends on the base waveform.
Some other part of the answer
depends on capital T. Well,
that's nice because now if I
think about taking the limit
as capital T goes to infinity,
I have a prayer of interpreting
things because part of my answer
is changing with t and part
of it isn't.
So all I need to do now is focus
on the part that is changing
with t and separate it
from the part that's
not changing with t.
So now, I can think
about taking--
so I just plug in
this expression
here for this integral.
And what I get then
is something that
looks a lot like a
Fourier series or even
a Laplace transform.
I get an integral-- ignore
the limit part for a moment.
I get an integral
of something times
some sort of a
weighting function.
And I get something over
here where the integral
was over time, but
the function over here
doesn't have time in it.
It only has omega in it.
That's the sense
in which it sort of
looks like the analysis
formula for either Fourier
series or Laplace transforms.
It looks like the
analysis formula
because I'm calculating a T ak,
the components of the series,
or this new thing, E of
omega, that doesn't depend--
neither of those depended on t.
OK, so the idea
is that when I do
this kind of a
limiting operation
on the periodic
extension, I get something
that ends up looking like
a transform relationship.
And if I think about
going the other way,
doing a synthesis
operation, I can
think about how
I would construct
x of t out of the
Fourier coefficients
But now, there's a
simple relationship
between the Fourier
series coefficients,
a sub k, and this thing Ew,
which I've represented here.
And I don't like the t.
So I'll do a
substitution from here.
t can be written as
omega 0 over 2pi.
1 over t can be written
as omega 0 ever 2pi.
And now, I've got
everything I need
to think about how that
sum approaches an integral
in a Riemann sum kind of sense.
Think about as I
add more and more--
as I make capital T
get bigger and bigger,
omega 0 gets
smaller and smaller.
As I make the capital
T bigger and bigger,
the spacing gets
smaller and smaller.
Increasingly, I can think
about this function E of omega
as being smooth and
increasingly constant
over the small interval
between the bars.
So I can think about the
sum as a Riemann sum passed
to the integral as the limit.
So when I do that,
omega 0 is the spacing
between the two adjacents.
It's the region
over which I want
to think about that
integrand being constant.
And so that in the limit, this
omega 0 passes to d omega.
And I'm left with something
that looks like a synthesis
equation.
So if I just write those
equations here and think
about this Ew thing being some
kind of a transform, which I'll
mysteriously write
as x of j omega,
then the result for
the aperiodic case
has a structure that looks very
much like the Fourier series,
or for that matter like
the Laplace transform.
What it says is that I can
synthesize an arbitrary x of t
by adding together a
whole bunch of components
that already depend on omega
weighted by some weighting
function.
s this looks like a
synthesis equation
very much like the synthesis
equation for Fourier series
or for Laplace transforms.
And I get an analysis
equation that similarly
has the same form again.
I take the x of t and
figure out the component
that should be at
omega by multiplying
by a complex exponential
and integrating.
OK, I have to emphasize
this is not a proof.
All I wanted to do
was kind of motivate
the way you can think
about an aperiodic signal
as being periodic in
some time interval
and passed to the limit.
And if you do that,
you can sort of
see where the equations
are coming from.
OK?
So the idea then--
whoops.
So the idea then
is that we will use
these relationships to define
an analysis and synthesis
of aperiodic signals.
And we'll refer to that
as a Fourier transform.
The Fourier
transform will let us
have insights that
are completely
analogous to the Fourier
series, except they now
apply for aperiodic signals.
So in particular,
we'll be able to think
about a signal being
composed of a bunch
of sinusoidal components.
And we'll be able to think
about systems as filters.
OK, so I've already
alluded to the fact
that the Fourier
transform relations
look very similar in form to
the Laplace transform relations.
And so I've illustrated the
analysis equations here just
to emphasize the similarity.
The Laplace transform,
you'll remember, had the--
we integrated some
signal that was
a function of time times a
complex exponential integrated
over time to get a
Laplace transform that was
a function of s, not time.
It was a way of having an
alternative representation
for the signal.
There was no new information.
The same information
was contained
in s of x as was
contained in x of t.
Except now, where it
was organized by time,
now, it's organized by s.
We get the same sort of thing
with a Fourier transform.
