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This deals about the digital processing programs
www.jntuworld.com Pictorial View Point of Spectra at Various Stages in 2-Channel QMF x [n] v [n] y0[n] x [n] 0 0 H (Z) 2 2 F (Z) 0 0 0 x[n] x[n] H (Z) F (Z) 1 2 2 1 x [n] v [n] y [n] x [n] 1 1 1 1 www.jntuworld.com Upper Channel (Low Pass) X(e jw ) 1 0 2 w 2 H0(e jw ) 1 0 2 w X0 (ejw ) 1 2 0 2 w V0 (ejw ) 1 2 0 2 w 2 www.jntuworld.com Y0 (ejw ) 1 2 0 2 w 2 F 0 (ejw ) 1 0 2 w X0 (ejw ) 1 2 0 2 w aliasterm at output F (Z) 1 0 2 0 2 w 2 www.jntuworld.com Lower Channel (High Pass) X (ejw ) 1 0 2 w 2 H1(e jw ) 1 0 2 w X1 (ejw ) 1 2 0 2 w V1 (ejw ) 1 2 0 2 w 2 www.jntuworld.com Y1 (ejw ) 1 2 0 2 w 2 F 1 (ejw ) 1 0 2 w X1 (ejw ) 1 2 0 2 w alias term at output F (Z) 1 1 2 0 2 w 2 www.jntuworld.com Perfect Reconstruction QMF Bank • If a QMF bank is free from aliasing, amplitude and phase distor- tions, then it is said to have the perfect reconstruction property. Mathematically: • QMF: 1 1 X(z) = 2 [H0(z)F0(z) + H1 (z)F1(z)]X(z) + 2 [H0(−z)F0(z) + H1(−z)F1(z)]X(−z) • Let X(z) = T (z)X(z) + A(z)X(−z) where, T (z) = 1 [H0(z)F0(z) + H1(z)F1(z)] is the T.F representing ampli- 2 tude and phase distortion. • A(z) = 1 [H0(−z)F0(z) + H1(−z)F1(z)] 2 is the T.F representing aliasing. www.jntuworld.com • Alias free QMF: A(z) = 0 ˆ X(z) = T (z)X(z) • Alias and Distortion free QMF: (T (ejw )) T (ejw ) = |T (ejw )|ej |T (ejw )| = c and (T (ejw )) = a + bw T (ejw ) = cej(a+bw) = ceja ejbw T (ejw ) = ce−jn0 w T (z) = cz −n0 & c=0 • Therefore, for a perfect reconstruction QMF ˆ X(z) = cz −n0 X(z) www.jntuworld.com x(n) = c.x[n − n0] ˆ & c=0 ∀ possible inputs x[n]. • In other words, x[n] is merely a scaled and delayed version of x[n]. ˆ A Simple Alias free QMF system: • In the earliest known QMF banks the analysis ﬁlters were related as: H1(z) = H0 (−z) (1) |H1(ejω | = |H0 (ej(Π−w))| (2) • This ensures that H1(z) is a good high pass ﬁlter if H0(z) is a good low pass ﬁlter. www.jntuworld.com • In fact, |H1(e(jω))| is a mirror image of |H0(ejω )| with respect to the Π quadrature frequency 2 , justifying the name quadrature mirror ﬁlters. • For alias cancelation we have: H0 (−z)F0(z) + H1(−z)F1(z) = 0 ⇒ F0(z) = H1(−z) & F1(z) = −H0(−z) (3) (1)&(3) ⇒ F0(z) = H0(z) & F1(z) = −H1(z) (4) • Thus all four ﬁlters are completely determined by a single ﬁlter H0 (z). The designer has to concentrate on the design of only this www.jntuworld.com ﬁlter. • The system with four ﬁlters related as above was known as QMF Bank. • But the term QMF has since been used to indicate generalized versions for ex., M -channel system. • From (4) =⇒ F0(z) and F1(z) are low pass and high pass ﬁlters, re- spectively. • Therefore, the distortion function is T (z) = 1 [H0(z).F0(z) + H1(z).F1(z)] 2 1 = 2 [H0(z).H0(z) + H1(z).