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.