Fundamental limitations in passive time-delay estimation--Part II: Wide-band systems

Abstract

This is the second part of a study which deals with the problem of passive time delay estimation. The focus here is on systems employing wide-band signals and/or arrays of very widely separated receivers. A modified (improved) version of the Ziv-Zakai lower bound (ZZLB) is used to analyze the effect of additive noise and signal ambiguities on the attainable mean-square estimation errors. When the lower bound is plotted as a function of signal-to-noise ratio (SNR), one observes two distinct threshold phenomena dividing the SNR domain into three disjointed segments. At high SNR, the lower bound coincides with the Cramér-Rao lower bound (CRLB). This is the ambiguity-free mode of operation where differential delay estimation is subject only to local errors. At moderate SNR (between the two thresholds), the lower bound exceeds the CRLB by a factor of 12(ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> /w) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{0}</tex> and w are, respectively, the center frequency and signal bandwidth. In this region, the ambiguities in the received signal phases cannot be resolved; however, a useful estimate of the differential delay can still be obtained using the received signal envelopes. At low SNR, the lower bound approaches a constant level depending only on the a priori search domain of the unknown delay parameter. In this region, signal observations are subject to envelope ambiguities as well, and are thus essentially useless for the delay estimation.