#### Exercises in Wave Digital Modeling

1. Comparing digital and analog frequency formulas. This first exercise verifies that the elementary tank circuit'' always resonates at exactly the frequency it should, according to the bilinear transform frequency mapping , where denotes analog frequency'' and denotes digital frequency''.
1. Find the poles of Fig.F.35 in terms of .

2. Show that the resonance frequency is given by

where denotes the sampling rate.
3. Recall that the mass-spring oscillator resonates at . Relate these two resonance frequency formulas via the analog-digital frequency map .
4. Show that the trig identity you discovered in this way is true. I.e., show that

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Energy-Preserving Parameter Changes (Mass-Spring Oscillator)