(−H1(z))] = 1 [H0 (z) − H1 (z)] = 1 [H0 (z) − H0 (−z)] 2 2 2 2 2 2 www.jntuworld.com Polyphase Representation: • Let H0(z) = E0(z 2 ) + z −1 .E1(z 2 ) H1(z) = H0(−z) H1(z) = E0(z 2) − z −1 .E1(z 2) • That is, in matrix-vector notation, 2 H0 (z) 1 1 E0(z ) = −1 2 H1 (z) 1 −1 z .E1(z ) • Similarly, F0(z) = H0(z) F0(z) = E0(z 2) + z −1 .E1(z 2) F1(z) = −H1(z) www.jntuworld.com F1(z) = −(E0(z 2 ) − z −1 E1(z 2)) ˆ ˆ ˆ X(z) = X0(z) + X1(z) ˆ X(z) = Y0(z)F0(z) + Y1(z)F1(z) ˆ X(z) = Y0(z)(z −1 E1(z 2 ) + E0(z 2 )) + Y1(z)(z −1 E1(z 2) − E0(z 2)) ˆ X(z) = z −1 E1(z 2)(Y0(z) + Y1(z)) + E0(z 2)(Y0(z) − Y1(z)) • That is, in matrix-vector notation, 1 1 F0(z) F1(z) = z −1 .E1(z 2) E0(z 2) 1 −1 www.jntuworld.com x(n) E (Z2) 2 2 E (Z2) 0 1 −1 −1 Z Z E (Z2) 2 2 E (Z2) 1 0 −1 −1 x(n) www.jntuworld.com • By using the noble identities, G(z M )(↓ M ) ≡ (↓ M )G(z) (↑ L)G(z L) ≡ G(z)(↑ L) • We have the modiﬁed structure as x(n) 2 E (Z ) E (Z ) 2 0 1 −1 −1 Z Z 2 E (Z ) E 0(Z ) 2 1 −1 −1 x(n) • The polyphase components are now operating at the lowest pos- sible rate, so that the number of multiplications and additions per unit time is minimized. www.jntuworld.com Noble Identities: 1. ((↓ M )X[z])G(z) ≡ (X(z).G(z M ))(↓ M ) x[n] v1[n] y1[n] x[n] v2[n] y2[n] M M G(Z) G(Z ) M RHS: y2[n] = (↓ M )v2[n] Y2(z) = (↓ M )V2(z) = (↓ M )(X(z)G(z M )) (M−1) j2πl j2πl 1 1 1 = M X(z M e M )G((z M e M )M ) l=0 www.jntuworld.com (M−1) j2πl 1 1 Y2(z) = M X(z M e M )G(z) l=0 LHS: v1[n] = (↓ M )x[n] (M−1) j2πl 1 1 V1(z) = M X(z M .e M ). l=0 Y1(z) = V1(z)G(z) (M−1) j2πl 1 1 Y1(z) = M X(z M .e M ).G(z) l=0 www.jntuworld.com 2. (X(z)G(z))(↑ L) ≡ ((↑ L)X(z))G(z L) x[n] v1[n] y3[n] x[n] x4[n] y4[n] L G(Z) L L G(Z ) RHS: x4[n] = (↑ L).x[n] X4(z) = (↑ L).X(z) X4(z) = X(z L ) −→ Compression in the Frequency Domain Y4(z) = X(z L )G(z L) www.jntuworld.com LHS: V1(z) = X(z)G(z) Y3(z) = (↑ L)V1(z) = V1(z L ) Y3(z) = X(z L )G(z L) www.jntuworld.com M-Channel Filter Bank: v (n) y (n) x(n) x (n) 0 0 x (n) 0 M M F (Z) 0 H (Z) 0 0 x (n) v (n) y (n) x 1(n) 1 1 1 F (Z) H (Z) M M 1 1 x (n) v (n) y (n) x (n) H (Z) M−1 M−1 M−1 M−1 M−1 M M F (Z) M−1 x(n) • x[n] is split into M subband signals xk [n] by the M analysis ﬁlters Hk (z). • Each signal is then decimated by M to obtain vk [n]. • The decimated signals are eventually passed through M -fold ex- www.jntuworld.com