LTI System Analysis

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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)

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Example 1

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

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Example 2 Laplace is Easier

• Transfer function 1/(s2) and input u(t)

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Example 3 Laplace Cannot Work

• Transfer function 1/(s2) and input et u(-t)

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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

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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.

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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