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ocw.mit.edu. we he doesn't
he's got n e t at you

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PROFESSOR STRANG: OK, so.
 

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These are our two topics.
 

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And, Thanksgiving is
coming up, of course.

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With convolutions,
where are we?

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I'm probably starting
a little early.

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Oh, I am.
 

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Is that right?
 

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Yeah, OK.
 

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So, with convolutions, I
feel we've got a whole

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lot of formulas.
 

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We practiced on some specific
examples, but we didn't

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see the reason for them.
 

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We didn't say the use of them.
 

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And I'm unwilling to let a
whole topic, important topic

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like convolutions, go
on with just formulas.

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So I want to talk about
signal processing.

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So that's a nice, perfect
application of convolutions.

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And you'll see it.
 

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You'll see the point, and
it's something that you're

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going to end up doing.
 

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And it's quite a simple
idea, once you understand

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convolutions you've got it.
 

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OK, but then I do want to go on
to the Fourier integral part.

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And basically, today I'll
just give the formulas.

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So you've got the Fourier
integral formulas that

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take from a function f,
defined for all x now.

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It's on the whole line, like
some bell-shaped curve or

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some exponential decaying.
 

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And then you get its transform,
I could call that c(k), but a

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more familiar notation
is f hat of k.

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So that will be the Fourier
integral transform, or just for

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short, Fourier transform of
f(x), and it'll involve

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all frequencies.
 

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So we are getting away from the
periodic case, and integer

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frequencies to the whole
line case with a whole

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line of frequencies.
 

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OK, so is that alright?
 

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Finishing, I'll have more to
say about 4.4 convolutions,

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but I want to say this much.
 

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OK, let's move to an example.
 

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So here's a typical
block diagram.

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In comes the signal.
 

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Vector x, values x_k.
 

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Often, in engineering and
EE, people tend to write

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that x[k], these days.
 

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OK, so, as being
easier to type.

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And sort of better.
 

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But I'll stay with
the subscript.

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So that signal comes in.
 

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And it goes through a filter.
 

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And I'm taking the simplest
filter I can think of, the one

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that just averages the current
value with the previous value.

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And we want to see what's
the effect of doing that.

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So we want to see, we want to
understand the outputs, the

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y_k's, which are just the
current value and the

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previous value averaged.
 

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We want to see that as
a convolution, and

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see what it's doing.
 

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OK, so that's the - and I guess
that here, it's a frequent

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convention in signal processing
to basically pretend that the

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signal is infinitely long.
 

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That there's no start
and no finish.

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Of course, in reality there has
to be a start and a finish.

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But if it's a long, you know
if it's a cd or something,

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and you're sampling it
every, thousands and

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thousands of times.
 

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And the input signal is so long
and you're not really caring

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about the very start
and the very end.

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It's simpler to just
pretend that you've got

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numbers for all of k.
 

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OK, then let's see what
kind of a filter this is.

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What does it do to the signal?
 

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OK.
 

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Well, one way filters are -
well, first of all, let's

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see it as a convolution.
 

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So I want to see that this
formula is a convolution that

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the output y, that output of
all the averages y, is the

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convolution of some filter,
some, and I'll give it a proper

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name, but with the input.
 

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And notice that, again, there's
no circle around here.

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We're not doing the cyclic
case, we're just pretending

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infinitely long.
 

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So OK, now I just want to ask
you what is the h, I want to

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recognize this as a filter.
 

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So that convolution, let's
remember the notation.

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And then we can
match this to that.

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So remember the notation
for that would be, for a

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convolution is, I take a sum.
 

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Of all x, of h_l.
 

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x_(k-l), right?
 

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That's what convolution is.
 

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This is our famous, and
it would be in principle

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the sum over f.
 

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Sum over l's.
 

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So I the filter is defined
by these numbers h, these

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h's, h_0, h_1, so on.
 

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Those numbers h multiply the
x's, where this familiar rule

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to give the k'th output.
 

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OK, let's have no mystery here.
 

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What are the h's if
this is the output?

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Well, I want to match
that with that.

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So here I see that this takes
x_k, multiplies by half.

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So that tells me that when
l is zero, I'm getting h_0

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times x_k, so what's h_0?
 

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So what's the h_0?
 

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It's 1/2.
 

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That's what this
formula is saying.

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When l is zero, so l is zero,
take h_0, 1/2 times x_k, ta-da.

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Now, what's the other h
that's showing up here?

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This is a very,
very short filter.

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It's only going to have two
coefficients, h_0 and h what?

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One.
 

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And what's the coefficient h_1?
 

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What's the number h_1?
 

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Now you've told me
everything about h when

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you tell me that number.
 

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Everybody sees it.
 

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It's also 1/2, right?
 

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Because I'm taking
1/2 of x_(k-1).

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So when l is that one, we have
h, the h is 1/2, times x_(k-1).

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Do you see that that simple
averaging, running average,

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you could call it.
 

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Running average, it's the most,
the first thing you would think

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of to - why would you
do such a thing?

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Why is filtering done?
 

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This filter, this averaging
filter, would smooth the data.

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So the data comes with
noise, of course.

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And what you'd like, so noise
is high frequency stuff.

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So what you want to do is like
damp those high frequencies a

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little bit, because much of it
is not, it hasn't got

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information in it.
 

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It's just noise, but you
want to keep the signal.

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So it's always this
signal to noise ratio.

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That's the key - SNR.
 

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PSNR.
 

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That's the constant expression,
signal to noise ratio.

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And we're sort of expecting
here that the signal to

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noise ratio is pretty good.
 

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High.
 

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Mostly signal.
 

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But there's some noise.
 

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This is a very simple,
extremely short, filter.

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So this vector h, it's
a proper convolution.

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You could say h has
infinitely many components.

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But they're all zero,
except for those two.

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Right, do you see it?
 

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Another way, just at the end
of last time, I asked you

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to think of a matrix that's
doing the same thing.

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Why do I bring a matrix in?
 

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Because anytime I see
something linear, and that's

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incredibly linear, right?
 

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I think OK, there's
a matrix doing it.

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So these y's, like
y_k, y_(k+1), all the

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y's are coming out.
 

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The x's are going in, x_k,
x_(k+1), x_(k-1), bunch of x's.

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And there's a matrix
doing exactly that.

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And what does that matrix have?
 

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Well, it has 1/2
on the diagonal.

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So that y_k will have a 1/2
of x_k, and what's the

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other entry in that row?
 

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I want 1/2 of x_k, and I
want 1/2 of x_(k-1), right?

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So I just put 1/2 next to it.
 

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So there is the main diagonal
of the halves, and there is

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the sub-diagonal of the half.
 

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So it's just constant diagonal.
 

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Now, yeah, let me
tell you the word.

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When an engineer, electrical
engineer looks at this, first

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thing, or this, any of these.
 

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First letters he uses is
linear time invariant.

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LTI.
 

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So linear we understand, right?
 

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What does this time
invariant mean?

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Time invariant means that
you're not changing the filter

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as the signal comes through.
 

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You're keeping a
half and a half.

