1. Concise guide on numerical methods
Grzegorz Pytel, Warsaw University of Technology Fourier Transform and its applications
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

2. Fourier Transform and its application
Talk scheme 1
Theoritcal
introduction 2
Numerical
methods 3
Applications
Fourier transform and its application in review
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Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

3. Theoretical introduction1
Theoretical introduction t
Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

4. Fourier Transform and its application
Fourier analysis
Symmetry
Periodic
Phenomena
Linear
Numerical methods
Group theory
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Fourier Transform and its application

5. Periodicity Periodicity sin⁡(2Π𝑡) Period 1 Freq 1 In space In time
Number of repetitions of a patttern in a second
Uses period. Measurement of how big the pattern is that repeats
Periodicity arises from symmetry
sin⁡(6Π𝑡)
Period 1 3
Freq 3 t?
You’re fixed in a period. Period of sum=1
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Fourier Transform and its application

6. Fourier Transform and its application
𝐿 𝑏𝑐 1 = 𝐹:𝑅 𝐶,𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠,𝑏𝑜𝑢𝑛𝑑𝑒𝑑, 𝑓 1 = −∞ ∞ 𝑓(𝑥) 𝑑𝑥<∞
Theorem:
𝑓,𝑔∈ 𝐿 𝑏𝑐 1 𝑡ℎ𝑒𝑛 𝑐𝑜𝑛𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓∗𝑔 𝑥 = −∞ ∞ 𝑓 𝑦 𝑔 𝑥−𝑦 𝑑𝑦 𝑎𝑛𝑑 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 ∀𝑥∈𝑅:
𝑓∗𝑔∈ 𝐿 𝑏𝑐 1 𝑅
f*g=g*f
Iff ℎ∈ 𝐿 𝑏𝑐 1 𝑅 𝑡ℎ𝑒𝑛
𝑓∗ 𝑔∗ℎ = 𝑓∗𝑔 ∗ℎ
f*(g+h)=f*g+f*h
Definition
If 𝑓∈ 𝐿 𝑏𝑐 1 𝑅 then we define Fourier Transform as:
𝑓 𝑦 = −∞ ∞ 𝑓(𝑥) 𝑒 −2Π𝑖𝑦𝑥 𝑑𝑥
Time
Frequency
More, including properties:
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Fourier Transform and its application

7. Discrete Fourier Transform
Let 𝑎 𝑖 ∈𝑅,𝑖=0,..,𝑁−1 be discretized values of signal
DFT: 𝑎 0 , 𝑎 1 ,.., 𝑎 𝑁− 𝐴 0 , 𝐴 1 ,.., 𝐴 𝑁−1 𝑤ℎ𝑒𝑟𝑒 𝐴 𝑖 ∈𝐶 𝑎𝑛𝑑 𝐴 𝑘 = 𝑛=0 𝑁−1 𝑎 𝑛 𝜔 𝑁 −𝑘𝑛 ,𝑘=0,..,𝑁−1
Where 𝜔 𝑁 = 𝑒 2Π𝑖 𝑁
𝐴 𝑘 = 𝑛=0 𝑁−1 𝑎 𝑛 𝑒 − 2Π𝑖𝑘𝑛 𝑁 ,𝑘=0,..,𝑁−1
Inverse Discrete Fourier Transform:
𝑎 𝑛 = 1 𝑁 𝑘=0 𝑁−1 𝐴 𝑘 𝜔 𝑁 𝑘𝑛 ,𝑛=0,..,𝑁−1
Complexity: 𝑂( 𝑁 2 )
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Fourier Transform and its application

8. Fast Fourier Transform
Wolfgang Bangerth, Guido Kanschat, and Ralf Hartmann, a software library for computational solution of partial differential equations using adaptive finite elements
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.
FFTs was constructed by J. W. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer
2007
Jonathan Shewchuk for Triangle, a two-dimensional mesh generator and Delaunay Triangulator
2003
1999
Chris Bischof and Alan Carle for ADIFOR 2.0, an automatic differentiation tool for Fortran 77 programs
1995
On Thursday, 8 July 1999, the third Wilkinson Prize
for Numerical Software was awarded to Matteo Frigo
and Steven Johnson of Massachusetts Institute of Technology. The winning entry, FFTW (the "Fastest Fourier Transform in the West"), is a library of C routines for the efficient computation of the discrete Fourier transform of real and complex data. The unparalleled efficiency over a wide range of computer platforms is by automatically determining the best computational strategy for the particular hardware.
Linda Petzold for DASSL, a differential algebraic equation solver
1991
Complexity: 𝑂(𝑁𝑙𝑜𝑔𝑁)
Wilkinson Prize axis
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Fourier Transform and its application

