The Fourier Transform
1
• Fourier Series
• Fourier Transform
• The Basic Theorems and Applications
• Sampling
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill,
2000.
Eric W. Weisstein. "Fourier Transform.“ http://mathworld.wolfram.com/FourierTransform.html
Fourier Series
• Fourier series is an expansion of a periodic function f(x)
(period 2L, angular frequency π/L) in terms of a sum of sine
and cosine functions with angular frequencies that are integer
multiples of π/L
• Any piecewise continuous function f(x) in the interval [-L,L]
may be approximated with the mean square error converging
to zero
2
3
Fourier Series
• Sine and cosine functions {sin (nx ), cos(nx )} form a complete
orthogonal system over [-π,π]
π
• Basic relations (m,n ≠ 0):
sin (mx )sin (nx )dx = πδ mn
∫π
−
π
∫π cos(mx )cos(nx )dx = πδ
mn
−
π
∫π sin(mx )cos(nx )dx = 0
−
• δmn: Kronecker delta
• δmn=0 for m ≠ n, δmn=1 for m = n
π
∫π sin(mx )dx = 0
−
π
∫π
−
cos(mx )dx = 0
4
Fourier Series: Sawtooth
• Example: Sawtooth wave
1
0.8
f(x)
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
x / (2L)
1.2
1.4
1.6
1.8
5
Fourier Series
• Fourier series is given by
1
f ( x ) = a0 +
2
• where
∞
∑a
bn =
cos(nx ) +
n =1
a0 =
an =
n
1
π
1
π
π
∞
∑b
n
sin (nx )
n =1
1
π
π
∫π f (x )dx
−
∫π f (x )cos(nx )dx
−
π
∫π f (x )sin(nx )dx
−
• If the function f(x) has a finite number of discontinuities and a
finite number of extrema (Dirichlet conditions): The Fourier
series converges to the original function at points of continuity
or to the average of the two limits at points of discontinuity
6
Fourier Series
• For a function f(x) periodic on an interval [-L,L] instead of
[-π,π] change of variables can be used to transform the
interval of integration
x≡
πx '
L
dx ≡
πdx '
L
• Then
∞
1
 nπ x '  ∞
 nπx ' 
f ( x ) = a0 + ∑ an cos
b
sin
+
 ∑ n 

2
 L  n =1
 L 
n =1
• and
L
a0 =
1
f ( x ' )dx '
∫
L −L
1
 nπx ' 
(
)
f
x
'
cos

dx '
∫
L −L
 L 
L
an =
1
 nπx ' 
(
)
f
x
'
sin

dx '
∫
L −L
 L 
L
bn =
Fourier Series
• For a function f(x) periodic on an interval [0,2L] instead of
[-π,π]:
an =
1
an =
L
2L
1
bn =
L
2L
∫
0
∫
0
1
L
2L
∫ f (x')dx'
0
 nπx ' 
f ( x ' ) cos
dx '
 L 
 nπ x ' 
f ( x ' )sin
dx '
 L 
7
8
Fourier Series: Sawtooth
• Example: Sawtooth wave
1
a0 =
L
2L
1
an =
L
2L
f (x ) =
x
∫0 2 Ldx = 1
x
2L
1 1 ∞ 1  nπ x 
f ( x ) = − ∑ sin 

2 π n =1 n  L 
x
 nπx 
cos
∫0 2 L  L dx
[2nπ cos(nπ ) − sin (nπ )]sin (nπ )
=
Finite Fourier series
n 2π 2
=0
1
0.8
x
 nπx 
sin
∫0 2 L  L dx
2L
− 2nπ cos(2nπ ) + sin (2nπ )
2n 2π 2
1
=−
nπ
=
f(x)
1
bn =
L
0.6
0.4
0.2
0
0
0.2
0.4
0.6
x / (2L)
0.8
1
1.2
Fourier Series: Sawtooth
x
f (x ) =
2L
1 1 m 1  nπx 
g (m, x ) = − ∑ sin 

