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Register a new userAnalysis and interpretation of the Cramer-Rao lower-bound in astrometry: One dimensional case
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In this paper we explore the maximum precision attainable in the location of a point source imaged by a pixel array detector in the presence of a background, as a function of the detector properties. For this we use a well-known result from parametric estimation theory, the so-called Cramer-Rao lower bound. We develop the expressions in the 1-dimensional case of a linear array detector in which the only unknown parameter is the source position. If the object is oversampled by the detector, analytical expressions can be obtained for the Cramer-Rao limit that can be readily used to estimate the limiting precision of an imaging system, and which are very useful for experimental (detector) design, observational planning, or performance estimation of data analysis software: In particular, we demonstrate that for background-dominated sources, the maximum astrometric precision goes as $B/F^2$, where $B$ is the background in one pixel, and $F$ is the total flux of the source, while when the background is negligible, this precision goes as $F^{-1}$. We also explore the dependency of the astrometric precision on: (1) the size of the source (as imaged by the detector), (2) the pixel detector size, and (3) the effect of source de-centering. Putting these results into context, the theoretical Cramer-Rao lower bound is compared to both ground- as well as spaced-based astrometric results, indicating that current techniques approach this limit very closely. Our results indicate that we have found in the Cramer-Rao lower variance bound a very powerful astrometric benchmark estimator concerning the maximum expected positional precision for a point source, given a prescription for the source, the background, the detector characteristics, and the detection process.
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In this paper we use the Cramer-Rao lower uncertainty bound to estimate the maximum precision that could be achieved on the joint simultaneous (or 2D) estimation of photometry and astrometry of a point source measured by a linear CCD detector array. We develop exact expressions for the Fisher matrix elements required to compute the Cramer-Rao bound in the case of a source with a Gaussian light profile. From these expressions we predict the behavior of the Cramer-Rao astrometric and photometric precision as a function of the signal and the noise of the observations, and compare them to actual observations - finding a good correspondence between them. We show that the astrometric Cramer-Rao bound goes as $(S/N)^{-1}$ (similar to the photometric bound) but, additionally, we find that this bound is quite sensitive to the value of the background - suppressing the background can greatly enhance the astrometric accuracy. We present a systematic analysis of the elements of the Fisher matrix in the case when the detector adequately samples the source (oversampling regime), leading to closed-form analytical expressions for the Cramer-Rao bound. We show that, in this regime, the joint parametric determination of photometry and astrometry for the source become decoupled from each other, and furthermore, it is possible to write down expressions (approximate to first order in the small quantities F/B or B/F) for the expected minimum uncertainty in flux and position. These expressions are shown to be quite resilient to the oversampling condition, and become thus very valuable benchmark tools to estimate the approximate behavior of the maximum photometric and astrometric precision attainable under pre-specified observing conditions and detector properties.
Context. The best precision that can be achieved to estimate the location of a stellar-like object is a topic of permanent interest in the astrometric community. Aims. We analyse bounds for the best position estimation of a stellar-like object on a CCD detector array in a Bayesian setting where the position is unknown, but where we have access to a prior distribution. In contrast to a parametric setting where we estimate a parameter from observations, the Bayesian approach estimates a random object (i.e., the position is a random variable) from observations that are statistically dependent on the position. Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimum mean square error (MMSE) of the best estimator of the position of a point source on a linear CCD-like detector, as a function of the properties of detector, the source, and the background. Results. We quantify and analyse the increase in astrometric performance from the use of a prior distribution of the object position, which is not available in the classical parametric setting. This gain is shown to be significant for various observational regimes, in particular in the case of faint objects or when the observations are taken under poor conditions. Furthermore, we present numerical evidence that the MMSE estimator of this problem tightly achieves the Bayesian CR bound. This is a remarkable result, demonstrating that all the performance gains presented in our analysis can be achieved with the MMSE estimator. Conclusions The Bayesian CR bound can be used as a benchmark indicator of the expected maximum positional precision of a set of astrometric measurements in which prior information can be incorporated. This bound can be achieved through the conditional mean estimator, in contrast to the parametric case where no unbiased estimator precisely reaches the CR bound.
Channel state information (CSI) is of vital importance in wireless communication systems. Existing CSI acquisition methods usually rely on pilot transmissions, and geographically separated base stations (BSs) with non-correlated CSI need to be assigned with orthogonal pilots which occupy excessive system resources. Our previous work adopts a data-driven deep learning based approach which leverages the CSI at a local BS to infer the CSI remotely, however the relevance of CSI between separated BSs is not specified explicitly. In this paper, we exploit a model-based methodology to derive the Cramer-Rao lower bound (CRLB) of remote CSI inference given the local CSI. Although the model is simplified, the derived CRLB explicitly illustrates the relationship between the inference performance and several key system parameters, e.g., terminal distance and antenna array size. In particular, it shows that by leveraging multiple local BSs, the inference error exhibits a larger power-law decay rate (w.r.t. number of antennas), compared with a single local BS; this explains and validates our findings in evaluating the deep-neural-network-based (DNN-based) CSI inference. We further improve on the DNN-based method by employing dropout and deeper networks, and show an inference performance of approximately $90%$ accuracy in a realistic scenario with CSI generated by a ray-tracing simulator.
In collisional thermometry, a system in contact with the thermal bath is probed by a stream of ancillas. Coherences and collective measurements were shown to improve the Fisher information in some parameter regimes, for a stream of independent and identically prepared (i.i.d.) ancillas in some specific states [Seah et al., Phys. Rev. Lett., 180602 (2019)]. Here we refine the analysis of this metrological advantage by optimising over the possible input ancilla states, also for block-i.i.d.~states of block size b=2. For both an indirect measurement interaction and a coherent energy exchange channel, we show when the thermal Cramer-Rao bound can be beaten, and when a collective measurement of $N>1$ ancilla may return advantages over single-copy measurements.
To achieve the joint active and passive beamforming gains in the reconfigurable intelligent surface assisted millimeter wave system, the reflected cascade channel needs to be accurately estimated. Many strategies have been proposed in the literature to solve this issue. However, whether the Cramer-Rao lower bound (CRLB) of such estimation is achievable still remains uncertain. To fill this gap, we first convert the channel estimation problem into a sparse signal recovery problem by utilizing the properties of discrete Fourier transform matrix and Kronecker product. Then, a joint typicality based estimator is utilized to carry out the signal recovery task. We show that, through both mathematical proofs and numerical simulations, the solution proposed in this letter can in fact asymptotically achieve the CRLB.
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