A Frequency-domain Treatment of Partial Coherence

Abstract

The phenomenon of partial coherence is treated in the frequency domain, rather than in the time domain. The starting-point of the description in the frequency domain is the set of mutual and auto power spectra of N signals, arranged in the form of an N × N Hermitian matrix. The necessary and sufficient condition for a Hermitian matrix to be a matrix of power spectra is that the Hermitian matrix is positive (semi-) definite. Apart from the temporal degree of coherence, which is the normalized coherence function, a spectral degree of coherence is defined as a suitably normalized power spectrum. Due to the positive (semi-) definiteness of the matrix of power spectra, the absolute value of the spectral degree of coherence is bounded by 1. Complete coherence is equivalently defined by ‘rank of the matrix of power spectra equals 1’ and ‘absolute value of the spectral degree of coherence equals 1’. Signals which are coherent in the above sense, seem to emanate from one single point source; furthermore, when propagating through linear, time-invariant systems, they remain coherent and behave like deterministic signals. Apart from the well-known definitions in the time and the frequency domain, spectral purity can be defined equivalently by ‘equality of the absolute value of the spectral degree of coherence (which value must be frequency independent) and the maximum absolute value of the temporal degree of coherence’. Furthermore, the absolute values of the temporal and spectral degree of coherence are constant and equal if and only if the signals are monochromatic, which is a special case of being spectrally pure.