REALIZABLE LOWER BOUNDS FOR TIME-DELAY ESTIMATION .2. THRESHOLD PHENOMENA

In time delay estimation of narrowband signals the mean square error (MSE) plotted as a function of the signal to noise ratio (SNR) exhibits threshold phenomena. The thresholds divide the domain of SNR values into several disjoint segments. For very low SNR values the observations are dominated by noise and are essentially useless for delay estimation. Here the MSE is determined largely by the available a priori information. For intermediate SNR values, time delay estimates are possible, but are subject to ambiguities resulting from the oscillatory nature of the signal sample correlation. For very high SNR values, these ambiguities are resolvable and the Cramer-Rao bound (CRB) yields a realistic bound for the attainable performance. A previous paper [5] used the Barankin bound to obtain a realizable lower bound for the intermediate SNR region but left open the question where the transitions from low to medium and medium to high SNR operation occur. Since the attainable MSE in the three regions can be very different, the location of the SNR thresholds is of considerable practical interest. The present paper addresses this issue. It obtains threshold values for the single and two echo problems, with emphasis on the effect of echo separation in the two echo case. It concludes that the location of the thresholds is only weakly dependent on echo separation, even though the attainable MSE in both the intermediate and high SNR regions varies drastically as the echo separation changes.