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HERMITE INTERPOLATION in LOOP GROUPS and CONJUGATE QUADRATURE FILTER APPROXIMATION Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sg http://math.nus.edu.sg/~matwml Conjugate Quadrature Filters Z, R, C, T integer, real, complex, unit circle C (T ) C - valued, continuous functions P (T ) Laurent polynomials m 2 fixed integer i 2 / m e primitive m-th root of unity PQ (T ), CQ (T ) conjugate quadrature filters m 1 k 0 | F ( z ) | 1 , z T k 2 Applications and Requirements CQF’s are used to construct paraunitary filter banks and orthonormal wavelet bases PQ (T ) FIR filters, compactly supp. wavelets CQ (T ) \ PQ (T ) linear phase filters Factor U ( z ) [(1 z m ) /(1 z )] d for regularity needed for stable filterbanks & smooth wavelets Design Approaches Much more difficult to design polynomial CQF’s Jorgensen describes an approach based on factorizing their polyphase representations (Notices of the AMS, 50(8)(2003),880-894) We describe an alternate approach that is based on approximating elements in CQ (T ) by elements in PQ (T ) This approach can preserve specified factors Two-Step Approximation Method Problem given U P (T ), H C (T ) UH CQ (T ) construct P P (T ) UP PQ (T ), P H Solution Step One construct Q P(T) UQ PQ (T), | Q || H | Step Two construct P P (T ) UP PQ (T ), P H Table of Contents Introduction Polyphase Representations and Loop Groups Spectral Factorization and Bezout Identities Phase Transformations of Modulation Matrices Hermite Interpolation and CQF Approximation Polyphase Representations Theorem The functors T YT f Y (τ f)(z) f ( z), z T f T Y T Y ( h)(z) h ( z ), z T h m h satisfy τ f f h : T Y f h Corollary v : T C is a modulation vector m v(F) [F, F,..., F] for F C(T) Cv v m 1 T 0 1 0 0 where 0 0 1 0 is the m x m C circulant matrix 0 0 0 1 1 0 0 0 0 Polyphase Representations Proposition w(F) : T C is polyphase vector m for F C(T) v(F) σ w(F) where (Fourier transform) (z ) 1 1 1 1 1 0 0 0 1 2 1 0 z 02 0 1 1 2 4 2 0 0 z 0 m 1 1 2 0 0 0 0 z m1 Corollary F(z) z k 1w( F ) k ( z m ), m k 1 F P(T) w( F ) k P(T), 2 m1 F CQ (T) w(F ) : T S C m Winding Number Definition The winding number of f : T T iθ 2 df(e ) W(f) 1 i2 iθ if f is differentiable 0 f(e ) ~ ~ ~ W(f) W(f ) f is differentiable and || f - f || 2 Remark W(f) is well defined, takes values in Z, is a continuous function of f, is a special case of the Brouwer degree of a map of sphere to itself Lemma Given f:TT there exists h:TiR with f exp(h) iff W(f) 0 Homotopy and Matrix Extension Definition Maps f i : S S , i 0,1 are homotopic n n iff F : [0,1] Sn Sn F(j,) f j , j 0,1 Theorem (H. Hopf) Map of a sphere into itself are homotopic iff their Brouwer degrees are equal Corollary f is homotopic to constant iff W(f)=0 Proposition f : T S , g : T SU(m) g , 1 f 2m-1 Proof Let e1 [1,0,..., 0] then g p(g) ge1 T is a fiber bundle, and hence a fibration 2m 1 p : SU(m) S SU(m)/SU(m 1) and the result follows from the homotopy lifting property Definition g is a polyphase matrix for f Algebra and Matrix Extension Proposition If entries f : T C in P(T) m and have no common zeros in C \ {0} then g : T SL(m), with entries in P(T) and g , 1 f Proof Follows from the Smith form for f Proposition If entries f : T S2m -1 in P(T) then g : T SU(m), with entries in P(T) and g , 1 f Proof Follows