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Algebraic Signal Processing TheoryMarkus Püschel
Electrical and Computer EngineeringCarnegie Mellon University
This work was funded by NSF under awards SYS-9988296 and SYS-310941
Collaborators:José Moura (ECE, CMU)Martin Rötteler (NEC, Princeton)Jelena Kovacevic (BME, CMU)
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PreliminariesAlgebra (as used in this talk) is the theory of groups, rings, and fields
The scope of the algebraic theory is linear signal processing (SP)
In this talk we focus on the discrete case (infinite and finite signals)
Background papers:Basic theory (main paper):Püschel and Moura, “Algebraic Theory of Signal Processing,” submittedFast algorithms:Püschel and Moura, “Algebraic Theory of Signal Processing: 1-D Cooley-TukeyType Algorithms,” submittedPüschel and Moura (SIAM J. Comp 03) and earlier work (Egner and Püschel)New lattice transforms:Püschel and Rötteler (ICASSP ‘04, DSP ‘04, ICASSP ‘05, ICIP ‘05)Sampling:Kovacevic and Püschel (ICASSP ’06)
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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The Basic IdeaSP is built around the key concepts: signals, filters (convolution), z-transform, spectrum, Fourier transform
infinitetime
finitetime
Fouriertransform
infinitespace
finitespace
set ofsignals
set offilters
z-transform
Laurent seriesin z -n
Laurent seriesin z -n
DTFT DFT
finitez-transform C-transform finite
C-transform
polynomialsin z -n
polynomialsin z -n
seriesin Cn
seriesin Tn
DSFT
polynomialsin Cn
polynomialsin Tn
DCTs/DSTs
Algebraic theory: All are instantiations of the same theory
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The Basic Idea (cont’d)Key concept in the algebraic theory:
Infinite and finite time and infinite and finite space are signal modelsBut many others are possibleOnce the signal model is defined, all other concepts follow
Signal model:
Signal model
SignalFilter“z-transform”SpectrumFourier transformFrequency response…
algebra of filters signal module associated “z-transform”
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Why Algebraic Theory?Identifies the filtering (convolution), “z-transform,” spectrum, etc., that goes with the DCTs/DSTs and other existing transforms
Explains boundary conditions for finite signal modelsE.g., why periodic for DFT and symmetric for the DCTs.
New signal models beyond timeSpaceSpace in higher dimension (nonseparable hexagonal lattice, quincunx lattice)
A comprehensive theory of fast transform algorithmsCurrent state: Hundreds of publications, but …Algebraic theory: Concise derivation, classification, reason for existence, many new fast algorithms found for DCTs/DSTs and new lattice transforms
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What we are Not Trying to do
Restate existing knowledge in a more complicated way
Do math for the math’s sake
Provide a theory that is purely “descriptive,”i.e., cannot be applied
The algebraic theory is “operational:”Enables the derivation of new signal modelsEnables the derivation of new fast algorithms for existing and new transforms
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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The Algebraic Structure of Signal and Filter SpaceSignal space, available operations:
signal + signal = signalα • signal = signal
Filter space, available operations:filter + filter = filterα • filter = filterfilter • filter = filter
Filters operate on signals:filter • signal = signal
filtersignal signal
Set of filters = an algebraSet of signals = an -module
vector space
ring
filter
filterfilterfilter filterα
vector space
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(Algebraic) Signal ModelSignals arise as sequences of numbers
To obtain a notion of filtering, Fourier transform, etc., one needs to assign module and algebra
Example: infinite discrete time:
Signal model (definition):
algebra of filtersan -module of signalslinear mapping
z-transform:signal model
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Algebras Occurring in SP: Shift-InvarianceWhat is the shift?
A special filter x (=z-1) = an element ofFilters expressible as polynomials/series in x
Shift-invariance
Shift-invariant + finite-dimensional (+ one shift only):
signal model is shift-invariant is commutative
shift(s) = generator(s) of
polynomial algebra
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Example: Finite Time Model and DFTFinite signals:
Signal model:
Spectrum and Fourier transform from Chinese remainder theorem
Filtering = cyclic convolution
Finite z-transform
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Summary so farSignal model
Shift-invariance: is commutativein addition finite makes a polynomial algebra
Infinite and finite time are special cases of signal models
shift signal model
How to go beyond time?Answer: Derivation of signal model from shift
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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Time Space
(time) marks… …shift
k-fold shift ……
realizationof (time) marks
signals
filters
Operation of filters on signals is automatically defined (the linear extension of the shift operation)
… …
…… ……
Chebyshev polynomials
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Time and Space (cont’d)
Time: we are done
Space:
z-transform
… …linearly independenteach a linear combination
of Cn, n ≥ 0
Chebyshev polynomials
C-transform
Signal model only for right-sided sequences:
but:
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Left Signal Extension
Infinite space model:
Simplest signal extension: monomial
Monomial if and only if
linearly independent
… …left signal extension
depends on choice of C
Chebyshev polynomials
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Visualization
Infinite discrete time (z-transform)
Infinite discrete space (C-transform, C=T,U,V,W)
… …
……-1
……
left boundary
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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Derivation: Finite Time Model… ?n time marks • not closed under shift
• no module
Solution: Right boundary condition
Monomial signal extension:(a = 1: finite z-transform)
Visualization:
…
periodic
…
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Derivation: Finite Space Model
Monomial signal extension: For each four cases
16 finite space models 16 DCTs/DSTs as Fourier transforms
n space marks… ? • not closed under shift
• no module
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16 Finite Space Models
Example: Signal model for DCT, type 2:
Visualization: …
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1D Trigonometric Transforms
Signal models for all existing (and some newly introduced) trigonometric transforms (~30)
Explains all existing trigonometric transforms
Gives for each transform associated “z-transform”filters, etc.
