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Now we're going to talk a little bit about the meaning of those Fourier
Coefficient Integrals. Now, what you're doing is we're starting
with some original complex waveform. And we're going to take a simple example
of a square wave here just to, to illustrate the idea.
So we have the amplitude of this waveform in time.
And this is our starting. Signal that we want to analyze.
Now, what we need to do is we're going to take and do a comparison of this waveform
to all of the different so called basis functions in the Fourier analysis.
Now the first one that we're looking at is the lowest frequency sine wave and so
this will give us the b1 coefficient from that the the Fourier Integral to compute
that. Now the meaning of that integral is we
multiply those two signals together. The, the sine wave basis function, and
our, our original, signal, y of t. And then, you sum up the area, under the
curve, of that product. So after I've multiplied those two
together, I get a curve that, is going to have, two humps.
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Because over here, the original signal is positive and the basis function is
positive, and so their product is positive.
Over here, the basis function, the sine wave is negative but the signal was also
negative and so negative times negative, that gives me positive.
And so I get this double-humped, it's like two half-cycles of sine waves, both
positive, and the meaning of the integral is to just sum up the area under the
curve. And so if I add up all the area under
this curve it turns out that you get 1 for that.
Now, I want to do the same thing with the cosine wave, and so I also have the
lowest frequency cosine, which has the same period as the total duration of my
input signal. And so I'm going to multiply those two
together. And then sum up the area of the resulting
curve. So you see, it's the cosine function is
positive, and the signal is positive. And so I get a positive region here.
Then the cosine goes negative, but the signal, the red, is still positive, so
that's negative. Then they're both negative.
The cosine function is negative, and the signal is negative, so their product is
positive. And then the cosine goes positive but the
signal's still negative so their product is negative.
Now, when we add up the area, we count area above the axis as positive.
Area below the axis is negative. And so if I add this up, you can see
these two are the same as those two, so this cancels that and this cancels that.
So if I add that altogether, I get zero. So that's telling me that the cosine wave
is not like the original input signal. They're not alike, they're too dissimilar
for the recipe to call for any of this cosine wave.
And in fact, the reason for that is because of the symmetry.
if we look at the symmetry of the original signal it's symmetric, or
anti-symmetric, about this point, as positive to the left and negative to the
right. the sine wave has exactly the right
symmetry. That's positive to the left and negative
to the right of that point. The trouble with the cosine is that is
has the opposite symmetry. It looks the same on to the left as it
does to the right. So, this is the cosine basis fun-,
function is symmetric about this point; the sine basis function is antisymmetric.
The signal is antisymmetric. And so, there's going to be similarity
between the signal and the sine, but there's no similarity between the signal
and the cosine. So that's why you have to, check your
signal against both the sine and the cosine basis functions.
So let's carry on, and look at, a couple of more examples, of the next higher
frequency in the Fourier series. So here's the same original basis
function, the red. And so now I want to calculate the b2,
which is the sine component. And so I draw the sine.
And now I'm going to multiply these. So it's going to positive over here,
negative, positive, and negative. And then the, I want to then add up the
area under the curve, counting above the axis as positive and below the axis as
negative. And so you can see that this is going to
all cancel out and we'll get 0. So there's none of this second harmonic
sine wave in that square wave, and we knew that because when we built the
square wave, a little while ago, we saw that it only took on harmonics.
There were no even numbers. in the Fourier series for that.
Now let's just take a look at one more. we'll, we'll see what the Fourier
coefficient a2 is for the cosine. At the second harmonic of that
fundamental. And, so, here's the cosine curve.
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And you can already see the cosine curve has, is symmetrical above this crossing
point. The signal is anti-symmetric.
And so that's not going to give us anything.
So if I multiply everything out I get, positive.
Two positive, quarter cycles, and this negative half cycle.
So this all cancels out. And the same thing happens here.
So if I add it all up, I get zero. So, again, these Fourier Coefficient
Integrals, or these integrals to compute the Fourier coefficients, are just a way
of measuring how much alike your signal is to the different basis functions.
And those basis functions are sine and cosines.
At all of the multiples of the fundamental one, which is exactly at the
frequency equal to 1 over the duration of your original signal.
Robert ClarkProvost and Senior Vice President for Research and Professor
Mark BockoDistinguished Professor and Chair
