A simple 2-D oscillator to determine the correct decomposition of perturbations into amplitude and phase noise

This paper presents a simple, analytically solvable example showing that the so-called "orthogonal decomposition" of noise in an oscillator yields the wrong result. The orthogonal decomposition assumes that the right eigenvectors of a matrix are orthogonal, and hence the projection along one of them may be computed by a simple inner product. However, this assumption is not always valid; for the example in this paper, the eigenvectors are not orthogonal and the projection must instead be determined by solving a linear system, or equivalently, computing an inner product with the left eigenvector.