fft lecture

8/6/2019 Fft Lecture

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Fast Fourier Transform

Key Papers in C

Dima Batenkov

Weizmann Institute of Science

[email protected]


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Fast Fourier Transform -

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Fast Fourier Transform - Overv

J. W. Cooley and J. W. Tukey. An algorithmcalculation of complex Fourier series. MathComputation, 19:297301, 1965


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J. W. Cooley and J. W. Tukey. An algorithmcalculation of complex Fourier series. MathComputation, 19:297301, 1965 A fast algorithm for computing the Discre

Transform


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Transform

(Re)discovered by Cooley & Tukey in 196adopted thereafter


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Transform

(Re)discovered by Cooley & Tukey in 196adopted thereafter

Has a long and fascinating history


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urier Analysis

Fourier Series

Continuous Fourier Transform

Discrete Fourier Transform

Useful properties 6

Applications Fourier Analysis


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urier Analysis

Fourier Series



Useful properties 6

Applications

Fourier Series

Expresses a (real) periodic function x(t)trigonometric series (L < t < L)

x(t) =1

2a0 +

n=1

(an cosn

Lt + b

Coefficients can be computed by

an =1

L

LL

x(t) cosn

Lt

bn =1

L

LL

x(t) sinn

Lt


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urier Analysis

Fourier Series



Useful properties 6

Applications

Fourier Series

Generalized to complex-valued functions

x(t) =

n=

cnei nL

t

cn =1

2L

L

L

x(t)ein

L


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urier Analysis

Fourier Series



Useful properties 6

Applications

Fourier Series


x(t) =

n=

cnei nL

t

cn =1

2L

L

L

x(t)ein

L

Studied by D.Bernoulli and L.Euler

Used by Fourier to solve the heat equatio


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urier Analysis

Fourier Series



Useful properties 6

Applications

Fourier Series


x(t) =

n=

cnei nL

t

cn =1

2L

L

L

x(t)ein

L

Studied by D.Bernoulli and L.Euler

Used by Fourier to solve the heat equatio

Converges for almost all nice functions

L2 etc.)


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urier Analysis

Fourier Series



Useful properties 6

Applications


Generalization of Fourier Series for infinit

x(t) =

F(f)e2i f

F(f) =

x(t)e2i f t dt

Can represent continuous, aperiodic sign

Continuous frequency spectrum


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urier Analysis

Fourier Series



Useful properties 6

Applications


If the signal X(k) is periodic, band-limitedNyquist frequency or higher, the DFT rep

exactly14

A(r) =N1

k=0

X(k)WrkN

where WN = e

2i

N and r = 0 , 1 , . . . , N

The inverse transform:

X(j) =1

N

N1

k=0

A(k)W

N


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urier Analysis

Fourier Series



Useful properties 6

Applications

Useful properties 6

Orthogonality of basis functions

N1

k=0

Wrk

N WrkN = NN(r

Linearity

(Cyclic) Convolution theorem

(x y)n =N1

k=0

x(k)y(

F {(x y)} = F {X}F {Y

Symmetries: real hermitian symmetrichermitian antisymmetric

Shifting theorems


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urier Analysis

Fourier Series



Useful properties 6

Applications

Applications

Approximation by trigonometric polynomi Data compression (MP3...)

Spectral analysis of signals

Frequency response of systems

Partial Differential Equations

Convolution via frequency domain Cross-correlation Multiplication of large integers Symbolic multiplication of polynomials


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ethods known by 1965

Available methods

Goertzels algorithm 7

Methods known by


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Available methods


Available methods

In many applications, large digitized dataavailable, but cannot be processed due torunning time of DFT

