EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin LTI System Analysis

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(  ) = F(  ) H(  ) Laplace transform: Y(s) = F(s) H(s)

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

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

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

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

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 -  ) Y(  ) = k e - j  F(  ) H(  ) = Y(  )/F(  ) = k e - j  Magnitude response: |H(  )| = k Phase response (linear):  H(  ) = –  Time delay is the negative of the derivative of  H(  ) w/r to  Channels have distortion Receiver needs to know the channel distortion Receiver uses an equalizer.

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