The statistics of time–frequency analysis

This paper discusses time–frequency analysis from a statistical signal processing perspective. Certain nonstationary stochastic processes, known as “locally stationary processes”, have covariance functions which yield nonnegative Wigner distributions. An extension to this class of processes to include signals with frequency modulation is presented. For such processes, time–frequency spectra may be defined without invoking “local-” or “quasi-” stationarity. The bilinear class of time–frequency distributions are estimators of these time–frequency spectra. An analysis of the statistical properties of these estimators, including moments and distributional properties, is presented. A derivation of an asymptotic Cramer–Rao Lower Bound on the variance, together with results on the bias and variance, demonstrates that a trade-off exists between time–frequency resolution and the variance of the estimated representation. While perfect time–frequency resolution is possible, it can only be achieved through a corresponding increase in variance.