www.jntuworld.com 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 www.jntuworld.com • 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) www.jntuworld.com • 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. www.jntuworld.com • 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) www.jntuworld.com • 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) www.jntuworld.com • 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 www.jntuworld.com • 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. www.jntuworld.com 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 www.jntuworld.com 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. www.jntuworld.com 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). www.jntuworld.com • 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) www.jntuworld.com 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) www.jntuworld.com 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) ∗ www.jntuworld.com 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) www.jntuworld.com 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) www.jntuworld.com 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) www.jntuworld.com • 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 + h.e−jw www.jntuworld.com ⇒ H † (ejw ) = h† + h†.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). www.jntuworld.com • 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- www.jntuworld.com 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 www.jntuworld.com → 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 www.jntuworld.com 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)). www.jntuworld.com 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. www.jntuworld.com 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 www.jntuworld.com • 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 www.jntuworld.com 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. www.jntuworld.com • 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 www.jntuworld.com 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) www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com − 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.