All (publicly known) methods for efficient

the symmetry of trigonometric functions,

Goertzels method7

is fastest known


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Available methods


Goertzels algorithm7

Given a particular 0 < j < N 1, we wan

A(j) =N1

k=0

x(k) cos

2j

Nk

B(j) =

N

k=


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Available methods




A(j) =N1

k=0

x(k) cos

2j

Nk

B(j) =

N

k=

Recursively compute the sequence {u(s)

u(s) = x(s) + 2cos

2j

N

u(s +

u(N) = u(N+ 1) = 0


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Available methods




A(j) =N1

k=0

x(k) cos

2j

Nk

B(j) =

N

k=

Recursively compute the sequence {u(s)

u(s) = x(s) + 2cos

2j

N

u(s +

u(N) = u(N+ 1) = 0

Then

B(j) = u(1) sin

2j

N

A(j) = x(0) + cos

2j

N

u(1


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Available methods



Requires N multiplications and only one

Roundoff errors grow rapidly5

Excellent for computing a very small num(< log N)

Can be implemented using only 2 interme

locations and processing input signal as Used today for Dual-Tone Multi-Frequenc

dialing)


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spiration for the FFT

Factorial experiments

Goods paper 9

Goods results - summary

Inspiration for the


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Goods paper 9



Purpose: investigate the effect of n factors quantity Each factor can have t distinct levels. Fo

t = 2, so there are 2 levels: + and .

So we conduct tn experiments (for every combination of the n factors) and calcula Main effect of a factor: what is the ave

output as the factor goes from to Interaction effects between two or mo

the difference in output between the cfactors are present compared to whenis present?


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Goods paper 9



Example of t = 2, n = 3 experiment Say we have factors A, B, C which can be

example, they can represent levels of 3 dto patients

Perform 23 measurements of some quantpressure

Investigate how each one of the drugs anaffect the blood pressure


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Goods paper 9



Exp. Combination A B C Bl

1 (1)

2 a +

3 b + 4 ab + +

5 c +

6 ac + +

7 bc + +

8 abc + + +


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Goods paper 9



The main effect of A is(ab b) + (a (1)) + (abc bc) + (ac

The interaction of A, B is(ab b) (a (1)) ((abc bc) (ac

etc.


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Goods paper 9



The main effect of A is(ab b) + (a (1)) + (abc bc) + (ac

The interaction of A, B is(ab b) (a (1)) ((abc bc) (ac

etc.

If the main effects and interactions are arrathen we can write y = Ax where

A =

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 11 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1


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Goods paper 9



F.Yates devised15 an iterative n-step algosimplifies the calculation. Equivalent to dA = Bn, where (for our case of n = 3):

B =

1 1 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

0 0 0 0 0 0

1 1 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1

0 0 0 0 0 0

so that much less operations are required


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Goods paper 9


Goods paper9

I. J. Good. The interaction algorithm and pranalysis. Journal Roy. Stat. Soc., 20:3613 Essentially develops a prime-factor FFT

The idea is inspired by Yates efficient algproves general theorems which can be aefficiently multiplying a N-vector by certa

The work remains unnoticed until Cooley

Good meets Tukey in 1956 and tells him method

It is revealed later4 that the FFT had not much earlier by a coincedence


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Goods paper 9


Goods paper9

Definition 1 Let M(i) be t t matrices, i =direct product

M(1) M(2) M(n

is atn tn matrix C whose elements are

Cr,s =n

i=1

M(i)ri,si

wherer = (r1, . . . , rn) ands = (s1, . . . sn) arepresentation of the index


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Goods paper 9


Goods paper9

Definition 1 Let M(i) be t t matrices, i =direct product

M(1) M(2) M(n

is atn tn matrix C whose elements are

Cr,s =n

i=1

M(i)ri,si

wherer = (r1, . . . , rn) ands = (s1, . . . sn) arepresentation of the index

Theorem 1 For every M, M[n] = An where

Ar,s = {Mr1,sns1r2

s2r3 . . .

snrn

Interpretation: A has at most tn+1 nonzero

a given B we can find M such that M[n] = Bcompute Bx = Anx in O(ntn) instead of O(


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Goods paper 9


Goods paper9

Goods observations For 2n factorial experiment:

