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Introduction to Fourier transform and signalanalysis
Zong-han, [email protected]
January 7, 2015
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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License of this document
Introduction to Fourier transform and signal analysis by Zong-han,Xie ([email protected]) is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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Orthogonal condition
Any two vectors a , b satisfying following condition aremutually orthogonal.
a∗
·b = 0 (1)
Any two functions a(x ), b (x ) satisfying the followingcondition are mutually orthogonal.
a∗(x ) ·b (x )dx = 0 (2)* means complex conjugate.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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Complete and orthogonal basis
cos nx and sinmx are mutually orthogonal in which n and mare integers.
π
−π cosnx
·sinmxdx
= 0
π
−πcos nx ·cos mxdx = πδ nm
π
−πsin nx
·sin mxdx = πδ nm (3)
δ nm is Dirac-delta symbol. It means δ nn = 1 and δ nm = 0when n = m.
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Fourier series
Since cosnx and sinmx are mutually orthogonal, we can expandan arbitrary periodic function f (x ) by them. we shall have a seriesexpansion of f (x ) which has 2π period.
f (x ) = a0 +∞
k =1(ak cos kx + b k sinkx )
a0 = 12π
π
−πf (x )dx
ak = 1
π π
−πf (x )cos kxdx
b k = 1
π π
−πf (x )sin kxdx (4)
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
C i i f
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Fourier series
If f (x ) has L period instead of 2π , x is replaced with πx / L.
f (x ) = a0 +∞
k =1
ak cos 2k πx
L + b k sin
2k πx L
a0 = 1
L L
2
−L
2
f (x )dx
ak = 2
L
L
2
−L
2
f (x )cos 2k πx
L dx , k = 1 , 2,...
b k = 2
L L
2
−L
2
f (x )sin 2k πx
L dx , k = 1 , 2,... (5)
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
C ti F i t f
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Fourier series of step function
f (x ) is a periodic function with 2π period and it’s dened asfollows.
f (x ) = 0 , −π < x < 0f (x ) = h, 0 < x < π (6)
Fourier series expansion of f (x ) is
f (x ) = h
2 +
2h
π
sin x
1 +
sin 3x
3 +
sin 5x
5 + ... (7)
f (x ) is piecewise continuous within the periodic region. Fourierseries of f (x ) converges at speed of 1/ n.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Fourier series of step function
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Fourier series of triangular function
f (x ) is a periodic function with 2π period and it’s dened asfollows.
f (x ) = −x , −π < x < 0f (x ) = x , 0 < x < π (8)
Fourier series expansion of f (x ) is
f (x ) = π2 −
4π
n =1 ,3,5...
cos nx n2
(9)
f (x ) is continuous and its derivative is piecewise continuous withinthe periodic region. Fourier series of f (x ) converges at speed of 1/ n2.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Fourier series of triangular function
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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Continuous Fourier transform
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Discrete Fourier transformReferences
Fourier series of full wave rectier
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Discrete Fourier transformReferences
Complex Fourier series
Using Euler’s formula, Eq. 4 becomes
f (x ) = a0 +∞
k =1
ak −ib k 2
e ikx + ak + ib k
2 e −ikx
Let c 0 ≡a0, c k ≡ a k
−ib k
2 and c −k ≡ a k + ib k
2 , we have
f (x ) =∞
m = −∞c m e imx
c m = 12π π
−πf (x )e −imx dx (12)
e imx and e inx are also mutually orthogonal provided n = m and itforms a complete set. Therfore, it can be used as orthogonal basis.Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Discrete Fourier transformReferences
Complex Fourier series
If f (x ) has T period instead of 2π , x is replaced with 2πx / T .
f (x ) = ∞m = −∞
c m e i 2πmx
T
c m = 1
T T
2
−T
2
f (x )e −i 2π mx
T dx , m = 0 , 1, 2... (13)
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transform
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Discrete Fourier transformReferences
Fourier transform
from Eq. 13, we dene variables k ≡ 2πm
T , f̂ (k ) ≡ c m T √ 2π and
k ≡ 2π (m +1)
T − 2πm
T = 2πT .We can have
f (x ) = 1√ 2π
∞
m = −∞f̂ (k )e ikx k
ˆf (k ) =
1
√ 2π T
2
−T
2 f (x )e −ikx
dx
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDi F i f
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Discrete Fourier transformReferences
Fourier transform
Let T −→ ∞f (x ) =
1
√ 2π ∞
−∞f̂ (k )e ikx dk (14)
f̂ (k ) = 1√ 2π ∞−∞ f (x )e −ikx dx (15)
Eq.15 is the Fourier transform of f (x ) and Eq.14 is the inverse Fourier transform of f̂ (k ).
