052600 VU Signal and Image Processing Torsten Möller ...vda.univie.ac.at/Teaching/SIP/17s/LectureNotes/11_FFT.pdf · Signal and Image Processing Torsten Möller + Hrvoje Bogunović+

052600 VUSignal and Image Processing

Torsten Möller + Hrvoje Bogunović + Raphael Sahann

[email protected]

[email protected]

[email protected]

vda.cs.univie.ac.at/Teaching/SIP/17s/

2

3

4© Prof. Dennis Freeman, MIT 2011.

All Fourier Transforms

5© Prof. Branko Jeren, FER, 2010

ContinuousAperiodic

DiscreteAperiodic

ContinuousPeriodic

DiscretePeriodic

CTFT

DTFT

CTFS

DTFS

SIGNAL

DFT - Example

6

N=10

DFT - Example

7

N=5

2D DFT

• DFT of image with M rows and N columns

• IDFT:

8

2D DFT: Separability property

• First across rows then across columns

• Alternatively: First across columns then across rows

• Equivalent for the inverse IDFT

9

where

2D DFT: Centering of DFT

10

DFT – propertiesTime-shifting• Inherent periodicity as the

byproduct of DFT

11

DFT – properties: Circular convolution

• Circular convolution: IDFT(DFT(x)*DFT(h))

12

DFT – properties: Circular convolution

• Linear convolution with aliasing

13

Linear convolution using DFT

14


15


• Need for zero-padding

• Length N of at least L+P-1

16


• Need for zero-padding

• Length N of at least L+P-1

17


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

DFT and FFT: Introduction

20© Prof. I. Selesnick

FFT: Radix-2

• Divide and conquer algorithm


FFT: Radix-2

22

Symmetry and

Periodicity

© Prof. I. Selesnick

FFT: Radix-2


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

• Substitute variables n=2r for n even and n=2r+1 for odd

• G[k] and H[k] are the N/2-point DFT’s of each subsequence

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 25

1N

odd n

knN/2j1N

even n

knN/2j1N

0n

knN/2j e]n[xe]n[xe]n[xkX

kHWkG

W]1r2[xWW]r2[x

W]1r2[xW]r2[xkX

kN

12/N

0r

rk2/N

kN

12/N

0r

rk2/N

12/N

0r

k1r2N

12/N

0r

rk2N



• 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• N2/2+N complex multiplications

• N2/2+N complex additions

• More efficient than direct DFT

• Repeat same process • Divide N/2-point DFTs into

• Two N/4-point DFTs

• Combine outputs



• After two steps of decimation in time

• Repeat until we’re left with two-point DFT’s

• Butterfly structure



• Final flow graph for 8-point decimation in time

• Complexity:• Nlog2N complex multiplications

and additions

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)log2N complex multiplications and additions

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

111x111X7x7X

011x110X3x6X

101x101X5x5X

001x100X1x4X

110x011X6x3X

010x010X2x2X

100x001X4x1X

000x000X0x0X

00

00

00

00

00

00

00

00

FFT: Decimation in frequency

• The DFT equation

• Split the DFT equation into even and odd frequency indexes

• Substitute variables to get

• Similarly for odd-numbered frequencies


1N

0n

nkNW]n[xkX

1N

2/Nn

r2nN

12/N

0n

r2nN

1N

0n

r2nN W]n[xW]n[xW]n[xr2X

12/N

0n

nr2/N

12/N

0n

r22/NnN

12/N

0n

r2nN W]2/Nn[x]n[xW]2/Nn[xW]n[xr2X

12/N

0n

1r2n2/NW]2/Nn[x]n[x1r2X

FFT: Decimation in frequency


• Final flow graph for 8-point decimation in frequency

Complexity: FFT vs direct DFT

33

LTI system analysis using FFT

• Short impulse response h[n]

• Very long signal x[n]

34


• 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

1

0

)()()(M

m

mnxmhny

• y(n) is of length L+M-1

• To find y(n) it is enough to applyDFT of length: N L+M-1

• Both h and u need to be zero-padded to length N)()()( kXkHkY

L L L

x1(n)

M-1 zeros

x2(n)

M-1 zeros

x3(n)

M-1 zeros

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

0,...0,0),1(),...,1(),0()(1 Lxxxnx

0,...0,0),12(),...,1(),()(2 LxLxLxnx

0,...0,0),13(),...,12(),2()(3 LxLxLxnx


• Ym(k)=H(k)Xm(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

y1(n)

L

+

y n y y y L( ) ( ), ( )..., ( ), 1 1 10 1 1

y2(n)

, ( ) ( ), ( ) ( ),...y L y y L y1 2 1 20 1 1

+

. . . , ( ) ( )y N y M1 21 1

y3(n)

, ( ), . . .y M2



• To apply FFT with radix-2 the length of the input block L has to beadjusted 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 onceas it is time-invariant.


• 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

• H(k) is computed only once so it can be neglected

• Every FFT needs multiplications

• We perform two of them: 1 x DFT and 1 x IDFT

• To compute Ym(k) we further need N complex multiplications

N N N

N M

N

M

log log2 2 1

• Ratio of number of multiplications by filteringwith FFT vs. linear convolution

log2 1024 1

128

11

128

• E.g. for N=1024 i M=128

LTI system analysis using FFT: Complexity

( / ) logN N2 2







052600 VU Signal and Image Processing Torsten Möller ...vda.univie.ac.at/Teaching/SIP/17s/LectureNotes/11_FFT.pdf · Signal and Image Processing Torsten Möller + Hrvoje Bogunović+

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