The Fourier Transform and its applications goals: present the Fourier ...

The Fourier Transform and its applications 

goals: 

present the Fourier Theory and its applications in a concise overview 

show its fundamental relevance for many field of physics 

encourage use of FT-based reasoning and approaches 

addresses: 

students of advanced semesters (post Vordiplom, master-level) 

literature: 

Jean Baptiste Joseph Fourier, 

* March 21st, 1768, + May 16th, 1830 

Bracewell, The Fourier Theory and Its Applications, McGraw-Hill 

The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 

Page 1


contents: 

● mathematical basis 

● signal processing: filtering, sampling, reconstruction 

● statistics: mean/variance,central limit theorem, noise, correlation and drift 

● optics: diffraction, antenna theorem, interferometry 


Page 2


contents: 


– definition, properties, cos-, sine-transform 

– convolution, autocorrelation, power spectrum 

– often used functions (hat, sinc, ...), δ-function 

– FT theorems, properties of moments 

– periodic functions, Fourier series 

– discrete Fourier transform, FFT 

– FT in higher dimensions, cylinder-, spherical coordinates 





Page 3


contents: 


– definition, properties, cos-, sine-transform 

– convolution, autocorrelation, power spectrum 

– often used functions (hat, sinc, ...), δ-function 

– FT theorems, properties of moments 

– periodic functions, Fourier series 

– discrete Fourier transform, FFT 

– FT in higher dimensions, cylinder-, spherical coordinates 





Page 4


contents: 


● signal processing 

– filtering 

– sampling (Nyquist theorem) 

– reconstruction 




Page 5


contents: 



● statistics 

– mean/variance in time and frequency domain, 

– Heisenberg relation 

– central limit theorem 

– noise, correlation and drift 



Page 6


drift, natural shapes: 

fractional Brownian motion 

Page 7


contents: 




● optics 

– diffraction theory 

– antenna theorem 

– interferometry 


Page 8

diffraction image of 

circular aperture 

interferometry: 

left: baseline-tracks of telescope array with earth rotation 

right: point-spread-function 

The Fourier Transform and its Applications, Jürgen Stutzki, Sommersemester 2007 Page 9

1. Mathematical Basics 

Note: no rigorous proofs here, rather a “pragmatic” approach 

1.1. Definition of the Fourier Transform (FT) 

Function f �x� 

Fourier Transform F �s�=∫ −∞ 


∞ 

f �x� e −2� i xs dx (“ minus-i ” transform) 

note on notation: function f lower capital; Fourier transform F upper capital 

spatial: x �� s f � x� �� F �s� 

time: t �� f �t � �� F �� 

Backtransform, inverse Fourier Transform 

Interpretation: 

∞ 

f �x �=∫ F �s� e 

−∞ 

2� i xs ds (“ plus-i “-transform) (prove given below) 

Fourier Transform: filters out the (complex) amplitude F �s�=∣F �s�∣e i � s of 

e −2� i xs -component, i..e. oscillation with frequency s 

Back-Transform: express f �x � as sum over e 2�i xs -oscillations with (complex) amplitudes, 

i.e. amplitudes and phase F �s�=∣F �s�∣e i � s 

1 

math_ground_1.odt 

Page 10

Remark: usual abbreviation: �=2�� (time: angular frequency) or 

gives FT: 

and back-Trafo f �t � = ∫ −∞ 

k =2�s (spatial: wavenumber) 

F� � 

�� =F � 

2� �= ∞ 

∫ −∞ 

� 

−2� i 

2� 

f �t � e t 

∞ 

dt = ∫ 

−∞ 


∞ 

∞ 

F �� e 2�i �t d �=∫ −∞ 

F � � 

2� � ei� t d � 

2� 

f �t� e −i �t dt 

1 

= 

2� ∫ ∞ 

−∞ 

�F �� e i �t d � 

2 

math_ground_1a.odt 

Page 11

or, for symmetry, define: �F ��= 1 

� 2� ∫ ∞ 

−∞ 

f �t � e −i�t dt and back-transform f �t�= 1 

�2� ∫ ∞ 

−∞ 


�F �� e i�t d � . 