2 π n =1 n  L 
• Example: Sawtooth wave
• Fourier series:
1
f(x)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
x / (2L)
1.2
1.4
1.6
1.8
9
10
Fourier Series: Sawtooth
1
0
0
0.2
0.4
0.6
0.8
1
x / (2L)
1.2
1.4
1.6
1.8
1
0
Amplitude a0/2, bn
0.5
0.4
0.2
1
0
2
3
4
5
0
0.2
0.4
0.6
0.8
1
x / (2L)
1.2
1.4
1.6
1.8
0
5
10
15
n : Frequency ω / (π/L)
20
11
Fourier Series: Properties
• Around points of discontinuity, a "ringing“, called Gibbs
phenomenon occurs
• If a function f(x) is even, f (x )sin (nx ) is odd and thus
π
∫π f (x )sin (nx )dx = 0
−
⇒ Even function: bn = 0 for all n
• If a function f(x) is odd, f (x ) cos(nx ) is odd and thus
⇒ Odd function: an = 0 for all n
π
∫π f (x )cos(nx )dx = 0
−
Complex Fourier Series
• Fourier series with complex coefficients
f (x ) =
∞
inx
A
e
∑ n
n = −∞
• Calculation of coefficients:
1
An =
2π
π
∫π
f ( x )e −inx dx
−
• Coefficients may be expressed in terms of those in the real
Fourier series
1
(
1
An =
2π
π
∫π
−
)
 2 a n + ib n for n < 0

1
f ( x )[cos (nx ) − i sin (nx )]dx =  a0
for n = 0
2
1
 2 (an − ibn ) for n > 0

12
13
Fourier Transform
• Generalization of the complex Fourier series for infinite L and
continuous variable k
• L → ∞, n/L → k, An → F(k)dk
• Forward transform F (k ) =
∞
∫
f (x )e − 2πikx dx
or F (k ) = Fx ( f ( x ))(k )
−∞
∞
• Inverse transform
2πikx
(
)
(
)
f x = ∫ F k e dk
−∞
• Symbolic notation
f (x )
F (k )
f ( x ) = Fk
−1
(F (k ))(x )
Forward and Inverse Transform
•
If
1.
14
∞
∫ f (x ) dx
exists
−∞
2. f(x) has a finite number of discontinuities
3. The variation of the function is bounded
•
Forward and inverse transform:
∞
•
∞


−1
− 2πikx
2πikx
dx dk
For f(x) continuous at x f (x ) = Fk (Fx f (x ))(x ) = ∫ e  ∫ f (x )e
−∞
 −∞

•
For f(x) discontinuous at x Fk −1 (Fx f (x ))(x ) = ( f (x+ ) + f (x− ))
1
2
Fourier Cosine Transform and Fourier Sine
Transform
• Any function may be split into an even and an odd function
1
1
(
)
[
(
)
(
)
]
f x = f x + f − x + [ f ( x ) − f (− x )] = E ( x ) + O ( x )
2
2
• Fourier transform may be expressed in terms of the Fourier
cosine transform and Fourier sine transform
F (k ) =
∞
∞
−∞
−∞
∫ E (x ) cos(2πkx )dx − i ∫ O(x )sin (2πkx )dx
15
16
Real Even Function
• If a real function f(x) is even, O(x) = 0
F (k ) =
∞
∞
∞
−∞
−∞
0
∫ E (x )cos(2πkx )dx = ∫ f (x )cos(2πkx )dx =2∫ f (x )cos(2πkx )dx
• If a function is real and even, the Fourier transform is also
real and even
F(k)
f(x)
x
k
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
17
Real Odd Function
• If a real function f(x) is odd, E(x) = 0
∞
∞
∞
−∞
−∞
0
F (k ) = −i ∫ O(x )sin (2πkx )dx = − i ∫ f ( x )sin (2πkx )dx = − 2i ∫ f ( x )sin (2πkx )dx
• If a function is real and odd, the Fourier transform is
imaginary and odd
f(x)
F(k)
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
Symmetry Properties
f(x)
F(k)
Real and even
Real and even
Real and odd
Imaginary and odd
Imaginary and even
Imaginary and even
Imaginary and odd
Real and odd
Real asymmetrical
Complex hermitian
Imaginary asymmetrical
Complex antihermitian
Even
Even
Odd
Odd
Hermitian: f(x) = f*(-x)
Antihermitian: f(x) = -f*(-x)
18
19
Symmetry Properties
f(x)
F(k)
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
20
Fourier Transform: Properties and Basic Theorems
• Linearity
af ( x ) + bg ( x )
aF (k ) + bG (k )
• Shift theorem
f ( x − x0 )
e −2πix0 k F (k )
• Modulation theorem
f ( x ) cos(2πk0 x )
1
(F (k − k0 ) + F (k + k0 ))
2
• Derivative
f ′( x )
i 2πkF (k )
21
Fourier Transform: Properties and Basic Theorems
• Parseval's / Rayleigh’s theorem
∞
∫
−∞
∞
f ( x ) dx = ∫ F (k ) dk
2
2
−∞
• Fourier Transform of the complex conjugated
f * (x )
F * (− k )
• Similarity (a ≠ 0 : real constant)
f (ax )
−1  k 
a F 
a
• Special case (a = -1): f (− x )
F (− k )
22
Similarity Theorem
1
f (x )
1
1
1
x
1
f (2 x )
1
1
x
1
x
f (5 x )
1
F (k )
k
1 k
F 
2 2
k
1 k
F 
5 5
k
Convolution
• Convolution describes an action of an observing instrument
(linear instrument) when it takes a weighted mean of a
physical quantity over a narrow range of a variable
• Examples:
• Photo camera: “Measured” photo is described by real image
convolved with a function describing the apparatus
• Spectrometer: Measured spectrum is given by real spectrum
convolved with a function describing limited resolution of the
spectrometer
• Other terms for convolution: Folding (German: Faltung),
composition product
• System theory: Linear time (or space) invariant systems are
described by convolution
23
Convolution
• Definition of convolution
y(x ) = f ∗ g =
24
∞
∫ f (τ )⋅ g (x − τ )dτ
−∞
Eric W. Weisstein.
“Convolution. http://mathworld.
wolfram.com/Convolution.html
• JAVA Applets: http://www.jhu.edu/~signals/
Properties of the Convolution
• Properties of the convolution
f ∗g = g∗ f
commutativity
f ∗ ( g ∗ h) = ( f ∗ g ) ∗ h
associativity
f ∗ ( g + h) = ( f ∗ g ) + ( f ∗ h )
distributivity
a( f ∗ g ) = (af ) ∗ g = f ∗ (ag )
d
df
dg
( f ∗ g)= ∗ g = f ∗
dx
dx
dx
• The integral of a convolution is the product of integrals of the
∞
∞
∞
factors
∫ ( f ∗ g )dx = ∫ f (x )dx ∫ g (x )dx
−∞
−∞
−∞
25
The Basic Theorems: Convolution
26
• The Fourier transform of the convolution of f(x) and g(x) is
given by the product of the individual transforms F(k) and G(k)
F (k )G (k )
f (x ) ∗ g (x )
f (x )∗ g (x )
∞ ∞
∫ ∫e
− 2πikx
− ∞− ∞
∞ ∞
=
∫ ∫ [e
f ( x')g (x − x')dx' dx
− 2πikx '
][
f ( x ' )dx ' e −2πik ( x − x ' ) g ( x − x ' )dx
− ∞− ∞
 ∞ −2πikx '
  ∞ −2πikx ''

