Abstract
Since time-delay, Doppler effect and phase estimation are fundamental tasks in a plethora of engineering fields, tractable lower performance bounds for this problem are key tools of broad interest for a large variety of remote sensing applications. In the large sample regime and/or the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér–Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this contribution, we introduce firstly a new compact CRB expression for the joint time-delay and Doppler stretch estimation, considering a generic delayed and dilated band-limited signal. This generalizes known results for both wideband signals and the standard narrowband signal model where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift. General compact closed-form CRB expressions for the amplitude and phase are also provided. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. The new CRB expressions are validated in a positioning case study, both using synthetic and real data. These results show that the maximum likelihood estimator converges to the CRB at high signal-to-noise ratios, which confirms the exactness of the CRB. The CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion where the perfect match also proves its exactness.
Highlights
Time-delay, Doppler stretch and phase estimation appear in a plethora of engineering fields such as navigation, radar, reflectometry, sonar or communications, to name a few [1,2,3,4,5,6,7,8,9,10,11,12], being the estimation of such parameters a key first stage of the receiver [7,10,11,12]
Since (9) belongs to the class of conditional signal models [16], the maximum likelihood estimator (MLE) converges to the Cramér–Rao bounds (CRB) at high SNR [17], which confirms the exactness of the CRB
A new compact closed-form CRB expression for the delay-Doppler estimation of a generic band-limited signal has been derived, which may be exploited in a variety of remote sensing applications, i.e., navigation, radar, sonar, Global Navigation Satellite Systems (GNSS)-R
Summary
Time-delay, Doppler stretch and phase estimation appear in a plethora of engineering fields such as navigation, radar, reflectometry, sonar or communications, to name a few [1,2,3,4,5,6,7,8,9,10,11,12], being the estimation of such parameters a key first stage of the receiver [7,10,11,12]. In addition to the historical remote sensing systems as sonar and radar, in more recent Global Navigation Satellite Systems (GNSS), due to the very high velocity of the transmitter located on a satellite, it is essential to incorporate the baseband signal dilatation due to the Doppler effect into the MLE formulation to reach the minimum achievable MSE, for instance in carrier phase-based precise positioning techniques [40]. This is the case of GNSS-based reflectometry (GNSS-R) applications such as altimetry [12,41,42]. The matrix/vector transpose is indicated by a superscript (·) as in A , and the transpose conjugate (·)H as in AH
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