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You're keeping the formula.
 

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The formula doesn't depend
on k, the numbers are

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just 1/2 and 1/2.
 

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They're the same, so if I shift
the whole signal by a thousand,

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the output shifts by
a thousand, right?

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If I take the whole signal
and delay it, delay it

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by a thousand times?
 

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Clock times?
 

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Then the same output will come
a thousand clock times delayed.

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So linear time invariant.
 

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That would be - I mean, linear
time invariant is just

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talking convolution.
 

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I mean, that's what it comes to
if we're in discrete problems.

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It's just that, for some h.
 

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Now, our h deserves like,
the other initials you see.

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OK, that was linear
time invariant.

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Now, the next initials
you'll see will be F-I-R.

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It's an FIR filter.
 

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So that's finite
impulse response.

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What's the impulse
response mean?

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It means the h.
 

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The vector h is the
impulse response.

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The vector h is what you
get if I put an impulse

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in, what comes out?
 

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Just tell me, what
happens here.

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Suppose an impulse, by an
impulse I mean I stick a one in

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one position and all zeroes.
 

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Our usual delta.
 

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Our impulse, our spike, is
just, suppose the x's have

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a one here, otherwise all
zero, what comes out?

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Well, suppose there's
just x_0 is one.

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What is y?
 

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Suppose the only input is
boom, you know a bell

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sounds at time zero.
 

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What comes out from the filter?
 

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Well, y_0 will be what?
 

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If the input has just a single
x, and it's one, then at that

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time zero, so x_0 is one.
 

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Then y_0 will be?
 

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1/2, and what will be y_1?
 

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Also 1/2, right?
 

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Because of y_1 will take
x_1, that's already

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dropped back to zero.
 

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Plus x_0, that's the bell.
 

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It'll be 1/2.
 

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In other words, the
output is this.

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No big deal.
 

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The impulse responses
is exactly h.

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So you can say they've
created a long word for

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a small idea, true.
 

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And the idea, the word finite
is the important number.

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00:14:00 --> 00:14:04
Finite, meaning that it's
finite length, I only have a

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finite number of h's, and
in this case two h's.

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So that's - part of
every subject is just

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learning the language.
 

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So LTI, it means you've
got a convolution.

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FIR means that the convolution
has finite length.

242
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OK, now for the question.
 

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What is this filter
doing to the signal?

244
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It's certainly averaging.
 

245
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That's clear.
 

246
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But we want to be more precise.
 

247
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Well, let me take
some examples.

248
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Suppose the input is all ones.
 

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All x_k are one.
 

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So a constant input.
 

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Natural thing to test.
 

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That's the constant input,
that's zero frequency.

253
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Zero frequency.
 

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What's the output?
 

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00:15:12 --> 00:15:15
From all ones going in wow,
sorry to ask you such

256
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a trivial question.
 

257
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You came in for some good
math here, and I'm just

258
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taking 1/2 and 1/2.
 

259
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So the output is all
y's equal, right?

260
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So, to me, that's telling, just
to introduce an appropriate

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word, low frequencies; in fact,
bottom frequencies, zero

262
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frequency, is passed
straight through.

263
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That's a low pass filter.
 

264
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That's telling me I have
a low pass filter.

265
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So that's an expression.
 

266
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That's so simple that you might
as well know those words.

267
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Low pass means that the lowest
frequencies pass through

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virtually unchanged.
 

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In this case, the very zero
frequency the DC term, the

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constant term, passes through
completely unchanged.

271
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Now, what about another input
all - well, now I want

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00:16:17 --> 00:16:18
high frequencies.
 

273
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Top frequencies.
 

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What's the the highest
oscillation I can get would

275
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be x equal, say, it starts
one minus one, one,

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00:16:34 --> 00:16:37
minus one, so on.
 

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Both directions.
 

278
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Oscillating as
fast as possible.

279
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I couldn't get a faster
frequency of oscillation

280
00:16:44 --> 00:16:48
in a discrete signal
than up, down, up, down.

281
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What's the output for that?
 

282
00:16:53 --> 00:16:58
So that's really oscillation.
 

283
00:16:58 --> 00:17:01
That's the fastest oscillation.
 

284
00:17:01 --> 00:17:03
What would be the output
from my averaging

285
00:17:03 --> 00:17:06
filter for this input?
 

286
00:17:06 --> 00:17:08
Zero.
 

287
00:17:08 --> 00:17:12
At every step, I'm averaging
this with the guy before

288
00:17:12 --> 00:17:13
and they add to zero.
 

289
00:17:13 --> 00:17:16
I'm averaging this with the guy
before, this with the guy -

290
00:17:16 --> 00:17:20
output is, y equals all zeroes.
 

291
00:17:20 --> 00:17:26
OK, so that confirms in my mind
that I have a low pass filter.

292
00:17:26 --> 00:17:30
The high frequencies
are getting wiped out.

293
00:17:30 --> 00:17:34
OK, so that's two examples.
 

294
00:17:34 --> 00:17:36
Now, what about
frequencies in between?

295
00:17:36 --> 00:17:41
Because ultimately we want
to see what's happening to

296
00:17:41 --> 00:17:43
frequencies in between.
 

297
00:17:43 --> 00:17:45
OK, so what's an
in-between frequency?

298
00:17:45 --> 00:17:54
So in between x_k could
be e^(ikn), let's

299
00:17:54 --> 00:17:55
say. e^(ik*omega).
 

300
00:17:57 --> 00:17:58
e^(ik*omega).
 

301
00:18:00 --> 00:18:09
Where this omega is somewhere
between minus pi and pi.

302
00:18:09 --> 00:18:13
OK, why do I say
minus pi and pi?

303
00:18:13 --> 00:18:16
If the frequency - so
that's the frequency.

304
00:18:16 --> 00:18:19
If omega is zero,
what's my signal?

305
00:18:19 --> 00:18:22
All ones, right?
 

306
00:18:22 --> 00:18:25
If omega is zero,
everything is my all ones.

307
00:18:25 --> 00:18:26
This is this k.
 

308
00:18:26 --> 00:18:29
So I now have a letter
for it, omega=0.

309
00:18:29 --> 00:18:32
 
 

310
00:18:32 --> 00:18:34
What's this top frequency?
 

311
00:18:34 --> 00:18:38
One, minus one, one, minus
one, what omega will give

312
00:18:38 --> 00:18:41
me alternating signs?
 

313
00:18:41 --> 00:18:43
Omega equal?
 

314
00:18:43 --> 00:18:44
Pi, right?
 

315
00:18:44 --> 00:18:44
Omega=pi.
 

316
00:18:45 --> 00:18:50
Because if this is pi, I have
e^(i*pi), which is minus one.

317
00:18:50 --> 00:19:01
So when omega=pi, my inputs are
e^(i*omega), to the k'th power.

318
00:19:01 --> 00:19:06
But this is minus one.
e^(i*pi), to the k'th power,

319
00:19:06 --> 00:19:09
and that's minus one.
 

320
00:19:09 --> 00:19:12
So that's the top frequency.
 

