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.