And in fact, this gives
away the mysterious reason
for calling it x of j omega
in the previous slide.
You can see that a different
way to think about the Fourier
transform is that it's simply--
a trivial way to think
about it, it's the value--
the Fourier transform is the
value of the Laplace transform
evaluated s equals j omega.
All you do is you take this
expression for the Laplace
transform, and every place there
was an s, make s equal j omega.
And you get this equation.
So that's the reason
we like the notation.
The Fourier transform
is x of j omega.
There are confusions
that arise by that.
And I'll talk about
those in a moment.
But for the time being,
the important thing
is that the Fourier
transform can
be viewed as a special
case looking at the j omega
axis of the Laplace transform.
OK?
So that view points
out two things.
There's a lot of similarities,
and there are some differences.
First, the similarities--
because you
can regard the Fourier transform
as kind of a special case--
that's not really true.
And I will say something about
that by the end of the hour
as well.
But because it's kind of a
special case of the Laplace
transform, the Fourier
transform inherits
a lot of the important
properties of a Laplace
transform.
In particular, the two
things that we looked at most
has been linearity.
Because the Laplace
transform is linear,
we can do all manner
of things with it.
The same as we use the
properties of linear systems
to simplify our view of how
to think about a system,
we could, for example,
because systems are linear,
we can look at the
response of a system
to a sum of inputs as
the sum of the responses
to the individuals.
That's a very important
property that we used of systems
as a result of linearity.
We did the same thing
with Laplace transforms.
The Laplace transform
of a sum is the sum
of a Laplace transforms.
And in conjunction with
the differentiation roll
by which we knew
that the Laplace
transform of a derivative is
s times the Laplace transform
the function, the
combination of linearity
and the differentiation
role allowed
us to apply Laplace transforms
to turn differential equations
into algebraic equations.
Precisely the same thing will
work with Fourier transforms.
For reasons that
should be clear,
if the Laplace transform has
the property of linearity,
so does the Fourier.
And if the Laplace
transform is simply
related to the
Fourier transform,
then there's a
simple relationship
between the Fourier
transform of a derivative
and the Fourier transform
of the underlying function.
So in the Laplace transform,
you multiply by s.
Not very surprisingly,
in the Fourier transform,
you multiply by j omega.
So there's enormous similarity.
And in fact, most of what
you know about Laplace,
you can immediately
carry over into Fourier.
There are some differences.
And if there
weren't differences,
we probably wouldn't bother
with talking about both of them.
Right?
There are some things that
will be easy to think about
with Fourier transforms.
And that's the reason we do it.
There are some things that
are easy to think about
with Laplace transforms.
Otherwise, we would have just
skipped straight to Fourier.
So there are some things that
Fourier and Laplace share.
There are some things
that are different.
One of the biggest
differences is the domain.
When we think about
a Laplace transform,
we think about x of s.
The domain or the Laplace
transform is the domain of s.
The domain of s, s
is a complex number.
For that reason, when we
thought out Laplace transforms,
we always talked about what
does the Laplace transform
look like in the s plane.
And we thought about
the real part of the s
and the imaginary part of s.
When we think about
Fourier transforms,
we're thinking about a
transform with real domain.
Rather than thinking about
x of s as a complex number,
we're going to think
of x of j omega, omega
a real number that's a
little confusing, right?
Just sort of to confuse
you, we rewrite the one
that is a complex number as s--
no indication whatever
that it's complex.
And the one that
is a real number,
we put a j in front of it to
remind you that it's real.
I apologize, I don't
know why we do this.
So just remember that s, which
looks kind of real isn't.
It's complex.
And j omega, which
looks kind of complex,
well, it's the omega
part that matters.
It's real.
OK, so the important thing
is the Laplace transform,
the domain of a
Laplace transform
is complex number s,
real and imaginary parts,
characterized by a plane.
The domain of the Fourier
transform is real.
That's enormously important.
And we'll come back to
that over and over again.
But just to drive
home the point,
one of the things
we thought about
with the Laplace transform was
this idea of eigenfunctions
and eigenvalues.
It was an idea of linearity.
It was the idea that we
can think about a system
by how you put in a
function, like E to the st,
and calculate the output.
Well, if the output, if the
system is linear time invariant
and can be characterized by
a Laplace transform h of s,
what's the output of that system
when the input is e to the st?