= ∫e
f ( x ' )dx '  ∫ e
g ( x ' ' )dx ' '
−∞
  −∞

= F (k )G (k )
• Convolution in the spectral domain
f (x )g (x )
F (k ) ∗ G (k )
]
Cross-Correlation
• Definition of cross-correlation
rfg (t ) = f (t ) g (t ) =
∞
∫
f * (τ )g (t + τ )dτ
−∞
f (t ) g (t ) = f * (− t ) ∗ g (t )
•
f
g≠g
f
!
• Fourier transform:
f (t ) g (t ) = f * (− t ) ∗ g (t )
⇒
f (t ) g (t )
, f * (− t )
F * (k )G (k )
F * (k ) (see FT rules)
27
Example: Cross-Correlation
Radar (Low Noise)
Signal sent by radar antenna
0
Time
Signal received by radar antenna
0
Time
Cross correlation of the signals
0
Time
28
Example: Cross-Correlation
Radar
Signal sent by radar antenna
0
Time
Signal received by radar antenna
0
Time
Cross correlation of the signals
0
Time
29
Example: Cross-Correlation
30
Ultrasonic Distance Measurement
Signal sent by ultrasonic transducer
Signal received by ultrasonic transducer
Time
Time
Cross correlation of the two signals
Pulse
Start
Θ
Transmitter
Timer
Object
Stop
Receiver
d
d=
vt flight
2
(Θ≈0)
0
tflight
Time
Autocorrelation Theorem
• Autocorrelation
f (t ) f (t ) =
∞
∫
f * (τ ) f (t + τ )dτ
−∞
• Autocorrelation theorem:
F (k )
f (t ) f (t )
2
• Proof
f (t ) f (t ) = f * (− t ) ∗ f (t )
f (t ) f (t )
,
f * (− t )
F * (k )F (k ) = F (k )
2
F * (k )
31
Example: Autocorrelation
Signal: Noise
Signal
Detection of a Signal in the Presence of Noise
Autocorrelation function:
Peak at t = 0 : Noise
rff
Time
0
Time
32
Example: Autocorrelation
Signal: Sinusoid plus noise
Signal
Detection of a Signal in the Presence of Noise
Autocorrelation function:
Peak at t = 0 : Noise
Period of rff reveals that signal is
periodic
rff
Time
0
Time
33
34
http://en.wikipedia.org/wiki/Cross-correlation
The Delta Function
35
• Function for the description of point sources, point masses,
point charges, surface charges etc.
• Also called impulse function: Brief (unit area) impulse so that
all measuring equipment is unable to resolve pulse
• Important attribute is not value at a given time (or position) but
the integral
• Definition of the delta function (distribution)
1.
δ ( x ) = 0 for x ≠ 0
∞
2.
∫ δ (x )dx = 1
−∞
36
The Delta Function
• The sifting (sampling) property of the delta function
∞
∫ δ (x ) f (x )dx = f (0)
−∞
• Sampling of f(x) at x = a
∞
∫ δ (x − a ) f (x )dx = f (a )
−∞
• Fourier transform:
δ ( x − x0 )
∞
− 2πikx
− 2πikx
(
)
δ
x
−
x
e
dx
=
e
0
∫
0
−∞
37
Fourier Transform Pairs
• Cosine function
1
1 
1 
1 
II(x ) =  δ  x +  + δ  x −  
2 
2 
2 
cos(πx )
1
II(x )
x
1
• Sine function
1
k
1 
1 
1 
I I (x ) =  δ  x +  − δ  x −  
2 
2 
2 
sin (πx )
1
x
i I I (x )
1
k
38
Fourier Transform Pairs
• Gaussian function
1
e
−πx 2
1
1
e
−πk 2
x
1
k
• Rectangle function of unit height and base Π(x)

