In a recent post, we looked at which Toeplitz real matrices gave useful quadrature oscillators by the recurrence . There, we computed their eigenvalues and solved the recurrence in terms of them. Of course, there are many other ways to approach this problem. Here we look at another approach that gives a good geometric picture of what happens, can be applied to general matrices, and may be used as a starting point for the case.
Two sinusoidal signals are said to be in quadrature if they have a constant phase difference of 90º. Quadrature signals are widely used in signal processing. A digital quadrature oscillator is just an algorithm that computes the sequence , , or a similar sequence of sinusoids in quadrature. Here is the oscillator frequency in radians per sample. Direct computation of this sequence is very time consuming, because the trigonometric functions have to be evaluated for each sample. Therefore, it is a good idea to use a linear recurrence scheme to compute . Using basic trigonometric identities, we see that
However, to actually perform these computations in a digital processor, one has to quantize , meaning that one has to replace by approximations. It is easy to see that if one replaces by some perturbation, then the eigenvalues of are no longer in the unit circle, so the oscillation can grow or decay exponentially and one would need to apply an AGC scheme to keep this method stable.
Since I'm away from my usual QTH, I decided to join yesterday's FreeDV net by listening through the University of Twente WebSDR and giving signal reports on the QSO Finder to the stations transmitting.