Relationship between Fourier and Laplace transforms

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Lecture 17

• Relationship between Fourier and Laplace
  transforms
• Filtering
• Lowpass, highpass and bandpass filters.
• Ideal filters and real approximations.

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Fourier and Laplace
Laplace
Fourier
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Examples
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Filtering
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Ideal lowpass filter
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Is it physically realizable? Looking at the
impulse response
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Causal approximation to ideal lowpass
This h(t) may be difficult to implement. Easier
alternative?
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Butterworth filter of order 10.
This is a rational H(s), with denominator of
order 10.
For more on filter design, see signal processing
courses. Here we will focus on ideal filters.
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Ideal highpass filter
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Ideal bandpass filter
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Bandpass as cascade of lowpass and highpass
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Lowpass (B2)
Highpass (B1)
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An equalizer
Adjustable gains for each frequency band.
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Both these systems do not affect the magnitude of
the signals (i.e. the input and output Fourier
transforms have the same magnitude). They do,
however, affect the phase.