321
00:19:12 --> 00:19:15
And also the bottom
frequency is pi.

322
00:19:15 --> 00:19:20
And the zero frequency
is the all one.

323
00:19:20 --> 00:19:24
And this is what happens - ah.
 

324
00:19:24 --> 00:19:29
Now, comes the point.
 

325
00:19:29 --> 00:19:32
What's the output if
this is the input?

326
00:19:32 --> 00:19:34
What's the output when
this is the input?

327
00:19:34 --> 00:19:36
We can easily figure that out.
 

328
00:19:36 --> 00:19:38
We can take that average.
 

329
00:19:38 --> 00:19:44
OK, so let me do that input.
 

330
00:19:44 --> 00:19:49
Input x_k is e^(ik*omega).
 

331
00:19:49 --> 00:19:52
 
 

332
00:19:52 --> 00:20:01
And now what's the output?
y_k is the average of that.

333
00:20:01 --> 00:20:05
And the one before.
 

334
00:20:05 --> 00:20:09
Divided by two, right?
 

335
00:20:09 --> 00:20:14
OK, now you're certainly going
to factor out, anybody who

336
00:20:14 --> 00:20:18
sees this is going to factor
out e^(ik*omega), right?

337
00:20:18 --> 00:20:22
I mean that's sitting there,
that's the whole point of these

338
00:20:22 --> 00:20:25
exponentials is they factor
out of all linear stuff.

339
00:20:25 --> 00:20:33
So if I factor that out, I get
a very very important thing.

340
00:20:33 --> 00:20:37
I get, well, it's over
two, I get a one.

341
00:20:37 --> 00:20:43
And I get, what's this term?
e^(ik*omega) is here.

342
00:20:43 --> 00:20:44
So I only want e^(-i*omega).
 

343
00:20:44 --> 00:20:48
 
 

344
00:20:48 --> 00:20:54
OK, that is called the
frequency response.

345
00:20:54 --> 00:20:58
So that's telling me the
response of what the filter

346
00:20:58 --> 00:21:01
does to frequency omega.
 

347
00:21:01 --> 00:21:04
It multiplies the signal.
 

348
00:21:04 --> 00:21:09
If I have a signal that's
purely with frequency omega,

349
00:21:09 --> 00:21:11
that signal is getting
multiplied by that

350
00:21:11 --> 00:21:14
response factor.
 

351
00:21:14 --> 00:21:14
1+e^(i*omega).
 

352
00:21:15 --> 00:21:21
When omega is zero,
what is this quantity?

353
00:21:21 --> 00:21:24
So let me call this
cap h(omega).

354
00:21:24 --> 00:21:27
 
 

355
00:21:27 --> 00:21:27
what.
 

356
00:21:27 --> 00:21:31
Is this factor, if
omega is zero?

357
00:21:31 --> 00:21:35
Then h(omega)=0 is?
 

358
00:21:35 --> 00:21:37
One.
 

359
00:21:37 --> 00:21:42
That's telling me again that
at zero frequency the output

360
00:21:42 --> 00:21:44
is the same as the input.
 

361
00:21:44 --> 00:21:45
Multiplied by one.
 

362
00:21:45 --> 00:21:52
And at omega equal to pi, what
is this frequency response?

363
00:21:52 --> 00:21:53
Zero, right.
 

364
00:21:53 --> 00:21:58
At omega=pi, this is
minus one so I get zero.

365
00:21:58 --> 00:22:02
And it's telling me again
that this is the response.

366
00:22:02 --> 00:22:08
And now it's also telling me
what the response factor is for

367
00:22:08 --> 00:22:10
the frequencies in between.
 

368
00:22:10 --> 00:22:15
And everybody would draw a
graph of the darn thing, right?

369
00:22:15 --> 00:22:19
So this was simple, let me
do its graph over here.

370
00:22:19 --> 00:22:21
So I'm going to graph h(omega).
 

371
00:22:23 --> 00:22:27
Well, I've a little problem.
h(omega)'s a complex number.

372
00:22:27 --> 00:22:31
I'll graph the
magnitude response.

373
00:22:31 --> 00:22:36
So here I'm going to do a
graph from minus pi to pi.

374
00:22:36 --> 00:22:38
This is the picture.
 

375
00:22:38 --> 00:22:41
This is the picture
people look at.

376
00:22:41 --> 00:22:46
This is the picture of
what the filter is doing.

377
00:22:46 --> 00:22:50
All the information about
the filter is in here.

378
00:22:50 --> 00:22:53
All the information
is in there.

379
00:22:53 --> 00:22:56
So if I graph that, I know
what the filter's doing.

380
00:22:56 --> 00:23:01
So you said at omega=0,
I get a value of one.

381
00:23:01 --> 00:23:03
At omega=pi, I get
a value of zero.

382
00:23:03 --> 00:23:06
At omega equal minus pi,
I get a value of zero.

383
00:23:06 --> 00:23:10
And I think if you figure
out the magnitude,

384
00:23:10 --> 00:23:12
it's just a cosine.
 

385
00:23:12 --> 00:23:15
It's just an arc of a cosine.
 

386
00:23:15 --> 00:23:19
OK, for that really,
really simple filter.

387
00:23:19 --> 00:23:23
So any engineer, any signal
processing person, looks at

388
00:23:23 --> 00:23:31
this graph of h(omega) and says
that is a very fuzzy filter.

389
00:23:31 --> 00:23:37
A good, an ideal filter, an
ideal low pass filter, would

390
00:23:37 --> 00:23:38
do something like this.
 

391
00:23:38 --> 00:23:44
An ideal filter would
stay at one up to some

392
00:23:44 --> 00:23:47
frequency - say, pi/2.
 

393
00:23:48 --> 00:23:50
And drop instantly to zero.
 

394
00:23:50 --> 00:23:52
There is a really good filter.
 

395
00:23:52 --> 00:23:55
I mean, people would pay
money for that filter.

396
00:23:55 --> 00:23:58
Because what happens when
you send a signal through

397
00:23:58 --> 00:24:00
that ideal filter?
 

398
00:24:00 --> 00:24:03
It completely wipes out
the top frequencies.

399
00:24:03 --> 00:24:06
Let's say, up after pi/2.
 

400
00:24:06 --> 00:24:09
And it completely saves
the in-between ones.

401
00:24:09 --> 00:24:11
So that's really
a sharp filter.

402
00:24:11 --> 00:24:16
Actually, what people would
like to do would be to have

403
00:24:16 --> 00:24:18
that filter available.
 

404
00:24:18 --> 00:24:22
And then also to have a perfect
ideal high pass filter.

405
00:24:22 --> 00:24:26
What would be an ideal
high pass filter?

406
00:24:26 --> 00:24:29
Yeah, let's talk about high
pass filters just a moment.

407
00:24:29 --> 00:24:34
Because this is, you're seeing
the reality of what people do,

408
00:24:34 --> 00:24:39
and how they - and that little
easy bit of math they do.

