Approximate information content of a signal with bispectral constraints

The information content of a signal constrained only by its autocorrelation function or by the total power at each of its Fourier frequencies is well known. This paper addresses the information content of a signal constrained by both the power at each Fourier frequency and by its bispectrum. It is found that the amount by which the information content of a signal with a known power spectrum is reduced when bispectral constraints are imposed is primarily determined by its bicoherence. Approximate solutions for the signal distribution function and signal entropy are obtained in the limit of small bicoherence values using a perturbation theory approach. In the limit of bicocoherence values approaching unity, approaches based on a "mean field" argument also yield expressions for the signal information content. Estimates of the reduction in information content by a quadratic distortion are made. Signal distribution functions are used to develop a non-linear signal detection algorithm similar to the North filter.