Barankin Bound Research Articles

On the limitations of Barankin type bounds for MLE threshold prediction

Various approximations of the Barankin bound (BB), called Barankin type bounds (BTBs), are usually used to predict the threshold region of the maximum likelihood estimator (MLE). In this paper, we show that BTBs are not always appropriate for predicting this threshold region. We prove that in some common estimation problems the BB is infinite, and thus, its approximations cannot reliably predict the MLE threshold region. This result is illustrated for two general signal processing models, in which BTBs can become arbitrarily large, and consequently, cannot reliably predict the MLE threshold region.

Unbiased Estimation of a Sparse Vector in White Gaussian Noise

The problem studied in this paper is unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. It is shown that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, the focus is placed on locally minimum variance unbiased (LMVU) estimators. Simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB) are derived. These bounds allow an estimation of the threshold region separating the low-signal-to-noise ratio (SNR) and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. In addition, numerical lower and upper bounds are derived; these are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare the proposed characterizations of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of this analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero entry of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators.

Barankin-Type Lower Bound on Multiple Change-Point Estimation

We compute lower bounds on the mean-square error of multiple change-point estimation. In this context, the parameters are discrete and the Cramér-Rao bound is not applicable. Consequently, we focus on computing the Barankin bound (BB), the greatest lower bound on the covariance of any unbiased estimator, which is still valid for discrete parameters. In particular, we compute the multi-parameter version of the Hammersley- Chapman-Robbins, which is a Barankin-type lower bound. We first give the structure of the so-called Barankin information matrix (BIM) and derive a simplified form of the BB. We show that the particular case of two change points is fundamental to finding the inverse of this matrix. Several closed-form expressions of the elements of BIM are given for changes in the parameters of Gaussian and Poisson distributions. The computation of the BB requires finding the supremum of a finite set of positive definite matrices with respect to the Loewner partial ordering. Although each matrix in this set of candidates is a lower bound on the covariance matrix of the estimator, the existence of a unique supremum w.r.t. to this set, i.e., the tightest bound, might not be guaranteed. To overcome this problem, we compute a suitable minimal-upper bound to this set given by the matrix associated with the Loewner-John Ellipsoid of the set of hyper-ellipsoids associated to the set of candidate lower-bound matrices. Finally, we present some numerical examples to compare the proposed approximated BB with the performance achieved by the maximum likelihood estimator.

Signal-to-noise ratio threshold effect in track before detect

Track before detect (TBD) refers to simultaneous detection and tracking using unthresholded sensor responses over time. The motivation for TBD is its capacity to deal with low signal-to-noise ratio (SNR) targets. Previously, the achievable error for TBD has been established using Cramer–Rao analysis. Although computationally simple the Cramer–Rao bound is not useful at low SNR as it does not predict the threshold effect. A more accurate notion of the achievable performance at low SNRs is provided by the computationally more complicated Barankin bound. The computational complexity of the Barankin bound arises from the need to optimise over a number of test points, with the tightness of bound increasing with the number of test points. An approximation to the Barankin bound is proposed which permits the use of multiple test points with reasonable computational expense. The improvements in threshold SNR prediction offered by the proposed bound are demonstrated in numerical examples.

A New Barankin Bound Approximation for the Prediction of the Threshold Region Performance of Maximum Likelihood Estimators

It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain signal-to-noise ratio (SNR) or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the Cramer-Rao bound (CRB). The search of the SNR threshold value (where the CRB becomes unreliable for prediction of maximum likelihood estimator variance) can be achieved with the help of the Barankin bound (BB), as proposed by many authors. The major drawback of the BB, in comparison with the CRB, is the absence of a general analytical formula, which compels one to resort to a discrete form, usually the Mcaulay-Seidman bound (MSB), requesting the search of an optimum over a set of test points. In this paper, we propose a new practical BB discrete form that provides, for a given set of test points, an improved SNR threshold prediction in comparison with existing approximations (MSB, Abel bound, Mcaulay-Hofstetter bound) at the expense of the computational complexity increased by a factor les ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> +1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> , where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> is the number of unknown parameters. We have derived its expression for the general Gaussian observation model to be used in place of existing approximations.

Harmonic retrieval in the presence of non-circular Gaussian multiplicative noise: performance bounds

We address the problem of harmonic retrieval in the presence of multiplicative and additive noise sources. In the new context of a complex-valued non-circular Gaussian multiplicative noise, we express the Cramér–Rao bound (CRB) as well as the asymptotic (large sample) CRB in closed form. Below a certain SNR threshold and/or when the number of samples is not large enough, the CRB becomes too optimistic and therefore we also derive the Barankin bound (BB). The new theoretical expressions for the CRB and BB are then used to study the behavior of the performance bound with respect to the signal parameters. We especially describe the region (in terms of SNR and number of samples) for which the CRB and the BB differ. Finally we compare the performance of the square-power-based frequency estimate, which is equivalent to the non-linear least-squares-based estimate, to these bounds.

Low Angle Estimation: Models, Methods, and Bounds

Boman, K. and Stoica, P., Low Angle Estimation: Models, Methods, and Bounds. Digital Signal Processing11 (2001), 35–79.In this work we study the performance of elevation estimators and lower bounds of the estimation error variance for a low angle target in a smooth sea scenario using an antenna array. The article is structured around some key assumptions on multipath knowledge, signal parameterization and noise covariance. We prove that the Cramér–Rao bound is highly dependent on the multipath model, while it is the same for the different signal parameterizations, and that it is independent of the noise covariance. The Cramér–Rao bound is sometimes too optimistic and not achievable. The tighter Barankin bound is derived to predict the threshold behavior seen at low SNR. Simulations show that the maximum likelihood methods are statistically efficient and achieve the theoretical lower bound on error variance, in the case of high enough SNR. Finally we show that the bounds can be used to design an improved array structure and study the influence of multiple frequencies.

Performance limits on underwater acoustic matched-field processing

Sensor array processing methods which use full-wave models of complex multipath propagation to facilitate estimation of underwater source locations and channel parameters are known to be prone to ambiguities and environmental variability. Particularly at low signal-to-noise ratios, the performance of matched-field methods may therefore deviate radically from the Cramer–Rao bound which assumes estimation errors are small. To study matched-field processing in the presence of anomalies, this paper considers the Barankin bound. The Barankin bound is particularly useful for determining the threshold signal-to-noise ratio below which any unbiased estimate of the source location or environmental parameters will be prone to large errors. In this paper, the Barankin bound is compared with a maximum likelihood (ML) estimator of source location in an uncertain shallow-water scenario. The results indicate that the Barankin bound predicts the threshold signal-to-diffuse-noise ratio of the ML method to within 3 dB. Further, in the presence of surface shipping, the impact of both vertical and horizontal spatial decorrelation of the interference on matched-field performance is studied for different array geometries. [Work supported by ONR.]

Delay estimation using narrow-band processes

Array processing of narrow-band Gaussian signals is studied with emphasis on delay estimation. The Barankin bound is used to examine the effect of ambiguity on mean-square measurement error. When the bound is plotted as a function of signal-to-noise ratio one observes a distinct threshold. Above the critical signal-to-noise ratio the lower bound on mean-square error is given by the Cramér-Rao inequality, which is approached by the Barankin inequality under these conditions. Below the threshold the Barankin bound can exceed the Cramér-Rao bound by large factors. The relative magnitude of the bounds in that region depends critically on the ratio of signal center frequency to signal bandwidth.