Chernoff Upper Bound Research Articles

On the Angular Resolution Limit uncertainty under compound Gaussian noise

The Angular Resolution Limit (ARL) is a fundamental statistical metric to quantify our ability to resolve two closely-spaced narrowband far-field complex sources. This statistical quantity, is defined as the minimal angular deviation between the two sources to be separated for a prefixed detection-based performance. In this work, we assume that the sources of interest are corrupted by a compound-Gaussian noise. In the standard literature, denoting with δ the true distance between the two sources, the derivation of the ARL is based on the statistical distribution of the Generalized Likelihood Ratio Test (GLRT) for a binary test where there is only one source under the null hypothesis (i.e., δ=0) and two sources under the alternative hypothesis δ ≠ 0. In literature, the true angular distance (TAD) is generally considered as an unknown deterministic parameter, then a maximum likelihood-based estimation of δ is exploited in the GLRT. In this paper, breaking away from existing contributions, we suppose that the TAD is a random variable, Gaussian distributed, meaning that δ∼N(δ0,σδ2). The TAD uncertainty can have many causes as for instance moving sources or/and platform, antenna calibration error, etc. In this work, a generic and flexible (but common) statistical model of the uncertain knowledge of the TAD is preferred instead of a too much specified error model. The degree of randomness (or uncertainty) is quantified by the ratio ξ=δ02/σδ2. The standard framework of the GLRT is no longer feasible for our problem formulation. To cope with the compound Gaussian noise modeling and the random model of the TAD, a powerful upper bound from information/geometry theory is exploited in this work. More precisely, a new expected Chernoff Upper Bound (CUB) on the minimal error probability is introduced. Based on the analysis of this upper bound, we show that the expected-CUB is highly dependent on the degree of uncertainty, ξ. As a by-product, the optimal s-value in the Chernoff divergence for which the expected-CUB is the tightest upper bound is analytically studied and the role of the mean value δ0 in the ARL context is analyzed.

Computational Information Geometry for Binary Classification of High-Dimensional Random Tensors.

Evaluating the performance of Bayesian classification in a high-dimensional random tensor is a fundamental problem, usually difficult and under-studied. In this work, we consider two Signal to Noise Ratio (SNR)-based binary classification problems of interest. Under the alternative hypothesis, i.e., for a non-zero SNR, the observed signals are either a noisy rank-R tensor admitting a Q-order Canonical Polyadic Decomposition (CPD) with large factors of size , i.e., for , where with converge towards a finite constant or a noisy tensor admitting TucKer Decomposition (TKD) of multilinear -rank with large factors of size , i.e., for , where with converge towards a finite constant. The classification of the random entries (coefficients) of the core tensor in the CPD/TKD is hard to study since the exact derivation of the minimal Bayes’ error probability is mathematically intractable. To circumvent this difficulty, the Chernoff Upper Bound (CUB) for larger SNR and the Fisher information at low SNR are derived and studied, based on information geometry theory. The tightest CUB is reached for the value minimizing the error exponent, denoted by . In general, due to the asymmetry of the s-divergence, the Bhattacharyya Upper Bound (BUB) (that is, the Chernoff Information calculated at ) cannot solve this problem effectively. As a consequence, we rely on a costly numerical optimization strategy to find . However, thanks to powerful random matrix theory tools, a simple analytical expression of is provided with respect to the Signal to Noise Ratio (SNR) in the two schemes considered. This work shows that the BUB is the tightest bound at low SNRs. However, for higher SNRs, the latest property is no longer true.

Effects of Erroneous CSI on the Performance of Multiple-Transmit Antenna Selection

The paper assumes that the channel state information (CSI) is available at the receiver and is known partially at the transmitter through a feedback channel. The bit error rate (BER) performance of multiple transmit antenna selection (MTAS) for the multiple-input and single-output (MISO) system with STBC (MTAS/MISO-STBC) will be investigated in detail. Meanwhile, the selection criterion that maximizes the channel Frobenius norm or minimizes the error probability of MTAS/MISO-STBC system is employed. In two cases of erroneous CSI (ECSI) and inerrable CSI (ICSI), wireless channels of the MTAS systems are modeled and their analytical expressions are derived. For the case of ICSI, the exact BER expressions of the Chernoff upper bound (CUB) for both the MISO-STBC system of full complexity and MTAS system are evaluated, respectively. Next, for the ECSI case, a comprehensive analytical expression of BER CUB for the same system is derived in detail. Finally, extensive Monte-Carlo simulations are presented to support and validate our numerical analysis proposed in this paper. Both the simulating results and the numerical results show that MTAS can reduce effectively the effect of the erroneous CSI.

Simplified derivation of the modified Chernoff upper bound

Modified forms of the Chernoff bound were first presented in 1982, offering improvements on the Chernoff bound for systems corrupted by additive interference and independent zero-mean Gaussian noise. Both upper and lower bounds were developed, with the upper bound generally being significantly tighter than the Chernoff bound. However, despite this it has not been as widely adopted in practice as one might expect given its accuracy and inherent simplicity. It is conjectured that this is due, at least in part, to the relatively obtuse, nonintuitive character of the derivation, involving analytic continuation and contour integration in the complex plane. A new, direct derivation is presented that enables the essential underlying simplicity and ease of application to be readily appreciated. To highlight this simplicity details are provided of example applications, as well as a prescriptive demonstration of the formulation of the MCB from first principles. As validation of the use of this technique comparisons are made of the performance and complexity of the MCB with other commonly used error rate calculation techniques.

Arbitrarily tight PBE bounds for optical CDMA communications

Arbitrarily tight upper and lower bounds on the probability of bit error (PBE) for direct-detection avalanche photodiode detector (APD) based code-division multiple-access (CDMA) communication systems are presented. Previously, only the Chernoff upper bound (CB), modified Chernoff upper bound (MCB), and Gaussian approximation (GA) of the PBE have been available. The MCB is calibrated. >

Binary orthogonal signaling over the Gaussian channel with unknown phase/fading: new results and interpretations

The problem of binary orthogonal signaling over a Gaussian noise channel with unknown phase/fading is considered. By viewing the problem in a rotated coordinate system, the orthogonal signal structure is considered as the combination of an antipodal signal set and a pilot tone for channel measurement. For data detection the optimum matched-filter envelope-detector is shown to be identical to a novel detector-estimator receiver in which the detector performs partially coherent detection, using an absolute coherent reference generated by the estimator from the channel measurement provided by the pilot-tone component of the orthogonal signal structure. This detector-estimator interpretation shows that it is incorrect to refer to the optimum receiver as a noncoherent receiver. It also leads to the development of new approaches for analyzing the error probability of the receiver. An exponential Chernoff upper bound is obtained for the Rician channel. >