Eigenvalues in the Undamped Case

When $g=1$, the eigenvalues reduce to

$${\lambda_i}= c\pm j\sqrt{1-c^2}$$

Assuming $\left\vert c\right\vert<1$, the eigenvalues can be expressed as

$${\lambda_i}= c\pm j\sqrt{1-c^2} = \cos(\theta) \pm j\sin(\theta) = e^{\pm j\theta} \protect$$ (O.17)

where $\theta=\omega T$ denotes the angular advance per sample of the oscillator. Since $c\in(-1,1)$ corresponds to the range $\theta\in(-\pi,\pi)$, we see that $c$ in this range can produce oscillation at any digital frequency.

For $\left\vert c\right\vert>1$, the eigenvalues are real, corresponding to exponential growth and/or decay. (The values $c=\pm 1$ were excluded above in deriving Eq. (O.17).)

In summary, the coefficient $c$ in the digital waveguide oscillator ($g=1$) and the frequency of sinusoidal oscillation $\omega$ is simply

$$\fbox{$\displaystyle c= \cos(\omega T)$}.$$