Compact CRB for delay, Doppler, and phase estimation – application to GNSS SPP and RTK performance characterisation

Abstract

The derivation of tight estimation lower bounds is a key tool to design and assess the performance of new estimators. In this contribution, first, the authors derive a new compact Cramér-Rao bound (CRB) for the conditional signal model, where the deterministic parameter's vector includes a real positive amplitude and the signal phase. Then, the resulting CRB is particularised to the delay, Doppler, phase, and amplitude estimation for band-limited narrowband signals, which are found in a plethora of applications, making such CRB a key tool of broad interest. This new CRB expression is particularly easy to evaluate because it only depends on the signal samples, then being straightforward to evaluate independently of the particular baseband signal considered. They exploit this CRB to properly characterise the achievable performance of satellite-based navigation systems and the so-called real-time kinematics (RTK) solution. To the best of the authors' knowledge, this is the first time these techniques are theoretically characterised from the baseband delay/phase estimation processing to position computation, in terms of the CRB and maximum-likelihood estimation.

Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals

“…Such a CRB is very convenient because it only depends on the signal sample, and it is summarized in the sequel for completeness. Considering the joint time-delay and phase estimation resorting to Model ( 6 ), the CRB is given by [ 37 ]: where Im represents the imaginary operator, , and the energy of the signal. and are defined as (for ): …”

GNSS data demodulation over fading environments: Antipodal andM‐ary CSK modulations

“…μ h = 1 and σ 2 h ¼ 0) the LLR in Equation 24reduces to the Gaussian LLR solution in Equation 7. Notice that Equation 24requires the knowledge of σ 2 , as well as μ h and σ 2 h . The latter values can be obtained by resorting to the maximum likelihood (ML) estimates [17]:…”

“…Note from Equation (25) that only simple arithmetical operations are required. Again, the complexity of the method depends on the number of samples K used to estimate μ h and σ 2 h , but for equal number of samples, the Bayesian solution is computationally less expensive than the best linear one.…”

“…In contrast with Section 3.2, where the underlying pilot signal assumption allowed to compute the ML estimates in (25) which are in turn used to compute the LLR approximation (26), in this case, we do not have access to such training sequence. Therefore, we propose a method to derive the μ h and σ 2 h ML estimates when no learning sequence is available. The marginal likelihood p(y n ) = p(y n |x n = 1)p(x n = 1) + p (y n |x n = −1)p(x n = −1) is a mixture of two Gaussian distributions:…”

“…The previous expression avoids the use of log/exp function evaluations, reducing the computational burden at the receiver. Notice that the first-order μ h and second-order σ 2 h moments of p (h) are assumed to be known, that is, partial statistical CSI is assumed. However, these parameters might not be available at the receiver and therefore they must be estimated online.…”