Fourier Transform a Notes

8/19/2019 Fourier Transform a Notes

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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, Xi [email protected] Introduction to Fourier transform and signal analysis

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References

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.


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References

Outline

1 Continuous Fourier transform

2 Discrete Fourier transform

3 References


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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.




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References

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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References

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)


C i i f



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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 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)


C ti F i t f



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References

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.


Continuous Fourier transform



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References

Fourier series of step function





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References

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.





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References

Fourier series of triangular function




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Discrete Fourier transformReferences

Fourier series of full wave rectier





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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




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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 = −∞

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)





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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


Continuous Fourier transformDi F i f



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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 ).


Continuous Fourier transformDi t F i t f



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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 )|.



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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.





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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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References

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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References

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




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References

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]



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.



R f



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References


Signals with different width.



References



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References


The bandwidth of the signals are different as well.



References



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Outline



3 References



References



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References

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


References



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References

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)



References



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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)


Continuous Fourier transformDiscrete Fourier transformReferences



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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.





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Aliasing example: under sampling of original signal

If we sampled the original sinusiodal signal with period 12, aliasinghappens.





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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.





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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





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Aliasing example: spectrum of two frequency signal

and we have its spectrum





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Aliasing example: downsampled two frequency signal

Doing same undersampling with interval 12.





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Aliasing example: DFT of downsampled signal

We have the spectrum of downsampled signal.





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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)





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Filtering example: ltering window and signal spectrum





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Filtering example: ltered signal



l



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Outline



3 References



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


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

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Fourier Transform a Notes