409
00:24:39 --> 00:24:41
Do you want to suggest a
high pass filter, let

410
00:24:41 --> 00:24:43
me come back to this?
 

411
00:24:43 --> 00:24:46
And just change it a little?
 

412
00:24:46 --> 00:24:53
So I plan to do not - I'm now
going to do a different filter.

413
00:24:53 --> 00:24:56
That's going to be a
high pass filter.

414
00:24:56 --> 00:24:58
And what do I mean by that?
 

415
00:24:58 --> 00:25:04
A high pass filter
will kill the x_k=1.

416
00:25:05 --> 00:25:07
I now want the output from
- this is now going to

417
00:25:07 --> 00:25:09
be, I'm going to change.
 

418
00:25:09 --> 00:25:11
Can I just erase, change
a lot of things?

419
00:25:11 --> 00:25:17
I'm now going to produce
a high pass filter.

420
00:25:17 --> 00:25:19
Sorry, pi.
 

421
00:25:19 --> 00:25:21
And what's the difference?
 

422
00:25:21 --> 00:25:26
When all x's are one, the
output is going to be?

423
00:25:26 --> 00:25:27
Zero.
 

424
00:25:27 --> 00:25:31
And when I have the highest
frequency, the output

425
00:25:31 --> 00:25:32
is going to be?
 

426
00:25:32 --> 00:25:37
The input.
 

427
00:25:37 --> 00:25:40
And what am I - and then
in between, I'll do

428
00:25:40 --> 00:25:42
something in between.
 

429
00:25:42 --> 00:25:47
OK, what do you think would
be a high pass filter, like

430
00:25:47 --> 00:25:52
the simplest high pass
filter we can think of?

431
00:25:52 --> 00:25:53
Anybody think of it?
 

432
00:25:53 --> 00:25:58
You're only getting, like, 15
seconds to think in this class.

433
00:25:58 --> 00:26:01
That's a small
drawback, 15 seconds.

434
00:26:01 --> 00:26:06
But, the high pass filter
that I think of first

435
00:26:06 --> 00:26:12
is, take the difference.
 

436
00:26:12 --> 00:26:13
Take the difference.
 

437
00:26:13 --> 00:26:18
Put minus 1/2s on
the sub-diagonal.

438
00:26:18 --> 00:26:22
This is the same, this is
also a convolution, but now

439
00:26:22 --> 00:26:24
what h_0 is still a half.
 

440
00:26:24 --> 00:26:27
But now h_1 is?
 

441
00:26:27 --> 00:26:29
Minus 1/2.
 

442
00:26:29 --> 00:26:31
We're still convolving.
 

443
00:26:31 --> 00:26:34
We're still convolving with
- it's still linear time

444
00:26:34 --> 00:26:37
invariant, that just means
it's a convolution.

445
00:26:37 --> 00:26:39
It's still a finite
impulse response.

446
00:26:39 --> 00:26:46
But the response, the impulse
response is now 1/2 minus 1/2.

447
00:26:46 --> 00:26:50
So what happens if I, in my
picture over here, if I send

448
00:26:50 --> 00:26:55
in any pure frequency, I'm
now doing minus 1/2 here.

449
00:26:55 --> 00:26:57
So I'll just keep the plus.
 

450
00:26:57 --> 00:27:02
But I'll also add in the minus.
 

451
00:27:02 --> 00:27:04
So now I'm looking at
1-e^(-1*omega/2).

452
00:27:08 --> 00:27:11
And again, let's plot a
few points for that guy.

453
00:27:11 --> 00:27:15
So what, at x at omega
- so this is omega

454
00:27:15 --> 00:27:16
in this direction.
 

455
00:27:16 --> 00:27:19
And this is h in
this direction.

456
00:27:19 --> 00:27:24
So at omega=0, what's
my high pass guy?

457
00:27:24 --> 00:27:29
When I send in a zero
frequency, constant,

458
00:27:29 --> 00:27:32
I get what output?
 

459
00:27:32 --> 00:27:37
Zeroes, because now - I I'll
call it a differencing filter.

460
00:27:37 --> 00:27:44
So I'll just, instead of
averaging I'm differencing.

461
00:27:44 --> 00:27:50
OK, so now for this one, maybe
I'll put an x to indicate I'm

462
00:27:50 --> 00:27:53
now doing, I'll do x's
for the high pass.

463
00:27:53 --> 00:27:57
So this now, the high pass guy,
kills the low frequency and

464
00:27:57 --> 00:28:00
preserves the high frequency.
 

465
00:28:00 --> 00:28:05
And you won't be surprised to
find it's some cosine or

466
00:28:05 --> 00:28:12
something that, well yeah,
it's got so sorry that's

467
00:28:12 --> 00:28:15
not much of a cosine.
 

468
00:28:15 --> 00:28:26
It's the mirror image
of the low pass guy.

469
00:28:26 --> 00:28:29
And maybe the sum of
squares adds to one

470
00:28:29 --> 00:28:30
or two or something.
 

471
00:28:30 --> 00:28:32
One, probably.
 

472
00:28:32 --> 00:28:34
The sum of squares
probably adds to one.

473
00:28:34 --> 00:28:39
And they're kind of
complementary filters.

474
00:28:39 --> 00:28:41
But they're very poor.
 

475
00:28:41 --> 00:28:47
Very crude, I mean that's so
far from the ideal filter.

476
00:28:47 --> 00:28:52
So how would we create a
closer to ideal filter?

477
00:28:52 --> 00:28:57
Well, we need more h's.
 

478
00:28:57 --> 00:29:00
With two h's, we're doing
the best we can, with

479
00:29:00 --> 00:29:01
just h_0 and h_1.
 

480
00:29:02 --> 00:29:05
With a longer filter, for which
we're going to have to pay a

481
00:29:05 --> 00:29:08
little more to use, but
we'll get a lot more.

482
00:29:08 --> 00:29:12
We'll get something, we could
get a filter that stays

483
00:29:12 --> 00:29:14
pretty close to this,
drops pretty fast.

484
00:29:14 --> 00:29:15
There's a whole world.
 

485
00:29:15 --> 00:29:22
Bell Labs had a little
team of filter experts.

486
00:29:22 --> 00:29:27
Creating, and now MATLAB will
create it for you, The

487
00:29:27 --> 00:29:30
coefficients, h, that would
give you a response, a

488
00:29:30 --> 00:29:34
frequency response, it'll stay
up toward, up close to

489
00:29:34 --> 00:29:36
one as long as possible.
 

490
00:29:36 --> 00:29:41
And drop as fast as possible,
and bounce around there.

491
00:29:41 --> 00:29:45
So next week, if I come back
to that topic, I can say a

492
00:29:45 --> 00:29:50
little more about these
really good filters.

493
00:29:50 --> 00:29:53
What was I trying to do today?
 

494
00:29:53 --> 00:29:59
Trying to see how
convolution is used.

495
00:29:59 --> 00:30:02
And this is a use you
will really make.

496
00:30:02 --> 00:30:04
So now I just have, I think,
about two more things to

497
00:30:04 --> 00:30:06
say about this example.
 

