VU Signal and Image Processing. Torsten Möller + Hrvoje Bogunović + Raphael Sahann

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1 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/17s/

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4 Prof. Dennis Freeman, MIT

5 All Fourier Transforms CTFT SIGNAL Continuous Aperiodic DTFT Discrete Aperiodic CTFS Continuous Periodic DTFS Discrete Periodic Prof. Branko Jeren, FER,

6 DFT - Example N=10 6

7 DFT - Example N=5 7

8 2D DFT DFT of image with M rows and N columns IDFT: 8

9 2D DFT: Separability property First across rows then across columns Alternatively: First across columns then across rows Equivalent for the inverse IDFT where 9

10 2D DFT: Centering of DFT 10

11 DFT properties Time-shifting Inherent periodicity as the byproduct of DFT 11

12 DFT properties: Circular convolution Circular convolution: IDFT(DFT(x)*DFT(h)) 12

13 DFT properties: Circular convolution Linear convolution with aliasing 13

14 Linear convolution using DFT 14

15 Linear convolution using DFT 15

16 Linear convolution using DFT Need for zero-padding Length N of at least L+P-1 16

17 Linear convolution using DFT Need for zero-padding Length N of at least L+P-1 17

18 Prof. Dennis Freeman, MIT

19 DFT and FFT: Introduction Allows to numerically compute the spectrum of a signal Efficient implementation of convolution You can also identify impulse response of unknown black-box LTI system h(t) = IDFT( DFT[y(t)] / DFT[x(t)] ) Signal coding. With the usage of e.g. DCT (a variant of DFT with real numbers only) JPEG In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime" and it was included in Top 10 Algorithms of 20th Century by the IEEE journal Computing in Science & Engineering. 19

20 DFT and FFT: Introduction Prof. I. Selesnick 20

21 FFT: Radix-2 Divide and conquer algorithm Prof. I. Selesnick 21

22 FFT: Radix-2 Symmetry and Periodicity Prof. I. Selesnick 22

23 FFT: Radix-2 Prof. I. Selesnick 23

24 FFT: Decimation in time Prof. I. Selesnick 24

25 FFT: Decimation in time Separate x[n] into two sequences of length N/2 Even indexed samples in the first sequence Odd indexed samples in the other sequence N 1 N 1 N 1 j 2 /N kn j 2 / N kn j 2 x[n]e x[n]e x[n]e X k n 0 n even Substitute variables n=2r for n even and n=2r+1 for odd n odd /N kn G[k] and H[k] are the N/2-point DFT s of each subsequence N / 2 1 N / 2 1 2rk 2r x[2r]w N x[2r 1]W N X k r 0 N / 2 1 r 0 G k x[2r]w N rk N / 2 k W Hk r 0 W N / 2 1 k N r 0 1 k x[2r 1]W rk N / 2 Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 25

26 FFT: Decimation in time 8-point DFT example using decimation-in-time Two N/2-point DFTs 2(N/2) 2 complex multiplications 2(N/2) 2 complex additions Combining the DFT outputs N complex multiplications N complex additions Total complexity N 2 /2+N complex multiplications N 2 /2+N complex additions More efficient than direct DFT Repeat same process Divide N/2-point DFTs into Two N/4-point DFTs Combine outputs Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 26

29 FFT: Butterfly Computation Flow graph constitutes of butterflies We can implement each butterfly with one multiplication Final complexity for decimation-in-time FFT (N/2)log 2 N complex multiplications and additions Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 29

30 FFT: In-Place Computation Note the arrangement of the input indices to allow in-place computation Input is overwritten by the output. no auxiliary data structure. One physical register needed for both input and output Bit reversed indexing X X X X X X X X x 0 X0 000 x x 4 X0 001 x x 2 X0 010 x x 6 X0 011 x x 1 X0 100 x x 5 X0 101 x x 3 X0 110 x x 7 X 111 x Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 30

