Stochastic Cramer-Rao Bound Research Articles

Derivation of multidimensional stochastic Cramer‐Rao bound for mixture of noncoherent and coherent near/far field signals

AbstractThe existing Cramer‐Rao bound (CRB) expressions in the literature are not applicable to the problem of localization of mixed near‐field (NF) and far‐field (FF) sources, including noncoherent and coherent signals, using two‐dimensional arrays. Given the unique properties of circular arrays, in this article, the mathematical model of received signals in multipath environments, including a mixture of noncoherent and coherent NF/FF signals, is introduced. Then, the stochastic CRBs associated with the above problem are derived. The compact closed‐form expression of CRB extracted in this article can be used for the algorithms under the above scenario as a benchmark to evaluate the estimation accuracy of azimuth DOA, elevation DOA, and range parameters as well as fading coefficient parameter. The simulation results show that the CRBs of the azimuth DOAs, elevation DOAs, ranges, and FCs decrease, on average, at a rate of approximately O(M−1.9), O(M−1.9), O(M−1.2), and O(M−1.8) as M (number of sensors) goes to infinity, respectively, where O is the notation of the asymptotic growth rate.

Cramér-Rao Bounds on Angle Estimating in Bistatic MIMO Radar With Non-Orthogonal Waveforms

Angle estimation is a hot topic in multiple input multiple output (MIMO) radar. However, most of the existing algorithms are only focus on the scenario of ideal orthogonal transmission waveform. In engineering applications, however, non-orthogonal waveforms scenario is more practical. In this paper, we focus on the Cramer-rao bound (CRB) on angle estimation in MIMO radar with non-orthogonal waveforms. Firstly, the angle estimation model in bistatic MIMO radar with non-orthogonal transmit waveforms is given, and the general expression of CRB is deduced. Thereafter, the CRBs in some special cases are discussed, e.g., monostatic geometry, orthogonal transmit waveforms scenario. In addition, theoretical comparisons between the stochastic CRBs and the deterministic ones are presented. Finally, the computer simulations are carried out to verify the theoretically ones.

Stochastic Cramer-Rao Bound for DOA Estimation With a Mixture of Circular and Noncircular Signals

The Cramer-Rao bound (CRB) offers insights into the inherent performance benchmark of any unbiased estimator developed for a specific parametric model, which is an important tool to evaluate the performance of direction-of-arrival (DOA) estimation algorithms. In this paper, a closed-form stochastic CRB for a mixture of circular and noncircular uncorrelated Gaussian signals is derived. As a general one, it can be transformed into some existing representative results. The existence condition of the CRB is also analysed based on sparse arrays, which allows the number of signals to be more than the number of physical sensors. Finally, numerical comparisons are conducted in various scenarios to demonstrate the validity of the derived CRB.

Slepian–Bangs Formula and Cramér–Rao Bound for Circular and Non-Circular Complex Elliptical Symmetric Distributions

This paper is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (CCES) distributions. Some general relations between the Cramer Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closedform expressions for the SCRBs of the parameters of interest are derived.

Further Results on the Cramér–Rao Bound for Sparse Linear Arrays

Sparse linear arrays, such as co-prime and nested arrays, can identify up $O(M^2)$ uncorrelated sources with only $O(M)$ sensors by utilizing their difference coarray model. In our previous work, we derived closed-form asymptotic mean-squared error (MSE) expressions for two coarray based MUSIC algorithms and analyzed the Cramer-Rao bound (CRB) in high signal-to-noise ratio (SNR) regions, under the assumption of uncorrelated sources. In this paper, we provide further analysis of the CRB presented in our previous work. We first establish the connection between two CRBs, the CRB derived with the assumption that the sources are uncorrelated, and the classical stochastic CRB derived without this assumption. We show that they are asymptotically equal in high SNR regions for uncorrelated sources. Next, we analyze the behavior of the former CRB for co-prime and nested arrays with a large number of sensors. We investigate the effect of configuration parameters on this CRB. We show that, for co-prime and nested arrays, this CRB can decrease at a rate of $O(M^{-5})$ for large values of $M$ , while this rate is only $O(M^{-3})$ for an $M$ -sensor ULA. This finding analytically demonstrates that co-prime and nested arrays can achieve better asymptotic estimation performance when the number of sensors is a limiting factor. We also show that for a fixed aperture, co-prime and nested arrays require more snapshots to achieve the same performance as ULAs, showing the trade-off between the number of spatial samples and the number of temporal samples. Finally, we demonstrate our analytical results with numerical experiments.

