Transf Fourier

Kuliah Minggu - 2

1 Transformasi Fourier

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 2 What do we hope to achieve with theFourier Transform?We desire a measure of the frequencies present in a wave. the spectrumIt will be nice if our measure also tells us when each frequency occurs.

Light electric fieldTime

Plane waves have only one frequency, w.

This light wave has many frequencies. And the frequency increases in time (from red to blue).

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 3 Fourier Cosine Series

Because cos(mt) is an even function (for all m), we can write an even function, f(t), as: where the set {Fm; m = 0, 1, } is a set of coefficients that define the series. And where well only worry about the function f(t) over the interval (,).

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 4 The Kronecker delta function

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 5 Finding the coefficients, Fm, in a Fourier Cosine Series

Fourier Cosine Series: To find Fm, multiply each side by cos(mt), where m is another integer, and integrate:

But since:

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 6 Finding the coefficients, Fm, in a Fourier Cosine Series_2

So: only the m = m term contributes

Dropping the from the m: yields the coefficients for any f(t)!

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 7 Fourier Sine SeriesBecause sin(mt) is an odd function (for all m), we can write any odd function, f(t), as:where the set {Fm; m = 0, 1, } is a set of coefficients that define the series.where well only worry about the function f(t) over the interval ( ,).

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 8 Finding the coefficients, Fm, in a Fourier Sine SeriesFourier Sine Series:

To find Fm, multiply each side by sin(mt), where m is another integer, and integrate:

But since:

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 9 Finding the coefficients, Fm, in a Fourier Sine Series_2So: only the m = m term contributes

Dropping the from the m:

yields the coefficients for any f(t)!

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 10 Fourier Series

even component odd componentwhere and

So if f(t) is a general function, neither even nor odd, it can be written:

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 11 Fourier Transform

F(m) Fm i Fm = Lets now allow f(t) to range from to , so well have to integrate from to , and lets redefine m to be the frequency, which well now call :

The FourierTransform F() is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We say that f(t) lives in the time domain, and F() lives in the frequency domain. F() is just another way of looking at a function or wave.

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 12 The Inverse Fourier TransformThe Fourier Transform takes us from f(t) to F(w). How about going back?

Recall our formula for the Fourier Series of f(t) :

Now transform the sums to integrals from to , and again replace Fm with F(). Remembering the fact that we introduced a factor of i (and including a factor of 2 that just crops up), we have:

Inverse Fourier Transform

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 13 The Fourier Transform and its InverseSo we can transform to the frequency domain and back. Interestingly, these functions are very similar. There are different definitions of these transforms. The 2 can occur in several places, but the idea is generally the same.

Inverse Fourier TransformFourierTransform

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 14 Fourier Transform NotationThere are several ways to denote the Fourier transform of a function.If the function is labeled by a lower-case letter, such as f, we can write:f(t) F(w)If the function is labeled by an upper-case letter, such as E, we can write:or:

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 15 The SpectrumWe define the spectrum of a wave E(t) to be:

This is our measure of the frequencies present in a light wave.

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 16 Example: the Fourier Transform of arectangle function: rect(t)

Imaginary Component = 0

F(w)

w

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 17 Sync(x) and why it's importantSinc(x/2) is the Fourier transform of a rectangle function.

Sinc2(x/2) is the Fourier transform of a triangle function.

Sinc2(ax) is the diffraction pattern from a slit.

It just appears everywhere...

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 18 The Fourier Transform of the trianglefunction, D(t), is sinc2(w/2)

w0

1

t0

1

1/2-1/2Well prove this when we learn about convolution.Sometimes people use L(t), too, for the triangle function.

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution

TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 19 Example: the Fourier Transform of adecaying exponential: exp(-at) (t > 0)

A complex Lorentzian!

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 20 Example: the Fourier Transform of aGaussian, exp(-at2) is itself!

t0

w0

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 21 Existence of Fourier Transform Some functions dont have Fourier transforms.The condition for the existence of a given F(w) is:

Functions that do not asymptote to zero in both the + and directions generally do not have Fourier transforms.

So well assume that all functions of interest go to zero at .

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 22 Fourier Transform Symmetry Properties

Expanding the Fourier transform of a function, f(t):Expanding further:

Re{F(w)}Im{F(w)}= 0 if Re or Im{f(t)} is odd = 0 if Re or Im{f(t)} is evenEven functions of wOdd functions of w

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution 23

Thank YouQuestions ???

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TF091309 Fisika GelombangDosen: Dr.rer.nat. Aulia M T Nasution