EE 313 Linear Systems and Signals Fall 2010

EE 313 Linear Systems and Signals Fall 2010 LTI System Analysis Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Initial conversion of content to Power. Point by Dr. Wade C. Schwartzkopf

Zero-State LTIC System Response • LTIC: Linear Time-Invariant Continuous-time • System response = zero-input response + zero-state response • Approaches to find the zero-state response Time-domain solutions to the differential equation Convolution in time domain: y(t) = f(t) * h(t) Continuous-time Fourier transform: Y(w) = F(w) H(w) Laplace transform: Y(s) = F(s) H(s) 2

Example #1 • Transfer function 1/(s+2) and input e -t u(t) • Either way takes about the same amount of work 3

Example #2: Laplace is Easier • Transfer function 1/(s+2) and input u(t) 4

Example #3: Laplace Cannot Work • Transfer function 1/(s+2) and input et u(-t) 5

Signal Distortion • Total distortion Y(w) = F(w) H(w) is the frequency response • Magnitude distortion |Y(w)| = |F(w)| |H(w)| • Phase distortion Y(w) = (F(w) H(w)) = F(w) + H(w) • All-pass filter: magnitude response of 1 Example: Phase shifting by 90° to convert cos(2 pfct) into sin(2 pfct) known as a Hilbert transformer 6

Distortionless Transmission • Want distortionless transmission: input and output signals have identical shapes, differ by a multiplicative constant, and may be delayed y(t) = k f(t - t) Y(w) = k e - j w t F(w) H(w) = Y(w)/F(w) = k e - j w t Magnitude response: |H(w)| = k Phase response (linear): H(w) = – w t Time delay is the negative of the derivative of H(w) w/r to w • Channels have distortion Receiver needs to know the channel distortion Receiver uses an equalizer. 7

Importance of Linear phase • Speech signals • Linear phase crucial – Use phase differences to – Audio locate a speaker – Images – Once locked onto a – Communication systems speaker, our ears are • Linear phase response relatively insensitive to – Need FIR filters phase distortion in – Realizable IIR filters speech from that speaker cannot achieve linear (underlies speech phase response compression systems, e. g. digital cell phones. ) 8