Eigenvalues in the Undamped Case

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When , the eigenvalues reduce to

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

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

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

(C.143)

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

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.144).)