Starting with the defining equation for an eigenvector and its corresponding eigenvalue ,
Equation (G.23) gives us two equations in two unknowns:
Thus, we have found both eigenvectors
They are linearly independent provided and finite provided .
We can now use Eq.(G.24) to find the eigenvalues:
Assuming , the eigenvalues are
Let us henceforth assume . In this range is real, and we have , . Thus, the eigenvalues can be expressed as follows:
Equating to , we obtain , or , where denotes the sampling rate. Thus the relationship between the coefficient in the digital waveguide oscillator and the frequency of sinusoidal oscillation is expressed succinctly as
We see that the coefficient range (-1,1) corresponds to frequencies in the range , and that's the complete set of available digital frequencies.
We have now shown that the system of Fig.G.3 oscillates sinusoidally at any desired digital frequency rad/sec by simply setting , where denotes the sampling interval.