Post on 06-Sep-2019
1 The Fourier Transform
• 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
2 Fourier Series
• Fourier series is an expansion (German: Entwicklung) of a
periodic function f(x) (period 2L, angular frequency /L) in
terms of a sum of sine and cosine functions with angular
frequencies that are integer multiples of /L
• Any piecewise continuous function f(x) in the interval [-L,L]
may be approximated with the mean square error converging
to zero
3 Fourier Series
• Sine and cosine functions form a complete
orthogonal system over [-,]
• Basic relations (m,n 0):
• mn: Kronecker delta
• mn=0 for m n, mn=1 for m = n
nxnx cos,sin
mndxnxmx
sinsin
mndxnxmx
coscos
0cossin
dxnxmx
0sin
dxmx
0cos
dxmx
4 Fourier Series: Sawtooth
• Example: Sawtooth wave
x / (2L)
f(x)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.2
0.4
0.6
0.8
1
5 Fourier Series
• Fourier series is given by
• where
• 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
nxbnxaaxfn
n
n
n sincos2
1
11
0
dxxfa
10
dxnxxfan cos1
dxnxxfbn sin1
6 Fourier Series
• For a function f(x) periodic on an interval [-L,L] instead of
[-,] change of variables can be used to transform the
interval of integration
• Then
• and
L
xx
'
L
dxdx
'
L
xnb
L
xnaaxf
n
n
n
n
'sin
'cos
2
1
11
0
''
cos'1
dxL
xnxf
La
L
L
n
'
'sin'
1dx
L
xnxf
Lb
L
L
n
''
10 dxxf
La
L
L
7 Fourier Series
• For a function f(x) periodic on an interval [0,2L] instead of
[-,]:
''
cos'1
2
0
dxL
xnxf
La
L
n
''
sin'1
2
0
dxL
xnxf
Lb
L
n
''1
2
0
dxxfL
a
L
n
8 Fourier Series: Sawtooth
• Example: Sawtooth wave
L
xxf
2
0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
x / (2L)
f(x)
Finite Fourier series
12
12
0
0 dxL
x
La
L
dxL
xn
L
x
La
L
n
cos
2
12
0
0
sinsincos222
n
nnnn
dxL
xn
L
x
Lb
L
n
sin
2
12
0
n
n
nnn
1
2
2sin2cos222
L
xn
nxf
n
sin
11
2
1
1
9 Fourier Series: Sawtooth
• Example: Sawtooth wave
• Fourier series:
L
xxf
2
x / (2L)
f(x)
m
n L
xn
nxmg
1
sin11
2
1,
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.2
0.4
0.6
0.8
1
10 Fourier Series: Sawtooth
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.5
1
0 5 10 15 200
0.2
0.4
n : Frequency w / (/L)
Am
plit
ud
e a
0/2
, b
n
x / (2L)
x / (2L)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85
4
3
2
1
0
1
11 Example: Approximation by Fourier-Series
m=1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.85 10
4
0
5 104
0.001
0.002
0.002
nxbnxaaxfm
n
n
m
n
n sincos2
1
11
0
12 Approximation by Fourier-Series
m=7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.82 10
4
0
2 104
4 104
6 104
8 104
0.001
0.0012
0.0014
0.0016
0.0018
0.002
nxbnxaaxfm
n
n
m
n
n sincos2
1
11
0
13 Approximation by Fourier-Series
m=10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.82 10
4
0
2 104
4 104
6 104
8 104
0.001
0.0012
0.0014
0.0016
0.0018
0.002
nxbnxaaxfm
n
n
m
n
n sincos2
1
11
0
14 Fourier Series: Properties
• Around points of discontinuity, a "ringing“, called Gibbs
phenomenon occurs
• If a function f(x) is even, is odd and thus
Even function: bn = 0 for all n
• If a function f(x) is odd, is odd and thus
Odd function: an = 0 for all n
nxxf sin 0sin
dxnxxf
nxxf cos 0cos
dxnxxf
15
inx
n
neAxf
Complex Fourier Series
• Fourier series with complex coefficients
• Calculation of coefficients:
• Coefficients may be expressed in terms of those in the real
Fourier series
dxexfA inx
n2
1
dxnxinxxfAn sincos2
1
nn
nn
iba
a
iba
2
1
2
1
2
1
0
for n < 0
for n = 0
for n > 0
16 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 or
• Inverse transform
• Symbolic notation
dxexfkF ikx2 kxfkF xF
dkekFxf ikx2 xkFxf k
1 F
xf kF
17 Forward and Inverse Transform
• If
1. exists
2. f(x) has a finite number of discontinuities
3. The variation of the function is bounded
• Forward and inverse transform:
• For f(x) continuous at x
• For f(x) discontinuous at x
