Barankin Bounds on Parameter Estimation

The Schwarz 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 embedded 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 Cramér-Rao bound when the signal-to-noise ratio (SNR) is large. However, as the SNR is reduced beyond a critical value, the Barankin bound deviates radically from the Cramér-Rao bound, exhibiting the so-called threshold effect. The bounds were applied to the linear FM waveform, and within the resulting class of bounds it was possible to select one that led to a closed-form expression for the lower bound on the variance of an unbiased range estimate. This expression clearly demonstrates the threshold behavior one must expect when using a nonlinear modulation system. Tighter bounds were easily obtained, but these had to be evaluated numerically. The sidelobe 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- \mu s duration and 40-MHz bandwidth, the radar must operate at an SNR greater than 10 dB if meaningful unbiased range estimates are to be obtained.