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Paraunitary PR Filter Banks

• PR   System:

– Filter   bank system without aliasing, amplitude and phase dis-
tortions.

• Causality   of PR System:

– Hk (Z)   is a causal system. ⇒ E(Z) is a causal system. ⇒ R(Z) can
be made a causal system. ⇒ Fk (Z) can be made a causal system
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• For   PR:
P (Z) = I                         (1)

R(Z).E(Z) = I                        (2)

T (Z) = Z −(M−1)                      (3)

• Eqn.(2)     is suﬃcient for PR, whether system is FIR or IIR.

• Further,     if we modify Eqn.(2) as

R(Z).E(Z) = C.Z −m0 .I                   (4)

• System      satisfying Eqn.(4) still have prefect reconstruction but now

T (Z) = C.Z −Mm0 .Z −(M−1)                (5)

T (Z) = C.Z −(Mm0 +M−1)                  (6)
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• Let, n0 = (M m0 + M − 1)

⇒ T (Z) = C.Z −n0                        (7)

⇒ x[n] = C.X[n − n0]
ˆ                                       (8)

More General Condition for PR:

•A   PR system is an alias free system with T (Z) = delay.

• For   the system to be alias free P (Z) should be pseudocirculant.

•A   matrix is said to be circulant if every row is obtained using a
circular shift (by one position) of the previous row.
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• Example:
                     
 P0(Z) P1(Z) P2(Z) 
                   
                   
 P2(Z) P0(Z) P1(Z)                    (9)
                   
                   
P1(Z) P2(Z) P0(Z)

•A   matrix is said to be pseudocirculant if it is circulant with ele-
ments below the main diagonal are multiplied with Z −1.
                                
 P0(Z)    P1(Z) P2(Z) 
                       
 −1                    
 Z P2(Z) P0(Z) P1(Z)                      (10)
                       
                       
−1      −1
Z P1(Z) Z P2(Z) P0(Z)

• Thus   all rows in the M × M pseudocirculant matrix P (Z) are deter-
mined by the 0th row, which is

P0(Z) P1(Z) · · · PM−1 (Z)              (11)
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• M -channel     maximally decimated ﬁlter bank is free from aliasing iﬀ
the P (Z) is pseudocirculant.

• Under      this condition T (Z) can be expressed as

T (Z) = Z −(M−1)    P0(Z M ) + z −1 P1(Z M ) + · · · + z −(M−1) PM−1 (Z M )   (12)

where Pm(Z) are the elements of the 0th row of P (Z).

• This   is a delay only if Pm(Z) = 0 for all but one value of m in the
range 0 ≤ m ≤ (M − 1). And this nonzero Pm(Z) must have the form
C.Z −m0 .

• Therefore,     an alias free system has prefect reconstruction iﬀ the
pseudocirculant P (Z) has 0th row equal to

0, · · · , 0, C.Z −m0 , 0, · · · , 0 .                     (13)
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• In   other words, P (Z) has the form
            
−m0       0  IM−r 
R(Z).E(Z) = C.Z                            (14)
−1
Z Ir 0

for some r in 0 ≤ r ≤ (M − 1), some integer m0, and some constant
C = 0.

• The    reconstructed signal then will be

x[n] = C.x[n − n0]
ˆ                                   (15)

where n0 = M m0 + r + M − 1

• Example: M =2 −→     2-channel QMF

0 ≤ r ≤ (M − 1) ⇒ r = 0, 1 ⇒ IM−r = I2   or I1
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• P (Z)   has the form
                                         
−m0   1 0                         −m0    0 1
C.Z                       or      C.Z                    (16)
0 1                                 Z −1 0

• Every    QMF bank satisfying equation (14) for some r can be ob-
tained by starting from a QMF bank satisfying equation (2) and
inserting a delay Z −r in front of each synthesis ﬁlter.

• Condition    on Determinant: Equation (6) ⇒

det R(Z). det E(Z) = C0.Z −k0                (17)

for some C0 = 0 and some integer k0.