Everybody shout.
It will make me
feel much better.
If you all shout
at once, I won't
be able to understand
a word you said,
and I'll assume you
said the right thing.
OK I didn't understand
a thing you said.
So I assume you all said
h of s e to the st. Right,
e to the st is an eigenfunction
of a linear time invariant
system.
Eigenfunction means
the function in
is the same form
as the function out
except it could be
multiplied by a constant.
The constant is the eigenvalue.
The eigenvalue is h of s.
If we wanted to know,
for example, if we wanted
to characterize a
very simple system,
we might have a system of
the form 1 over 1 plus s.
We might have a signal of
the form 1 over 1 plus s.
So let's say we
have a system now.
Let's say that x represents
some kind of a system.
Then we would have said
that that's a pole.
Where's the pole?
Minus 1-- we would have
said we have a system
with a single pole at minus 1.
I would never have drawn
this complicated picture
at the bottom because
it would be frightening.
I would always draw
something friendly
like the picture over here.
Right, the entire system can
be understood by a single x.
OK well, if you were computing
eigenfunctions and eigenvalues,
you would like to know what's
the magnitude and phase.
Sorry, the x of s
is a complex valued
function of complex domain.
s is a complex number, and the
answer is a complex number.
So we'd like to know the
real and imaginary parts
of h of s or the magnitude
and phase equivalently.
If we want to know the magnitude
and phase, for example,
of h of s, in principle, we need
to know what magnitude could it
be for all the different s's.
So what's plotted
here is a picture
of the magnitude
of this function
as a function of all
the different s's
that can be an eigenfunction.
Right, so for all
of the-- so any
s is an eigenfunction
of the system.
And that plot
plots the magnitude
of the associated eigenvalue.
The point is that I have
to tell you a complex plane
number of values.
Right?
There a value for s equals 1,
s equals minus 1, s equals 2,
s equals minus 2, s equals j, s
equals 2j, s equals 17 plus 5j.
All the different values,
all the different points
in the s plane have a different
associated eigenvalue.
And to completely
characterize this system,
I have to tell you all of those.
By contrast, if I think
about the Fourier transform,
the Fourier transform
maps a function
of time to a function of omega.
The complete characterization
of the Fourier transform
is showed here.
All I need to
worry about is what
are all the possible
values of omega.
I'm thinking now
instead of thinking s,
I'm thinking how
would I compose x of t
by summing together a
bunch of sine waves.
The reason I want
to think about that
is because I want to think
about systems in terms
of frequency responses.
So I want to know which
frequencies are amplified,
which ones are attenuated,
which ones are phase delayed,
which ones are phase advanced.
And in order to do that
kind of construction,
all I need to know is
what's the magnitude
and angle of the system
function for all possible values
of omega.
So that's an
enormous difference.
Instead of having,
in the previous case,
I had a function of time turning
into a function of two space.
Function of one space turned
into a function of two space.
Here I have a function
of one space turning
into a function of one space.
So that is conceptually
a whole lot simpler.
Even more importantly,
it is going
to give rise to
something that we'll
spend most of the time for
the rest of the term on--
the notion of signal processing
where we can alternatively
represent a signal x
not by its time samples,
but instead by its
frequency samples.
It would be very difficult
to use that technique.
Although it would
work perfectly,
there would be an
explosion of information
if we tried to use a signal
processing technique with this
where we represent this
one-dimensional signal
by a two-dimensional transform
because we would be exploding
the amount of information.
We would be increasing
substantially
the amount of information
required to specify the signal.
When we do the Fourier,
there is no such explosion.
It was a one-dimensional
function of time.
It is a one-dimensional
function of omega.
OK, OK, I've been
talking too much.
I would like you to
make sure that you
understand the mechanics
of what I've just said.
So here's a signal, x1 of t.
Which of these,
if any, represents
the Fourier transform?
You're all very quiet.
Look at your neighbor.
Don't be quiet.
And then start.
And then you can go
back to being quiet.
[SIDE CONVERSATIONS]
So it's quiet.
So I assume that
means convergence.
So which function represents the
Fourier transform of x1 of t?
Everybody raise your hand.
Indicate by a number of fingers.
