Algorithms of Scientific Computing

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1 Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Michael Bader Technical University of Munich Summer 2018

2 Fast Fourier Transform Outline Discrete Fourier transform Fast Fourier transform Special Fourier transform: real-valued FFT sine/cosine transform Applications: Fast Poisson solver (FST) Computergraphics (FCT) Efficient Implementation Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

3 Discrete Fourier Transform (DFT) Definition: For a vector of complex numbers (f 0,..., f 1 ) T, the discrete Fourier transform (DFT) is given by the vector (F 0,..., F 1 ) T, where Interpretation: The DFT can be derived as F k = 1 1 f n e i2πnk/. n=0 trigonometric interpolation/approxmiation approximation of the coefficients of the Fourier series Applications: JPEG, MPEG, MP3, etc. signal processing in general, solvers in science and engineering Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

4 DFT as Interpolation (1) Interpolation problem: ansatz functions: g k (x) := e ikx in the interval [0, 2π], k = 0,..., 1 supporting points: x n := 2πn/, n = 0,..., 1 interpolation value f n, n = 0,..., 1 find weights F k such that at all supporting points 1 1 f n = F k g k (x n ) f n = F k e i2πnk/. trigonometric interpolation Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

5 DFT as Interpolation (2) Interpolation problem: ansatz functions: g k (z) := z k (complex unit polynomials), k = 0,..., 1 supporting points: z n := e i2πn/ = ω n, where ω := e i2π/ interpolation values f n, n = 0,..., 1, respectively. find the weights F k such that at all supporting points 1 1 f n = F k g k (z n ) f n = F k e i2πnk/. Polynomial interpolation at the complex unit roots ω n Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

6 Interpretation of the Interpolation Problem Starting from the first formulation, 1 f n = F k g k (x n ), g k (x n ) = e i2πnk/, we look for a representation of the signal f n or of a function f (x) of the form 1 f (x) = F k g k (x), g k (x) = e i2πkx. The ansatz functions are sine or cosine oscillations: e ikx = cos(kx) + i sin(kx) Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

7 Interpretation of the Interpolation Problem (2) Conclusions: we look for the representation of a periodic function as a sum of sines and cosines the F k are, thus, called Fourier coefficients: k represents the wave number the value of F k represents the amplitude of the corresponding frequency the Fourier transform leads to a frequency spectrum useful when a problem is easier to solve in the frequency domain than in the spatial domain. Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

8 Example: Spatial vs. Frequency Domain 3D Kurve des Signals s 1 3D Kurve des Signals s 2 3D Kurve des Signals s 3 Im(s 1 ) Im(s 2 ) Im(s 3 ) t Werte Re(s ) t Werte Re(s 1 2 ) t Werte Re(s ) 3 s 1 = e 3it, s 2 = 0.2 i e it, s 3 = e it e 12it Frequenzspektrum f Frequenzspektrum f 2 3 Frequenzspektrum f 1 Anteile Anteile Anteile Frequenzen Frequenzen Frequenzen Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

9 Solution of the Interpolation Problem Both interpolation problems lead to the identical linear systems of equations: 1 f n = F k ω nk, for all n = 0,..., 1; where ω := e i2π/, i.e. ω nk := ei2πnk/ ( unit roots : ω = 1) If we write the vectors of the f n and F k as f := (f 0,..., f 1 ) and F := (F 0,..., F 1 ), the linear system of equations can be formulated in matrix-vector notation WF = f, where the entries of the Fourier matrix W are given by W nk := ω nk. Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

10 Properties of the Fourier Matrix W W is symmetric: W = W T, and has the form W = ω 0 ω 0 ω 0... ω 0 ω 0 ω 1 ω 2... ω ( 1) ω 0 ω 2 ω 4... ω 2( 1). ω 0. ω ( 1) W ( W T ) = WW H = I, since [ WW H] 1 ( = ω kj ω lj kl.. ω 2( 1)... ω ( 1)( 1) ) 1 = { ω (k l)j if k = l = 0 if k l. Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

