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.37 in terms of $\rho$ .

   2. Show that the resonance frequency is given by
      $$f_s\arccos\left(\rho\right)$$
      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
      $$f_s \arccos\left[\frac{k-m}{k+m}\right] = 2f_s \arctan\left[\sqrt{\frac{m}{k}}\right].$$