POLYPHASE GEOMETRY Wayne Lawton Department of Mathematics National University of Singapore S , Chinese-French-Singaporean Joint Workshop on Wavelet Theory and Applications (9 – 13 June 2008)

Algebra numbers circle groups Euler’s Isomorphism subalgebra of functions defined by Laurent polynomials (dense) matrices algebra of continuous functions algebra of denotes involution, norm-complete

Signals, Shifts, Rotations Signal Space Shift Operator Rotation Operators* *also called modulations and frequency shifts

Invariant Linear Operators Definition is shift invariant if Theorem Every linear shift invariant operator Corollary They form a commutativealgebra that we call the convolution algebra

Toeplitz Matrices Banded finite support

Laurent Polynomial Algebra The z-transform algebra onto maps this operator norm and hence the completion of Hilbert subspace is called self(skew)-adjoint if and unitary if An element A and clearly preserves the is a hasmany unitary elements but has very few! algebra of operators on the of aalgebra

Spectrum if The spectrum of an operator on is called an eigenvalue of is for some if then and ifthenfor defined bySuch signals can be interpreted as generalized eigenvectors for

Invariant Linear Operators Theorem For every the subset of linear operatorsthat satisfy algebra. that we call the q-Multirate Algebra form a Interesting than the Convolution Algebra ( whenever q > 1 )

Rotation Algebra andThe unitary operators satisfy the commutation relation since they and their adjoints generate the rotation of operators onalgebra Theorem q-multirate algebra

q-Block Toeplitz Matrices

Algebraic Polyphase Matrix [B(z)] is obtained from the blocked z-transform an isomorphism from q-multirate algebra onto algebra of q x q matrices over Advantage: B is FIR iff entries of [B(z)] in P(T) Disadvantage: can’t see the geometry – how the operator was built from convolutions and rotations - but terrific for approximation non FIR by FIR especially for applying Pressley-Segal Theorem: polynomial loops are dense in U(q) for q > 1

Matrix Algebra is rotation algebra generated by matrices and that satisfy Observation

Geometric Polyphase Matrix satisfies Theorem The algebrais an Azumaya algebra. viz. elements are sections of a (nontrivial) matrix bundle over Observation Over P(T) replace circle by a toric variety and bundle by scheme. Several intractible problems may soften up.

Geometric Tools Exist introduced Azumaya algebras in algebraic geometry applied Azumaya algebras to diophantine geometry