All 1 , 2 , and 3 

are common notations, used in different fields and contexts; we prefer the first one. 

Other notations often used: 

● instead of F �s� : �f �s� or � f �s� . 

This is useful to note down certain relations, e.g.� f ⋅g= � f ∗�g (FT of convolution, see below) 

● F �s� = FT � f � x �� , operator FT generates Fourier-Trafo of function f �x � 

3 


Page 12

1.2. Existence of the FT 

intuitively, and from physical insight: each time series has spectrum, 

i.e. contains a distribution of frequencies 

this applies to all functional dependencies occurring in the real world physics 

but: even simple examples (which don't as such occur in the real world) show problems 

● Sine,Cosine: a 0 sin �2��t � , a 0 cos�2�� t� s (needs to be switched on and off!) 

● 

0, 

Heavyside Step-Function: H �t �={ 1, 

t�0 

t�0 

● 

0, 

delta-Funktion, Impuls-Fkt.: � �t �={ ∞ , 

t≠0 

t=0 (more specifically �∞ 

∫ −∞ 

do not have FT in the proper sense 


(needs to be switched off!) 

��t� dt=1 ) 

(never gets infinitely sharp!) 

Fourier transform in the proper sense 

the Fourier-Transform exists (i.e. the Fourier-Integral converges for all values of s) if: 

∞ 

1. ∣ f � x �∣ is integrable, i.e. ∫ ∣ f � x�∣ dx exists 

−∞ 

2. f �x � has only finite discontinuities 

3. and f �x � shows “limited variations” (Lipshitz-Bedingung) 


Page 13

improper Fourier Transform 

Many functions allow FT only in the “improper” sense: FT in the limit: 

∞ 

if ∫ −∞ 

∞ 

if ∫ −∞ 

−� x2 

∣f � x�∣ dx does not exist, one considers a modified function f �� x�=e 

−� x2 

∣e 

note: lim 

��0 

and name lim 

��0 

f ��x�= f � x � ; �≪1 : slowly switching f �x � on and off 

f � x�∣ dx exists, consider the series of varying � , � �0 


∞ 

F ��s�=∫ −∞ 

e −2� i xs 2 

−� x 

e 

f � x � dx 

F ��s�=F �s� the (improper) Fourier Transform 

Note: such improper FTs are not really functions, but distributions (see below). 

f � x� . 


Page 14

1.3 properties of the FT 

1.3.0 linearity 

with h � x �=a f � x ��b g � x � , we get H �s�=a F �s��b G �s� 

(proof: linearity of integration) 

1.3.1 symmetries 

e.g. f �x � real valued, i.e. f * �x �= f �x � , and with the FT F �s�=∫ e −2� ixs f �x� dx 

we get: F * �s� =∫e 2� i xs f * −2� i x �−s� 

� x � dx=∫e f � x� dx= F �−s� ,i.e. F �s� is hermitean. 

or, more specifically, separating F �s� into real and imaginary part: 

F �s� = ℜ F �s��i ℑ F �s� 

F * �s� = ℜ F �s�−i ℑ F �s� 

F �−s� = ℜ F �−s��i ℑ F �−s� 

and hence 

ℜ F �s�=ℜ F �−s� , i.e. ℜ F �s� is even 

ℑ F �s�=−ℑ F �−s � , i.e. ℑ F �s� is odd; q.e.d. 

Thus: f �x � real valued ⇔ F �s� hermitean 

and vice versa: f �x � hermitean ⇔ F �s� real valued 

Similarly: f �x � imaginary, i.e. f * � x �=− f �x � : 

we get F * �s� =∫ e 2� i xs f * � x� dx=−∫ e −2� i x �− s� f �x� dx= −F �−s� , 

Thus: f �x � imaginary ⇔ F �s� anti-hermitean 

and vice versa f �x � anti-hermitean ⇔ F �s� imaginary 



Page 15

and last: f �x � even, i.e. f �x�= f �−x� : 

we get 

F �s�=∫ e 

−∞ 

−2� i x s f � x� d x = 

� 


�∞ 

�∞ 

=∫ e 

−∞ 

−2� i u�−s � Thus: 

f �u� du=F �−s� 

f �x � even ⇔ F �s� even 

and vice versa f �x � odd ⇔ F �s� odd 

−∞ �∞ 

�∞| 

u=−x |−∞ ;du=−dx 

−∞ 

−∫ e 

�∞ 

2� i us f �u� d u 

, 


Page 16

Note: 

any function f �x � can be separated into its odd and even parts, o � x� and e � x � : 

with e � x �=e �−x � , o � x�=−o�−x � we have: 

f � x� = e� x ��o �x � 

f �−x� = e� x �−o �x � 


, and hence e �x � = 1 

2 

similarly: separation into real and imaginary part: 

f �x� = ℜ f � x��i ℑ f � x� 

f * �x� = ℜ f � x�−i ℑ f � x� 

o � x � = 1 

2 

� f �x �� f �−x�� 

. 

� f �x �− f �−x�� 

, and ℜ f � x� = 1 

2 � f � x�� f * � x�� 

ℑ f � x� = 1 

2i � f � x�− f * � x�� 

With this, we can separate the hermitean and anti-hermitean parts: 

so that 

f �x �=e � x ��o� x �=ℜe � x��i ℑ e � x ��ℜ o� x ��i ℑo � x � 

=[ ℜe � x ��i ℑo � x �] 

hermitean 

anti−hermitean 

�[ ℜo �x ��i ℑ e � x� ] 

=h� x ��a � x � 

f * �−x �= =[ ℜe � x �−�−�i ℑ o �x �]�[−ℜ o� x �−i ℑ e �x �]=h � x �−a �x � 

h� x� = 1 

2 � f � x�� f * �−x�� 

a� x� = 1 

2 � f � x�− f * �−x�� 


Page 17

to summarize: 

Symmetry properties of FT 

or 

f �x � = e � x � � o� x � = ℜ f � x � � i ℑ f �x � = h � x � � a � x � 

� 

� 

� 

� 


� 

� 

� 

� 

F �s� = E �S � � O �s� = H �s� � A �s� = ℜ F �s� � i ℑ F �s� 

proof: e.g. 

f �x � = ℜe � x � � i ℑ e � x � � ℜ o� x � � i ℑo � x � 

� 

� 

� 

� 

� 

� 

F �s� = ℜ E �s� � i ℑ E �s� � i ℑO �S � � ℜO�s� 

FT �e� x��= � e� x�= 1� 

1 

� f �x�� f �−x��= 

2 

2 ��f � x��f �−x�� 

= 1 

�∞ 

2 [∫ e 

−∞ 

−2�i xs �∞ 

f � x� dx�∫ e 

−∞ 

−2� i xs ] f �−x� dx 

= 1 [ 2 F �S ��∫ 

�∞ 

e 

−∞ 

−2� i u�−s� u] f �u� d 

= 1 [ F �s��F �−s�]=E �s� 

2 

� 

� 

� 

� 

� � 

� 

� 

q.e.d. 

� 

� 


Page 18

1.3.2 FT of the complex conjugate function 

for the Fourier Transform of f * � x � we get: 

FT � f * � x ��= � f * �x �=∫e −2�i xs f * �x � dx 

−2� i x �−s 

=∫ 

�] 

* 

[e f * � x � dx=F * �−s� 

with this, and with FT � f �−x ��=F �−s� , we can derive all the above symmetry relations: 

e.g. 

FT � ℜ f � x ��=FT� 1 

2 [ f � x �� f * 1 

� x �]� = 

2 [ F �S ��F * �−s�]=H �s� , 

or 

etc. 

FT �a� x��=FT � 1 

2 [ f � x�− f * �−x�]� =1 

2 [ F �S�− F* �s�]=i ℑ F �s� 

Thus, all symmetry relations above can be derived from the (easy to memorize) three 

relations: 

1) FT � f � x ��=F �s� 

2) FT � f �−x ��=F �−s� 

3) FT � f * � x ��=F * �−s� 



Page 19

1.4 Sine- and Cosine-Transformation 

Recall Euler's relation: e i � =cos ��i sin � . Inserting this into the definition of the FT, we get: 

�∞ 

F �s�=FT � f �x��=∫ e 

−∞ 

−2� i xs f �x� dx=∫ cos�2� xs� f � x � dx – i ∫ sin�2� xs� f �x� dx 

−∞ 

−∞ 

∞ 

=∫ 0 

f � x�cos�2� xs� dx�∫ −∞ 


0 

�∞ 

f � x�cos�2� x s� d x 

∞ 

−i∫ 

0 

f � x�sin�2� xs� dx−i ∫ 

−∞ 

f � x�sin�2� x s� d x 

Substituting in the second and forth expression: 

0 0 

| x �−u|∞ ; dx �−du , we get: 

∞ 

=∫ 0 

∞ 

=∫ 0 

f � x�cos�2� xs� dx�∫ 0 

∞ 

−∞ 

f �−u�cos�2�u s� d u 

∞ 

−i∫ 0 

[ f � x�� f �−x�] cos�2� xs� dx−i∫ 0 

∞ 

=2∫ 0 

e� x� cos�2� xs� dx −i 2∫ 0 

∞ 

�∞ 

f � x�sin�2� xs� dx� i∫ 0 

∞ 

[ f �x�− f �−x�]sin�2� xs� dx 

o� x� sin�2� xs� dx 

We define: Cosine-Transformation CT � f � x ��=2∫ 0 

Sine-Transformation 

∞ 

∞ 

0 

∞ 

f �−u�sin�2�u s� d u 

f � x � cos�2� xs� dx , and 

ST � f � x��=2∫ f � x� sin�2� xs� dx 

0 

note: only x�0 counts in CT and ST ! 


Page 20

F �s�=FT � f � x ��=CT �e �x �� – i ST �o �x �� 

i.e. Fourier Transform ⇔ 

Cosine-Transform of the even and Sine-Transform of the odd part of f �x � 

Note: as e � x � and o � x� need to be defined only for x�0 , the continuation to x�0 being 

given by symmetry ( e �−x �=e � x � ; o �−x �=−o� x � ), the Fourier transform carries the 

full information on f �x � , both for x�0 and x�0 , although the integration in the CT 

and ST only use e � x � and o � x� for x�0 . 

We can thus, like always, get f �x � back from F �s� through the "+i" transform, giving: 

f �x�=FT �i � F �s��=CT � E �s��i ST �O�s�� 

i.e. Fourier Backtransform ⇔ 

Cosine-Transform of the even and Sine-Transform of the odd part of F �s� 

Note: in the CT , ST -world (no FT known), 

given F �s� , calculate E �s� and O�s� , from this, throughCT resp. ST , calculate 

f �x � 

Symmetries of Cosine- and Sine-Transform: with F C �s�=CT � f �x �� , F S �s�=ST � f �x�� , 

we have: F C �−s�=F C �s� Cosine-Transform is even in s, 

F S �−s�=−F S � s� Sine-Transform is odd in s 



Page 21

● only the behaviour of f �x� for x�0 enters into to Cosine- and Sine-Transform 

● the symmetries of F C �s� and F S �s� imply that they are fully determined by their s�0 

behavior 

Thus, only the x�0 behavior of an arbitrary function f �x � determines its F C �s� and F S �s� 

Consider a new function f H � x � : 

f � x � , x �0 

f H � x �={ 

= f � x� H �x � , 

0, x �0 

obtained by "cutting-off" the negative 

part of f �x � , i.e. by multiplying f �x � with the Heavyside step-function defined above. 

Obviously, it has the same Cosine- and Sine transform as f �x � itself: 

F S , H �s�=ST � f H � x ��=2∫ 0 

∞ 

=2∫ 0 

and similarly F C , H �s�=F C �s� . 


∞ 

f H �x � sin �2� xs� dx 

f � x � sin �2� xs� dx=ST � f � x ��=F S �s� 

Now, e H � x�= 1 

2 

2 [ f H �x�� f H �−x�]={1 

and oH �x�= 1 

2 

2 [ f H � x�− f H �−x�]={1 

1 

2 

− 1 

2 

f � x� , x�0 

f �−x� , x�0 

f � x� , x�0 

f �−x� , x�0 


Page 22

FT � f � x � H � x ��=CT �e H �x ��−i ST �o H � x �� 

= 1 

2 [CT � f � x��−i ST � f � x�� ] . 

