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
Ideal lowpass Filter CT h(1) 兀t HgO Noncausal h(t<0)≠0 · Oscillatory response—e.g. step response Overshoot by 9% s(t) Gibbs phenomenon TdT 109 h(r)di 1/2 H(0) 0 Oc
Ideal Lowpass Filter CT ←⎯ → • Noncausal h(t <0) ≠ 0 • Oscillatory Response — e.g. step response Overshoot by 9%, Gibbs phenomenon
Nonideal Lowpass Filter Sometimes we dont want a sharp cutoff, e.g noise Often have specifications in time and frequency domain = Trade-offs IH(jo) s(t) Step esponse Freq Response 2T/or Passband Transition Stopband 62 Os
Nonideal Lowpass Filter • Sometimes we don’t want a sharp cutoff, e.g. • Often have specifications in time and frequency domain ⇒ Trade-offs Step respons e Freq. Response
CT Rational Frequency responses CT: If the system is described by lccdes then k Gu) HGu) bk (jo ∑kk(j) Hi(j)= First-or Second-order factors Prototypical Systems First-order system, has only one H1(1u jwt+1 energy storing element, e.g. L or C H (=)+2(=)+1 (ju)2+25n(ju)+ Second-order system, has two energy storing elements, e.g. L and C
CT Rational Frequency Responses CT: If the system is described by LCCDEs, then Prototypical Systems — First-order system, has only one energy storing element, e.g. L or C — Second-order system, has two energy storing elements, e.g. L and C
DT Rational Frequency responses If the system is described by LCCdE's linear-Constant-Coefficient Difference Equations), then n-k←→Y(e)e-1n,xmn-k←→X(e4)e ka jwk hlea k ke bk( ake Tk kle wk ∏互2 Hi(e first-or Second-order
DT Rational Frequency Responses If the system is described by LCCDE’s (Linear-Constant-Coefficient Difference Equations), then
DT First-Order Systems n-agIn-1=n, a< 1, initial rest H() 1-ae- frequency domain H(eu) +a 2-2a cos a asin w ∠H( 1 an 1-a cos w lme domain hn= aun n 刚=M*=∑ Qn2+1 k=0
DT First-Order Systems