And it's overwhelmingly
correct, which is wonderful.
That's the point.
The point is Fourier
transforms are easy.
And you've all got it.
So it's trivial to run
this kind of an integral.
It's not very different from
doing a Laplace transform.
Here I've indicated
the Laplace transform.
Right?
We do e to the minus
t. x of t is 1 or 0.
We change the limits to
indicate the 1 or zeroness.
Very trivial here, we
get a slightly different
looking answer because
instead of e to the st,
we have e to the j omega t.
But otherwise, it's
pretty much the same.
The big difference, though,
is again the domain.
So if you think
about the answer--
so the answer is four
like all of you said--
if you think about the
answer from the point of view
of eigenfunctions
and eigenvalues,
you have to think
about a two space.
The two space for even
that simple function,
sort of the least complicated
function I could think of,
is illustrated here.
And what you're
supposed to see there
is if I were to integrate
x of t e to the minus st
dt to get x of s, if I think
about s as sigma plus j omega--
it has a real part and
an imaginary part--
the real part, as I make the
real part big, e to the st
becomes something that explodes.
And you can see that manifest
here over in this region.
So this is the
real axis this way.
This is the imaginary
axis that way.
You can see that
as you go to bigger
numbers in the positive
real direction,
the magnitude explodes.
If you go in the
negative direction
because there was a sum here,
the magnitude explodes again.
You get this horrible
function that
spends a lot of its
time near infinity.
Right?
So that's a complicated picture
by comparison to the picture
that you get if you look
at Fourier transform.
So if you look at the
Fourier transform,
you get something that's
relatively simpler.
We're only looking along
the imaginary axis now.
Furthermore, there's an
easy way to interpret this.
This is explicitly telling us
if you put a certain frequency
into the system, say this
represented a system function,
if this represented
a system function,
it's telling you that there's
a simple way of thinking
about how it amplifies or
attenuates frequencies.
Right?
It likes frequencies
near the middle.
There's a lot of frequencies--
so if this represented
a system function,
it would pass with a gain
of two frequencies near 0.
And the magnitude would be
smaller for these others.
And there is a phase
relationship too.
So there's insights that you
can get from this Fourier
representation that are less
easy to get from the Laplace.
I mean the Laplace was
a complete specification
of a signal or a system, either.
So all the information is there.
It's just that it's
more apparent--
some of the information is
more apparent-- in the Fourier
representation.
OK, second question, what if
I stretched the time axis?
x1 was 1 between minus 1 and 1.
x2 is 1 between minus 2 and 2.
So all I'm doing,
stretching the axis.
What happens to the
Fourier transform?
Look at your neighbor.
Choose a number.
[SIDE CONVERSATIONS]
OK, which answer tells me what
happens when I stretch time?
So everybody raise
your hand and tell me
some number between 0
and 5, 1 and 5 actually.
OK, 20% correct.
S
So what's going to happen?
Well, it's pretty easy to simply
do out the integral again.
Right, so that's the sort
of most primitive way
you can think about it.
If you simply run
the integral, I've
written it in a
kind of funny way.
Right?
So a lot of you said
one for the answer.
This kind of looks like one.
Why is that not one?
That's actually three.
Why do I like to write it as--
instead of writing 2
since 2 omega over omega
I like to write 4 signed
2 omega over 2 omega.
Why do I like that?
Because I'm completely random.
AUDIENCE: Omega is
the same that way?
DENNIS FREEMAN: Excuse me.
AUDIENCE: Omega can
be the same-- like you
can have omega absorb the 2.
DENNIS FREEMAN:
That's kind of right.
So can you unscramble
the sentence slightly?
What is more-- yes.
AUDIENCE: Aren't they the
same form that we use?
DENNIS FREEMAN: It's the
same form in what sense?
I mean what's the same about it?
Yes.
Yes.
AUDIENCE: Like I thought
I was going to say omega
is near 0 when number two is 4.
DENNIS FREEMAN:
Correct, correct.
If you think about what
happens for omega near 0,
I've got the sine of
2 omega, which is--
what's the sine of 2
omega when you make it 0?
0.
So I have 0 over 0.
Bad.
So what do I do?
L'Hospital.
So if I do L'Hospital's
rule, then I
can make this thing
look like one.
