Exercises in Wave Digital Modeling

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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.Q.34 in terms of $\rho$ .
2. Show that the resonance frequency is given by
  
  $\displaystyle f_s\arccos\left(\rho\right)$
  where 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
  
  $\displaystyle f_s \arccos\left[\frac{k-m}{k+m}\right] = 2f_s \arctan\left[\sqrt{\frac{m}{k}}\right].$

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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