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The Fourier
Coefficients applet lets

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us explore the approximation
of periodic functions

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by Fourier series.

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This is the opening
screen of the applet.

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On the right, you see a
series of sliders controlling

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coefficients.

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And they're the coefficients
of the sine terms in a Fourier

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series, as you can see by
clicking this formula box.

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So at this point, what we're
studying is an odd function,

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a sine series, and the sliders
let us control the coefficients

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of sine of k*t.

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If I click the cosine
box, I'd see coefficients

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of the cosine functions.

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Let's go back to sine
and start to play

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with the tool a little bit.

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What will happen if I
move the b_2 slider?

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I should get multiples
of the sine of 2t

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showing up on my
graphing window.

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If I push b_2 to the right,
I get positive multiples

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of the sine of 2t.

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The sine of 2t goes
through two full periods

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when t ranges from 0 to 2pi.

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And if b_2 is negative,
I get negative multiples

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of the sine of 2t.

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What happens if I
change b_1 in this tool?

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This will add a multiple of
the sine of t to the function

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that I already have.

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If I push this to
the right, I'm going

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to get positive multiples
of the sine of 2t,

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and you can see the
alteration in the yellow graph

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when I do that.

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When I push b_1 to the left,
I'm adding negative multiples

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of the sine of t.

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You get quite a variety
of interesting functions

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by playing with these
various sliders.

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But it's even more
fun to try to match

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a target periodic function by
means of sine or cosine series.

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And the tool has built
into it a whole family

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of target functions to try for.

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Right now, the
target function is 0.

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But we can select target A
instead, and what we have now

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is a square wave.

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Actually, this isn't
the standard square wave

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that we deal with in
the course, quite.

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It is periodic, of period
2pi, and alternates

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between positive
and negative values,

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but those values are pi
over 4 and minus pi over 4,

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instead of plus 1 and
minus 1, as they are

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for the standard square wave.

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That's chosen so that the
Fourier coefficients come

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out more neatly, as we'll see.

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So let's try to
approximate this function

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by a finite trigonometric sum.

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I'm going to push the
Reset key here to reset all

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the coefficients to zero,
and start trying to match

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the function that I have at hand
with multiples of sine of k*t.

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Let's begin with the sine
function itself, sine of t.

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Well, I don't know what to do.

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Let's just set it so
the maxima coincide.

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Why not?

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Maybe that will be
an appropriate amount

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of the sine of t function
to put into the pot

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when I'm trying to mix
up the square wave.

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Let's go on to b_2 now.

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I'm going to push the
b_2 slider to the right.

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When I do that, you
see a hump forming

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on the left-hand side
of the positive part

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of the square wave.

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If push b_2 to the left,
the same kind of hump

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forms on the right-hand
side of that positive part

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of the square wave.

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Neither one of them seems to
be doing us very much good.

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And in fact, it's creating an
asymmetry in the Fourier series

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that's not present in
our target function.

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It's not respecting
the symmetry that you

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see around pi over 2 in
the square wave itself.

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So I suspect that b_2 is not
going to be useful to us,

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that the sine of 2t
is not going to be

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a constituent in the Fourier
series for square wave.

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In fact, I think that
none of the even sines

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will be of any use to us.

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And I can get rid of them and
look only at the odd terms

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by clicking this Odd Terms box.

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This leaves the sine function
that we had in place.

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It eliminates all the
odd terms in the series.

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It eliminates all the even
coefficients and leaves us only

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with the odd ones.

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But there are more
of them, so we

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should be able to do a
better job of approximating

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the square wave.

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So let's continue.

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Let's try changing b_3 to
get a good approximation.

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Well, I could push
b_3 to the left,

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and I get a sharper
spike in the middle.

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That doesn't look too good.

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It looks better to
push b_3 to the right

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and flatten out the top.

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But when I do that, the
top becomes too low.

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And so maybe I made a
mistake in choosing b_1 so

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that the maxima agreed.

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Maybe I should have made
b_1 a little bit bigger.

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Now you can see the
white curve there--

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that's giving me the multiple
of the sine wave itself,

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the one that I'm
controlling by this slider.

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And I guess I should push it
up so that my flat part comes

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closer to the top
of the square wave.

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Maybe that's a
better approximation.

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Well, I can continue
playing this game,

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trying out various multiples
of higher sine functions,

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and going back to try to
fix up what I had before.

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In fact, what we
should be trying to do

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is create a least squares fit.

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You should try to look at
the root-mean-square distance

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between the Fourier series
and the target function,

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and minimize that distance.

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That distance can be
read out in this tool

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by clicking the Distance
button down here.

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And so the
root-mean-square distance

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between the green curve
and the yellow one

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is 0.24 and so on, in this case.

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Let me come down here
and kill the formula.

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We know the formula
well now, and it

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makes the screen a little less
cluttered if we get rid of it.

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And let me reset the values
of the coefficients to zero,

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and start to try to
approximate this square wave

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again by watching the
root-mean-square distance

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between the Fourier
series that I'm creating

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and the target function.

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So I'll start with b_1 in
this case, If I move it down,

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the distance increases.

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If I move it up, the distance
decreases more and more slowly.

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But now when b_1 becomes
greater than one,

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it starts to increase.

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So let me push it down again.

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I can use these little arrow
keys to step the value of b_1.

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That's very convenient.

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I'm just watching the
root-mean-square distance,

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this number up here.

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I'm trying to make that
as small as possible.

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It's a measure of
the closeness of fit.

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Ah, it just started
to increase again.

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Let me step up.

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And I found the value of one.

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I didn't set it to one.

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I set it to the value that
minimized this number.

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And that's the right
quantity of the sine wave

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to put in, if you want to mix
up the square wave, or rather pi

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over 4 times the square wave.

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Let's look at b_3.

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Again, if I decrease
b3, things get worse.

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The root-mean-square
distance increases.

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But if I increase b_3
above the value of zero,

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that number is getting smaller.

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Now it's growing again.

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I think I'll stop there
and use these arrow keys.

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That's better, that's better,
that's better, that's worse.

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Let me go back.

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And I find the
value 0.33 for b_3.

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That's the optimal
amount of the sine of 3t

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to put in if you want
to mix up the formula

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for this square wave.

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I'll do one more.

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Any guesses for what the
optimal value of b_5 should be?

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It's not going to be negative.

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That will increase the
root-mean-square distance.

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Let's push it up in
the positive direction.

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That's about right.

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That's worse, that's
better, that's

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better, better, better, worse.

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I come down here.

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That's optimal.

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We've discovered by
experimentation with this

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applet that the optimal
value of b_5 is 0.200.

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Each one of these
coefficients seems

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to have optimal value equal to
the reciprocal of the index.

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1/3 in this case, 1 in this
case, 1/5 in this case.

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And I predict that the
optimal value of b_7

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is going to be 1/7,
which is about 0.14.

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There you go.

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it's right in there.

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So this applet lets you
understand the true meaning

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of the Fourier coefficients.

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They're the multiples of the
sine waves or the cosine waves

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which give you the optimal fit,
in terms of a finite Fourier

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series, to the target function.