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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 $ \omega_a = \tan(\omega_d T /
2)$, where $ \omega_a$ denotes ``analog frequency'' and $ \omega_d$ denotes ``digital frequency''.
    1. Find the poles of Fig.F.35 in terms of $ \rho$.


    2. Show that the resonance frequency is given by

      $\displaystyle f_s\arccos\left(\rho\right)
$

      where $ f_s$ denotes the sampling rate.
    3. Recall that the mass-spring oscillator resonates at $ \omega_0=\sqrt{k/m}$. Relate these two resonance frequency formulas via the analog-digital frequency map $ \omega_a = \tan(\omega_d T /
2)$.
    4. Show that the trig identity you discovered in this way is true. I.e., show that

      $\displaystyle f_s \arccos\left[\frac{k-m}{k+m}\right] =
2f_s \arctan\left[\sqrt{\frac{m}{k}}\right].
$



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Energy-Preserving Parameter Changes (Mass-Spring Oscillator)