Decimation-in-frequency Fast Fourier Transforms for the ...ericmalm.net/ac/research/ffts/emalm-2005-05-02-ffts.pdf · 5/2/2005 · The Discrete Fourier Transform Group-Theoretic

a·r·tharvey·mudd·college

The Discrete Fourier TransformGroup-Theoretic DFTs

Fast Fourier Transforms for SnComputational Results

Decimation-in-frequency Fast Fourier Transformsfor the Symmetric Group

Eric Malm

Presentation Days ’05Thesis Advisor: Michael Orrison

Harvey Mudd College

02 May 2005

Eric Malm FFTs for the Symmetric Group




Overview1 The Discrete Fourier Transform

Signal AnalysisExampleSignificance of DFT

2 Group-Theoretic DFTsSymmetries and GroupsWedderburn’s TheoremGeneralized DFTs

3 Fast Fourier Transforms for SnDecimation-In-Frequency FFTsRepresentation Theory of SnConstruction of Factorization

4 Computational Results






Signal Analysis

The SetupSuppose we want to analyze some periodic signal f

• Pick some full time period of f

• Take N samples f0, f1, . . . , fN-1 of f in this time period

1 2t

1

2

f HtL






Signal Analysis




1 2t

1

2

f HtL






Signal Analysis




1t

1

2

f HtL






Discrete Fourier Transforms

æçççççççççççè

f0f1¶

fN-1

ö÷÷÷÷÷÷÷÷÷÷÷ø

input----® f̂k =

1N

N-1

â

j=0

fje-2Πijk / N output

-----®

æççççççççççççè

f̂0f̂1¶

f̂N-1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø

• Process f0, . . . , fN-1 with the Discrete Fourier Transform

• Get N complex numbers f̂0, . . . f̂N-1 such that

f(t) »N-1

â

k=0

f̂k

amplitude

Kcos2ΠkN

t + i sin2ΠkN

tO

“pure” frequency








f0f1¶

fN-1

ö÷÷÷÷÷÷÷÷÷÷÷ø

input----® f̂k =

1N

N-1

â

j=0

fje-2Πijk / N

output-----®


f̂0f̂1¶

f̂N-1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø



f(t) »N-1

â

k=0

f̂k

amplitude

Kcos2ΠkN

t + i sin2ΠkN

tO









f0f1¶

fN-1

ö÷÷÷÷÷÷÷÷÷÷÷ø

input----® f̂k =

1N

N-1

â

j=0


-----®


f̂0f̂1¶

f̂N-1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø



f(t) »N-1

â

k=0

f̂k

amplitude

Kcos2ΠkN

t + i sin2ΠkN

tO









f0f1¶

fN-1

ö÷÷÷÷÷÷÷÷÷÷÷ø

input----® f̂k =

1N

N-1

â

j=0


-----®


f̂0f̂1¶

f̂N-1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø



f(t) »N-1

â

k=0

f̂k

amplitude

Kcos2ΠkN

t + i sin2ΠkN

tO









f0f1¶

fN-1

ö÷÷÷÷÷÷÷÷÷÷÷ø

input----® f̂k =

1N

N-1

â

j=0


-----®


f̂0f̂1¶

f̂N-1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø



f(t) »N-1

â

k=0

f̂k

amplitude

Kcos2ΠkN

t + i sin2ΠkN

tO







Example

ExampleOur original signal is secretly the sum of three “pure” frequencies:

1 2t

1

2

f HtL

=

1 2t

1

+1 2

t

1

+1 2

t

1






Significance of the DFT

Time-Shift InvarianceSuppose we sampled our signal f over a different time period

• The samples f0, . . . , fN-1 could be much different

• But the Fourier coefficients f̂0, . . . , f̂N-1 will not be

The DFT is therefore invariant under translational symmetry

3�21�2t

1

2

f HtL





Symmetries and GroupsWedderburn’s TheoremGeneralized DFTs

Symmetries and Groups


• Different spaces have different symmetries

• Write symmetries abstractly as groups

Space Symmetry

Group

time domain time translations

Z / NZ

sphere rotations

SO(3)

lists permutations

Sn

ABC

(1 3)---®

CBA

(1 3 2)----®

BAC

(2 3)---®

BCA








• Different spaces have different symmetries

• Write symmetries abstractly as groups

Space Symmetry Grouptime domain time translations Z / NZ

sphere rotations SO(3)lists permutations Sn

ABC

(1 3)---®

CBA

(1 3 2)----®

BAC

(2 3)---®

BCA






Group Algebras

Reformulation as Group Algebra

• Treat functions on spaces as functions on groups

• Rewrite functions on group as group algebra elements:

f : X ® C --® f : G ® C --® âgÎG

f(g) g

0��

1��

2��

3��

4��

5��

6��

7��

t

1

2

f HtL






Wedderburn’s Theorem

Theorem (Wedderburn)The group algebra CG of a finite group G is isomorphic to an algebraof block diagonal matrices:

CG @hÅj=1C

dj´dj

Example (CS3)

CS3 @ C1´1Å C2´2

Å C1´1=


ê

ê ê

ê ê

ê

ö÷÷÷÷÷÷÷÷÷÷÷ø






Generalized Discrete Fourier Transforms (DFTs)

Definition (Generalized DFT)Any such isomorphism D on CG is a generalized DFT for G

• Coefficients in matrix D(f): generalized Fourier coefficients

• Blocks along diagonal: smallest CG-invariant spaces in CG

Change of BasisDFT a change of basis into a symmetry-invariant basis

• Picking standard bases on CG, matrix algebra gives DFT matrix

• Naïve bound of O(|G|2) on complexity of DFT evaluation





Decimation-In-Frequency FFTsRepresentation Theory of SnConstruction of Factorization

Decimation-In-Frequency Fast Fourier Transforms (FFTs)

Decimation-In-Frequency FFT

• Fix chain of subgroups of G:

1 = G0 < G1 < × × × < Gn-1 < Gn = G.