Backtransform of Cosine- and Sine-transform 

With this, we now consider an arbitrary function f �x � , defined only at x�0 and with 

CT � f � x ��=F C �s� 

ST � f � x��=F S �s� 

Note that F C �s� and F S �s� , given their symmetry, have to be taken into account only at 

positive frequencies s�0 . 

We claim that 

the back-transform of the Cosine transform is the Cosine transform: 

f � x �=2∫ F C�s� cos �2� xs� ds 

0 

and the back-transform of the Sine transform is the Sine transform: 


∞ 

∞ 

f � x �=2∫ 0 

F S �s� sin�2 � xs� ds 


Page 23

Proof: 

1) for the Cosine back-trafo: 

define an even function for all x : f 1� x �= f �∣x∣�={ 

Its FT is 

FT � f 1 � x ��=F 1 �s�=CT �e 1 � x�� 

� CT� f 1� x�� 

−i ST � �o1 �x ��=CT 

� f � x��=F C �s� 

, 

=0 

which is also even, being a pure cosine transform: 

F 1 �−s�=F 1 �s� , resp. E 1 �s�=F 1 �s � , O 1 �s�=0 

Its back-transform reproduces f 1 � x � : 

f 1� x �=FT �i � F 1 �s��=CT � E 1�s�� 

� CT� F 1 �s�� 

�i ST �O 1 �s�� 

� =0 

� � 

=0 

ST�F 2 �s�� 


f � x � , x�0 

f �−x � , x�0 , i.e. e 1 � x �= f 1 � x� , o 1 �x �=0 . 

=CT �CT � f �∣x∣�� 

and for 

x�0 , where f �x � is defined: f � x �= f 1 � x �=CT �CT � f � x �� =CT � F C � s� � q.e.d. 

2) similarly for the Sine backtrafo, but with the odd function f 2 �x �={ f � x � , x�0 

− f �−x � , x�0 : 

FT � f 2� x��=F 2�s�=CT � �e 2� x�� 

=0 

−i ST ��o 2� x �� 

ST � f 2�x�� =−i ST � f �x ��=−i F S�s� , and 

f 2�x�=FT �i � F 2�s��=CT � E2�s�� i ST �O 2�s�� =i �−i�ST � F s�s��=ST �ST � f �∣x∣�� 

q.e.d. 

Page 24


pltfour.py 

Page 25

Summary: 


∞ 

● Fourier-Transform F �s�=∫ f �x� e 

−∞ 

−2� i xs ● 

dx (“ minus-i ” transform) 

∞ 

Fourier-Backtransform f �x �=∫ F �s� e 

−∞ 

2� i xs ● 

ds (“ plus-i “-transform) 

symmetry relations: 

f �x � = e � x � � o� x � = ℜ f � x � � i ℑ f �x � = h � x � � a � x � 

� 

� 

� 

� 

� 

� 

� 

� 

F �s� = E �S � � O �s� = H �s� � A �s� = ℜ F �s� � i ℑ F �s� 

f �x � = ℜe � x � � i ℑ e � x � � ℜ o� x � � i ℑo � x � 

● Cosine-, and Sine-Transform 

� 

� 

� 

� 

� 

� 

F �s� = ℜ E �s� � i ℑ E �s� � i ℑO �S � � ℜO�s� 

∞ 

CT � f � x ��=2∫ 0 

∞ 

� 

� 

f � x � cos�2� xs� dx 

ST � f � x ��=2∫ f � x� sin�2� xs� dx 

0 

which are their own back-transforms 

● Fourier Transform ⇔ 

Cosine-Transform of the even and Sine-Transform of the odd part of f �x � 

F �s�=FT � f � x ��=CT �e �x �� – i ST �o �x �� 

� 

� 

� � 

� 

� 

� 

� 


Page 26

The Fourier Transform and its applications goals: present the Fourier ...

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