Fundamental limitations in passive time delay estimation--Part I: Narrow-band systems

Abstract

Time delay estimation of a noise-like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar/sonar applications. A new method is presented to analyze the mean-square error performance of delay estimation schemes based on a modified (improved) version of the Ziv-Zakai lower bound (ZZLB). This technique is shown to yield the tightest results on the attainable system performance for a wide range of signal-to-noise ratio (SNR) conditions. For delay estimation using narrow-band (ambiguity-prone) signals, the fundamental result of this study is illustrated in Fig. 3. The entire domain of SNR is divided into several disjoint segments indicating several distinct modes of operation. If the available SNR does not exceed SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation. As a result, the attainable mean-square error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{\epsilon}^{2}</tex> is bounded only by the a priori parameter domain. If SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> < SNR < SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , the modified ZZLB coincides with the Barankin bound. In this regime differential delay observations are subject to ambiguities. If SNR > SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> the modified ZZLB coincides with the Cramer-Rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved. The transition from the ambiguity-dominated mode of operation to the ambiguity-free mode of operation starts at SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and ends at SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> . This is the threshold phenomenon in time delay estimation. The various deflection points SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and the various segments of the bound (Fig. 3) are given as functions of such important system parameters as time-bandwidth product (WT), signal bandwidth to center frequency ratio (W/ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) and the number of half wavelengths of the signal center frequency contained in the spacing between receivers. With this information the composite bound illustrated in Fig. 3 provides the most complete characterization of the attainable system performance under any prespecified SNR conditions.