Error Rates of the Maximum-Likelihood Detector for Arbitrary Constellations: Convex/Concave Behavior and Applications

Abstract

Motivated by a recent surge of interest in convex optimization techniques, convexity/concavity properties of error rates of the maximum likelihood detector operating in the AWGN channel are studied and extended to frequency-flat slow-fading channels. Generic conditions are identified under which the symbol error rate (SER) is convex/concave for arbitrary multi-dimensional constellations. In particular, the SER is convex in SNR for any one-and two-dimensional constellation, and also in higher dimensions at high SNR. Pairwise error probability and bit error rate are shown to be convex at high SNR, for arbitrary constellations and bit mapping.Universal bounds for the SER 1 st and 2 nd derivatives are obtained, which hold for arbitrary constellations and are tight for some of them. Applications of the results are discussed, which include optimum power allocation in spatial multiplexing systems, optimum power/time sharing to decrease or increase (jamming problem) error rate, an implication for fading channels ("fading is never good in low dimensions") and optimization of a unitaryprecoded OFDM system. For example, the error rate bounds of a unitary-precoded OFDM system with QPSK modulation, which reveal the best and worst precoding, are extended to arbitrary constellations, which may also include coding. The reported results also apply to the interference channel under Gaussian approximation, to the bit error rate when it can be expressed or approximated as a non-negative linear combination of individual symbol error rates, and to coded systems.

Optimum Power Allocation for Average Power Constrained Jammers in the Presence of Non-Gaussian Noise

“…R ECENTLY, signal and power randomization approaches have received considerable interest in the literature [1]- [5]. Several papers have addressed different aspects of the jammer power randomization/allocation problem.…”

“…On the other hand, jamming at the maximum available power limit is shown to be more advantageous beyond that critical value. This discussion is extended from binary modulations to arbitrary signal constellations in [5] by concentrating on the ML detection case in an additive white Gaussian noise (AWGN) channel. It is stated that the symbol error rate (SER) performance of the target receiver can be degraded via appropriate power/time sharing under a fixed average jammer noise power.…”

“…In [10], a game-theoretic framework is established to discuss the effects of jamming on the channel capacity by modeling the communicator and the jammer as the players of a two-person zero-sum game subject to certain constraints. Although various aspects of the jamming problem have been investigated in [6]- [10] and the optimum jammer power allocation problem has been studied for Gaussian PDFs and ML receivers in [4], [5], no studies have considered the case of arbitrary jammer and background noise PDFs and generic decision rules at the receivers. In this letter, we study the problem of determining the optimum power allocation policy for an average power constrained jammer operating over an arbitrary additive noise channel, where the aim is to minimize the detection probability of an instantaneously and fully adaptive receiver employing generic decision rules in the NP framework.…”

Secure Diversity-Multiplexing Tradeoff of Zero-Forcing Transmit Scheme at Finite-SNR

“…From a DMT perspective, the MIMO wiretap channel is equivalent to a reduced MIMO channel with (N t − N e ) and N m transmit and receive antennas, respectively. As the error probability is generally log-concave [26], [27], the diversity gain is an increasing function in SNR. Therefore, the finite-SNR secrecy diversity gain d s is equal to zero if N e ≥ N t .…”

Optimal stochastic signal design and detector randomization in the neyman-pearson framework

“…It is shown that the average probability of error is a nonincreasing convex func tion of the signal power when the channel has a continuously differentiable unimodal noise probability density function (PDF) with finite variance. This discussion is extended from binary modulations to arbitrary signal constellations in [4] by concentrating on the maximum likelihood (ML) detection for AWGN channels. It is proven that an average power-limited transmitter cannot improve its error performance via time sharing between different power levels in low dimensions (I-D and 2-D) as opposed to the situation for some M-D constellations, M � 3.…”