9. Fast Fourier Transform – Danielson Lenczos
Procedure FFT(N,ω,a(x))
If N=1 then 𝐴 0 𝑎 0
Else {
𝑏 𝑥 = 𝑖=0 𝑁 2 −1 𝑎 2𝑖 𝑥 𝑖 ;𝑐 𝑥 𝑖=0 𝑁 2 −1 𝑎 2𝑖+1 𝑥 𝑖
𝐵 𝐹𝐹𝑇 𝑁 2 , 𝜔 2 ,𝑏 𝑥 ;𝐶 𝐹𝐹𝑇 𝑁 2 , 𝜔 2 ,𝑐 𝑥
for i from 0 to 𝑁 2 −1 do {
𝐴 𝑖 𝐵 𝑖 + 𝜔 𝑖 𝐶 𝑖
𝐴 𝑁 2 +𝑖 𝐵 𝑖 + 𝜔 𝑖 𝐶 𝑖 } }
Return 𝐴 0 , 𝐴 1 ,…, 𝐴 𝑁−1
end
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Fourier Transform and its application

10. Fourier Transform and its application2
Numerical methods
World’s major methods used for computing FFT
Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

11. Fast Fourier Transform
Cooley–Tukey
Prime-factor
Bruun's
Rader's
Bluestein's
Winogard
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Fourier Transform and its application

12. Fourier Transform and its application
Cooley-Tukley
Radix 2-Decimation in Time, 1965
𝑋 𝑘=0..𝑁−1 = 𝑚=0 𝑁 2 −1 𝑥 2𝑚 𝑒 − 2Π𝑖 𝑁 2 2𝑚 𝑘 + 𝑒 − 2Π𝑖 𝑁 𝑘 𝑚=0 𝑁 2 −1 𝑥 2𝑚+1 𝑒 − 2Π𝑖 𝑁 2 𝑚𝑘
Danielson-Lanczos Lemma, 1942
C-T C source code:
Twiddle factor
The discrete Fourier transform of length (where is even) can be rewritten as the sum of two discrete Fourier transforms, each of length N/2. One is formed from the even-numbered points; the other from the odd-numbered points. Denote the th point of the discrete Fourier transform by . Then
𝐹 𝑛 = 𝑘=0 𝑁−1 𝑓 𝑘 𝑒 − 2Π𝑖𝑛 𝑁 = 𝑘=0 𝑁 2 𝑒 − 2Π𝑖𝑘𝑛 𝑁 𝑓 2𝑘 + 𝑊 𝑛 𝑘=0 𝑁 2 −1 𝑒 − 2Π𝑖𝑘𝑛 𝑁 2 𝑓 2𝑘+1 = 𝐹 𝑛 𝑒 + 𝑊 𝑛 𝐹 𝑛 0
More:
Danielson-Lanczos
Cooley-Tukley
Time [min]
140
0.02
Task’s size
64 DFT 3–5 significant digits, real numbers
2048 DFT 8 digits
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Fourier Transform and its application

13. Fourier Transform and its application
Cooley-Tukley
Can only be used to speed the calculation of DFTs of a size that is a power of two
Requires padding with 0
extra multiplications by roots of unity called twiddle factors
NlogN complexity
More inc. sourcecode at:
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Fourier Transform and its application

14. Cooley-Tukley general factorizations
DCT
DFT 1
𝑁 2 2
Transpose
Multiply by N twiddle factors
𝑁 1
More info and Python sourcecode:
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Fourier Transform and its application