M =

1 1

1 1


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Goods paper 9


Goods paper9

Goods observations n-dimensional DFT, size N in each dimen

A(s1, . . . , sn) =(N1,...,N1)

(r1,...,rn)

x(r)Wr1sN

can be represented as

A = [n]x, = {WrsN} r, s = 0

By the theorem, A = nx where is spa


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Goods paper 9


Goods paper9

Goods observations n-dimensional DFT, size N in each dimen

A(s1, . . . , sn) =(N1,...,N1)

(r1,...,rn)

x(r)Wr1sN

can be represented as

A = [n]x, = {WrsN} r, s = 0

By the theorem, A = nx where is spa

n-dimensional DFT, Ni in each dimension

A = ((1) (n))x = (1)

(i) = {WrsNi}

(i) = IN1 (i) INn


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Goods paper 9


Goods paper9

If N = N1N2 and (N1, N2) = 1 then N-poinrepresented as 2-D DFT

r = N2r1 + N1r2

(Chinese Remainder Thm.)s = N2s1(N12 ) mod N

s s1 mod N1, s

These are index mappings s (s1, s2) and

A(s) = A(s1, s2) =N

r=0

x(r)WrsN =N11

r1=0

(substitute r, s from above) =N11

r1=0

N21

r2=0

x(r1, r2)W

This is exactly 2-D DFT seen above!


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Goods paper 9


Goods paper9

So, as above, we can write

A = x

= P((1) (2))Q1

= P(1)(2)Q1x

where P and Q are permutation matrices. TN(N1 + N2) multiplications instead of N

2.


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Goods paper 9



n-dimensionanal, size Ni in each dimens

n

Ni

n

Ni

instead of

1-D, size N = n

Ni

where Nis are mutu

n

Ni

N instead of N

Still thinks that other tricks such as usinand sines and cosines of known angles museful


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e article

ames W.Cooley (1926-)

ohn Wilder Tukey

915-2000)

The story - meeting

The story - publication

What if...

Cooley-Tukey FFT

FFT propertiesDIT signal flow

DIF signal flow

The article


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

James W.Cooley (1926-)

1949 - B.A. in Mathematics(Manhattan College, NY)

1951 - M.A. in Mathematics(Columbia University)

1961 - Ph.D in Applied Math-

ematics (Columbia Univer-sity)

1962 - Joins IBM Watson Research Cent

The FFT is considered his most significanmathematics

Member of IEEE Digital Signal Processin

Awarded IEEE fellowship on his works on

Contributed much to establishing the term


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

John Wilder Tukey (1915-2000)

1936 - Bachelor in Chem-istry (Brown University)

1937 - Master in Chemistry(Brown University)

1939 - PhD in Mathemat-

ics (Princeton) for Denu-merability in Topology

During WWII joins Fire Control Researchcontributes to war effort (mainly invlolving

From 1945 also works with Shannon andBell Labs

Besides co-authoring the FFT in 1965, hacontributions to statistics


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

The story - meeting

Richard Garwin (Columbia University IBMcenter) meets John Tukey (professor of mPrinceton) at Kennedys Scientific Adviso1963


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

The story - meeting

Richard Garwin (Columbia University IBMcenter) meets John Tukey (professor of mPrinceton) at Kennedys Scientific Adviso1963 Tukey shows how a Fourier series of l

N = ab is composite can be expresse

of b-point subseries, thereby reducingcomputations from N2 to N(a + b)


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

The story - meeting




Notes that if N is a power of two, the arepeated, giving Nlog2 N operations


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

The story - meeting




Notes that if N is a power of two, the arepeated, giving Nlog2 N operations

Garwin, being aware of applications inwould greatly benefit from an efficient

realizes the importance of this and fropushes the FFT at every opportunity


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow


Garwin goes to Jim Cooley (IBM Watsonwith the notes from conversation with Tukimplement the algorithm


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow



Says that he needs the FFT for computin

transform of spin orientations of He3. It iswhat Garwin really wanted was to be ableexposions from seismic data


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow





Cooley finally writes a 3-D in-place FFT aGarwin


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow






The FFT is exposed to a number of math


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow







It is decided that the algorithm should be

domain to prevent patenting


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow








domain to prevent patenting Cooley writes the draft, Tukey adds some

Goods article and the paper is submittedComputation in August 1964


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow








domain to prevent patenting Cooley writes the draft, Tukey adds some

Goods article and the paper is submittedComputation in August 1964

Published in April 1965


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

What if...

