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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
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