Statistics of Complex Cross Spectrum Estimate for Sinusoidal Signals and Arbitrary Noise Spectra

The joint characteristic function of the real and imaginary parts of the complex cross spectrum estimate is derived in closed form for sinusoidal signals and arbitrary noise spectra. The corresponding joint probability density function for noise-only is also obtained in closed form, while that for signal present requires a numerical procedure involving a two-dimensional fast Fourier transform. These results are used to obtain the mean and deflection of the magnitude cross spectrum estimate. Comparisons are made with the deflection of an auto spectrum estimate available from the two channels by weighted summation. Potential limitations of restricting attention to the deflection criterion as a detectability measure are pointed out. Cross spectrum, Spectrum estimate, Sinusoidal signals, Arbitrary noise, Characteristic function, Probability density