And if I write it in the form
sine 2 omega over 2 omega,
that has a value near
0 that approaches 1.
So the amplitude is 4.
So that's a way
of separating out
the part that's unity
amplitude from the part that
is the constant that
multiplies the amplitude.
So the amplitude is 4.
And frequency, which had
been pi, moves to pi over 2.
So the point is that--
so the answer is number three.
The peak increases, and the
frequency spacing decreases.
But more generally, the point
is that if I stretched time,
I compress frequency.
But I compress frequency
in a very special way.
I compress frequency in
an area-preserving way.
That's why the peak popped up.
So what I'd like to do is think
about a general scaling rule.
If I wanted to think
about scaling x1 into x2,
such that x2 is a scaled
version of time compared to x1,
so if I wanted x2
of t to be x1 of at,
and if I wanted to stretch
x1 to turn it into x2,
should I make a1--
should I make a
bigger or less than 1?
I'm trying to generalize
the result that I just did.
Right?
So I stretched x1 into x2.
And what I saw is that frequency
shrunk, and amplitude went up.
So now, I'm thinking
about what would happen
if I did that in general.
If I took x1, and I
stretched it by setting
x2 equal to x1 of
at, would I want
a to be bigger or less than
1 if I want to stretch x1
to turn it into x2?
AUDIENCE: Less than 1.
DENNIS FREEMAN: Less than
1 because then the logic
is that if I wanted, for
example, x2 of 2 to be x1 of 1,
stretch x2--
stretch x1, sorry, so that
it's value x2 at the position 2
is the same as the
original function x1 at 1.
If I stretched
it, then I clearly
have to have a equal
to a half in that case.
And in general, stretching would
correspond to a less than 1.
And now, I can think
about where that fits
in the transform relationship.
Think about finding the
Fourier transform of x2,
and substituting
x1 of at for x2,
and then making this
relationship look
more like a Fourier transform.
So I don't want
the at to be here.
I want function of t.
So I can rewrite at as tau.
Now, this looks like a
Fourier transform except that
I've changed all
my t's to tau's.
And the point is that
that transformation of tau
equals at shows
up in two places.
There's an explicit time here.
And there's a time
dependence in the dt.
So the dt one is the one
that gives me the shrinking
and swelling of the axis.
And the 1 over a from here
is the one that gives me
the changing amplitude.
That's how you get the
area-preserving property.
However much it got
compressed-- however
much it stretched in time
so that it became compressed
in frequency, whatever the
factor is that compressed it
in frequency, it also
makes, by the same factor,
it makes it taller.
We'd like to build up intuition
for how the Fourier transform
works.
That's the reason for doing
these kinds of properties.
So now, there's another way of
thinking about that same thing
by thinking about what we
call the moment theorems.
Here what we think
about is what would
happen if we evaluated the
Fourier transform at omega
equals 0.
Well, omega equals 0 is
associated with a particularly
simple complex exponential.
If the frequency is
0, e to the j0 t is 1.
So what you see
is that the value
of the Fourier
transform at omega
equals 0 is the area
under the curve.
So the idea then
is that if I took
an x of t, which was x1, which
was 1 between minus 1 and 1,
there's an area of 2.
And that's a way of
directly saying, well,
the Fourier transform
at 0 better be 2.
And the intuitive
thing that you're
supposed to take
away from that is
when you look at a Fourier
transform, the value at 0
is the dc.
How much constant is
there in that signal?
So there's a very
explicit representation
for the frequency content.
I mean that's what the Fourier
transform is all about.
And in particular, the
zero frequency is dc.
It's the average value.
That kind of a relationship
works both ways.
If you were to use
the synthesis formula
and think about how do
you synthesize x of 0,
well, it's the
same sort of thing
except now the t is 0
instead of the omega being 0.
And what we get is
1 over 2pi times
the area under the transform.
So what that says is
whatever is going on
over in this wiggly thing, the
net area, the average value,
divided by 2pi has to
equal the value at x1 of 0.
x1 of 0 is clearly 1.
So that means the area under
this thing must be 2pi.
That wasn't particularly clear.
I mean I don't
know automatically
the area under that curve.
But that's such a
frequently recurring thing
that it's useful to notice that
the area under this funny curve
happens to be precisely the
area of this inscribed triangle.
