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Lecture 8Lecture 8

Fourier Analysis.Fourier Analysis.

Aims:Aims: Fourier Theory: Description of waveforms in terms of a

superposition of harmonic waves. Fourier series (periodic functions); Fourier transforms (aperiodic

functions). Wavepackets

Convolution convolution theorem.

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Fourier TheoryFourier Theory

It is possible to represent (almost) any function as a superposition of harmonic functions.

Periodic functions:Periodic functions: Fourier series

Non-periodic functions:Non-periodic functions: Fourier transforms

Mathematical formalismMathematical formalism Function f(x), which is periodic in x, can be

written:

where,

Expressions for An and Bn follow from the “orthogonality” of the “basis functions”, sin and cos.

1

2sin

2cos

21

n

nno lnx

Blnx

AAxf

...2,12

sin2

...2,1,02

cos2

2/

2/

2/

2/

ndxlnx

xfl

B

ndxlnx

xfl

A

l

l

n

l

l

n

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Complex notationComplex notation

Example: simple case of 3 termsExample: simple case of 3 terms

Exponential representation:Exponential representation:

with k=2n/l.

lxlxlxy 4cos2sin2cos

y

lx2sin

lx4cos

lx2cos

xexfl

C

eCxf

lnxil

ln

n

lnxin

d1 /2

2/

2/

/2

xexfl

kC

ekCxf

ikxl

l

n

ikx

d1 2/

2/

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ExampleExample

Periodic top-hat:Periodic top-hat:

N.B.

2/8/,8/2/0

8/8/

lxllxl

lxlAxf

Fourier transformFourier transform f(x)f(x)

4/sinc4

8/sinc4

8/sin2

1

8/8/

8/

8/

8/

8/

nAkl

A

klklA

eeiklA

ike

lA

dxAel

kC

iklikl

l

l

l

l

ikxikx

1

sin0

x

xxx

x

Zero when n is a multiple of 4

Zero when n is a multiple of 4

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Fourier transform variablesFourier transform variables

xx and and kk are conjugate variables. are conjugate variables. Analysis applies to a periodic function in any

variable.

tt and and are conjugate. are conjugate.

Example: Forced oscillatorExample: Forced oscillator Response to an arbitrary, periodic, forcing

function F(t). We can represent F(t) using [6.1]. If the response at frequency nf is R(nf), then

the total response is

2/

2/

/2

/2

1

]1.6[

T

T

Tntin

n

Tntin

etFT

C

eCtF

Tnn /2

n

tinnf

feCnR

Linear in both response and driving amplitudeLinear in both response and driving amplitudeLinear in both response and driving amplitudeLinear in both response and driving amplitude

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Fourier TransformsFourier Transforms

Non-periodic functions:Non-periodic functions: limiting case of periodic function as period .

The component wavenumbers get closer and merge to form a continuum. (Sum becomes an integral)

This is called Fourier Analysis. f(x) and g(k) are Fourier Transforms of each

other.

Example:Example:Top hatTop hat

Similar to Fourier series but now a continuous function of k.

dkexfkg

dkekgxf

ikx

ikx

)(21

)(

)(21

)(

2/0

2/2/)(

xx

xxxAxf

)2/sinc(2

)2/sin(22

221

)(

2/

2/

2/

2/

xkxA

xkkA

ikeA

dkAekg

x

x

ikxx

x

ikx

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Fourier transform of a GaussianFourier transform of a Gaussian

Gaussain with r.m.s. deviation Gaussain with r.m.s. deviation xx==..

Note

Fourier transform

Integration can be performed by completing the square of the exponent -(x2/22+ikx).

where,

22 2/

2)(

xe

Axf

Adxxf

)(

dxeeA

kg ikxx 22 2/

221

)(

2/;22

2dxdu

ikxu

2/

222

22222

222

2

2

ku

kikxikx

x

2/2/

2/

22222

222

22

2

22

)(

kuk

ku

eA

dueeA

dueeA

kg

==

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TransformsTransforms

The Fourier transform of a Gaussian is a Gaussian.

Note: k=1/. i.e. xk=1 Important general result: “Width” in Fourier space is inversely related to

“width” in real space. (same for top hat)

Common functions Common functions (Physicists crib-sheet) -function constant

cosine 2 -functions sine 2 -functions

infinite lattice infinite lattice of -functions of -functions

top-hat sinc functionGaussian Gaussian

In pictures………...

-function-function

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Pictorial transformsPictorial transforms

Common transformsCommon transforms

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Wave packetsWave packets

Localised wavesLocalised waves A wave localised in space can be created by

superposing harmonic waves with a narrow range of k values.

The component harmonic waves have amplitude

At time t later, the phase of component k will be kx-t, so

Provided /k=constant (independent of k) then the disturbance is unchanged i.e. f(x-vt).

We have a non-dispersive wave. When /k=f(k) the wave packet changes shape

as it propagates. We have a dispersive wave.

dkekgxf ikx)(21

)(

dxexfkg ikx)(21

)(

dkekgtxf tkxi )()(21

),(

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ConvolutionConvolution

Convolution: a central concept in Physics.Convolution: a central concept in Physics.

It is the “smearing” or “blurring” of one function by the other.

Examples occur in all experimental situations where the limited resolution of the apparatus results in a measurement “broader” than the original.

In this case, f1 (say) represents the true signal and f2 is the effect of the measurement. f2 is the point spread function.

duuxfufxfxfxh )()()(*)()( 2121

h is the convolution of f1 and f2h is the convolution of f1 and f2h is the convolution of f1 and f2h is the convolution of f1 and f2h is the convolution of f1 and f2h is the convolution of f1 and f2

Convolution symbol Convolution integralConvolution integral

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Convolution theoremConvolution theorem

Convolution and Fourier transforms

Convolution theorem:Convolution theorem: The Fourier transform of a PRODUCT of two

functions is the CONVOLUTION of their Fourier transforms.

Conversely:The Fourier transform of the CONVOLUTION of two functions is a PRODUCT of their Fourier transforms.

Proof:

duukgug

dxexfduug

dxexfdueug

dxexfxfkh

xuki

ikxiux

ikx

)()(21

)(21

)(21

)()(21

21

)()(21

)(

21

)(21

21

21

F.T.of

f1.f2

F.T.of

f1.f2

Convolutionof g1 and g2

Convolutionof g1 and g2

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Convolution………….Convolution………….

Summary:Summary:

If,

then

and

Examples:Examples: Optical instruments and resolution 1-D idealised spectrum of “lines” broadened

to give measured spectrum

2-D: Response of camera, telescope. Each point in the object is broadened in the image.

Crystallography. Far field diffraction pattern is a Fourier transform. A perfect crystal is a convolution of “the lattice” and “the basis”.

)()(2)(*)(

)(*)(21

)()(

)()(),()(

2121

2121

2211

kgkgxfxf

kgkgxfxf

kgxfkgxf

FT

FT

FTFT

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Convolution SummaryConvolution Summary

Must know….Must know…. Convolution theorem How to convolute the following functions. -function and any other function.

Two top-hats

Two Gaussians.

22

21

2