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LTI System Analysis

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Use phase differences to locate a speaker ... from that speaker (underlies speech compression systems, e.g. digital cell phones. ... – PowerPoint PPT presentation

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Title: LTI System Analysis


1
LTI System Analysis
2
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
  • Differential equations (Chap. 2, (2.1a) - eq
    (2.1c))
  • Convolution in time domain (Chap. 2,
    (2.65b)),y(t) f(t) h(t)
  • Laplace transform (Chap. 4, (4.41)) Y(s) F(s)
    H(s)
  • Continuous-time Fourier transform (Chap. 7,
    (7.54)) Y(w) F(w) H(w)

3
Example 1
  • Transfer function 1/(s2) and input e -t u(t)
  • Either way takes about the same amount of work

4
Example 2 Laplace is Easier
  • Transfer function 1/(s2) and input u(t)

5
Example 3 Laplace Cannot Work
  • Transfer function 1/(s2) and input et u(-t)

6
Signal Distortion
  • Total distortion
  • Y(w) F(w) H(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(2pfct) into sin(2pfct) known as a Hilbert
    transformer

7
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.

8
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
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