• Project into successively smaller subspaces in stagescorresponding to subgroups

• Goal: Obtain sparse factorization of change-of-basis matrix D

Sn an Ideal Proof-of-Concept GroupNonabelian, representation theory well understood, natural chain ofsubgroups

1 < S2 < S3 < × × × < Sn






Representation Theory of Sn

Each block in matrix algebra for CSn corresponds to a differentnon-increasing (proper) partition of n

Example (CS3)

CS3 @ I ê M Å K

ê ê

ê êO Å I ê M

3 #






Character Graph for 1 < S2 < S3

Graded diagram of proper partitions for 1 < S2 < × × × < Sn


ê

ê ê

ê ê

ê

ö÷÷÷÷÷÷÷÷÷÷÷ø

CS3

1

1 2 12

1 2 3 1 32







Each pathway through the diagram corresponds to a filled-in partitionand a row/column in matrix


ê

ê ê

ê ê

ê

ö÷÷÷÷÷÷÷÷÷÷÷ø

CS3

1

1 2

12

1 2 3

1 32







Each pathway through the diagram corresponds to a filled-in partitionand a row/column in matrix


ê

ê ê

ê ê

ê

ö÷÷÷÷÷÷÷÷÷÷÷ø

CS3

1

1 2

12

1 2 3

1 32







A pair of paths ending at same diagram specifies a 1-D Fourier space


ê

¯

® ê ê

ê ê

ê

ö÷÷÷÷÷÷÷÷÷÷÷ø

CS3

1

1 2 12

1 2 3 1 32






Construction of Factorization

Stages of Subspace Projections

• Partial paths give CSn subspaces

• At stage for Sk, project ontothese subspaces

• Build sparse factor fromprojections

• Full paths by stage for Sn

CS3 @ I ê MÅ Kê ê

ê êOÅ I ê M






Construction of Factorization

Stages of Subspace Projections

• Partial paths give CSn subspaces

• At stage for Sk, project ontothese subspaces

• Build sparse factor fromprojections

• Full paths by stage for Sn

CS3 @ I ê MÅ Kê ê

ê êOÅ I ê M





Factorization of CS3 DFT Matrix

Full DFT matrix for CS3:

æççççççççççççççççççççççççççè

1 1 1 1 1 1

1 -1 12 -

12

12 -

12

0 0 -32

32

32 -

32

0 0 -12 -

12

12

12

1 1 -12 -

12 -

12 -

12

1 -1 -1 1 -1 1

ö÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷ø





Factorization of CS3 DFT Matrix

Three factors (CS2 on rows, CS2 on columns, CS3 on rows):


1 11 -

12

11

1 12

1 -1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø


11 11 -1

1 -11

1 1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø


1 11 1

1 11 -1

1 -11 -1

ö÷÷÷÷÷÷÷÷÷÷÷÷ø

plus a permutation matrix and a row-scaling diagonal matrix





Factorization of CSn DFT Matrices

=

=





Factorization of CSn DFT Matrices

=





Results from Prototype Implementation

Prototypeþÿ ImplementationCan compute exact FFT up to n = 6

Operation Counts For Evaluationn tfull

n tDIFn tMaslen

n [1] 12n(n - 1)

3 4.7 2.7 2.7 3

4 18.8 5.3 5.4 6

5 87.9 8.8 9.1 10

6 486.4 13.8 13.6 15





Future Directions

Theory

• Prove O(n2|Sn|) bounds on operation counts

• Deduce better bases for blocks in factors

• Relate FFT on Sn to FFTs on Sn-1

Implementation

• Improve efficiency ofþÿ implementation

• Port to MATLAB or GAP

• Parallelize decimation-in-frequency FFT algorithm





References

[1] David K. Maslen.

The efficient computation of fourier transforms on the symmetric group.

Mathematics of Computation, 67(223):1121–1147, July 1998.

[2] David K. Maslen and Daniel N. Rockmore.

Generalized FFTs - a survey of some recent results.

DIMACS Series in Discrete Mathematics and Computer Science, 28:183–237,1997.

[3] Bruce E. Sagan.

The Symmetric Group: Representations, Combinatorial Algorithms, andSymmetric Functions.

Wadsworth & Brooks/Cole, Pacific Grove, CA, 1991.

On The WebSenior thesis website: Xhttp://www.math.hmc.edu/~emalm/thesis/\


Documents

Decimation-in-frequency Fast Fourier Transforms for the ...ericmalm.net/ac/research/ffts/emalm-2005-05-02-ffts.pdf · 5/2/2005 · The Discrete Fourier Transform Group-Theoretic