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

$${\lambda_i}= c\pm j\sqrt{1-c^2} = \cos(\theta) \pm j\sin(\theta) = e^{\pm j\theta} \protect$$ (C.164)

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.(C.165).)

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

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