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CLASSICAL INFORMATION THEORY by chenshu

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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 YT
    
    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 m1 
                                                     
Corollary F(z)   z k 1w( F ) k ( z m ),
                     m
                     k 1
            F  P(T)  w( F ) k  P(T),
                                           2 m1
            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:TT there exists h:TiR 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 mm , 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 mm is a (unitary) modulation
matrix if it maps T into U(m) and if CV  τV
                      mm
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
                    mm

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               
                                         hm1 
          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  hn1   n hn1
 2n  1  m  2n  hm2 j     m1 j
                                           hj ,   j  2,..., n
         Diagonal Stabilizer Subgroups
Definition D  {g  G : g is a diagonal matrix}
                       
           D  DG
           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,  j1 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,...,    m1
                                   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
                           mm
Definition Mr  { g : T  C     : U | gi,1 , i  2 }
                            mm
           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 mm 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

								
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