source: “Algebraic Theory of Signal Processing,” submitted
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More Exotic 1-D Model
Generic next neighbor shift
Space variant but shift invariant
Same procedure yields infinite and finite models
Connects to orthogonal polynomials
Applications?
… …
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Top-Down: 1-D Time (Directed) Models
……
… …
infinitefinite or compact
(periodic)
continuous
discrete
making compact
sampling
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Top-Down: 1-D Space (Undirected) Models
…
…
infinitefinite or compact(xx-symmetric)
continuous
discrete
making compact
sampling
…16 choices
2 choices 4 choices
4 choices
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Finite Signal Models in Two Dimensions
Visualization(without b.c.)
Signal Model Fourier Transform
time, separable
space, separable
time shifts: x, y
space shifts: x, y
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time, nonseparable
space, nonseparable
space, nonseparable
time shifts: u, v
space shifts: u, v, w
space shifts: u, v
ICIP ‘05
ICASSP ‘04
ICASSP ’05(see also Mersereau)
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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30G. Bi “Fast Algorithms for the Type-III DCT of Composite Sequence Lengths” IEEE Trans. SP 47(7) 1999
DCT, type III
Algorithmderivation Typical derivation
(More than hundred such papers)
Reason for existence?Underlying principle?All algorithms found?
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Fast Algorithms: Cooley-Tukey FFTSignal model: Finite z-transform
Fourier transform
DFT
Cooley-Tukey FFT
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Cooley-Tukey FFT Type Algorithmsassume p decomposes
coarse decomposition
complete decomposition
Example:
yields Cooley-Tukey FFT
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Application to DCTs/DSTs
Decomposition properties of Chebyshev polynomials
Induced Cooley-Tukey type algorithms (most not known before)
DCT/DST 3/4
DCT/DST 1/2
DCT/DST 5-8
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Algebraic Theory of Algorithms (Beyond DFT)
General Cooley-Tukey type algorithmsmany new algorithms for DCTs/DSTs, RDFT, DHT, DQT, DTT, …
General prime-factor type algorithms
General Rader type algorithms
Explains and easily derives practically all existing algorithms and relationships between transforms
Formulates general principle that accounts for all algorithms
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Organization
Overview
The algebraic structure underlying linear signal processing
From shift to signal model: Time and space
From infinite to finite signal models
Fast algorithms
Conclusions
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Related Work on Algebraic Methods in SPAlgebraic systems theory (Kalman, Basile/Marro, Wonham/Morse, Willems/Mitter, Fuhrmann, Fliess, …)
Focuses on infinite discrete time; different type of questions
Fourier analysis/Fourier transforms on groups G(Beth, Rockmore, Clausen, Maslen, Healy, Terras, …)
In the algebraic theory the special caseIf G non-commutative, necessarily non-shift-invariantAlgebraic theory provides associated filters etc., ties to SP concepts
Algebraic methods to derive DFT algorithms(Nicholson, Winograd, Nussbaumer, Auslander, Feig, Burrus, …)
Recognizes algebra/module for DFT, but only used for deriving algorithms
Origin of this workBeth (84), Minkwitz (93), Egner/Püschel (97/98)Helpful hints: Steidl (93), Moura/Bruno (98), Strang (99)
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Future Work
FastAlgorithms
Core Theory of Signal Processing
Sampling
Filterbanks
MultiresolutionAnalysis
HigherdimensionalSignal Processing
Frames
Algebraic Theory of Signal Processing
large parts donecurrent research
Collaborators: José Moura,Jelena Kovacevic, Martin Rötteler
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Algebraic Theory of Signal Processing: Conclusions
Signal model: One concept instantiating different SP methods
General (axiomatic) approach to linear SPFinite SP, understanding existing transformsFirst new applications:
New SP methods (non-separable 2-D)Comprehensive theory of fast algorithms
SMART project: www.ece.cmu.edu/~smart
Signal modelSignalFilter“z-transform”SpectrumFourier transformFrequency response…
Shift derivation
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Chebyshev Polynomials
Defining three-term recurrence:
Special cases:
Closed forms:
symmetry
back1back2back3
choice
n ≥ 0
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The General Fourier Transform
Given any signal model
…
(filters)
(signals)
operates on(filtering)
(filters)
spectrum = set of smallest invariant subspaces
Infinite discrete time:
projection onto
eigenspace for all filters
…
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Finite Shift-Invariant Signal Models
Finite signals:
Which finite-dimensional algebras are commutative?Answer: Polynomial algebras (focus on one variable)
Filtering (convolution): multiplication modulo
Signal model: