Barankin Bounds on Parameter Estimation Accuracy Applied to Communications and Radar Problems

The Schwartz Inequality is used to derive the Barankin lower bounds on the covariance matrix of unbiased estimates of a vector parameter, The bound is applied to communications and radar problems in which the unknown parameter is imbedded in a signal of known form and observed in the presence of additive white Gaussian noise. Within this context it is shown that the Barankin bound reduces to the Cramer-Rao bound when the signal -tonoise ratio (SNR) is large. However, as the SNR is reduced beyond a critical value the Barankin bound deviates radically from the Cramer-Rao bound thereby exhibiting the so-called threshold effect. A particularly interesting signal, which has been widely used in practice to estimate the range of a target, is the linear FM waveform. The bounds were applied to this signal and within the resulting class of bounds it was possible to select one which led to a closed form expression for the lower bound on the variance of the range estimate. This expression clearly demonstrates the threshold behaviour one must expect when using a nonlinear modulation system. Tighter bounds were easily obtained but these had to be evaluated using numerical techniques. It is shown that the side-lobe structure of the linear FM compressed pulse leads to a significant increase in the variance of the estimate. For a practical linear FM pulse of 1 microsecond duration and 40 megahertz bandwidth it is shown that the radar must operate at an SNR greater than 10 dB if meaningful range estimates are to be obtained. Accepted for the Air Force Franklin C. Hudson Chief, Lincoln Laboratory Office