• Any     PR system (FIR or IIR) has to satisfy this determinant con-
dition.
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FIR perfect Reconstruction Systems:
• Hk (Z)   and Fk (Z) −→ FIR ﬁlters.
⇒ E(Z)     and R(Z) are FIR.
⇒   determinants of E(Z) and R(Z) are FIR.
⇒   For PR, product of the two determinants should be a delay.
⇒ det E(Z) = α.Z −k      α = 0, k =integer.

⇒ det R(Z)    must have a similar form.

• Assuming E(z)     can be inverted, R(z) can be obtained as E −1(z)

• However,     computing E −1(z) is a complicated process. Can we have
a simple approach?

• Yes,     using Paraunitary property on E(z) matrix
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Paraunitary PR Filter Banks

• PR   ﬁlter banks in which E(Z) satisﬁes a special property called
lossless or paraunitary property.
⇒ E(Z)    is a causal FIR matrix called paraunitary matrix, which
satisﬁes the PR property with K = M degree of E(Z).

• Eventhough   paraunitary property is not a necessary condition for
PR, the ﬁlter bank based on E(Z) satisﬁes many other useful prop-
erties.

1. Fk (Z) can be found from Hk (Z) by inspection.
2. Exist good design techniques with fast convergence.
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3. Paraunitary property is basic to the design of cosine modulated
PR system.
4. Paraunitary property is basic to generation of orthonormal wavelet
basis.

Lossless Transfer Matrices:

r                         p
i/p s       Hkm (Z)       o/p s

• System    with r inputs and p outputs, with transfer function Hkm(Z)
from every input to every output.

• Entire   system is said to be a MIMO LTI system, and can be
characterized by the set of ‘pr’ transfer functions Hkm(Z).
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• Let, y[n] = [y0[n], y1[n], · · · , yp−1 [n]]   be the system output vector.

• Let, x[n] = [x0[n], x1[n], · · · , xr−1[n]]    be the system input vector.

• Compactly,        system can be represented as:

Y (Z) = H(Z).X(Z)                      (18)

• H(Z)    is called the transfer matrix of the system.

Analysis Filter Bank:
(M−1)
Hk (Z) =             z −l .Ekl (Z)          (19)
l=0

⇒          h[Z] = E(Z M ).e(Z)                (20)
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where,                                       
 H0(Z) 
          
          
 H1(Z) 
h[Z] = 



   .
.      
          
          
HM−1 (Z)

                                                   
     E00(Z)      E01(Z)    ···    E0(M−1) (Z)     
                                                  
                                                  
     E10(Z)      E11(Z)    ···    E1(M−1) (Z)     
E[Z] = 



        .
.                                         
                                                  
                                                  
E(M−1)0(Z) E(M−1)1 (Z) · · · E(M−1)(M−1) (Z)
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               
          1      
                 
                 
        z −1     
e[Z] = 

.

          .
.      
                 
                 
z −(M−1)

Synthesis Filter Bank:
(M−1)
Fk (Z) =           z −(M−1−l) .Rkl (Z)   (21)
l=0

⇒        f T (Z) = z −(M−1) .e(Z).R(Z M )    (22)

where, f T (Z) = [F0(Z), F1(Z), · · · , FM−1(Z)]

e(Z)   denotes eT (Z −1)
∗
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and
                                                   
      R00(Z)       R01(Z)    ···   R0(M−1) (Z)      
                                                    
                                                    
      R10(Z)       R11(Z)    ···   R1(M−1) (Z)      
R(Z) = 



         .
.                                          
                                                    
                                                    
R(M−1)0 (Z) R(M−1)1 (Z) · · · R(M−1)(M−1) (Z)

Note:

1. H(Z) is termed as paraconjugate of H(Z).

2. This is deﬁned such that, on the unit circle, H(Z) = [H(Z)]∗. i.e.
complex conjugation.

3. Let H(Z) = 1 + 2z −1, then H(Z) = 1 + 2Z .
−1                     ∗   ∗
+b
4. Let H(Z) = (a+bz−1) , then H(Z) = (a∗+d∗z) .
(c+dz )                (c    z)
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5. First conjugate coeﬃcients and then replace z with z −1 . Symboli-
cally, H(Z) = H∗ (Z −1).
T

(N−1)
6. If H(Z) =           h[n].z −n ,   then
n=0
(N−1)
H(Z) =             h∗[n].z n
n=0

z −(N−1) .H(Z) = h∗(N − 1) + h∗(N − 2).Z −1 + · · · + h∗(0).z −(N−1) .

i.e. the coeﬃcients are reversed and then conjugated.