498
00:30:06 --> 00:30:08
Let's see, what are they?
 

499
00:30:08 --> 00:30:11
Well, first, so all
the information is

500
00:30:11 --> 00:30:12
in this h(omega).
 

501
00:30:12 --> 00:30:15
 
 

502
00:30:15 --> 00:30:18
Oh, yeah.
 

503
00:30:18 --> 00:30:24
This simple example gives us a
way to visualize convolution.

504
00:30:24 --> 00:30:26
And I think we need that.
 

505
00:30:26 --> 00:30:27
Right?
 

506
00:30:27 --> 00:30:30
Because up to now, convolution
has been a formula.

507
00:30:30 --> 00:30:31
Right?
 

508
00:30:31 --> 00:30:34
It's been this formula.
 

509
00:30:34 --> 00:30:36
That's the formula for
convolution, and how

510
00:30:36 --> 00:30:39
do I visualize that?
 

511
00:30:39 --> 00:30:43
Just think of, may I
try to visualize that?

512
00:30:43 --> 00:30:50
Here I have, this
is the time line.

513
00:30:50 --> 00:30:54
The different k's. k equals
zero, one, two, minus one.

514
00:30:54 --> 00:30:57
 
 

515
00:30:57 --> 00:30:58
And I have x_-1,
x_0, x_1, x_2, x_3.

516
00:30:58 --> 00:31:03
 
 

517
00:31:03 --> 00:31:08
So that would be a little
bouncy up and down.

518
00:31:08 --> 00:31:11
And the averaging filter,
let me go back to

519
00:31:11 --> 00:31:12
the averaging one.
 

520
00:31:12 --> 00:31:17
The averaging filter would
smooth out the bumps.

521
00:31:17 --> 00:31:23
Because it would take the,
like, average neighbors.

522
00:31:23 --> 00:31:26
And that's a smoothing process.
 

523
00:31:26 --> 00:31:32
As we see here, it's a process
that kills high frequency.

524
00:31:32 --> 00:31:37
Now, what is this visualization
I want you to think of?

525
00:31:37 --> 00:31:41
I want you to just think
of, like, a moving window.

526
00:31:41 --> 00:31:44
So here is the input.
 

527
00:31:44 --> 00:31:47
Now, I move a window along.
 

528
00:31:47 --> 00:31:51
And that window, so let's
say here's the window,

529
00:31:51 --> 00:31:54
I should have another.
 

530
00:31:54 --> 00:31:55
So that's the window.
 

531
00:31:55 --> 00:32:00
When the window is there, it
takes the average of those two.

532
00:32:00 --> 00:32:02
That gives me the new output.
 

533
00:32:02 --> 00:32:06
Now, think of the window as
moving along here, taking

534
00:32:06 --> 00:32:08
the average of these.
 

535
00:32:08 --> 00:32:10
Move the window along, take
the average of these.

536
00:32:10 --> 00:32:12
Move the window along.
 

537
00:32:12 --> 00:32:18
Do you see the sort of, this is
what a convolution is doing.

538
00:32:18 --> 00:32:24
This is a picture of my
formula, sum of h_k*x_(l-k).

539
00:32:24 --> 00:32:28
 
 

540
00:32:28 --> 00:32:33
So the window is the h's, is
the width of the h's, and as

541
00:32:33 --> 00:32:35
that window moves along.
 

542
00:32:35 --> 00:32:41
I mean, you could write, you
could create, design a little

543
00:32:41 --> 00:32:44
circuit that would
do exactly this.

544
00:32:44 --> 00:32:46
That would do the convolution.
 

545
00:32:46 --> 00:32:51
You just have to put together
some multipliers, because

546
00:32:51 --> 00:32:54
you have these h's,
these, like, halves.

547
00:32:54 --> 00:32:59
And you have to put in an
adder, that'll add the pieces.

548
00:32:59 --> 00:33:07
And those are the essential
little electronic pieces

549
00:33:07 --> 00:33:12
of an actual filter.
 

550
00:33:12 --> 00:33:13
Then you just move it along.
 

551
00:33:13 --> 00:33:16
So it needs a delay.
 

552
00:33:16 --> 00:33:18
That's about the
content of a filter.

553
00:33:18 --> 00:33:26
Is multipliers that will
multiply by the h's, so in come

554
00:33:26 --> 00:33:35
the x, multiply by the h's, do
the addition, and do a shift

555
00:33:35 --> 00:33:37
to get onto the next one.
 

556
00:33:37 --> 00:33:39
You see how a filter works?
 

557
00:33:39 --> 00:33:43
I think that image, or
convoluted is a little

558
00:33:43 --> 00:33:44
bit vague, maybe?
 

559
00:33:44 --> 00:33:46
This window moving along?
 

560
00:33:46 --> 00:33:51
But it's quite meaningful.
 

561
00:33:51 --> 00:33:56
And then the final thing I'll
say about filters is this.

562
00:33:56 --> 00:34:02
That, what's the connection
between h(omega) and h(k)?

563
00:34:03 --> 00:34:05
Or h_k, let me call it h_k?
 

564
00:34:06 --> 00:34:11
What's the connection
between the numbers

565
00:34:11 --> 00:34:14
in impulse response?
 

566
00:34:14 --> 00:34:19
Just, which were the h's,
and the function, which is

567
00:34:19 --> 00:34:21
the frequency response?
 

568
00:34:21 --> 00:34:25
Which tells me what happens
to a particular frequency?

569
00:34:25 --> 00:34:30
Each frequency, e to the - you
notice how the frequency that

570
00:34:30 --> 00:34:34
went in is the frequency
that comes out.

571
00:34:34 --> 00:34:39
It's just amplified or
diminished by this

572
00:34:39 --> 00:34:41
h(omega) factor.
 

573
00:34:41 --> 00:34:46
So you see the h's are the
coefficients here of h(omega).

574
00:34:47 --> 00:34:51
In other words, h(omega)
is the sum of the

575
00:34:51 --> 00:34:53
h_k's, e to the, e^-ik.
 

576
00:34:53 --> 00:34:56
 
 

577
00:34:56 --> 00:35:00
Here's the beautiful formula.
 

578
00:35:00 --> 00:35:05
That's obvious, right?
 

579
00:35:05 --> 00:35:08
Here you're seeing the formula
in the simplest case, with

580
00:35:08 --> 00:35:10
just an h_0 and an h_1.
 

581
00:35:11 --> 00:35:15
But of course, it would have
worked if I had several h's.

582
00:35:15 --> 00:35:19
So this h(omega), this
factor that comes out,

583
00:35:19 --> 00:35:22
is just this guy.
 

584
00:35:22 --> 00:35:30
Now, if I look at that,
what am I saying?

585
00:35:30 --> 00:35:37
I've seen things that connect a
function of omega with a number

586
00:35:37 --> 00:35:41
of filter coefficients.
 

587
00:35:41 --> 00:35:45
I saw that in Section
4.1, in Fourier series.

588
00:35:45 --> 00:35:50
This is the Fourier series
for that function.

