# CLASSICAL INFORMATION THEORY by chenshu

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```									    HERMITE INTERPOLATION in
LOOP GROUPS and CONJUGATE

Wayne Lawton

Department of Mathematics
National University of Singapore
S14-04-04, matwml@nus.edu.sg
http://math.nus.edu.sg/~matwml
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
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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