Signals and Systems Fall 2003 Lecture #12 16 October 2003 Linear and nonlinear phase Ideal and nonideal frequency-Selective 234 Filters ct& dt rational frequency responses DT First-and Second-Order Systems
1. Linear and Nonlinear Phase 2. Ideal and Nonideal Frequency-Selective Filters 3. CT & DT Rational Frequency Responses 4. DT First- and Second-Order Systems Signals and Systems Fall 2003 Lecture #12 16 October 2003
Linear phase CT () HGu) 9(t) H(j)=e-1→|H(j)=1,∠H(ju) aw(Linear In w Y()=e yaX (w) time-shift g t=at-a Result: Linear phase e simply a rigid shift in time, no distortion Nonlinear phase o distortion as well as shift DT ym=xm-m←→Y(e) H(e) H(e)=1,∠H(e4)=-0 Question: What about H(eu)=e- 3wa, af integer?
Linear Phase Result: Linear phase ⇔ simply a rigid shift in time, no distortion Nonlinear phase ⇔ distortion as well as shift CT Question: DT
All-Pass systems →|H()=|H(e) CT Hlw)= e3 u Linear phase ()=a+ Nonlinear phase Hlu 2⊥,2 DT H(ej) Linear phase H(e°) Nonlinear phase H (1-1/2.c0s)2+(1/2.sin)2 (1-1/2.cosu)2+(1/2·sin)2
All-Pass Systems CT DT
Demo Imi pulse response and output of an all-pass system with nonlinear phase Principal Phase Input to Allpass System 0 0.5 0 02.557.51012.515 6 Decay Rate: 2857 Unwrapped Phase 10 Impulse Response 0 02.557.51012.515 0 6 Group Delay Ouput of Allpass System 0 2.557.51012.515 Frequency(Hz) Time(sec)
Demo: Impulse response and output of an all-pass system with nonlinear phase
How do we think about signal delay when the phase is nonlinear? Group delay ∠H(j0) When the signal is narrow-band and concentrated near wo, LH(w) linear with w near wo, then d∠H(j) instead ∠H(ju) reflects the time delay or frequencies " near Wo ∠H(j)≈LH(0)-r(u0)(u-w0)=0-r(uo): T(w)=ch(u))=Group Delay or w near wo H(ju)≈|H(juo)e t H
How do we think about signal delay when the phase is nonlinear? Group Delay φ