589
00:35:50 --> 00:35:52
Right?
 

590
00:35:52 --> 00:35:55
You might say, OK,
why that minus?

591
00:35:55 --> 00:35:57
I say, it's there
because the electrical

592
00:35:57 --> 00:35:59
engineers put it there.
 

593
00:35:59 --> 00:36:00
They liked it.
 

594
00:36:00 --> 00:36:04
And the rest of the world
has to live with it.

595
00:36:04 --> 00:36:08
So, you notice I
don't concede on i.

596
00:36:08 --> 00:36:11
I refuse to write j.
 

597
00:36:11 --> 00:36:14
But they all would.
 

598
00:36:14 --> 00:36:16
I speak about they,
but probably some

599
00:36:16 --> 00:36:17
you would write j.
 

600
00:36:17 --> 00:36:22
So I'm hoping it's OK if I
write i. i is for imaginary.

601
00:36:22 --> 00:36:25
I don't see how you could
say the word imaginary

602
00:36:25 --> 00:36:29
starting with a j.
 

603
00:36:29 --> 00:36:33
And what was the matter
with i, anyway?

604
00:36:33 --> 00:36:33
Current.
 

605
00:36:33 --> 00:36:35
Well, current used to be i.
 

606
00:36:35 --> 00:36:39
I mean who, is it still?
 

607
00:36:39 --> 00:36:41
Well, let's just accept it.
 

608
00:36:41 --> 00:36:44
OK, they can call the current
i, and the square root of minus

609
00:36:44 --> 00:36:48
one j, but not in 18.085.
 

610
00:36:48 --> 00:36:52
So OK, here we are.
 

611
00:36:52 --> 00:36:57
So my point is just that
we have a Fourier series.

612
00:36:57 --> 00:37:01
Here we a 2pi
periodic function.

613
00:37:01 --> 00:37:04
Here we have its
Fourier coefficients.

614
00:37:04 --> 00:37:10
The only difference is that we
started with the coefficients.

615
00:37:10 --> 00:37:12
And created the function.
 

616
00:37:12 --> 00:37:19
But otherwise, we're back to
Section 4.1, Fourier series.

617
00:37:19 --> 00:37:22
But that fact that we started
with the coefficients and built

618
00:37:22 --> 00:37:28
the function, sometimes you
could say, OK that sounds a

619
00:37:28 --> 00:37:30
little different from the
regular Fourier series,

620
00:37:30 --> 00:37:32
where you go the other way.
 

621
00:37:32 --> 00:37:36
So people give it the
name discrete time

622
00:37:36 --> 00:37:37
Fourier transform.
 

623
00:37:37 --> 00:37:40
You might see those
letters sometime.

624
00:37:40 --> 00:37:43
The discrete time Fourier
transform goes from the

625
00:37:43 --> 00:37:47
coefficients to the function.
 

626
00:37:47 --> 00:37:50
Where the standard Fourier
series starts with a function,

627
00:37:50 --> 00:37:51
goes to the coefficients.
 

628
00:37:51 --> 00:37:54
But really, it doesn't matter.
 

629
00:37:54 --> 00:37:56
The point is, yeah.
 

630
00:37:56 --> 00:38:01
So you could say, maybe we
have now a force transform.

631
00:38:01 --> 00:38:03
Like the first transform
was Fourier series.

632
00:38:03 --> 00:38:05
The second one was
the discrete.

633
00:38:05 --> 00:38:07
The third one is the
Fourier integral that's

634
00:38:07 --> 00:38:09
coming in one minute.
 

635
00:38:09 --> 00:38:12
And the fourth is this one.
 

636
00:38:12 --> 00:38:20
But hey, it's just the
coefficients and the function

637
00:38:20 --> 00:38:23
have switched places.
 

638
00:38:23 --> 00:38:30
In the, which one is the start
and which one is at the end.

639
00:38:30 --> 00:38:32
OK, let me pause a minute
because that's everything

640
00:38:32 --> 00:38:36
I wanted to say about
simple filters.

641
00:38:36 --> 00:38:41
And you can see that this
is a very simple filter,

642
00:38:41 --> 00:38:44
and could be improved.
 

643
00:38:44 --> 00:38:47
Better numbers would
give, I mean what would

644
00:38:47 --> 00:38:48
be better numbers?
 

645
00:38:48 --> 00:38:54
I suppose that 1/4, 1/2, 1/4
would probably be better.

646
00:38:54 --> 00:38:58
If I took those numbers,
I'm pretty sure that this

647
00:38:58 --> 00:39:06
thing would be closer to
ideal by quite a bit.

648
00:39:06 --> 00:39:11
If I plotted - so what do I
mean by, those are the h's.

649
00:39:11 --> 00:39:14
So I would take 1/4
plus 1/2 e^(-i*omega).

650
00:39:15 --> 00:39:18
Plus 1/4 e^(-2i*omega).
 

651
00:39:19 --> 00:39:22
This would be my
better h(omega).

652
00:39:23 --> 00:39:27
This would be my frequency
response to a better averaging

653
00:39:27 --> 00:39:32
filter, sort of this is like
averaged averaged, right?

654
00:39:32 --> 00:39:35
If I do a half an average and
then I do the average again.

655
00:39:35 --> 00:39:41
In other words, if I just send
these signals y_k through that

656
00:39:41 --> 00:39:45
same averaging filter, so
average again to get a z_k.

657
00:39:47 --> 00:39:51
I think probably the
coefficients would be 1/4, 1/2,

658
00:39:51 --> 00:39:54
1/4, and it would be, I've
taken out more noise.

659
00:39:54 --> 00:39:56
Right?
 

660
00:39:56 --> 00:40:00
Each time I did that averaging,
I damp the high frequencies, so

661
00:40:00 --> 00:40:03
if I do it twice I
get more damping.

662
00:40:03 --> 00:40:07
But I lose signal, of course.
 

663
00:40:07 --> 00:40:11
I mean, presumably there's some
information in the signal

664
00:40:11 --> 00:40:12
in these frequencies.
 

665
00:40:12 --> 00:40:15
And I'm reducing it.
 

666
00:40:15 --> 00:40:18
And if I average twice
I'm reducing it further.

667
00:40:18 --> 00:40:23
So a better one would be
to get a sharp cutoff.

668
00:40:23 --> 00:40:27
OK, that's filters.
 

669
00:40:27 --> 00:40:33
I guess what I hope is that we
have the idea of a convolution,

670
00:40:33 --> 00:40:36
and now we see what
we can use it for.

671
00:40:36 --> 00:40:37
Right?
 

672
00:40:37 --> 00:40:39
And there are many others.
 

673
00:40:39 --> 00:40:40
So we'll have another.
 

674
00:40:40 --> 00:40:44
We'll come back to convolutions
and de-convolution.

675
00:40:44 --> 00:40:49
Because if you have a CT
scanner, that doing a

676
00:40:49 --> 00:40:51
little convolution.
 

677
00:40:51 --> 00:40:54
I mean, you're the input,
right, to the CT scanner?