from the factorization theorem for m x 1 paraunitary matrices Loop Groups Remark Elements in C(T) C mm , called loops, may be regarded as matrix-valued functions on T or as matrices having values in C(T) Definition Loop groups G C(T) SU(m) G C (T) SU(m) G pol P(T) SU(m) their Lie algebras G C(T) su(m) G C (T) su(m) Gpol P(T) su(m) Exponential Function Proposition Let O su(m) be matrices whose spectral radius Then exp : su(m) SU(m) is a real-analytic diffeomorphism of O onto an open neigborhood O of I SU(m) Proposition (Trotter) Given h1 ,..., h M G lim exp L exp h1 L hM L L exp h1 h M Furthermore, if h1 ,..., h M G then convergence holds in the C (T ) topology Magic Basis Theorem For n 0, {1, i} define 0 z n a(n, , z ) , b(n, , z ) a (n, , z ) z n 0 1 z z n n z z n n c(n, , z ) n n 2 z z n z z n X {c(0, i, z ), a(0, i, z), a(0, i, z)} is basis for su(2) B2 X {a, b, c : n 0, 1, i} basis P(T) su(2) leads to basis B for Gpol and B B B I 2 Density Theorem G pol is dense in G , G Proof Euler’s formula implies that B B exp θ B cos θ I sin θ B G pol Trotter’s formula implies that every element in exp G pol is the limit of elements in G pol and every element in G is the product of elements in exp G pol Corollary PQ (T ) is dense in CQ (T ) Proof Approximate polyphase matrix of F C Q (T ) Spectral Factorization Definition Let H C (T) A function F C(T) is a spectral factor of H if | F |2 H Definition P P(T) is minimal phase if all its roots have modulus 1 Theorem (L. Fejer and F. Riesz) Every P P (T) has a minimal phase spectral factor Definition F C(T) is an outer function if c T, H C (T) log H L (T) and 1 2 1 e rz is F(z) c exp lim e rz is log H(e ) ds r 1 2 is 0 Bezout Identities Theorem If U1 ,..., U m P (T) have no common roots in C \ {0} and H1 ,..., H m C (T) satisfy the Bezout identity U1H1 U m H m 1 then 0, Q1 ,..., Q m P(T) U1Q1 U m Q m 1, || H k Qk || , k 1,.., m Proof Uses matrix extension in P(T) SL(m) and Weierstrass approximation Remark Extends the 1-dim version of a multi-dim result in W. Lawton and C. A. Micchelli, Bezout identities with inequality constraints, Vietnam Journal of Mathematics 28#2(2000),1-29 Step One Theorem If H C(T), U P(T) , UH C Q (T) then 0, Q P(T) Q has no zeros in T UQ PQ (T) and || | H | | Q | || Proof Uses previous theorem Modulation Matrices Definition V : T C mm is a (unitary) modulation matrix if it maps T into U(m) and if CV τV mm Proposition V : T C is a modulation matrix iff W : T U(m) V ΩΛσW Furthermore Vi, j P(T) Wi, j P(T) and F C Q (T) modulation matrix V V1,1 F and if F PQ (T) we may choose V Vi, j P(T) Proof Follows directly from previous results Stabilizer Subgroups 1 1 Definition Subroups Sr σ G, S S r Lie algebras S r {h G : exp h Sr }, S {h G : exp h S } Subroups Sr Sr G , S S G S G , S S r r 1 1 Lie algebras S S r G , S S G r Corollary V : T C a modulation matrix g G mm g S g V is a modulation matrix CgC g -1 g Sr Vg is a modulation matrix g g Analogous statements hold for C functions Bases for Stabilizer Subgroups Corollary σ B is a basis for S r G pol and B is a basis for S G pol 1 1 Furthermore, B 2 I if B is in either basis Corollary Sr G pol is dense in S and in S r r S G pol is dense in S and in S Proof Follows from density theorem and the fact that h G, exp h exp h Structure of Left Stabilizer Algebra Proposition If h G then h S h1 h2 