So the height of
the triangle is 2.
Half the base is pi.
So the area is 2pi.
I'm sure some Greek knew this.
But I don't.
So if somebody
can think of a way
to derive that answer without
using Fourier transforms--
I can do it with
Fourier transforms.
And I can look it up in
books where the authors also
use Fourier transforms.
But I'm sure some ancient
Greek can do this.
So the question is if
anybody can figure out
how the ancient Greeks would
have come to that conclusion,
I would be very
interested to know.
Does everybody get-- so
areas of inscribed whatevers,
right, that's what they did.
Right?
So I would like
to know how to get
the fact that the area under
this wiggly function 2 sine
omega over omega is 2pi without
knowing Fourier transforms.
So that's an open challenge.
So try to figure out how
to prove that without using
Fourier transforms.
So if we use the moment idea,
and we think about this scaling
thing, we come up with a
very interesting result
that if you were to
stretch x1, which
had been 1 just
between minus 1 and 1,
to turn it into x2, which
was 1 between minus 2 and 2,
and just keep stretching,
what would happen?
Well, it gets
skinnier and skinnier
and skinnier and skinnier.
But in a very special
way, the area is the same.
Even though it got skinnier
and skinnier and skinnier
and skinnier, the
area is the same.
If you keep doing that,
it turns into an impulse.
Well, that's pretty interesting.
That's an alternative way
of deriving an impulse.
An impulse, we think
about an impulse
as a generalized function.
Any function that
has the property
that in some kind of a limit,
the area shrinks towards 0,
but the area
doesn't change, that
turns into a delta function.
That's a different way of
thinking about the definition
of a delta function.
And so we just found
something very interesting.
The Fourier transform
of the constant 1
seems to be a delta
function as 0 of area 2pi.
Well, that's pretty interesting.
What's the Laplace
transform of 1?
Too shocking.
What's the Laplace
transform of 1?
AUDIENCE: It's a delta function.
DENNIS FREEMAN: Delta function.
What's the Laplace
transform of 1?
And So Laplace transform,
right, so x of s,
integral 1e to the minus st dt.
What's the Laplace
transform of 1?
AUDIENCE: 1 over s.
DENNIS FREEMAN: 1 over s.
How about 1 over s?
Yes?
No?
1 over s-- yes.
1 over s-- no.
Me.
1 over s, no.
Why not?
AUDIENCE: You would just
use the su of 1 u of t.
DENNIS FREEMAN: It's 1, yes.
So the Laplace transform
of u of t is 1 over s.
Right?
You remember there was a region
of convergence associated
with Laplace transforms.
The region of convergence,
we thought about
like if you had a time
function like u of t,
then it would converge
as long as you
multiplied by some factor
that generally attenuated.
So that bounded what
kinds of s's worked.
We needed the real part
of s bigger than 0.
Because if the real part
of s was the other way,
the interval would diverge--
bad.
Right so we could find the
Laplace transform of u of t--
real part of s bigger than 0.
Or we could find the Laplace
transform of a backward step.
The region of
convergence would flip.
And we got a sign change.
But the Laplace transform
of 1 doesn't exist.
There is no region of
convergence for the function 1.
That's a big difference between
Fourier and Laplace as well.
Even though Fourier, is in some
sense, a subset of Laplace,
there are some signals that
have Fourier transforms
and not Laplace transforms,
and so in that sense,
Laplace is a subset of Fourier.
So in fact, you
better think of them
as Venn diagrams that overlap.
So there are some
signals that have both,
but there are some signals that
have one and not the other.
OK, so the final and maybe
most important property
of Laplace transforms
is that they have
a simple, inverse relationship.
You may remember that
I talked about there
being an inverse
relationship for Laplace.
So you can think
about x of t being
1 over j 2pi, the integral over
some sort of a contour of x
of s e to the st ds.
And I told you don't
ever try to do that
without going over to math and
talking to those folks first.
That's complicated.
The interesting thing
about this relationship
is that it's really simple.
So there's a very simple
relationship between a Fourier
transform and its inverse.
OK, so I think I'll
defer talking about that
until next time, the
reason being that I want
to end a little earlier today.
So I'll finish talking
about the rest of the slides
on the next lecture.
