DSP: Minimum-Phase Systems Digital Signal Processing Minimum-Phase Systems D. Richard Brown III D. Richard Brown III 1 / 7 DSP: Minimum-Phase Systems Phase Response Characterization of Transfer Function Definition A causal stable LTI system H with transfer function H(z) with all zeros inside the unit circle is called minimum phase. Definition A causal stable system H with transfer function H(z) with all zeros outside the unit circle is called maximum phase. Definition A causal stable system H with transfer function H(z) with at least one zero inside the unit circle and at least one zero outside the unit circle is called mixed phase. Minimum phase systems are important because they have a stable inverse G(z)=1/H(z). You can convert between min/max/mixed-phase systems by cascading allpass filters. D. Richard Brown III 2 / 7 DSP: Minimum-Phase Systems Minimum Phase Systems When we say a system is “minimum phase”, we mean that it has the least phase delay (or least phase lag) among all systems with the same magnitude response. 7.5 H1(z) − − 7 H2(z) 1 2 H3(z) 6.5 H1(z)=6+ z − z H4(z) −1 −1 6 = 6(1+ z /2)(1 − z /3) 5.5 5 −1 −2 magnitude response H2(z)=1 − z − 6z 4.5 4 − − 0 0.5 1 1.5 2 2.5 3 3.5 =(1+2z 1)(1 − 3z 1) −1 −2 2 H3(z)=2 − 5z − 3z 0 −1 −1 = 2(1+ z /2)(1 − 3z ) −2 −1 −2 −4 H4(z)=3+5z − 2z −6 − − phase response (rad) −8 =(1+2z 1)(1 − z 1/3) −10 0 0.5 1 1.5 2 2.5 3 3.5 normalized freq (rad/sample) Minimum phase systems with real-valued impulse responses have ∠H(ej0)=0. D. Richard Brown III 3 / 7 DSP: Minimum-Phase Systems Phase Analysis Recall that the phase response of a rational H(z) can be written as M N jω b0 −jω −jω ∠H(e )= ∠ + ∠(1 − cme ) − ∠(1 − dne ) a0 mX=1 nX=1 M N b0 jω jω jω jω = ∠ + ∠(e − cm) − ∠e − ∠(e − dn) − ∠e a0 mX=1 nX=1 To have the least phase delay for a causal system, we must have ∠ jω ∠ jω ∗ (e − cm) > (e − 1/cm) for all |cm| < 1. We will illustrate this graphically... D. Richard Brown III 4 / 7 DSP: Minimum-Phase Systems im im im re re re im im im re re re D. Richard Brown III 5 / 7 DSP: Minimum-Phase Systems Minimum Phase ⇒ Minimum Group Delay Suppose we have a rational transfer function with the term (1 − cz−1) in the numerator with c = aejb and 0 <a< 1. We can write − − − ∠(1 − ce jω)= ∠(1 − ae j(ω b)) = ∠ (1 − a cos(ω − b)+ aj sin(ω − b)) a sin(ω − b) = arctan 1 − a cos(ω − b) Note that ∂ f(ω) f ′(ω)g(ω) − g′(ω)f(ω) arctan = ∂ω g(ω) g2(ω)+ f 2(ω) Omitting the algebraic details, the group delay then follows as 2 ∂ − a − a cos(ω − b) a − cos(ω − b) τ (ω)= − ∠(1 − ce jω)= = g ∂ω 1+ a2 − 2a cos(ω − b) a−1 + a − 2 cos(ω − b) Note that the denominator is unaffected by replacing a with a−1 > 1. But the numerator gets larger if we replace a with a−1 > 1. D. Richard Brown III 6 / 7 DSP: Minimum-Phase Systems Minimum Phase ⇒ Minimum Energy Delay Given a causal system with impulse response h[n], we can define the partial energy of the impulse response as n E[n]= |h[k]|2. kX=0 A minimum phase system with impulse response hmin[n] has the property n n 2 2 Emin[n]= |hmin[k]| ≥ |h[k]| = E[n] Xk=0 Xk=0 for all n ≥ 0 and all h[n] with the same magnitude response as hmin[n]. Recall our earlier example: −1 −2 H1(z)=6+ z − z E = {36, 37, 38} −1 −2 H2(z)=1 − z − 6z E = {1, 2, 38} −1 −2 H3(z)=2 − 5z − 3z E = {4, 29, 38} −1 −2 H4(z)=3+5z − 2z E = {9, 34, 38} D. Richard Brown III 7 / 7.