ECE310

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ECE310 Lecture 16

• Fourier Transform Analysis of Signals and Systems
• 03/29/01, 03/31/01

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Frequency Response

• Transfer function H(f)
• The CTFT of the impulse response

y(t) x(t)h(t)
h(t)
x(t)
Y(f) X(f)H(f)
H(f)
X(f)
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Filter

• A system that removes signal power in one
  frequency range while leaving the signal power in
  other frequency range
• Characterized by its impulse response and
  transfer function (or frequency response)
• Examples
• Ideal filters
• Practical passive filters
• Distortion

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Distortion

• The shape of the signal has been changed
• Amplitude scaling, shifting do not change the
  shape of the signal
• g(t) ? Ag(t-t0)
• Time scaling changes the shape of the signal
• g(t) ? g(at)
• g(t) ? lng(t)

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Distortionless System

• Response to a filter is the convolution of its
  excitation with its impulse response
• x(t) d(t) x(t)
• x(t) Ad(t) Ax(t)
• x(t) Ad(t-t0) Ax(t-t0)
• h(t) Ad(t-t0)
• Magnitude A (constant with frequency)
• Phase tan-1(Im/Re) -2pft0 (linear with
  frequency)
• P8-4

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Filter Types

• Lowpass filter
• Pass all frequencies from 0 to some maximum and
  blocks all others
• Highpass filter
• Blocks all frequencies from 0 to some maximum and
  passes all others
• Bandpass filter
• Passes a range of frequencies from some minimum
  frequency (not 0) to some maximum frequency (not
  infinity) and blocks all others
• Bandstop filter
• Blocks a range of frequencies from some minimum
  frequency (not 0) to some maximum frequency (not
  infinity) and passes all others

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Ideal Filter

• Causes no distortion of the signal in its
  passband and completely blocks signal power at
  all other frequencies
• Page 8-5

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Impulse Response of Ideal Filters

• Page 8-6, 7
• Ideal lowpass

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What Do They Look Like? (p8-7,8)

• All of the impulse responses of these ideal
  filters have a non-zero response before the
  impulse is applied at time zero
• Non-causal system Response begins before the
  input is applied
• All real physical systems are causal

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Examples Ideal Lowpass
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Examples Ideal Highpass
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Practical Passive Filters

• An approximation to the ideal filters
• RC lowpass filter
• Circuit
• Intuitive analysis
• H(f)
• h(t)
• RL lowpass filter
• Circuit
• Intuitive analysis
• H(f)
• RLC bandpass filter

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Frequency Response
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What Do They Look Like? (p8-12)
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Log-Magnitude Frequency Response Plot
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Bandwidth

• A range of frequencies present in a signal or the
  range of frequencies that the filter allows to
  pass
• A range of frequencies in positive frequency space