Frequency Response of Discrete-time LTI Systems

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Frequency Response of Discrete-time LTI Systems

• Prof. Siripong Potisuk

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For a discrete-time LTI system, the frequency
response is defined as
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In terms of transfer function,
The frequency response is just the transfer
function evaluated along the unit circle in the
complex z-plane.
w
H(ejw) periodic in w with period 2p
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Digital Frequency
X(ejw) is simply a frequency-scaled version of
X(jW)

• Normalization of the frequency axis so that W
  Ws
• in X(jW) is normalized to w 2p for X(ejw)

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For H(z) generated by a difference eq. with
real coefficients,
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For a given choice of H(ejw) as a function of w,
the frequency composition of the output can be
shaped - preferential amplification -
selective filtering of some frequencies
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Ex. Consider a 2nd order digital filter with
Plot the magnitude and phase responses of the
system.
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Effects of Pole Zero Locations

• A zero at indicates that
  the filter
• will fully reject spectral component of input
  at
• Effects of a zero located off the unit circle
  depends on its distance from the unit circle.
• A zero at origin has no effect.
• A pole on the unit circle means infinite gain at
  that frequency.
• The closer the poles to the unit circle, the
  higher the magnitude response.