Good reveals in 19934 that Garwin visite

Garwin told me with great enthusiasm aexperimental work... Unfortunately, not wthe subject..., I didnt mention my FFT woalthough I had intended to do so. He wro

9, 1976 and said: Had we talked about ihave stolen [publicized] it from you insteaTukey...... That would have completely cof the FFT


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Cooley-Tukey FFT

A(j) = N1k=0 X(k)WjkN N = N1N2


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Cooley-Tukey FFT


Convert the 1-D indices into 2-D

j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =


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Cooley-Tukey FFT



j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =

Decimate the original sequence into N2 N1-subsequ


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Cooley-Tukey FFT



j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =


A(j1, j0) =N1

k=0

X(k)WjkN


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Cooley-Tukey FFT



j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =


A(j1, j0) =N1

k=0

X(k)WjkN

=

N21

k0=0

N11

k1=0X(k1, k0)W

(j0+j1N1)(k0+k1N1N2


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Cooley-Tukey FFT



j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =


A(j1, j0) =N1

k=0

X(k)WjkN

=

N21

k0=0

N11

k1=0X(k1, k0)W

(j0+j1N1)(k0+k1N1N2

=N21

k0=0

Wj0k0N W

j1k0N2

N11

k1=0

X(k1, k0)Wj0kN1


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Cooley-Tukey FFT



j = j0 + j1N1 j0 = 0 , . . . , N1 1 j1 =

k = k0 + k1N2 k0 = 0 , . . . , N2 1 k1 =


A(j1, j0) =N1

k=0

X(k)WjkN

=

N21

k0=0

N11

k1=0X(k1, k0)W

(j0+j1N1)(k0+k1N2)N1N2

=N21

k0=0

Wj0k0N W

j1k0N2

N11

k1=0

X(k1, k0)Wj0k1N1

=


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Cooley-Tukey FFT

The sequences Xj0 have N elements and require NNcompute

Given these, the array A requires NN2 operations to

Overall N(N1 + N2)

In general, if N = ni=0 Ni then N(ni=0 Ni) operation


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Cooley-Tukey FFT

If N1 = N2 = = Nn = 2 (i.e. N= 2n):

j = jn12n1 + +j0, k = kn12

A(jn1, . . . ,j0) =1

k0=0

1

kn1=0

X(kn1, . . . , k0)WjN

= k0

Wjk0N

k1

Wjk12N

kn1

Wjkn1N

Xj0 (kn2, . . . , k0) = kn1

X(kn1, . . . , k0)Wj0kn12

Xj1,j0 (kn3, . . . , k0) = kn2

Wj0kn24

Xj0 (kn2, . . . , k0)Wj1k2

Xjm1,...,j0 (knm1, . . . , k0) = knm

W(j0++jm22m2)knm2m

Xjm2,

Wjm1knm2


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Cooley-Tukey FFT

Finally, A(jn1, . . . , j0) = Xjn1,...,j0 Two important properties:

In-place: only two terms are involved in the calculaintermediate pair Xjm1,...,j0(. . . ) jm1 = 0, 1: Xjm2,...,j0(0 , . . . )

Xjm2,...,j0(1 , . . . )So the values can be overwritten and no more addused

Bit-reversal It is convenient to store Xjm1,...,j0(knm1, . . . , k0

j02n1 + + jm12

nm + knm12mn1 +

The result must be reversed Cooley & Tukey didnt realize that the algorithm in fac

