Log-concavity property of the error probability with application to local bounds for wireless communications

A clear understanding the behavior of the error probability (EP) as a function of signal-to-noise ratio (SNR) and other system parameters is fundamental for assessing the design of digital wireless communication systems.We propose an analytical framework based on the log-concavity property of the EP which we prove for a wide family of multidimensional modulation formats in the presence of Gaussian disturbances and fading. Based on this property, we construct a class of local bounds for the EP that improve known generic bounds in a given region of the SNR and are invertible, as well as easily tractable for further analysis. This concept is motivated by the fact that communication systems often operate with performance in a certain region of interest (ROI) and, thus, it may be advantageous to have tighter bounds within this region instead of generic bounds valid for all SNRs. We present a possible application of these local bounds, but their relevance is beyond the example made in this paper.

💡 Research Summary

This paper addresses a fundamental problem in digital wireless communications: how the error probability (EP) behaves as a function of the signal‑to‑noise ratio (SNR) and other system parameters. While exact EP expressions are often cumbersome and require numerical evaluation, the authors propose an analytical framework that exploits a structural property of EP—log‑concavity—when the modulation constellation lies on a regular multidimensional grid and the disturbances are Gaussian (thermal noise, Gaussian‑modeled interference) possibly combined with fading.

Main Contributions

1. Proof of Log‑Concavity
   • The authors model the transmitted symbols as a set (X={x_i}_{i=1}^M) in (\mathbb{R}^d) with a priori probabilities (p_i). The received vector is (x_i+\sigma g), where (g\sim\mathcal N(0,I_d)) and (\sigma) is related to the SNR.
   • An error occurs when the noisy observation falls outside the decision region (R_i) associated with (x_i). The EP can be written as
     \