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ADSP-19-Paraunitary-EC623

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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.
  Cascade Structure:

                              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.
                                                                       www.jntuworld.com
• 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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