678
00:40:54 --> 00:40:57
You march in, hoping
for the best.

679
00:40:57 --> 00:41:02
OK, CT scanner convolves you
with their little filter.

680
00:41:02 --> 00:41:10
And then it does a
deconvolution, to an

681
00:41:10 --> 00:41:15
approximate deconvolution, to
have a better image of you.

682
00:41:15 --> 00:41:19
OK, let's leave that.
 

683
00:41:19 --> 00:41:23
Can I change direction and just
write down the formulas for the

684
00:41:23 --> 00:41:27
Fourier integral transform?
 

685
00:41:27 --> 00:41:29
And do one example?
 

686
00:41:29 --> 00:41:32
OK.
 

687
00:41:32 --> 00:41:38
I don't know what you think
about a lecture that stops

688
00:41:38 --> 00:41:41
and starts a new topic.
 

689
00:41:41 --> 00:41:46
Is it - maybe it's
tough on the listener?

690
00:41:46 --> 00:41:48
Or maybe it's a break.
 

691
00:41:48 --> 00:41:49
I don't know.
 

692
00:41:49 --> 00:41:51
Let's look at it positively.
 

693
00:41:51 --> 00:41:53
Alright, break.
 

694
00:41:53 --> 00:41:58
Alright.
 

695
00:41:58 --> 00:42:02
So let me remember the
Fourier series formulas.

696
00:42:02 --> 00:42:05
So I'm just going to break,
and now we go to the

697
00:42:05 --> 00:42:09
integral transform.
 

698
00:42:09 --> 00:42:13
OK, so let me remember the
formula for the coefficients,

699
00:42:13 --> 00:42:13
which was 1/2pi.
 

700
00:42:15 --> 00:42:22
The integral of
f(x)e^(-ikx)dx, right?

701
00:42:22 --> 00:42:26
And then when we added it
up to get f(x) back again,

702
00:42:26 --> 00:42:31
we added up a sum of
the c_k's e^(ikx)'s.

703
00:42:31 --> 00:42:34
 
 

704
00:42:34 --> 00:42:37
That's 4.1.
 

705
00:42:37 --> 00:42:39
We know those formulas.
 

706
00:42:39 --> 00:42:40
And we notice again.
 

707
00:42:40 --> 00:42:42
Complex conjugate.
 

708
00:42:42 --> 00:42:44
One direction is the
conjugate compared to

709
00:42:44 --> 00:42:45
the other direction.
 

710
00:42:45 --> 00:42:51
Now, all I plan to do is
write down the formula.

711
00:42:51 --> 00:42:55
And remember, I'm going to
use f hat of k, instead

712
00:42:55 --> 00:42:57
of the coefficients.
 

713
00:42:57 --> 00:43:00
Because it's a function of
k, all k's and not just

714
00:43:00 --> 00:43:02
integers are allowed.
 

715
00:43:02 --> 00:43:05
And then I'm going
to recover f(x).

716
00:43:05 --> 00:43:08
OK, now this integral went
from minus pi to pi,

717
00:43:08 --> 00:43:10
because that was periodic.
 

718
00:43:10 --> 00:43:12
But now all the integrals
are going to go from minus

719
00:43:12 --> 00:43:15
infinity to infinity.
 

720
00:43:15 --> 00:43:17
We've got every k, every x.
 

721
00:43:17 --> 00:43:19
So we take, what do you
expect here? f(x)?

722
00:43:21 --> 00:43:21
e^(-ikx)?
 

723
00:43:23 --> 00:43:23
dx?
 

724
00:43:24 --> 00:43:25
Yes.
 

725
00:43:25 --> 00:43:27
Fine.
 

726
00:43:27 --> 00:43:32
Same thing, f(x) is there, but
now any k is allowed so I

727
00:43:32 --> 00:43:34
have a function of all k's.
 

728
00:43:34 --> 00:43:35
And now I want to recover f(x).
 

729
00:43:36 --> 00:43:38
So what do I do?
 

730
00:43:38 --> 00:43:39
You can guess.
 

731
00:43:39 --> 00:43:41
I've got an integral now.
 

732
00:43:41 --> 00:43:43
Not a sum.
 

733
00:43:43 --> 00:43:47
Because a sum was when I
had only integer numbers.

734
00:43:47 --> 00:43:54
Now, I've got f hat of k, and
would you like to tell me what

735
00:43:54 --> 00:44:00
the magic factor is, there
in the integral formula?

736
00:44:00 --> 00:44:02
It's just what you hope.
 

737
00:44:02 --> 00:44:04
It's the e^(ik*omega).
 

738
00:44:05 --> 00:44:08
Where d what?
 

739
00:44:08 --> 00:44:14
Now this is where it's easy
to make a mistake. d,

740
00:44:14 --> 00:44:16
I'm integrating here.
 

741
00:44:16 --> 00:44:19
I'm putting the whole,
reconstructing the function.

742
00:44:19 --> 00:44:24
I'm putting back the harmonics
with the amount f hat of k

743
00:44:24 --> 00:44:29
tells me how much e^(ik*omega)
there is in the function.

744
00:44:29 --> 00:44:32
I put them all together,
so I integrate dk.

745
00:44:34 --> 00:44:36
I'm integrating over
the frequencies.

746
00:44:36 --> 00:44:39
This was the sum over k.
 

747
00:44:39 --> 00:44:41
From minus infinity
to infinity.

748
00:44:41 --> 00:44:44
Now this is an integral,
because we've got,

749
00:44:44 --> 00:44:46
it's all filled in.
 

750
00:44:46 --> 00:44:48
And it remains to
deal with this 2pi.

751
00:44:49 --> 00:44:54
And I see in the book
that the 2pi went there.

752
00:44:54 --> 00:44:55
I don't know why.
 

753
00:44:55 --> 00:44:56
Anyway, there it is.
 

754
00:44:56 --> 00:45:00
So let's follow
that convention.

755
00:45:00 --> 00:45:01
Put the 2pi here.
 

756
00:45:01 --> 00:45:05
So there's the formula.
 

757
00:45:05 --> 00:45:07
The pair of formulas,
the twin formulas.

758
00:45:07 --> 00:45:12
The transform, from f to f hat,
and the inverse transform,

759
00:45:12 --> 00:45:14
from f hat back to f.
 

760
00:45:14 --> 00:45:22
And it's just like the one
you've seen for Fourier series.

761
00:45:22 --> 00:45:28
Well, I think the only good way
to remember those is to put in

762
00:45:28 --> 00:45:32
a function and find
its transform.

763
00:45:32 --> 00:45:38
So my final thing for today
would be take a particular

764
00:45:38 --> 00:45:39
function, f(x).
 

765
00:45:39 --> 00:45:42
 
 

766
00:45:42 --> 00:45:48
Here, let me take ever f(x) to
be, here's one. f to be zero

767
00:45:48 --> 00:45:52
here, and then a jump to one.
 

768
00:45:52 --> 00:45:54
And then an exponential decay.
 

769
00:45:54 --> 00:45:55
So e^-ax.
 