7. Given,                                                           
      1       
              
              
     z −1     
e[Z] = 

.

      .
.       
              
              
z −(N−1)
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             T
          1       
                  
                  
          z       
e[Z] = e∗ (Z ) = 
T   −1

 .

           .
.      
                  
                  
z (N−1)
               T
−(N−1)
Z                
                 
 −(N−2)          
 z
(N−1) 

=Z                          .
   .             
   .             
                 
                 
1

Z −(N−1) .e(Z) = [Z −(N−1) , Z −(N−2) , · · · , 1]

• Thus   compactly, the analysis ﬁlter bank is represented as,

h(Z) = E(Z M ).e(Z)                      (23)
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• The    synthesis ﬁlter bank is represented as

f T (Z) = z −(M−1) .e(Z).R(Z M )                       (24)

Lossless Property:

• A p×r    causal matrix H(Z) is said to be lossless if
(a). Each entry Hkm(Z) is stable and                      (b). H(ejw ) is unitary,
i.e.,
H † (ejw ).H(ejw ) = d.Ir ,      ∀ ω & some d > 0                  (25)

Note:

1. A† is the transpose conjugate of A.
⇒ H † (ejw )   is [H(ejw )]†.
2. If H(ejw ) = h[0] + h[1].e−jw
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⇒ H † (ejw ) = h†[0] + h†[1].ejw .

• Additionally,     if H(Z) coeﬃcients are real, then H(Z) is lossless
bounded real (LBR).

• The    property given in equation (25) is the unitary property.

• Paraunitary     Property:

– For   rational transfer functions it can be shown that

H(Z).H(Z) = d I,   ∀ Z             (26)

– Whereas      equation (25) holds on the unit circle z = 1.

• The    property given in equation (26) is termed as paraunitary prop-
erty.

• Or   equation (26) =⇒ equation (25).
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• Therefore,     we can deﬁne a lossless system to be a causal, stable
paraunitary system.

• So   in order to prove that a channel system is lossless, it is suﬃcient
to prove (a) stability & (b) paraunitariness.

• Columnwise       Orthogonality:
→     If columns k & m are mutually orthogonal, then

Hk (Z)Hm (Z) = 0f ork = m                (27)

• Normalized      Systems:
−If   a loss less system has d =1 in equation (25), we say it is
a normalized lossless.
−Accordingly    we have normalized- unitary and normalized pa-
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raunitary.

• Square    Matrices:
−   For the case of square matrices, Equation (26) →

H −1 (z) = H(Z)/d       ∀ Z              (28)

• So   that the inverse is obtained by use of ’tilde’ operation.

• Power     complementary:
−For   a given a column k,

Hk (Z)Hk (Z) = d

• Moreover,    in case of square matrices,

H(Z)H(Z) = H(Z)H(Z) = dI
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→   Every row is a power complementary and every pair of rows is
orthogonal.

• Let H(z)   is a 2 × 2 matrix

                
 H00(Z) H01 (Z) 
H(Z) =                               (29)
H10(Z) H11 (Z)

• H(Z)H(Z) = H(Z)H(Z) = dI

⇒

H10(Z)H00 (Z) + H10(Z)H10(Z) = d,

H01(Z)H01(Z) + H11(Z)H11(Z) = d,    (8)

H00 (Z)H01(Z) + H10(Z)H11(Z) = 0
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Some Properties of Paraunitary Systems:

1. Determinant is all-pass:

• Let, p = r   and det H(Z) = A(Z)
• According     to equation (25)

A(Z)A(Z) = dr       ∀ Z

⇒ A(z)   is allpass.

• In   particular, if H(z) is FIR then A(z) is a delay, i.e.,

det H(Z) = a.Z −k     k ≥ 0, a = 0       (30)

(for FIR paraunitary H(z)).
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2. Power Complementary Property:

• For M × 1   transfer matrix h[Z] = [H0(Z), H1(Z), · · · , HM−1(Z)]T , the pa-
raunitary property implies power complementary property, i.e.,
(M−1)
2
H(ejω ) = C,     ∀ ω                     (31)
k=0

• This   follows directly from h[Z].h[Z] = C .