DFTs from smaller ones with some multiplications alo

The terms W(j0++jm22

m2)knm2m are later called

6 tw


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erview

e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

FFT properties

Roundoff error significantly reduced com

formula6

A lower bound of 12

n log2 n operations ov

algorithms is proved12, so FFT is in a sen

Two canonical FFTs

Decimation-in-Time: the Cooley & TukEquivalent to taking N1 = N/2, N2are the DFT coefficients of even-numbXj0(1) - those of odd-numbered. The fi

simply linear combination of the two. Decimation-in-Frequency (Tukey-Sand

N1 = 2, N2 = N/2. Then Y() = Xeven-numbered and odd-numbered saZ() = X1() is the difference. The evcoefficients are the DFT of Y, and theweighted DFT of Z.


63/90


erview

e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIT signal flow


64/90


erview

e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIT signal flow


65/90


erview

e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIT signal flow


66/90


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIF signal flow


67/90


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIF signal flow


68/90


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e article


ohn Wilder Tukey

915-2000)

The story - meeting


What if...

Cooley-Tukey FFT


DIF signal flow

DIF signal flow


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wly discovered history of FFT

What happened after the

blication

Danielson-Lanczos

Gauss Newly discovered histo


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blication

Danielson-Lanczos

Gauss

What happened after the publi

Rudnicks note13 mentions Danielson-La

19423. Although ... less elegant and is pterms of real quantities, it yields the samebinary form of the Cooley-Tukey algorithmnumber of arithmetical operations.

Historical Notes on FFT2

Also mentions Danielson-Lanczos Thomas and Prime Factor algorithm (

distinct from Cooley-Tukey FFT

A later investigation10 revealed that Gausdiscovered the Cooley-Tukey FFT in 1805


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Danielson-Lanczos

G. C. Danielson and C. Lanczos. Some improvementFourier analysis and their application to X-ray scatterFranklin Institute, 233:365380 and 435452, 1942


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Danielson-Lanczos


... the available standard forms become impractical fof coefficients. We shall show that, by a certain transfit is possible to double the number of ordinates with othan double the labor


73/90

Danielson-Lanczos



Concerned with improving efficiency of hand calculat


74/90

Danielson-Lanczos




We shall now describe a method which eliminates thcomplicated schemes by reducing ... analysis for 4n canalyses for 2n coefficients


75/90

Danielson-Lanczos




We shall now describe a method which eliminates thcomplicated schemes by reducing ... analysis for 4n canalyses for 2n coefficients

If desired, this reduction process can be applied twic(???!!)

Adoping these improvements the approximate timesfor 8 coefficients, 25 min. for 16, 60 min. for 32 and 1( 0.37Nlog2 N)


76/90

Danielson-Lanczos

Take the DIT Cooley-Tukey with N1 = N/2, N2 = 2

A(j0 +N

2j1) =

N/21

k1=0

X(2k1)Wj0k1N/2 + W

j0+N

2j1

N

N/21

k1=0


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Danielson-Lanczos


A(j0 +N

2j1) =

N/21

k1=0

X(2k1)Wj0k1N/2 + W

j0+N

2j1

N

N/21

k1=0

Danielson-Lanczos showed completely analogous re

framework of real trigonometric series: ...the contribution of the even ordinates to a Four

coefficients is exactly the same as the contributionordinates to a Fourier analysis of 2n coefficients

..contribution of the odd ordinates is also reducibscheme... by means of a transformation process i

phase difference ...we see that, apart from the weight factors 2cos

2sin(/4n)k, the calculation is identical to the coshalf the number of ordinates


78/90

Danielson-Lanczos


A(j0 +N

2j1) =

N/21

k1=0

X(2k1)Wj0k1N/2 + W

j0+N

2j1

N

N/21

k1=0

Danielson-Lanczos showed completely analogous re

framework of real trigonometric series: ...the contribution of the even ordinates to a Four

coefficients is exactly the same as the contributionordinates to a Fourier analysis of 2n coefficients