1
 1
Π(x ) = 
2
0

1
x<
2
1
x=
2
1
x>
2
sinc(x ) =
1
1
1
x
sinc(k )
1
k
sin πx
πx
39
Fourier Transform Pairs
• Triangle function of unit height and area
1 − x
Λ(x ) = 
 0
x ≤1
x >1
1
1
1
x
sinc 2 (k )
k
1
• Constant function (unit height)
δ(k)
1
1
x
1
k
Sampling
40
• Sampling: “Noting” a function at finite intervals
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
Sampling of a Signal
Signal 1
Samples taken at equal
intervals of time
Sampled signal
41
42
Sampling
Signal 1 + Signal 2
Signal 2
(very short)
Samples taken at equal
intervals of time
Sampled signal does
not contain the additional
Signal
Aliasing
Sampling with a low
sampling frequency
Sampled signal
Apparent signal
43
44
Sampling
• Sampling function (sampling comb) III(x) “Shah”
III( x ) =
∞
∑ δ (x − n )
n = −∞
1
III(ax ) =
a
∞
n

δx− 
∑
a
n = −∞ 
• Multiplication of f(x) with III(x)
describes sampling at unit intervals
III( x ) f ( x ) =
∞
∑ f (n )δ (x − n )
n = −∞
• Sampling at intervals τ:
∞
 x
III  f ( x ) = τ ∑ f (nτ )δ ( x − nτ )
τ 
n = −∞
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
45
Sampling
• Fourier transform of the sampling function
III( x )
III(k )
 x
III 
τ τ 
III(τk )
1
III( x )
III(k )
 x
III 
2
2 III(2k )
τ
here: τ = 2
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
τ−1
Sampling:
Band limited signal
46
• Band limited signal: F(k) = 0 for |k|>kc
f(x)
F(k)
x
kc
k
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
47
Sampling in the Two Domains
f(x)
×
τ
 x
III 
τ 
F(k)
x
kc
τ III(τk )
∗
|k|
τ-1
=
 x
III  f ( x )
τ 
kc
τ III(τk ) ∗ F (k )
=
kc
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
48
Reconstruction of the Signal
τ III(τk ) ∗ F (k )
 x
III  f ( x )
τ 
∗
kc
Multiplication with rectangular
function in order to remove
additional signal components
Convolution
1
=f(x)
×
|k|
=
F(k)
|k|
Original signal in
frequency domain
x
Fourier
Transform
kc
|k|
Critical Sampling and Undersampling
f(x)
F(k)
x
III(2kc x ) f ( x )
1
2kc
49
kc
|k|
 k 
 ∗ F (k )
2
k
 c
(2kc )−1 III
kc
Undersampling
τ>
1
2k c
kc
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
Summary: Sampling
50
• Sampling theorem
• A function whose Fourier transform is zero (F(k) = 0) for |k|>kc is fully
specified by values spaced at equal intervals τ < 1/(2kc).
• Alternative (simplified) statement: The sampling frequency
must be higher than twice the highest frequency in the signal
• Description of sampling at intervals τ: Multiplication of f(x) with
III(x/ τ)
• Reconstruction of the signal: Multiplication of the Fourier
transform (τ III(τk ) ∗ F (k )) with
 k 

Π
 2k c 
and inverse
transformation
Or: Convolution of the sampled function with a sinc-function