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Continuous Fourier transformDiscrete Fourier transform
References
Introduction to Fourier transform and signalanalysis
Zong-han, [email protected]
January 7, 2015
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
License of this document
Introduction to Fourier transform and signal analysis by Zong-han,Xie ([email protected]) is licensed under a Creative CommonsAttribution-NonCommercial 4.0 International License.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
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, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Complete and orthogonal basis
cos nx and sinmx are mutually orthogonal in which n and mare integers. ∫ π
−πcos nx · sinmxdx = 0∫ π
−πcos nx · cosmxdx = πδnm∫ π
−πsin nx · sinmxdx = πδnm (3)
δnm is Dirac-delta symbol. It means δnn = 1 and δnm = 0when n 6= m.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series
Since cos nx 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 + bk sin kx)
a0 =1
2π
∫ π
−πf (x)dx
ak =1
π
∫ π
−πf (x) cos kxdx
bk =1
π
∫ π
−πf (x) sin kxdx (4)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series
If f (x) has L period instead of 2π, x is replaced with πx/L.
f (x) = a0 +∞∑k=1
(ak cos
2kπx
L+ bk sin
2kπx
L
)
a0 =1
L
∫ L2
− L2
f (x)dx
ak =2
L
∫ L2
− L2
f (x) cos2kπx
Ldx , k = 1, 2, ...
bk =2
L
∫ L2
− L2
f (x) sin2kπx
Ldx , k = 1, 2, ... (5)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of step function
f (x) is a periodic function with 2π period and it’s defined asfollows.
f (x) = 0,−π < x < 0
f (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, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of step function
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of triangular function
f (x) is a periodic function with 2π period and it’s defined asfollows.
f (x) = −x ,−π < x < 0
f (x) = x , 0 < x < π (8)
Fourier series expansion of f (x) is
f (x) =π
2− 4
π
∑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 of1/n2.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of triangular function
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of full wave rectifier
f (t) is a periodic function with 2π period and it’s defined asfollows.
f (t) = − sinωt,−π < t < 0
f (t) = sinωt, 0 < t < π (10)
Fourier series expansion of f (x) is
f (t) =2
π− 4
π
∑n=2,4,6...
(cos nωt
n2 − 1
)(11)
f (x) is continuous and its derivative is piecewise continuous withinthe periodic region. Fourier series of f (x) converges at speed of1/n2.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier series of full wave rectifier
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Complex Fourier series
Using Euler’s formula, Eq. 4 becomes
f (x) = a0 +∞∑k=1
(ak − ibk
2e ikx +
ak + ibk2
e−ikx)
Let c0 ≡ a0, ck ≡ ak−ibk2 and c−k ≡ ak+ibk
2 , we have
f (x) =∞∑
m=−∞cme
imx
cm =1
2π
∫ π
−πf (x)e−imxdx (12)
e imx and e inx are also mutually orthogonal provided n 6= m and itforms a complete set. Therfore, it can be used as orthogonal basis.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Complex Fourier series
If f (x) has T period instead of 2π, x is replaced with 2πx/T .
f (x) =∞∑
m=−∞cme
i 2πmxT
cm =1
T
∫ T2
−T2
f (x)e−i2πmxT dx ,m = 0, 1, 2... (13)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Fourier transform
from Eq. 13, we define variables k ≡ 2πmT , f (k) ≡ cmT√
2πand
4k ≡ 2π(m+1)T − 2πm
T = 2πT .
We can have
f (x) =1√2π
∞∑m=−∞
f (k)e ikx 4 k
f (k) =1√2π
∫ T2
−T2
f (x)e−ikxdx
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier transform
Let T −→∞
f (x) =1√2π
∫ ∞−∞
f (k)e ikxdk (14)
f (k) =1√2π
∫ ∞−∞
f (x)e−ikxdx (15)
Eq.15 is the Fourier transform of f (x) and Eq.14 is the inverseFourier transform of f (k).
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Properties of Fourier transform
f (x), g(x) and h(x) are functions and their Fourier transforms aref (k), g(k) and h(k). a, b x0 and k0 are real numbers.
Linearity: If h(x) = af (x) + bg(x), then Fourier transform ofh(x) equals to h(k) = af (k) + bg(k).
Translation: If h(x) = f (x − x0), then h(k) = f (k)e−ikx0
Modulation: If h(x) = e ik0x f (x), then h(k) = f (k − k0)
Scaling: If h(x) = f (ax), then h(k) = 1a f (ka )
Conjugation: If h(x) = f ∗(x), then h(k) = f ∗(−k). With thisproperty, one can know that if f (x) is real and thenf ∗(−k) = f (k). One can also find that if f (x) is real and then|f (k)| = |f (−k)|.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Properties of Fourier transform
If f (x) is even, then f (−k) = f (k).
If f (x) is odd, then f (−k) = −f (k).
If f (x) is real and even, then f (k) is real and even.
If f (x) is real and odd, then f (k) is imaginary and odd.
If f (x) is imaginary and even, then f (k) is imaginary and even.
If f (x) is imaginary and odd, then f (k) is real and odd.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Dirac delta function
Dirac delta function is a generalized function defined 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 infinite at x = 0 and to zero else where.