..contribution of the odd ordinates is also reducibscheme... by means of a transformation process i

phase difference ...we see that, apart from the weight factors 2cos

2sin(/4n)k, the calculation is identical to the coshalf the number of ordinates

These weight factors are exactly the twiddle fac


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blication

Danielson-Lanczos

Gauss

Gauss

Herman H. Goldstine writes in a footnoteNumerical Analysis from the 16th through

(1977)8:This fascinating work of Gauss was nrediscovered by Cooley and Tukey in paper in 1965

This goes largely unnoticed until the rese

Johnson and Burrus10 in 1985 Essentially develops an DIF FFT for tw

sequences. Gives examples for N = Declares that it can be generalized to

factors, although no examples are giv Uses the algorithm for solving the proorbit parameters of asteroids

The treatise was published posthumothen other numerical methods were prnobody found this interesting enough.


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pact

mpact

Further developments

Concluding thoughts

Impact


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Impact

Certainly a classic

Two special issues of IEEE trans. on Aud

Electroacoustics devoted entirely to FFT

Arden House Workshop on FFT11

Brought together people of very divers

someday radio tuners will operate witunits. I have heard this suggested witbut one can speculate. - J.W.Cooley

Many people discovered the DFT via th


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Real-FFT (DCT...)

Parallel FFT

FFT for prime N

...


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Concluding thoughts

It is obvious that prompt recognition and significant achievments is an important g


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Concluding thoughts


However, the publication in itself may not


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Concluding thoughts



Communication between mathematiciansanalysts and workers in a very wide rang

can be very fruitful


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Concluding thoughts




can be very fruitful Always seek new analytical methods and

increase in processing speed


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Concluding thoughts






Careful attention to a review of old literaturewards


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Concluding thoughts







Do not publish papers in Journal of Frank


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Do not publish papers in Journal of Frank

Do not publish papers in neo-classic Lati


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References

1. J. W. Cooley and J. W. Tukey. An algorithm for the machine calculation of complex Fourier series.Mathematics of Computation, 19:297301, 1965.

2. J. W. Cooley, P. A. Lewis, and P. D. Welch. History of the fast Fourier transform. In Proc. IEEE,volume 55, pages 16751677, October 1967.

3. G. C. Danielson and C. Lanczos. Some improvements in practical Fourier analysis and their appli-cation to X-ray scattering from liquids. J. Franklin Institute, 233:365380 and 435452, 1942.

4. D.L.Banks. A conversation with I. J. Good. Statistical Science, 11(1):119, 1996.

5. W. M. Gentleman. An error analysis of Goertzels (Watts) method for computing Fourier coeffi-cients. Comput. J., 12:160165, 1969.

6. W. M. Gentleman and G. Sande. Fast Fourier transformsfor fun and profit. In Fall Joint ComputerConference, volume 29 ofAFIPS Conference Proceedings, pages 563578. Spartan Books, Washington,D.C., 1966.

7. G. Goertzel. An algorithm for the evaluation of finite trigonometric series. The American Mathe-matical Monthly, 65(1):3435, January 1958.

8. Herman H. Goldstine. A History of Numerical Analysis from the 16th through the 19th Century.Springer-Verlag, New York, 1977. ISBN 0-387-90277-5.

9. I. J. Good. The interaction algorithm and practical Fourier analysis. Journal Roy. Stat. Soc., 20:361372, 1958.

10. M. T. Heideman, D. H. Johnson, and C. S. Burrus. Gauss and the history of the Fast FourierTransform. Archive for History of Exact Sciences, 34:265267, 1985.

11. IEEE. Special issue on fast Fourier transform. IEEE Trans. on Audio and Electroacoustics, AU-17:65186, 1969.

12. Morgenstern. Note on a lower bound of the linear complexity of the fast Fourier transform. JACM:Journal of the ACM, 20, 1973.

13. Philip Rudnick. Note on the calculation of Fourier series (in Technical Notes and Short Papers).j-MATH-COMPUT, 20(95):429430, July 1966. ISSN 0025-5718.

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