jw H (e ) 0 2 (M−1) w M M M panders and recombined via the synthesis ﬁlters Fk (z) to produce x[n]. ˆ Expression for the Reconstructed Signal : • Each subband signal is given by Xk (z) = Hk (z).X(z) (5) www.jntuworld.com • The decimated signal vk [n] has Z-Transform (M−1) −j2πl 1 1 Vk (z) = M Xk (z M .e M ) l=0 (M−1) 1 1 −j2πl 1 −j2πl Vk (z) = Hk (z M .e M ).X(z M .e M ) (6) M l=0 • The outputs of the expander are therefore given by (M−1) 1 −j2πl −j2πl Yk (z) = Vk (z M ) = Hk (z.e M ).X(z.e M ) (7) M l=0 • Therefore, the reconstructed signal is (M−1) ˆ X(z) = Fk (z).Yk (z) k=0 (M−1) (M−1) −j2πl −j2πl 1 = Fk (z). M (Hk (z.e M ).X(z M )) k=0 l=0 www.jntuworld.com • Rearranging the terms we have M−1 M−1 ˆ 1 −j2πl −j2πl X(z) = X(z.e M ). Hk (z.e M ).Fk (z) (8) M l=0 k=0 • Let, M−1 1 −j2πl Al (z) = Hk (z.e M ).Fk (z) 0≤l ≤M −1 (9) M k=0 • Therefore, M−1 −j2πl ˆ X(z) = X(z.e M ).Al (z) (10) l=0 • The spectrum of reconstructed signal is therefore given by M−1 2πl ˆ X(ejw ) = X(ej(w− M )).Al (ejw ) l=0 www.jntuworld.com 2πl • Where, X(ej(w− M ) ) for l = 0 represents a shifted version of X(ejw ). • Therefore, ˆ the reconstructed spectrum X(ejw ) is a linear combina- tion of X(ejw ) and its (M − 1) uniformly shifted versions. Errors Created in M -Channel QMF Bank Aliasing Errors: • The output of synthesis ﬁlter bank i.e. reconstructed signal is given by M−1 −j2πl ˆ X(z) = X(z.e M ).Al (z) (11) l=0 −j2πl • X(z.e M ), l > 0 is due to the decimation and interpolation opera- tions. −j2πl • For l > 0, X(z.e M ) represents the lth aliasing term. www.jntuworld.com −j2πl • Each X(z.e M ) is associated with term Al (z). • Aliasing can be eliminated for every possible input x[n], iﬀ Al (z) = 0 1≤l ≤M −1 Illustration: M =3⇒ Three parallel ﬁlters. M−1 −j2πl ˆ X(z) = X(z.e M ).Al (z) (12) l=0 −j2π Let, W = e M M−1 ˆ X(z) = X(z.W l ).Al (z) l=0 2 ˆ X(z) = X(z.W l ).Al (z) l=0 ˆ X(z) = X(z).A0(z) + X(zW ).A1(z) + X(zW 2).A2(z) (13) www.jntuworld.com M−1 1 Al (z) = . Hk (z.W l ).Fk (z) (14) M k=0 2 1 A0(z) = . Hk (z).Fk (z) (15) 3 k=0 2 1 A1(z) = . Hk (z.W ).Fk (z) (16) 3 k=0 2 1 A2(z) = . Hk (z.W 2).Fk (z) (17) 3 k=0 • A1 (z) and A2(z) are the terms associated with aliasing. For the aliasing free condition, A1(z) = 0 www.jntuworld.com 2 1 3. Hk (z.W ).Fk (z) = 0 k=0 2 ⇒ Hk (z.W ).Fk (z) = 0 k=0 H0 (z.W ).F0(z) + H1 (z.W ).F1(z) + H2 (z.W ).F2(z) = 0 (18) Similarly, A2(z) = 0 2 1 3 . Hk (z.W 2).Fk (z) = 0 k=0 H0(z.W 2).F0(z) + H1(z.W 2).F1(z) + H2(z.W 2).F2(z) = 0 (19) www.jntuworld.com Plot of H0(z), H1(z) and H2(z): H0(Z) H1(Z) H2(Z) o π 2π 