dxxf
dkdxexfexxfxf ikxikx
xk
221FF
xfxfxxfxk
2
11FF
18 Fourier Cosine Transform and Fourier Sine
Transform
• Any function may be split into an even and an odd function
• Fourier transform may be expressed in terms of the Fourier
cosine transform and Fourier sine transform
xOxExfxfxfxfxf 2
1
2
1
dxkxxOidxkxxEkF 2sin2cos
19 Real Even Function
• If a real function f(x) is even, O(x) = 0
• If a function is real and even, the Fourier transform is also
real and even
0
2cos22cos2cos dxkxxfdxkxxfdxkxxEkF
F(k) f(x)
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
x k
20 Real Odd Function
• If a real function f(x) is odd, E(x) = 0
• If a function is real and odd, the Fourier transform is
imaginary and odd
0
2sin22sin2sin dxkxxfidxkxxfidxkxxOikF
F(k) f(x)
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
sin(..-k..) = -sin(..k..)
21 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)
22 Symmetry Properties
F(k) f(x)
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
23 Fourier Transform: Properties and Basic Theorems
• Linearity
• Shift theorem
• Modulation theorem
• Derivative
0xxf kFekix02
xkxf 02cos 002
1kkFkkF
kbGkaF xbgxaf
xf kkFi 2
24 Fourier Transform: Properties and Basic Theorems
• Parseval's / Rayleigh’s theorem
• Fourier Transform of the complex conjugated
• Similarity (a 0 : real constant)
• Special case (a = -1):
xf * kF *
xf kF
axf
a
kFa
1
dkkFdxxf22
25
xf kF
Similarity Theorem
1
1
1
1
1
1
1
xf 2
22
1 kF
xf 5
55
1 kF
x k
1 x k
x k 1
26 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
27 Convolution
• Definition of convolution
• JAVA Applets: http://www.jhu.edu/~signals/
dxgfgfxy
Eric W. Weisstein.
“Convolution. http://mathworld.
wolfram.com/Convolution.html
28 Properties of the Convolution
• Properties of the convolution
• The integral of a convolution is the product of integrals of the
factors
dxxgdxxfdxgf
fggf
hgfhgf )(
hfgfhgf )(
agfgafgfa
dx
dgfg
dx
dfgf
dx
d
commutativity
associativity
distributivity
29 The Basic Theorems: Convolution
• 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)
• Convolution in the spectral domain
xgxf kGkF
dxdxxxgxfe ikx '''2
dxxxgedxxfe xxikikx ''' '2'2
'''''' ''2'2 dxxgedxxfe ikxikx
kGkF
xgxf
xgxf kGkF
30 Cross-Correlation
• Definition of cross-correlation
• !
• Fourier transform:
, (see FT rules)
dtgftgtftrfg
*)(
tgtftgtf *)(
fggf
tgtftgtf *)(
kGkF* tgtf )(
tf * kF *
31 Example: Cross-Correlation Radar (Low Noise)
Signal sent by radar antenna Signal received by radar antenna
Time 0 Time 0
Cross correlation of the signals
Time 0
32 Example: Cross-Correlation Radar
Signal sent by radar antenna Signal received by radar antenna
Time 0
Cross correlation of the signals
Time 0
Time 0
33 Example: Cross-Correlation Ultrasonic Distance Measurement
Signal received by ultrasonic transducer
Time
Signal sent by ultrasonic transducer
Time
Transmitter
Receiver
TimerObject
d
Pulse
Start
Stop
Transmitter
Receiver
TimerObject
d
Pulse
Transmitter
Receiver
TimerObject
d
Pulse
Start
Stop
2
flightvtd (0)
Cross correlation of the two signals
Time 0 tflight
34 Autocorrelation Theorem
• Autocorrelation
• Autocorrelation theorem:
• Proof
dtfftftf
*)(
2* kFkFkF tftf )(
tf * kF *
2
kF tftf )(
tftftftf *)( ,
35 Example: Autocorrelation Detection of a Signal in the Presence of Noise
Time
Sig
na
l
Signal: Noise
Autocorrelation function:
Peak at t = 0 : Noise
Time
r ff
0
36 Example: Autocorrelation Detection of a Signal in the Presence of Noise
Sig
na
l r f
f
Signal: Sinusoid plus noise
Autocorrelation function:
Peak at t = 0 : Noise
Period of rff reveals that signal is
periodic
Time
Time 0
37
http://en.wikipedia.org/wiki/Cross-correlation
38 The Delta Function
• 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, also called Dirac
function) 0x 0x
1
dxx
for 1.
