Linear Shift-Invariant Systems

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Linear Shift-Invariant Systems
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Linear

• If x(t) and y(t) are two input signals to a
  system, the system is linear if
• Hax(t) by(t) aHx(t) bHy(t)
• Where H specifies the transformation performed on
  the signal by the system

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Shift Invariant

• If xO(t) Hx(t) then Hx(t-t) xO(t - t)

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Properties of an LSI

• If the input is x(t) A Cos(2pf0t ?)
• The response to x(t) is Hx(t)
• A HCos(2pf0t ?) Aout Cos(2pf0t ?out)

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Transfer function of an LSI

• Consider the response of an LSI to a complex
  sinusoid

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If input is periodic then
If input is finite duration
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• The Fourier Transform of the impulse response
  function is the transfer function of the linear
  system.
• The Inverse Fourier Transform of the transfer
  function is the impulse response function of the
  linear system.
• This is a very powerful result.
• The easiest way to design a filter is to select
  an impulse response function

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Designing a filter

• A bandpass filter can be designed by taking an
  impulse response function that starts at t0,
  reaches a single peak and declines to zero with
  time.
• The longer the impulse response, the narrower the
  filter.
• To set the center frequency,fc, of the filter,
  multiply it by Cos(2 p fc t).

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Predicting the filters output
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Transfer function with zero phase shift

• Consider a rectangular filter with no phase shift