h3 hm h h1 h2 hm1 m h hm 1 hm 2 2 h1 2 hm 2 2 m 1 h2 m 1 h3 m 1 h4 h1 m 1 where h1 , , hm G satisfy Structure Equations w(h1 )1 0, h1 i R m 2n hn1 n hn1 2n 1 m 2n hm2 j m1 j hj , j 2,..., n Diagonal Stabilizer Subgroups Definition D {g G : g is a diagonal matrix} D DG D {d G : exp(d ) D} D {d G : exp( d ) D } Lemma D {h G : h is a diagonal matrix} D D G Proposition h D Sr h i diag [b1 ,...,bm ], b j C(T) real, j1 b j 0 m h D S h i diag [a, a,..., a], a C(T) real, w(a) 1 0 m 1 Phase Transformations Corollary V modulation matrix f:T T, W(f) 0 d D S , d r D S r ((exp d ) V (exp d r )) 1,1 f V1,1 Proof Since W(f) 0 h : T iR exp h f Construct d i diag [a, a,..., m1 a ] d r i diag [b1 ,..., bm ] where and ib1 w(h)1 ia h w(h)1 b2 b1 hence w(a )1 0 b3 ,..., bm 0 Factor Preserving Transformations mm Definition Mr { g : T C : U | gi,1 , i 2 } mm M { g : T C : U | g1, j , j 2 } Subroups U r G M r U G M U U r Lemma The Lie algebras U r {h G : exp h U r } G M r U r G M r U {h G : exp h U } G M U G M Proposition If V : T C mm and U |V1,1 then g U U | (g V) 1,1 and g U r U | (V g)1,1 Definitions and assertions hold for C functions Proof Follows directly from the equations (g V)1,1 k 1 g1,k Vk ,1 (V g)1,1 k 1 V1,k g k ,1 m m Jets Definition C (T) space of infinitely differentiable complex-valued functions on T with topology of uniform convergence of N-derivatives for any N Dz : P(T) P(T) , Dz f f z i D : C (T) C (T) , D f f (e ) izDz f ( z j ) , d j 0, d j 1 d j s s For U ( z ) dj j 1 define U-jet maps J z : P(T) C , J : C (T) C d d J z f [ f ( 1 ),..., D d1 1 z f ( 1 ), f ( 2 ),..., D d s 1 z f (s ) J f [ f ( 1 ),..., D d1 1 f ( 1 ), f ( 2 ),..., D d s 1 f (s ) Parameterization of Jets Lemma P(T) C (T) and linear isomorphism L : C C J θ f L J z f , f P(T ) d d Proof Follows from D izDz Proposition ker (J z ) U P(T) is an ideal in P(T) and ker (J ) U C (T) is an ideal in C (T) J z P(T) P(T) / U P(T), J θ C (T) C (T) / U C (T) linear injection : C P(T) J z Φ v v, v C d d C space of algebraic polynomial s of degree d d Proof First two assertions are standard algebra, Shilov’s Linear Algebra proves third using CRT Extended Jets Definition The extended right and left jets and J : G C d(m-1) Jr : G C d(m-1) are C-linear maps of the loop algebra into Cd(m-1) J r h [J h2,1,...,J hm,1 ] , h G T J h [J h1,2 ,...,J h1,m ] , h G T Lemma U r {h G : J r h 0} U {h G : J h 0} Lemma Vr J r Sr J r (Sr Gpol ) V J S J (S Gpol ) d(m-1) are R-linear subspaces of C Cross Sections and Hermite Interpolation Lemma If d r D S r G pol there exists r : Vr S r G pol h r (h) d r is R-linear and diag (r (h)) d r , h Vr and J r Θ r : Vr Vr is the identity map on Vr Analogous assertions hold for d and Theorem If d r D S r then exp( d r ) is in the closure of U r Sr G pol Analogous assertions hold for d D S G pol Proof Let Ar : r (Vr ) G pol Trotter approx. exp Br J r log Ar r : Vr Vr approx. identity so result follows by Brouwer degree argument Step Two Theorem If H C(T), U P(T) , UH C Q (T) then 0, P P(T) P has no zeros in T UP PQ (T) and || H P || ~ Proof Compute H C(T) with no zeros in T ~ with H H then compute Q using Step One and multiplication by an integer power of z so that U Q PQ (T) , | Q | | H |, W ( phase ( f )) 0 ~ where f phase ( H Q ) : T T Now compute d r , d as in the Phase Modulation page and then apply the previous Theorem