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDi t F i t f
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Discrete Fourier transformReferences
Properties of Fourier transform
f (x ), g (x ) and h(x ) are functions and their Fourier transforms aref̂ (k ), ĝ (k ) and ĥ(k ). a, b x 0 and k 0 are real numbers.
Linearity: If h(x ) = af (x ) + bg (x ), then Fourier transform of h(x ) equals to ĥ(k ) = af̂ (k ) + b ̂g (k ).Translation: If h(x ) = f (x −x 0), then ĥ(k ) = f̂ (k )e −ikx 0Modulation: If h(x ) = e ik 0x f (x ), then ĥ(k ) = f̂ (k −k 0)Scaling: If h(x ) = f (ax ), then ĥ(k ) = 1a f̂ (
k a )
Conjugation: If h(x ) = f ∗(x ), then ĥ(k ) = f̂ ∗(
−k ). With this
property, one can know that if f (x ) is real and thenf̂ ∗(−k ) = f̂ (k ). One can also nd that if f (x ) is real and then|f̂ (k )| = |f̂ (−k )|.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Discrete Fourier transformReferences
Properties of Fourier transform
If f (x ) is even, then f̂ (−k ) = f̂ (k ).If f (x ) is odd, then f̂ (
−k ) =
−f̂ (k ).
If f (x ) is real and even, then f̂ (k ) is real and even.If f (x ) is real and odd, then f̂ (k ) is imaginary and odd.If f (x ) is imaginary and even, then f̂ (k ) is imaginary and even.If f (x ) is imaginary and odd, then f̂ (k ) is real and odd.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Discrete Fourier transformReferences
Dirac delta function
Dirac delta function is a generalized function dened as thefollowing equation.
f (0) =
∞
−∞
f (x )δ (x )dx
∞−∞ δ (x )dx = 1 (16)The Dirac delta function can be loosely thought as a functionwhich equals to innite at x = 0 and to zero else where.
δ (x ) =+ ∞, x = 00, x = 0
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Convolution theory
Considering two functions f (x ) and g (x ) with their Fouriertransform F (k ) and G (k ). We dene an operation
f ∗g = ∞−∞ g (y )f (x −y )dy (18)as the convolution of the two functions f (x ) and g (x ) over theinterval {−∞∼∞}. It satises the following relation:
f ∗g =
∞
−∞
F (k )G (k )e ikx dt (19)
Let h(x ) be f ∗g and ĥ(k ) be the Fourier transform of h(x ), wehaveĥ(k ) = √ 2πF (k )G (k ) (20)
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Parseval relation
∞−∞ f (x )g (x )∗dx = ∞
−∞
1√ 2π ∞−∞ F (k )e ikx dk
1√ 2π
∞
−∞G ∗(k )e −ik x dk dx
= ∞−∞1
2π ∞−∞ F (k )G ∗(k )e i (k −k )x dkdk By using Eq. 17, we have the Parseval’s relation.
∞−∞ f (x )g ∗(x )dx = ∞
−∞F (k )G ∗(k )dk (21)
Calculating inner product of two fuctions gets same result as theinner product of their Fourier transform.
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Cross-correlation
Considering two functions f (x ) and g (x ) with their Fouriertransform F (k ) and G (k ). We dene cross-correlation as
(f g )(x ) = ∞−∞ f ∗(x + y )g (x )dy (22)as the cross-correlation of the two functions f (x ) and g (x ) overthe interval {−∞∼∞}. It satises the following relation: Leth(x ) be f g and ĥ(k ) be the Fourier transform of h(x ), we have
ĥ(k ) = √ 2πF ∗(k )G (k ) (23)Autocorrelation is the cross-correlation of the signal with itself.
(f f )(x ) = ∞−∞ f ∗(x + y )f (x )dy (24)Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Uncertainty principle
One important properties of Fourier transform is the uncertaintyprinciple. It states that the more concentrated f (x ) is, the morespread its Fourier transform f̂ (k ) is.
Without loss of generality, we consider f (x ) as a normalizedfunction which means ∞−∞ |f (x )|2dx = 1, we have uncertainty
relation:
∞
−∞(x
−x 0)2
|f (x )
|2dx
∞
−∞(k
−k 0)2
|f̂ (k )
|2dk
1
16π2 (25)
for any x 0 and k 0 ∈R . [3]
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
R f
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References
Fourier transform of a Gaussian pulse
f (x ) = f 0e − x 2
2σ 2 e ik 0x
f̂ (k ) = 1
√ 2π ∞
−∞f (x )e −
ikx
dx
= f 01/σ 2
e − (k 0 − k )
2
2/σ 2
|̂f (k )|2 ∝ e − (k 0 − k )
2
1/σ 2
Wider the f (x ) spread, the more concentrated f̂ (k ) is.
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R f
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Fourier transform of a Gaussian pulse
Signals with different width.
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Fourier transform of a Gaussian pulse
The bandwidth of the signals are different as well.
