Signals and Systems Fall 2003 Lecture #12 16 October Linear and Nonlinear Phase 2. Ideal and Nonideal Frequency-Selective Filters 3. CT & DT Rational Frequency Responses 4. DT First- and Second-Order Systems

Linear Phase CT Result:Linear phase ⇔ simply a rigid shift in time, no distortion Nonlinear phase ⇔ distortion as well as shift DT Question: What about H (e jw ) = e -jwα, α ≠ integer?

All-Pass Systems CT DTDT Linear phase Nonlinear phase Linear phase Nonlinear phase

Demo:Impulse response and output of an all-pass system with nonlinear phase Principal PhaseInput to Allpass System Unwrapped Phase Impulse Response Output of Allpass System Phase (rad) Group Delay (sec) Frequency (Hz) Time (sec) Group Delay

How do we think about signal delay when the phase is nonlinear? Group Delay φ When the signal is narrow-band and concentrated near ω 0,  H (jw) ~ linear with ω near ω 0, then instead of reflects the time delay. For frequencise “near” ω 0 For w “near” ω 0

Ideal Lowpass Filter CTCT Noncausal h(t <0) ≠ 0 Oscillatory Response — e.g. step response Overshoot by 9%, Gibbs phenomenon

Nonideal Lowpass Filter Sometimes we don’t want a sharp cutoff, e.g. Often have specifications in time and frequency domain ⇒ Trade-offs Step response Freq. Response Passband Transition Stopband signalnoise

CT Rational Frequency Responses CT: If the system is described by LCCDEs, then Prototypical System — 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 First- or Second-order

DT First-Order Systems Frequency domain Time Domain initial rest

Demo: Unit-sample, unit-step, and frequency response of DT first-order systems Impulse Response Step Response Magnitude of Frequency Response Phase of Frequency Response (rad) n (samples) ω (rad/sec)

DT Second-Order System decaying oscillations where

Demo:Unit-sample, unit-step, and frequency response of DT second-order systems Impulse Response Step Response Phase of Frequency Response (rad) n (samples) ω (rad/sec)