15. Prime factor (Good-Thomas algorithm)
DFT:
Is a FFT that re-expresses the DFT of a 𝑁= 𝑁 1 𝑁 2 as two-dimensional 𝑁 1 𝑥 𝑁 2 DFT, but only for the case where 𝑁 1 and 𝑁 2 are relatively prime
𝑥 𝑘 = 𝑛=0 𝑁−1 𝑥 𝑛 𝑒 − 2Π𝑖 𝑁 𝑛𝑘 ,𝑘=0,…,𝑁−1
𝑛= 𝑛 1 𝑁 2 + 𝑛 2 𝑁 1 𝑚𝑜𝑑 𝑁
𝑘= 𝑘 1 𝑁 2 −1 𝑁 2 + 𝑘 2 𝑁 1 −1 𝑁 1
𝑋 𝑘 1 𝑁 2 −1 𝑁 2 + 𝑘 2 𝑁 1 −1 𝑁 1 = 𝑛 1 =0 𝑁 1 −1 𝑛 2 =0 𝑁 2 −1 𝑥 𝑛 1 𝑁 2 + 𝑛 2 𝑁 1 𝑒 − 2Π𝑖 𝑁 2 𝑛 2 𝑘 2 𝑒 − 2Π𝑖 𝑁 1 𝑛 1 𝑘 1
More at:
Example at:
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Fourier Transform and its application

16. Fourier Transform and its application
Bruun’s
Only real coefficients until the last computation stage
May be less accurate than Cooley–Tukey in the face of finite numerical precision (Storn, 1993)
𝜔 𝑁 𝑛 = 𝑛 𝑘 𝑒 − 2Π𝑖 𝑁 𝑛
𝑥 𝑧 = 𝑛=0 𝑁−1 𝑥 𝑛 𝑧 𝑛
x(z) – polynomial whose coefficients are 𝑥 𝑛
𝑋 𝑘 =𝑥 𝜔 𝑁 𝑘 =𝑥 𝑧 𝑚𝑜𝑑 𝑧− 𝜔 𝑁 𝑘
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Fourier Transform and its application

17. Fourier Transform and its application
Rader’s
Computes DFT of prime sizes by re-expressing the DFT as a cyclic convolution
Is a one-dimensional index-mapping scheme that turns a length-N DFT (N prime) into a length-(N−1) convolution and a few additions. Rader's conversion works only for prime-length N.
Most efficient for prime-length N
More at:
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Fourier Transform and its application

18. Fourier Transform and its application
Rader’s
Mapping 𝑘= 𝑟 𝑚 𝑚𝑜𝑑𝑁,𝑚=0,..,𝑁−2, N-prime, uniquely generates all elements k=[1,..,N-1]
N=5,r=2
2 0 𝑚𝑜𝑑5=1
2 1 𝑚𝑜𝑑5=2
2 2 𝑚𝑜𝑑5=4
2 3 𝑚𝑜𝑑5=3
N-prime, the inverse of r 𝑟 −1 𝑟𝑚𝑜𝑑𝑁=1
N=5,r=2, 𝑟 −1 =3
2𝑥3𝑚𝑜𝑑5=1
3 0 𝑚𝑜𝑑5=1
3 1 𝑚𝑜𝑑5=3
3 2 𝑚𝑜𝑑5=4
3 3 𝑚𝑜𝑑5=2
𝑚𝑛 𝑚=[0,..,𝑁−2 ]⋀𝑛=[1,..,𝑁−1])(𝑛= 𝑟 −𝑚 𝑚𝑜𝑑𝑁)
𝑝𝑘 𝑝=[0,..,𝑁−2 ]⋀𝑘=[1,..,𝑁−1])(𝑘= 𝑟 −𝑝 𝑚𝑜𝑑𝑁)
𝑋 𝑘 = 𝑛=0 𝑁−1 𝑥 𝑛 𝑊 𝑁 𝑛𝑘 = 𝑥 0 + 𝑛=1 𝑁−1 𝑥 𝑛 𝑊 𝑁 𝑛𝑘 𝑖𝑓 𝑘≢0 𝑛=0 𝑁−1 𝑥 𝑛 𝑖𝑓 𝑘=0
𝑊 𝑁 𝑛𝑘 = 𝑒 − 𝑖 2Π𝑖𝑛𝑘 𝑁
For 𝑘≢0:
𝑋 𝑟 𝑝 𝑚𝑜𝑑𝑁 = 𝑚=0 𝑁−2 𝑥 𝑟 −𝑚 𝑚𝑜𝑑𝑁 𝑊 𝑟𝑃𝑟−𝑚 +𝑥 0 = 𝑥 0 +𝑥 𝑟 −𝑙 𝑚𝑜𝑑𝑁 ∗ 𝑊 𝑟 𝑙 ,𝑙=0,1,..,𝑁−2
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Fourier Transform and its application

