5.4 Relationship between magnitude and phaseFor SY described by rational system functions, there is some constraint between magnitude and phase.magnitude magnitudethe number of poles and zeros⎛ ⎞ 1 ⎟ jω * | H (e jω ) |2 = H (e jω )H * (e jω ) = H(e )H ⎜ jω * ⎟ ⎜ e z = e jω ⎝ ⎠ * *Explore the possible choices of system function, given the square of the magnitudeC( z) = H ( z)H (1/ z )M( )1/rrθ* 1/ ckaafinite choices for the phase finite choices for the phasephase phasethe number of poles and zeros* (1 − ck z −1 ) (1 − ck z ) ⎛ b0 ⎞ ∏ ⎛ b0 ⎞ ∏ k =1 k =1 =⎜ ⎟ N ⋅⎜ ⎟ N ⎝ a0 ⎠ (1 − d z −1 ) ⎝ a0 ⎠ (1 − d * z )Mck∏kk =1 M∏k =1kaafinite choices for the magnitude finite choices for the magnitude⎛b ⎞ =⎜ 0 ⎟ ⎝ a0 ⎠2∏ (1 − c zk k =1 N k k =1−1)(1 − c z )* k * )(1 − d k z )∏ (1 − d z−1zeros: poles:* ck ,1/ ck* d k ,1 / d kOccur in conjugate Occur in conjugate reciprocal pairs reciprocal pairsExample 5.112(1 − z−1 )(1+ 0.5z −1 ) H1 ( z) = (1− 0.8e jπ / 4 z−1 )(1− 0.8e− jπ / 4 z −1)Geometric explanation(1− z−1)(1+ 2z−1) H2 ( z) = (1− 0.8e jπ / 4 z−1)(1− 0.8e− jπ / 4 z−1)z = e jωv1v3z = e jωv4−0.5w4−2w1w3w2v2− 0.5−2| H1 (e jω ) |= K1v3 v4 v1 v2Z2z = e jωw4| H2 (e jω ) |= K2θv4Z1ωw3 w4 w1 w2C1(z)C2 ( z)−2 − 0.5* C2 ( z) = H2 ( z)H2 (1/ z* )C1 ( z) = H1 ( z)H1* (1/ z* )r1/ r| H1 (e jω ) | K1 v4 K1 = ⋅ = ⋅ r — constant | H 2 (e jω ) | K 2 w4 K2C1 ( z ) = C2 ( z )| H1 (e jω ) |2 =| H 2 (e jω ) |2形状相同 形状相同So, | H1 (e jω ) |=| H 2 (e jω ) |, butH1 (e jω ) ≠ H 2 (e jω )Example 5.12C(z) = H(z)H* (1/ z* )how? how?H1(z) :{Z 1 , Z 2 } , Z 3z1H 2 ( z) :{Z1 , Z 2 } , Z 6z1 p1j Im(z)p1z3 p2 p3determine the zeros and poles to associate with H(z) H(z): causal and stable SY H(z): causal and stable SY Poles of H(z): P, P , P 1 2 3 Let H(z) be real SY, Let H(z) be real SY,Re (z)p3z6z2 p2z2H 1 ( e jω ) = H 2 ( e jω )H 3 ( z) :z4 p1{Z 4 , Z 5 } , Z 3H 4 ( z) :z4{Z 4 , Z 5 } , Z 6p1H 3 ( e jω ) = H 4 ( e j ω ){Z1, Z2} or {Z4 , Z5} , {Z3} or {Z6}p3z3 p2p3p2z5z6zeros and poles either are real or zeros and poles either are real or occur in complex conjugate pairs. occur in complex conjugate pairs.z5=(constant=1)=1If the number of poles and zeros of H(z) is unspecified, then, the number of choices for H(z) would be unlimited.j Im(z)Summary1) zeros and poles of H(z)expand ⎧H1(z)p1z3 p2Re (z)z1C1 ( z)1⎫ ⎧ 1⎫ ⎨ck , * ⎬ , ⎨ d k , * ⎬ ⎩ ck ⎭ ⎩ d k ⎭p3z21 a*C(z) z =ejω| H(ejω )|2⎧ H (z): c1k , d1k 2) Let ⎨ 1 , if ⎩H2 (z): c2k , d2k 3) C(z) = H(z)H* (1/ z* )* 1 = A —constant ⎧ c1k =1/ c2k ⎪ , then ⎨| H2 (e jω ) | ⎨ * d1k =1/ d2k ⎪ H (e jω ) ≠ H (e jω ) ⎩ ⎩ 1 2aj Im(z)⎧ | H (e jω ) |1 a*z1 ap1z3 p2Re (z)H ( z ) = H1 ( z ) ⋅z −1 − a* 1 − az −1p3z2C ( z)N* z−1 − ak In general, if H(z) = H1(z) ⋅ ∏ , then C( z) = C1 ( z) −1 k =1 1− ak zthe number of poles and zerosCausal and stable H(z), not unique, same magnitude5.5 All-pass systemsDefinition: | H (e ) |= 1jω2) General AP-SY:H ap ( z ) = A∏* z −1 − a k ——N th-order AP-SY Let −1 k =1 1 − ak z * 1/ ak conjugate reciprocal pairs ak Na = re jθ— all-pass system (AP-SY)H ap ( z ) = z −1 − a * 1 − az −11) AP factor (First-order AP-SY):1 a*zeros: poles:H ap (e jω ) = AN ⎛ ⎡ r sin(ω − θ k ) ⎤ ⎞ H ap (e jω ) = ∑ ⎜ −ω − 2 arctan ⎢ ⎥⎟ ⎜ ⎟ k =1 ⎝ ⎣1 − r cos(ω − θ k ) ⎦ ⎠ For real Nth-order AP-SYazero: 1/ a pole: a*conjugate reciprocal pairH ap (e jω ) = 1⎡ r sin(ω − θ ) ⎤ jθ H ap (e jω ) = −ω − 2 arctan ⎢ ⎥ (Let a = re ) ⎣1 − r cos(ω − θ ) ⎦H ap ( z ) = A ⋅ ∏* z −1 − d k M c ( z −1 − ek )( z −1 − ek ) ( N = M r + 2M c ) ⋅ −1 ∏ −1 * −1 k =1 1 − d k z k =1 (1 − ek z )(1 − ek z ) Mrreal poles/zeros complex conjugate poles/zerosPropertiesThe zeros and poles occurs in conjugate reciprocal pairs. For a stable and causal AP-SY, the group delay is always positive, and the continuous phase is always nonpositive for 0 < ω < π.grd [ H ap (e jω )] > 0 arg[ H ap (e )] ≤ 0, 0 < ω < πjωExample: a second-order and real AP-SY.H ap ( z ) = ( z −1 − a * ) ( z −1 − a ) ⋅ (1 − az −1 ) (1 − a * z −1 )conjugate reciprocal pairsa∗Leta = re jθ1zeros: 1/ a * , 1/ a poles: a,H ap (e jω ) = 1a*a*a1a⎡ r sin(ω − θ ) ⎤ ⎡ r sin(ω + θ ) ⎤ Hap (e jω ) = −2ω − 2arctan ⎢ ⎥ − 2arctan ⎢1− r cos(ω + θ ) ⎥ ⎣1 − r cos(ω −θ ) ⎦ ⎣ ⎦2ApplicationsAny rational system can be expressed as: H(z) = Hmin(z)Hap(z) Used as compensators for phase (group-delay) distortion (相位失真的补偿器)H d ( z ) ：非线性相位系统H ap ( z )：全通系统线性相位Example. Transform an unstable SY into stable SY by cascading a AP-SY1 d*1 d*H ( z ) = H d ( z ) H ap ( z )幅度关系：？×H ( z)d=H s ( z)dH (e jω ) = H d (e jω )H ap ( z )相位关系： ϕ (ω ) = ϕ d (ω ) + ϕ ap (ω )τ 群延迟关系： (ω ) = τ d (ω ) + τ ap (ω )要使H(z)具有线性相位，只需构造 τ ap (ω ) ，使H s ( z ) = H ( z ) H ap ( z )H s (e ) = H (e )jω jω—— stableτ (ω ) = τ d (ω ) + τ ap (ω ) = τ 0 （常数）Homework5.4, 5.6, 5.8, 5.24, 5.363。