A Stochastic CRB for Non-unitary Beam-space Transformations and its Application to Optimal Steering Angle Design

Many of modern radar systems employ antenna arrays with thousands of anetnnas. These antennas are usually grouped into a few subarrays or beams through the process of beamspace transformation, to estimate the angle of a target. In this letter, we consider a simple but pragmatic beamspace transformation that only allows beam steering, and present a technique of finding optimal steering angles that minimize the Cramer-Rao bound (CRB) for angle estimation. The CRB and mean squared error (MSE) of the angle estimate using the resulting optimal steering angles are compared with those using the widely used DFT-based beam steering.

Closed-Form Expressions of the Exact Cramer-Rao Bound for Parameter Estimation of BPSK, MSK, or QPSK Waveforms

This letter addresses the stochastic Cramer-Rao bound (CRB) pertaining to the joint estimation of the carrier frequency offset, the carrier phase and the noise and signal powers of binary phase-shift keying (BPSK), minimum shift keying (MSK), and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive white circular Gaussian noise. Because the associated models are governed by simple Gaussian mixture distributions, an explicit expression of the Fisher information matrix is given and an explicit expression for the stochastic CRB of these four parameters are deduced. Specialized expressions for low and high SNR are presented as well. Finally, these expressions are related to the modified CRB and our proposed analytical expressions are numerically compared with the approximate expressions previously given in the literature.

Gaussian Cramer-Rao bound for direction estimation of noncircular signals in unknown noise fields

This paper focuses on the stochastic Cramer-Rao bound (CRB) on direction of arrival (DOA) estimation accuracy for noncircular Gaussian sources in the general case of an arbitrary unknown Gaussian noise field parameterized by a vector of unknowns. Explicit closed-form expressions of the stochastic CRB for DOA parameters alone are obtained directly from the Slepian-Bangs formula for general noncircular complex Gaussian distributions. As a special case, the CRB under the nonuniform white noise assumption is derived. Our expressions can be viewed as extensions of the well-known results by Stoica and Nehorai, Ottersten et al., Weiss and Friedlander, Pesavento and Gershman, and Gershman et al. Some properties of these CRBs are proved and finally, these bounds are numerically compared with the conventional CRBs under the circular complex Gaussian distribution for different unknown noise field models.

Stochastic Cramér–Rao bound for direction estimation in unknown noise fields

The stochastic Cramer-Rao bound (CRB) plays an important role in array processing because several advanced high-resolution direction-of-arrival (DOA) estimation methods are known to achieve this bound asymptotically. In the paper, the stochastic CRB on the DOA estimation accuracy is studied in the general case of an arbitrary unknown noise field parameterised by a vector of unknowns. Explicit closed-form expressions for the CRB are derived and its properties are examined theoretically and by practically relevant numerical examples.

The stochastic CRB for array processing: a textbook derivation

The stochastic Cramer-Rao bound (CRB) for direction estimation in array processing applications was indirectly derived some ten years ago as the (asymptotic) covariance matrix of the maximum likelihood (ML) estimator. Attempts to obtain the stochastic CRB directly via the CRB theory fell short of providing a simple derivation and consequently, no direct derivation of this useful performance bound was available in the open literature. we correct this situation by providing a textbook-like direct derivation of the stochastic CRB.