δ(x) =
{+∞, x = 0
0, x 6= 0
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Dirac delta function
From Eq.15 and Eq.14
f (k) =1
2π
∫ ∞−∞
∫ ∞−∞
f (k ′)e ik′xdk ′e−ikxdx
=1
2π
∫ ∞−∞
∫ ∞−∞
f (k ′)e i(k′−k)xdxdk ′
Comparing to ”Dirac delta function”, we have
f (k) =
∫ ∞−∞
f (k ′)δ(k ′ − k)dk ′
δ(k ′ − k) =1
2π
∫ ∞−∞
e i(k′−k)xdx (17)
Eq.17 doesn’t converge by itself, it is only well defined as part ofan integrand.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Convolution theory
Considering two functions f (x) and g(x) with their Fouriertransform F (k) and G (k). We define 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 satisfies the following relation:
f ∗ g =
∫ ∞−∞
F (k)G (k)e ikxdt (19)
Let h(x) be f ∗ g and h(k) be the Fourier transform of h(x), wehave
h(k) =√
2πF (k)G (k) (20)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Parseval relation
∫ ∞−∞
f (x)g(x)∗dx =
∫ ∞−∞
1√2π
∫ ∞−∞
F (k)e ikxdk1√2π∫ ∞
−∞G ∗(k ′)e−ik
′xdk ′dx
=
∫ ∞−∞
1
2π
∫ ∞−∞
F (k)G ∗(k ′)e i(k−k′)xdkdk ′
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.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Cross-correlation
Considering two functions f (x) and g(x) with their Fouriertransform F (k) and G (k). We define 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 satisfies the following relation: Leth(x) be f ? g and h(k) be the Fourier transform of h(x), we have
h(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, [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 f (k) is.Without loss of generality, we consider f (x) as a normalizedfunction which means
∫∞−∞ |f (x)|2dx = 1, we have uncertainty
relation:(∫ ∞−∞
(x − x0)2|f (x)|2dx)(∫ ∞
−∞(k − k0)2|f (k)|2dk
)=
1
16π2(25)
for any x0 and k0 ∈ R. [3]
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Fourier transform of a Gaussian pulse
f (x) = f0e−x2
2σ2 e ik0x
f (k) =1√2π
∫ ∞−∞
f (x)e−ikxdx
=f0
1/σ2e−(k0−k)2
2/σ2
|f (k)|2 ∝ e−(k0−k)2
1/σ2
Wider the f (x) spread, the more concentrated f (k) is.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Fourier transform of a Gaussian pulse
Signals with different width.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Fourier transform of a Gaussian pulse
The bandwidth of the signals are different as well.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Nyquist critical frequency
Critical sampling of a sine wave is two sample points per cycle.This leads to Nyquist critical frequency fc .
fc =1
2∆(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 fc .h(t) is completely determined by its samples hn. In fact, h(t) isgiven by
h(t) = ∆∞∑
n=−∞hn
sin[2πfc(t − n∆)]
π(t − n∆)(27)
It’s known as Whittaker - Shannon interpolation formula.
Zong-han, [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(tk) and tk ≡ 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
fn ≡n
N∆, n = −N
2, ...,
N
2(28)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
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Discrete Fourier transform
Fourier transform of Signal h(t) is H(f ). We have discrete Fouriertransform Hn.
H(fn) =
∫ ∞−∞
h(t)e−i2πfntdt ≈ ∆N−1∑k=0
hke−i2πfntk
= ∆N−1∑k=0
hke−i2πkn/N
Hn ≡N−1∑k=0
hke−i2πkn/N (29)
Inverse Fourier transform is
hk ≡ 1
N
N−1∑n=0
Hnei2πkn/N (30)
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Periodicity of discrete Fourier transform
From Eq.29, if we substitute n with n + N, we have Hn = Hn+N .Therefore, discrete Fourier transform has periodicity of N.
Hn+N =N−1∑k=0
hke−i2πk(n+N)/N
=N−1∑k=0
hke−i2πk(n)/Ne−i2πkN/N
= Hn (31)
Critical frequency fc corresponds to 12∆ .
We can see that discrete Fourier transform has fs period wherefs = 1/∆ = 2 ∗ fc is the sampling frequency.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Aliasing
If we have a signal with its bandlimit larger than fc , we havefollowing spectrum due to periodicity of DFT.
Aliased frequency is f − N ∗ fs where N is an integer.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
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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
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Aliasing example: spectrum of original signal
and we have its spectrum
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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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, [email protected] Introduction to Fourier transform and signal analysis
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Aliasing example: under sampling of original signal
If we sampled the original sinusiodal signal with period 12, aliasinghappens.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
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Aliasing example: DFT of under sampled signal
fc of downsampled signal is 12∗12 , aliased frequency is
f − 2 ∗ fc = −0.03333 and it has symmetric spectrum due to realsignal.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Aliasing example: two frequency signal
Let’s say we have a signal containing two sinusoidal signal offrequency 0.05 and 0.0125. The sampling interval is 1. We havethe signal
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Aliasing example: spectrum of two frequency signal
and we have its spectrum
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Aliasing example: downsampled two frequency signal
Doing same undersampling with interval 12.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Aliasing example: DFT of downsampled signal
We have the spectrum of downsampled signal.
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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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 filter 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, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Filtering example: filtering window and signal spectrum
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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Filtering example: filtered signal
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References
Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
References
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_transform
http://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: 978-0120598762
Numerical Recipes 3rd Edition: The Art of ScientificComputing by William H. Press (Author), Saul A. Teukolsky.ISBN-13: 978-0521880688Zong-han, [email protected] Introduction to Fourier transform and signal analysis
Continuous Fourier transformDiscrete Fourier transform
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References
Chapter 12 and 13 inhttp://www.nrbook.com/a/bookcpdf.php
http://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, [email protected] Introduction to Fourier transform and signal analysis