3. Submatrices of Paraunitary H(Z):

• From the    deﬁnition of paraunitary, every column of a paraunitary
transfer matrix is itself paraunitary.
• In   fact, any P × L system matrix of H(Z) is paraunitary.
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Interconnections and Examples:
Operations Preserving Paraunitary and Lossless Properties:

• If H(Z)   is square and paraunitary
⇒ H(Z M ), H T (Z)   and H(Z) are also square and para unitary.

• If H(Z)   is lossless then H(Z M ) and H T (Z) are lossless.

H0 (Z)         H1 (Z)

• Overall   transfer function

H(Z) = H1(Z)H0(Z)

• H(Z)   is paraunitary if H0(Z) and H1(Z) are paraunitary.
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• If H0(Z)   and H1(Z) are lossless, then H(Z) is also lossless.
⇒   The operation of cascading (or product) preserves losslessness.
Givens Rotation:
                
 Cosθm Sinθm 
Rm =               ,      θm   real
−Sinθm Cosθm

c = cos θm

s = sin θm

C
−S

S

C
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• If y = Rm x,   then y is obtainable by rotating x by θm, clockwise.
x

r.sin(φ)

r
r.sin(φ − θm )
θm
r              y
φ

0                r.cos(φ)   r.cos(φ − θm )

• The   operator Rm is known as the Givens rotation, planar rotation
or simply rotation.

• Rm   is unitary.
i.e         T
Rm Rm = I
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          
1 0 
• Let,       ∧(Z) =         
−1
0 Z
                     
 1 0  1 0 
⇒ ∧(Z). ∧ (Z)=             =1
0 Z    0 Z −1

⇒ ∧(Z)    is paraunitary.

R0                      R1                     RN

Z −1                Z −1   Z −1

(Z)

• Figure   shows a cascade of paraunitary systems, which is therefore
paraunitary.
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• The   transfer matrix is

HN (Z) = RN ∧ (Z) RN−1 ∧ (Z) · · · R1 ∧ (Z) R0

• Example: N =1 ⇒ HN (Z) = R1 ∧ (Z) R0.

with θ0 = θ1 = π , the transfer function of cascaded system is
4

                                                
π         π
 cos(θ0 ) sin(θ0)   cos( 4 ) sin( 4 )    1  1 1
R0 =                   =                    =√       .
−sin(θ0) cos(θ0)     −sin( π ) cos( π )     2 −1 1
4       4

    
1 0 
∧(Z)=        
0 Z −1
             
1     1 0  1 1 
∧(Z)R0 =   √
2
              
0 Z −1    −1 1
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                
     1      1 
1
=    √
2
                
−Z −1   Z −1

                         
 1 1  1
1
1 
R0 ∧ (Z)R0 =    2              
−1 1    −Z −1 Z −1
                                   
          1 − Z −1         1 + Z −1   
1
Hm (Z) = 
2                                       
−(1 + Z −1 ) −(1 − Z −1)
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Paraunitary Vectors:
                               
 1                          1 
1.        e(Z) =        ⇒ e(Z).e(Z) = [1 Z]        = 2.
Z −1                         Z −1

Z −1

⇒   e(Z)   is a paraunitary.

cosθ

sinθ
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         
 cos(θ) 
2.        P0 =         
sin(θ)
       
T                        cos(θ) 
P0 .P0 = [cos(θ) sin(θ)]          = 1.
sin(θ)

⇒   P0   is normalized lossless.

3. Since all the building blocks are paraunitary, the cascaded system
is paraunitary.

R1                          RN

Z −1        Z −1            Z −1

P0
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−   It has transfer matrix

PN (Z) = RN ∧ (Z) RN−1 ∧ (Z) · · · R1 ∧ (Z) R0

4. N represents M × M DFT matrix.

Z −1

Z −1
W∗

Z −1

e(Z)        h(Z)

⇒ W∗   is unitary.
⇒   Moreover, e(Z).e(Z) = M , so that e(Z) is paraunitary.
⇒ h[Z] = W ∗.e[Z]   is —–paraunitary.

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