770
00:45:55 --> 00:46:00
 
 

771
00:46:00 --> 00:46:03
OK, so that's the input.
 

772
00:46:03 --> 00:46:06
It's not odd, it's not even.
 

773
00:46:06 --> 00:46:12
So I expect sort of a complex f
hat of k, which I can compute.

774
00:46:12 --> 00:46:14
So f hat of k is what?
 

775
00:46:14 --> 00:46:18
Now, let's just figure out
f hat of k and look at

776
00:46:18 --> 00:46:21
the decay rate and all
the other good stuff.

777
00:46:21 --> 00:46:23
So what do I do?
 

778
00:46:23 --> 00:46:27
I'm just doing this integral.
 

779
00:46:27 --> 00:46:28
For practice.
 

780
00:46:28 --> 00:46:31
OK, so f is zero in
the first half.

781
00:46:31 --> 00:46:34
So I really only integrate
zero to infinity.

782
00:46:34 --> 00:46:41
And in that region it's e^-ax,
and I multiply by e^(-ikx),

783
00:46:41 --> 00:46:46
and I integrate dx,
and what do I get?

784
00:46:46 --> 00:46:52
I'll get this, is an integral
we can do, and it's easy

785
00:46:52 --> 00:46:53
because this is e^-(a+ik)x.
 

786
00:46:53 --> 00:46:58
 
 

787
00:46:58 --> 00:47:02
You're always going to
see it that way, right?

788
00:47:02 --> 00:47:04
That we're integrating.
 

789
00:47:04 --> 00:47:07
And then the integral of an
exponential is the exponential

790
00:47:07 --> 00:47:14
divided by the factor that
will come down when we

791
00:47:14 --> 00:47:15
take the derivative.
 

792
00:47:15 --> 00:47:19
So I think we just
have this, right?

793
00:47:19 --> 00:47:23
Don't you think, to integrate
that exponential, we just get

794
00:47:23 --> 00:47:26
the exponential divided
by its little factor.

795
00:47:26 --> 00:47:29
And now we have to
stick in the limits.

796
00:47:29 --> 00:47:31
And what do I get
at the limits?

797
00:47:31 --> 00:47:36
This is like, a fun part of
Fourier integral formulas.

798
00:47:36 --> 00:47:41
What do I get at the upper
limit, x equal infinity?

799
00:47:41 --> 00:47:47
If x is very large, what
does this thing do?

800
00:47:47 --> 00:47:50
Goes to zero.
 

801
00:47:50 --> 00:47:52
It's gone.
 

802
00:47:52 --> 00:47:57
The e^(ikx) is oscillating
around, it's of size one.

803
00:47:57 --> 00:48:01
But the e^(-ax), so I needed
a to be positive here.

804
00:48:01 --> 00:48:03
That picture had to
be the right one.

805
00:48:03 --> 00:48:07
A positive.
 

806
00:48:07 --> 00:48:10
Then at infinity, I get zero.
 

807
00:48:10 --> 00:48:13
So now I just plug in this
lower limit, that comes

808
00:48:13 --> 00:48:14
with a minus sign.
 

809
00:48:14 --> 00:48:15
So what do I get?
 

810
00:48:15 --> 00:48:20
The minus sign will make
this a+ik, and what does

811
00:48:20 --> 00:48:21
it thing equal at x=0?
 

812
00:48:23 --> 00:48:27
One. e^0 is one.
 

813
00:48:27 --> 00:48:31
So there is the
Fourier transform.

814
00:48:31 --> 00:48:36
Of my one-sided exponential.
 

815
00:48:36 --> 00:48:39
Now, just a quick look at that
and then I'll do some more,

816
00:48:39 --> 00:48:46
this example's Example one in
Section 4.5, and we'll

817
00:48:46 --> 00:48:49
do more examples.
 

818
00:48:49 --> 00:48:53
But let's just
look at that one.

819
00:48:53 --> 00:48:58
I see a jump in the function.
 

820
00:48:58 --> 00:49:05
What do I expect in the decay
rate of the transform?

821
00:49:05 --> 00:49:10
So a jump in the function, I
expect a decay rate of 1/k.

822
00:49:12 --> 00:49:15
Decay rate, right?
 

823
00:49:15 --> 00:49:17
Just as for Fourier
coefficients, so for the

824
00:49:17 --> 00:49:19
integral transform.
 

825
00:49:19 --> 00:49:23
So a decay rate in f hat.
 

826
00:49:23 --> 00:49:25
And it's here.
 

827
00:49:25 --> 00:49:28
1/k in the denominator.
 

828
00:49:28 --> 00:49:29
Yeah.
 

829
00:49:29 --> 00:49:34
So that's a a good example.
 

830
00:49:34 --> 00:49:38
You might say, wait a minute,
OK that's fine but what

831
00:49:38 --> 00:49:40
about the second one?
 

832
00:49:40 --> 00:49:47
Could I put in 1/(a+ik)
and get back the pulse?

833
00:49:47 --> 00:49:49
The exponential pulse?
 

834
00:49:49 --> 00:49:52
The answer is yes, but
maybe I don't know how

835
00:49:52 --> 00:49:53
to do that integral.
 

836
00:49:53 --> 00:49:58
So I'm sort of fortunate that
these formulas are proved for

837
00:49:58 --> 00:50:00
any function including
this function.

838
00:50:00 --> 00:50:08
So this example shows
the decay rate.

839
00:50:08 --> 00:50:12
The possibility sometimes of
doing one integral but maybe

840
00:50:12 --> 00:50:16
the other integral going the
other direction is not so easy.

841
00:50:16 --> 00:50:17
And that's normal.
 

842
00:50:17 --> 00:50:20
So that's, like, Fourier
transforms and inverse

843
00:50:20 --> 00:50:26
transforms, we don't expect to
be able to do them all by hand.

844
00:50:26 --> 00:50:33
I mean, I'll just say that
actually, anybody who studies

845
00:50:33 --> 00:50:38
complex variables and residues,
I don't know if you know any of

846
00:50:38 --> 00:50:42
this, heard these words about,
there are ways to integrate.

847
00:50:42 --> 00:50:47
I could put 1/(a+ik) in here.
 

848
00:50:47 --> 00:50:50
And, actually, and do
this integral for minus

849
00:50:50 --> 00:50:51
infinity to infinity.
 

850
00:50:51 --> 00:50:54
By stuff that's in Chapter 5.
 

851
00:50:54 --> 00:50:59
Can I just point ahead without
any plan to discuss it.

852
00:50:59 --> 00:51:06
That some integrals can be
done by x+iy tricks, by

853
00:51:06 --> 00:51:08
using complex numbers.
 

854
00:51:08 --> 00:51:10
But I won't do more.
 

855
00:51:10 --> 00:51:11
OK, thanks.
 

856
00:51:11 --> 00:51:15
So that's the formulas.
 

857
00:51:15 --> 00:51:17
And that's one example.
 

858
00:51:17 --> 00:51:21
Wednesday there will be more
examples and then no review

859
00:51:21 --> 00:51:23
session Wednesday evening.