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Exercise, Cont.

For the exercise,

a)
Find the reflection coefficient $ k$ of the induced scattering junction in terms of $ L$ and $ C$ .
b)
Find the poles in terms of $ k$ .
c)
Find the resonance frequency in terms of the sampling interval $ T$ and the reflection coefficient $ k$ .
d)
Recall that an analog LC loop resonates at $ 1/\sqrt{LC}$ , and relate these two resonance frequency formulas via the analog-digital frequency map $ \omega_a = \tan(\omega_d T / 2)$ .
e)
Show that the trig identity you discovered in this way is true.

This exercise verifies that the elementary ``tank circuit'' always resonates at exactly the frequency it should, according to the bilinear transform mapping.


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Download WaveDigitalFilters.pdf
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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-25 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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