11 Properties of the Fourier Matrix W (details) important property #1: (ω k ) = e i2πk/ = e i2πk = 1 important property #2: (ω k ) = ( e i2πk/) = (cos(2πk/) + i sin(2πk/)) = cos(2πk/) i sin(2πk/) = cos( 2πk/) + i sin( 2πk/) = e i2πk/ = ω k W ( W T ) = WW H = I, since if k = l then 1 if k l then 1 [ WW H] 1 ( = ω kj ω lj kl ω (k l)j ω (k l)j 1 (1 ξ) = 1 ω 0 = = 1 1 ξ j = ) 1 = ξ j where ξ = ω (k l) 1 ξ j ω kj ω lj and then: ξ (j+1) = 1 ξ, thus Remember that (ω k ) = 1 for any k ( unit roots!), thus 1 1 = ω (k l)j 1 ξ j = 1 ξ 1 ξ ω (k l)j = 0 Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

12 Computation of the Fourier Coefficients F k Since WW H = I, the inverse of W is W 1 = 1 WH : ω 0 ω 0 ω 0... ω 0 W 1 = 1 ω 0 ω 1 ω 2... ω ( 1) ω 0 ω 2 ω 4... ω 2( 1).... ω 0 ω ( 1) ω 2( 1)... ω ( 1)( 1) the vector F of the Fourier coefficients can be computed easily as a matrix-vector product with computational effort O( 2 ): F = 1 WH f or F k = 1 1 f n ω nk. n=0 Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

13 Inverse Discrete Fourier Transform (IDFT) Definition: The inverse Discrete Fourier Transform (IDFT) of the vector (F 0,..., F 1 ) is given by the vector (f 0,..., f 1 ), where 1 f n = F k e i2πnk/. Observation: DFT and IDFT are inverse operations: F k = f n e i2πnk/, f n = F k e i2πnk/. n=0 F = DFT(IDFT(F)) or f = IDFT(DFT(f)). Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

14 The Pair DFT/IDFT as Matrix-Vector Product With the notation ω := e i2π/, i.e. ω nk := e i2πnk/, we formulate the DFT/IDFT as F k = f n ω nk f n = F k ω nk n=0 With the vectors f := (f 0,..., f 1 ) T and F := (F 0,..., F 1 ) T, we denote (and compute) the DFT und IDFT as matrix-vector products F = 1 WH f, f = WF, where the elements of the Fourier matrix W are W nk := ω nk. Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

15 Properties of the DFT DFT and IDFT are (as a matrix-vector product) linear: since ω nk DFT(αf + βg) = α DFT(f ) + β DFT(g) IDFT(αf + βg) = α IDFT(f ) + β IDFT(g) = ωn(k+) = ω (n+)k, the f n and the F k are periodic: f n+ = f n F k+ = F k for all k, n Z Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

16 Alternative Forms of the DFT Possible variants (in all imaginable combinations): Scaling factor 1 in the IDFT instead of the DFT; 1 alternatively a factor in DFT and IDFT. switched signs in the exponent of the exponential function in DFT and IDFT use j for the imaginary unit (electrical engineering) Shift of indices: periodic data: F k = F k+ aliasing of frequencies: e i2πnk/ = e i2πn(k±)/ Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

17 DFT with Shifted Indices Data and frequencies symmetric : In general: F k = f n e i2πnk/, f n = F k e i2πnk/ n= 2 +1 k= 2 +1 F k = 1 P+ n=p+1 f n e i2πnk/, f n = Q+ k=q+1 F k e i2πnk/ Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

18 DFT in Program Libraries Matlab, IMSL (Int. Math. and Stat. Library): F k+1 = 1 f n+1 e i2πnk/ k = 0,..., 1 n=0 f n+1 = 1 1 F k+1 e i2πnk/ n = 0,..., 1 Maple: 1 as factor for DFT and IDFT. Index shift by +1, since: Data/coefficients start at index 0 Arrays to store the numbers start at index 1 Michael Bader Algorithms of Scientific Computing Discrete Fourier Transform (DFT) Summer

Fast Fourier Transform

Fast Fourier Transform December 8, 2016 FFT JPEG RGB Y C B C R (luma (brightness), chroma 2 (color)) chroma resolution is reduced image is split in blocks 8 8 pixels JPEG RGB Y C B C R (luma (brightness),