2.
39 The Delta Function
• The sifting (sampling) property of the delta function
• Sampling of f(x) at x = a
• Fourier transform:
0fdxxfx
afdxxfax
0xx 022
0
ikxikx edxexx
40
x
k
Fourier Transform Pairs
• Cosine function
• Sine function
1
1
1
1
1
1 x k
xcos xII
2
1
2
1
2
1II xxx
2
1
2
1
2
1II xxx
xi II xsin
41
x k
2xe 2ke
Fourier Transform Pairs
• Gaussian function
• Rectangle function of unit height and base P(x)
P
2
10
2
1
2
12
11
x
x
x
x
1
1
1
1
1
1
1
1
x k
x
xx
sinsinc
ksinc
42 Fourier Transform Pairs
• Triangle function of unit height and area
10
11
x
xxx
x k 1
1
1
1
k2sinc
• Constant function (unit height)
x 1
1
k 1
(k)
43 Sampling
• Sampling: “Noting” a function at finite intervals
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
44 Sampling of a Signal
Signal 1
Samples taken at equal
intervals of time
Sampled signal
45 Sampling
Sampled signal does
not contain the additional
Signal
Samples taken at equal
intervals of time
Signal 1 + Signal 2
Signal 2
(very short)
46 Aliasing
Sampling with a low
sampling frequency
Sampled signal
Apparent signal
47 Sampling
• Sampling function (sampling comb) III(x) “Shah”
n
nxx III
n a
nx
aax
1III
• Multiplication of f(x) with III(x)
describes sampling at unit intervals
n
nxnfxfx III
• Sampling at intervals :
n
nxnfxfx
IIIBracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
48 Sampling
xIII kIII
• Fourier transform of the sampling function
xIII
1 kIII xIII
2III
x k2III2
kIII
1
here: = 2
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
49 Sampling:
Band limited signal
• Band limited signal: F(k) = 0 for |k|>kc
x k kc
f(x) F(k)
Bracewell, R. The Fourier Transform and Its Applications,
3rd ed. New York: McGraw-Hill, 2000.
50 Sampling in the Two Domains
kc
xIII k III
-1
kc
xfx
III kFk III
f(x) F(k)
|k| kc x
= =
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
51 Reconstruction of the Signal
kc
xfx
III kFk III
|k|
|k|
f(x) F(k)
|k| kc x
= =
Multiplication with rectangular
function in order to remove
additional signal components
Original signal in
frequency domain
1 Convolution
Fourier
Transform
52 Critical Sampling and Undersampling
kc
f(x) F(k)
|k| kc x
kFk
kk
c
c
2III2
1
ck2
1
xfxkc2III
kc
Undersampling
ck2
1
Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.
53 Sampling
• Sampling theorem
• A function whose Fourier transform is zero (F(k) = 0) for |k|>kc is fully
specified by values spaced at equal intervals < 1/(2kc).
• Alternative (simplified) statement: The sampling frequency
must be higher than twice the highest frequency in the signal • Remark: This does not imply that a reconstruction is impossible for all cases if
the sampling frequency is lower. If additional knowledge about the signal is
available (e.g. lower limit of Fourier transform, single frequency signal),
reconstruction may be possible.
• Description of sampling at intervals : Multiplication of f(x) with
III(x/ )
• Reconstruction of the signal: Multiplication of the Fourier
transform with and inverse
transformation
Or: Convolution of the sampled function with a sinc-function
kFk III
P
ck
k
2