31 FFT: Decimation in frequency The DFT equation X k N 1 n 0 nk x[n]w N Split the DFT equation into even and odd frequency indexes Substitute variables to get Similarly for odd-numbered frequencies N 1 N / 2 1 N 1 n2r n2r x[n]w N x[n]w N X 2r n 0 n N / 2 N / 2 1 n 2r 1 X 2r 1 x[n] x[n N / 2] W N / 2 n 0 Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 31 n 0 x[n] W N / 2 1 N / 2 1 N / 2 1 n2r n N / 2 2r x[n]w N x[n N /2]W N x[n] x[n N / 2] X 2r n 0 n 0 n 0 n2r N W nr N / 2

33 Complexity: FFT vs direct DFT 33

34 LTI system analysis using FFT Short impulse response h[n] Very long signal x[n] 34

35 LTI system analysis using FFT Important application when filtering long signals Signal x(n) of length L if filtered with FIR filter with impulse response of length M. Output is given by a convolution y( n) M 1 m 0 h( m) x( n Y ( k) H ( k) X ( k) m) y(n) is of length L+M-1 To find y(n) it is enough to apply DFT of length: N L+M-1 Both h and u need to be zero-padded to length N

36 LTI system analysis using FFT: Overlap and add L L L x 1 (n) M-1 zeros x 2 (n) M-1 zeros x 3 (n) M-1 zeros

37 LTI system analysis using FFT: Overlap and add The length of input block is L samples Length of DFT and IDFT is N=L+M-1 To every block we add M-1 zeros and compute N-point DFT x1( n) x(0), x(1),..., x( L 1),0,0,...0 x2( n) x( L), x( L 1),..., x(2l 1),0,0,...0 x3( n) x(2l), x(2l 1),..., x(3l 1),0,0,...0

38 LTI system analysis using FFT: Overlap and add Y m (k)=h(k)x m (k) k=0,1,,n-1 We obtain output blocks of N where there is no aliasing as the length of DFT-a (IDFT) is N=L+M-1 We need to overlap the last M-1 samples of each output block with the first M-1 samples of the next block

39 LTI system analysis using FFT: Overlap and add y( n) y ( 0), y ( 1)..., y ( L 1), L, y ( L) y ( 0), y ( L 1) y ( 1), , y ( N 1) y ( M 1), y ( M), y 1 (n) + y 2 (n) + y 3 (n)

40 LTI system analysis using FFT To apply FFT with radix-2 the length of the input block L has to be adjusted so that N (N=L+M-1) is exponent of 2 Signals x(n) and h(n) have to be zero padded to the lenght of N Transformation H(k) of the impulse response is performed only once as it is time-invariant.

41 LTI system analysis using FFT Block convolution: overlap and add Input is split into non-overlapping segments Perform linear convolution Need for zero-padding (P-1 zeros) Overlap at output: last P-1 samples of the previous output segment are added to the current segment 41

42 LTI system analysis using FFT: Complexity H(k) is computed only once so it can be neglected Every FFT needs ( N / )log N multiplications We perform two of them: 1 x DFT and 1 x IDFT To compute Y m (k) we further need N complex multiplications Ratio of number of multiplications by filtering with FFT vs. linear convolution E.g. for N=1024 i M= N log2 N N log2 N 1 N M M log

43 Prof. Dennis Freeman, MIT

44 Prof. Dennis Freeman, MIT

45 Prof. Dennis Freeman, MIT

46 Prof. Dennis Freeman, MIT

47 Prof. Dennis Freeman, MIT

48 Prof. Dennis Freeman, MIT

! Circular Convolution. " Linear convolution with circular convolution. ! Discrete Fourier Transform. " Linear convolution through circular

! Circular Convolution. Linear convolution with circular convolution. ! Discrete Fourier Transform. Linear convolution through circular Previously ESE 531: Digital Signal Processing Lec 22: April 18, 2017 Fast Fourier Transform (con t)! Circular Convolution " Linear convolution with circular convolution! Discrete Fourier Transform " Linear