3 3 π ω → Uniform ﬁlter bank. → H0 (z) is a Low Pass Filter. → H1 (z) and H2(z) are Band pass Filters. www.jntuworld.com Plot of H0(z), H0(zW ) and H0(zW 2): −j2π −j2π 1. H0(zW ) = H0 (z.e M ) = H0 (z.e 3 ) −j4π 2. H0(zW 2) = H0 (z.e 3 ) H0 (Z) −π − 2π 3 −π 3 0 π 3 2π 3 π ω H0 (ZW ) 0 ω H0(ZW 2) 0 ω www.jntuworld.com Plot of H1(z), H1(zW ) and H1(zW 2): −j2π 1. H1(zW ) = H1 (z.e 3 ) −j4π 2. H1(zW 2) = H1 (z.e 3 ) H0(Z) −π − 2π 3 −π 3 0 π 3 2π 3 π ω H1(ZW ) 0 ω H1(ZW 2 ) 0 ω www.jntuworld.com Plot of H2(z), H2(zW ) and H2(zW 2): −j2π 1. H2(zW ) = H2 (z.e 3 ) −j4π 2. H2(zW 2) = H2 (z.e 3 ) www.jntuworld.com • Signal which enters F0(z) is coming through the channel having the analysis ﬁlter H0(z). • Due to the decimation and interpolation the signal which enters F0(z) contains the terms H0(z)X(z), H0(zW )X(zW ) and H0(zW 2)X(zW 2). → The ﬁrst term is the required one, and other represents alias terms. • F0(z) is used to eliminate the alias terms X(zW ) and X(zW 2). For this response |F0(ejw )| should resemble H0(ejw ). • Responses of F1(z) and F2(z) also follow the responses of H1(z) and H2 (z) for the similar reasoning. • Since the ﬁlters Fk (z) and Hk (z) are not ideal in practice, they do www.jntuworld.com not eliminate the shifted replicas Hk (zW ) and Hk (zW 2) completely. • Responses of H0(zW ).F0(z), H1(zW ).F1(z) and H2(zW ).F2(z) overlap. • Objective of alias cancelation is to chose the synthesis ﬁlter such that these overlapping terms cancel out. Amplitude and Phase Distortions: Output of Synthesis Bank: M−1 M−1 ˆ X(z) = ˆ Xk (z) = Vk (z).Fk (z) k=0 k=0 M−1 M−1 ˆ X(z) = 1 (M X(zW l ).Hk (zW l )).Fk (z) k=0 l=0 M−1 M−1 l 1 = X(zW ).( M Fk (z).Hk (zW l )) l=0 k=0 www.jntuworld.com M−1 ˆ X(z) = X(zW l ).Al (z) 0≤l ≤M −1 l=0 For Alias Free M -Channel QMF System: Al (z) = 0 1 ≤ l ≤ M − 1. • Therefore, ˆ X(z) = X(z).A0 (z) ˆ X(z) = X(z).T (z) • Where, T (z) = A0(z) is the distortion function. M−1 . 1 T (Z) = A0(z) = M Hk (z).Fk (z). k=0 • Substituting Z = ejw , we have T (ejw ) T (ejw ) = |T (ejw )|.ej • If |T (ejw )| = C = 0, ˆ then X(z) will not have any amplitude distortion. www.jntuworld.com • Similarly, T (ejw ) = a + bω , ˆ a linear phase factor, then X(z) will not have any phase distortion. Perfect Reconstruction System: If Hk (z) and Fk (z) are such that aliasing is completely canceled and T (z) is a pure delay (i.e. T (z) = C.z −n0 , C = 0), then the system is free from aliasing, amplitude and phase distortions. Such system will have x[n] = C.x[n − n0] and is called a prefect reconstruction system. ˆ www.jntuworld.com