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References
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References
Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
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Nyquist critical frequency
Critical sampling of a sine wave is two sample points per cycle.This leads to Nyquist critical frequency f c .
f c = 12∆
(26)
In above equation, ∆ is the sampling interval.Sampling theorem: If a continuos signal h(t ) sampled with interval∆ happens to be bandwidth limited to frequencies smaller than f c .h(t ) is completely determined by its samples hn . In fact, h(t ) isgiven by
h(t ) = ∆∞
n = −∞hn
sin[2π f c (t −n∆)]π (t −n∆)
(27)
It’s known as Whittaker - Shannon interpolation f ormula.Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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Discrete Fourier transform
Signal h(t ) is sampled with N consecutive values and samplinginterval ∆. We have hk ≡h(t k ) and t k ≡k ∗∆,k = 0 , 1, 2, ..., N
−1.
With N discrete input, we evidently can only output independentvalues no more than N. Therefore, we seek for frequencies withvalues
f n
≡ n
N ∆, n =
−N
2 , ...,
N
2 (28)
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Discrete Fourier transform
Fourier transform of Signal h(t ) is H (f ). We have discrete Fouriertransform H n .
H (f n ) = ∞−∞ h(t )e −i 2π f n t dt ≈∆N −1
k =0
hk e −i 2π f n t k
= ∆N
−1
k =0
hk e −i 2π kn / N
H n ≡N −1
k =0
hk e −i 2π kn / N (29)
Inverse Fourier transform is
hk ≡ 1
N
N −1
n =0
H n e i 2π kn / N (30)
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Periodicity of discrete Fourier transform
From Eq.29, if we substitute n with n + N , we have H n = H n + N .Therefore, discrete Fourier transform has periodicity of N .
H n + N =N −1
k =0
hk e −i 2π k (n + N )/ N
=N −1
k =0
hk e −i 2π k (n )/ N e −i 2π kN / N
= H n (31)
Critical frequency f c corresponds to 12∆ .We can see that discrete Fourier transform has f s period wheref s = 1 / ∆ = 2 ∗f c is the sampling frequency.
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Aliasing
If we have a signal with its bandlimit larger than f c , we havefollowing spectrum due to periodicity of DFT.
Aliased frequency is f −N ∗f s where N is an integer.
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Aliasing example: original signal
Let’s say we have a sinusoidal sig-nal of frequency 0.05. The sampling interval is 1. We have the signal
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Aliasing example: spectrum of original signal
and we have its spectrum
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Aliasing example: critical sampling of original signal
The critical sampling interval of the original signal is 10 which ishalf of the signal period.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: under sampling of original signal
If we sampled the original sinusiodal signal with period 12, aliasinghappens.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: DFT of under sampled signal
f c of downsampled signal is 12∗
12 , aliased frequency isf −2∗f c = −0.03333 and it has symmetric spectrum due to realsignal.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: two frequency signal
Let’s say we have a signal containing two sinusoidal signal of frequency 0.05 and 0.0125. The sampling interval is 1. We havethe signal
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: spectrum of two frequency signal
and we have its spectrum
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: downsampled two frequency signal
Doing same undersampling with interval 12.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Aliasing example: DFT of downsampled signal
We have the spectrum of downsampled signal.
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Filtering
We now want to get one of the two frequency out of the signal.Wewill adapt a proper rectangular window to the spectrum.Assuming we have a lter function w (f ) and a multi-frequencysignal f (t ), we simply do following steps to get the frequency bandwe want.
F −1{w (f )F{f (t )}} (32)
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Filtering example: ltering window and signal spectrum
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
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Filtering example: ltered signal
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
l
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Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
R f
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References
Supplementary Notes of General Physics by Jyhpyng Wang,http://idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdf
http://en.wikipedia.org/wiki/Fourier_series
http://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Aliasing
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
MATHEMATICAL METHODS FOR PHYSICISTS by GeorgeB. Arfken and Hans J. Weber. ISBN-13: 978-0120598762Numerical Recipes 3rd Edition: The Art of ScienticComputing by William H. Press (Author), Saul A. Teukolsky.ISBN-13: 978-0521880688Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transformReferences
R f
http://idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdfhttp://idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdfhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Aliasinghttp://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theoremhttp://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/[email protected]://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theoremhttp://en.wikipedia.org/wiki/Aliasinghttp://en.wikipedia.org/wiki/Fourier_transformhttp://en.wikipedia.org/wiki/Fourier_serieshttp://idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdfhttp://idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdfhttp://find/
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References
Chapter 12 and 13 in
http://www.nrbook.com/a/bookcpdf.phphttp://docs.scipy.org/doc/scipy-0.14.0/reference/fftpack.html
http://docs.scipy.org/doc/scipy-0.14.0/reference/signal.html
Zong-han, Xi [email protected] Introduction to Fourier transform and signal analysis
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