19. Fourier Transform and its application
Bluestein’s (1968)
Computes FFT at arbitrary sizes by re-expressing the DFT as a convolution
can be performed with a pair of FFTs (plus the pre-computed FFT of bn) via the convolution theorem. The key point is that these FFTs are not of the same length N: such a convolution can be computed exactly from FFTs only by zero-padding it to a length greater than or equal to 2N–1
O(N log N)
𝑋 𝑘 = 𝑛=0 𝑁−1 𝑥 𝑛 𝑒 − 2Π𝑖 𝑁 𝑛𝑘 ,𝑘=0,..,𝑁−1
𝑛𝑘= − 𝑘−𝑛 2 + 𝑛 2 + 𝑘 2 2
𝑋 𝑘 = 𝑒 − Π𝑖 𝑁 𝑘 2 𝑛=0 𝑁−1 𝑥 𝑛 𝑒 − Π𝑖 𝑁 𝑛 2 𝑒 Π𝑖 𝑁 (𝑘−𝑛) 2 ,𝑘=0,..,𝑁−1
𝑎 𝑛 = 𝑥 𝑛 𝑒 − Π𝑖 𝑁 𝑛 2
𝑏 𝑛 = 𝑒 Π𝑖 𝑁 𝑛 2
𝑋 𝑘 = 𝑏 𝑘 𝑛=0 𝑁−1 𝑎 𝑛 𝑏 𝑘−𝑛 ,𝑘=0,..,𝑁−1
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Fourier Transform and its application

20. Fourier Transform and its application
Winogard
𝑋= 𝐹 1 𝐹 2 𝑥
𝐹= 𝐴 𝑇 𝐷𝐴
𝑋= 𝐴 2 𝑇 𝐷 2 𝐴 2 𝐴 1 𝑇 𝐷 1 𝐴 1 𝑥
X= 𝐴 2 𝑇 𝐴 1 𝑇 𝐷 2 𝐷 1 𝐴 2 𝐴 1 𝑥
𝐷= 𝐷 2 𝐷 1
𝑋= 𝐴 1 𝑇 𝐴 2 𝑇 𝐷 𝐴 2 𝐴 1 𝑥
DFT’s row
DFT’s column
Winogard need O(N) multiplications to compute the FFT. However, at the cost of asymptotically many more additions, and thus
they represent a tradeoff that is typically unfavorable on modern CPUs
with hardware multipliers.
Fastest, smallest number of operations
Extremely efficient for small N (N<20)
The number of adds becomes huge for large N
Most difficult
More at:
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Fourier Transform and its application

21. Industrial applications3
Industrial applications
Examples of industrial applications of FFT and DFT
Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

22. Industrial products/companies reliable on FFT
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Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

23. Fourier Transform and its application
Voice power in matlab
DEMO
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Fourier Transform and its application

24. Fourier Transform and its application
DEMO Source code
function [n,fs,waveData] = waveRecordFFT()
n=20000;
fs=11025;
waveData=wavrecord(n,fs);
wavplay(waveData,fs);
frequency(waveData,n,fs);
End
function[] = frequency(waveData,n,fs)
s1=waveData(:,1);
p=fft(s1);
nUniquePts=ceil((n+1)/2);
p=p(1:nUniquePts);
p=abs(p);
p=p/n;
p=p.^2;
if rem(n,2)
p(2:end) = p(2:end)*2;
else
p(2:end -1) = p(2:end - 1)*2;
end
freqArray=(0:nUniquePts-1)*(fs/n);
plot(freqArray/1000,10*log10(p),'k')
xlabel('Frequency (kHz)')
ylabel('Power (dB)')
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Fourier Transform and its application

25. Fourier Transform and its application
DIGITIZING NATURE
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Fourier Transform and its application
Fourier Transform and its applications. Grzegorz Pytel. Warsaw University of Technology

26. Fourier Transform and its application
End remarks
FFT length
Multiplies
Adds
Mults+Adds
Radix 2
1024
10248
30728
40976
Split radix
7172
27652
34842
Prime factor
1008
5804
29100
34904
Winograd
3548
34416
37964
The most commonly used FFT algorithms by far are the power-of-two-length FFT algorithms
Prime Factor and Winograd require somewhat fewer multiplies, but the overall difference usually isn't sufficient to warrant the extra difficulty
Changing radix can speed up, up to %
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Fourier Transform and its application

27. Fourier Transform and its application
Bibliography
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Fourier Transform and its application