Minimum Average Error Probability Research Articles

Novel Fano type lower bounds on the minimum error probability of list $M$-ary hypothesis testing

he problem of list $M$-ary hypothesis testing with fixed list size $L< M$ is considered. Based on some random observation, the test outputs a list of $L$ candidates out of $M$ possible hypotheses. The probability of list error is defined as the probability of the event that the list output by the test does not contain the true hypothesis that has generated the observation. An identity is derived that relates the minimum average probability of error of the optimal list hypothesis test to the minimum average probability of error of an optimal maximum a posteriori probability decision rule. The latter decides among an alternative set of hypotheses corresponding to all possible $L$-component mixtures of the distributions that characterize the observation under the original $M$ candidate hypotheses. As an application, the proposed identity is employed to obtain novel Fano type lower bounds on the minimum error probability of list $M$-ary hypothesis testing.

Belief propagation with quantum messages for quantum-enhanced classical communications

For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.

Optimal measurements for symmetric quantum states with applications to optical communication

The minimum probability of error (MPE) measurement discriminates between a set of candidate quantum states with the minimum average error probability allowed by quantum mechanics. Conditions for a measurement to be MPE were derived by Yuen, Kennedy and Lax (YKL). MPE measurements have been found for states that form a single orbit under a group action, i.e., there is a transitive group action on the states in the set. For such state sets, termed geometrically uniform (GU) by Forney, it was shown that the `pretty good measurement' (PGM) attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the PGM on a GU set involves inverting large matrices, and is not easy in general. Our first contribution is a formula for the MPE and conditional probabilities of GU sets, using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). MPE measurements for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the MPE measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many common group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs---which is what is needed to solve the YKL conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states for some examples relevant to quantum-limited classical optical communication.

Optimal Detector Randomization in Cognitive Radio Systems in the Presence of Imperfect Sensing Decisions

In this study, optimal detector randomization is developed for secondary users in a cognitive radio system in the presence of imperfect spectrum sensing decisions. It is shown that the minimum average probability of error can be achieved by employing no more than four maximum a-posteriori probability (MAP) detectors at the secondary receiver. Optimal MAP detectors and generic expressions for their average probability of error are derived in the presence of possible sensing errors. Also, sufficient conditions are presented related to improvements due to optimal detector randomization.

Optimal Ultrasmall Block-Codes for Binary Discrete Memoryless Channels

Optimal block-codes (in the sense of minimum average error probability, using maximum likelihood decoding) with a small number of codewords are investigated for the binary asymmetric channel (BAC), including the two special cases of the binary symmetric channel (BSC) and the Z-channel (ZC), both with arbitrary cross-over probabilities. For the ZC, the optimal code structure for an arbitrary finite blocklength is derived in the cases of two, three, and four codewords and conjectured in the case of five codewords. For the BSC, the optimal code structure for an arbitrary finite blocklength is derived in the cases of two and three codewords and conjectured in the case of four codewords. For a general BAC, the best codebooks under the assumption of a threshold decoder are derived for the case of two codewords. The derivation of these optimal codes relies on a new approach of constructing and analyzing the codebook matrix not rowwise (codewords), but columnwise. This new tool leads to an elegant definition of interesting code families that is recursive in the blocklength n and admits their exact analysis of error performance. This allows for a comparison of the average error probability between all possible codebooks.

Data-aided efficient synchronization for UWB signals based on minimum average error probability

One of the biggest challenges in ultra-wideband (UWB) radio is the accurate timing acquisition for the receiver. In this article, we develop a novel data-aided synchronization algorithm for pulses amplitude modulation (PAM) UWB systems. Pilot and information symbols are transmitted simultaneously by an orthogonal code division multiplexing (OCDM) scheme. In the receiver, an algorithm based on the minimum average error probability (MAEP) of coherent detector is applied to estimate the timing offset. The multipath interference (MI) problem for timing offset estimation is considered. The mean-square-error (MSE) and the bit-error-rate(BER) performances of our proposed scheme are simulated. The results show that our algorithm outperforms the algorithm based on the maximum correlator output (MCO) in multipath channels.

A new universal random coding bound for the multiple-access channel

The minimum average error probability achievable by block codes on the two-user multiple-access channel is investigated. A new exponential upper bound is found which can be achieved universally for all discrete memoryless multiple-access channels with given input and output alphabets. It is shown that the exponent of this bound is greater than or equal to those of previously known bounds. Moreover, examples are given where the new exponent is strictly larger.

Optimum quantum decision between a pure and a mixed state with application to detection of TCS signals

A method is given for calculating the minimum average error probability attainable in deciding between a mixed state and a pure state of a quantum system. It is applied, first, to the detection of a noisy coherent pulse represented by a state with a Gaussian P-representation and, second, to the detection of a signal represented at the transmitter by a two-photon coherent state (TCS). In both problems the field of the receiver is free of background noise and, when no signal was transmitted, is in the ground state.

On Minimum Average Probability of Error Expression for a Binary Pulse-Communication System With Intersymbol Interference

Results useful in the calculation of the exact closed-form expression for the minimum average probability of error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_e</tex> of a binary pulse linear communication system with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> intersymbol interferences and additive colored Gaussian noise are given. By first performing statistical averaging and then using the variational method, a compact form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_e</tex> that depends only on the overall system impulse response taken over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> time instants is obtained. These <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> unknowns are given as the solution of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> simultaneous nonlinear equations. Specific examples illustrating this approach are considered.

Intersymbol interference and error probability

Using a criterion of minimum average error probability we derive a method for specifying an optimum linear, time invariant receiving filter for a digital data transmission system. The transmitted data are binary and coded into pulses of shape <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\pm s(t)</tex> . The linear transmission medium introduces intersymbol interference and additive Gaussian noise. Because the intersymbol interference is not Gaussian and can be correlated with the binary digit being detected, our problem is one of deciding which of two waveforms is present in a special type of correlated, non-Gaussian noise. For signal-to-noise ratios in a range of practical interest, the optimum filter is found to be representable as a matched filter followed by a tapped delay line--the same form as that of the least mean square estimator of the pulse amplitude. The performance (error probability vs. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N</tex> ) of the optimum filter is compared with that of a matched-filter receiver in an example.