[Paper Review] A New Representation for the Symbol Error Rate

Adithya Rajan, Cihan Tepedelenlioğlu|arXiv (Cornell University)|Jun 14, 2012

Advanced Wireless Communication Techniques19 references3 citations

TL;DR

This paper introduces a novel representation of the symbol error rate (SER) for arbitrary multi-dimensional constellations under additive white Gaussian noise (AWGN) and compound Gaussian noise, expressing SER as a product of a completely monotone function and a non-negative power of the signal-to-noise ratio (SNR). The key contribution is a necessary and sufficient condition for SER complete monotonicity based on the rank of the constellation matrix, which enables stochastic ordering and simplifies SER analysis over fading channels.

ABSTRACT

The symbol error rate of the minimum distance detector for an arbitrary multi-dimensional constellation impaired by additive white Gaussian noise is characterized as the product of a completely monotone function with a non-negative power of the signal to noise ratio. This representation is also shown to apply to cases when the impairing noise is compound Gaussian. Using this general result, it is proved that the symbol error rate is completely monotone if the rank of its constellation matrix is either one or two. Further, a necessary and sufficient condition for the complete monotonicity of the symbol error rate of a constellation of any dimension is also obtained. Applications to stochastic ordering of wireless system performance are also discussed.

Motivation & Objective

• To characterize the symbol error rate (SER) of arbitrary multi-dimensional constellations under AWGN and compound Gaussian noise using a unified analytical framework.
• To establish conditions under which the SER is completely monotone, a property critical for stochastic ordering and average SER analysis over fading channels.
• To generalize existing results on SER convexity and complete monotonicity from 1D and 2D constellations to arbitrary dimensions.
• To develop a new stochastic order, $\mathscr{G}_p$, for comparing average SERs over quasi-static fading channels using the SER's complete monotonicity.
• To provide a canonical representation of SER that facilitates the derivation of average SER expressions without requiring closed-form SER expressions.

Proposed method

• Represent the SER as $ P_{\text{e}}(\rho) = \rho^p \int_0^\infty e^{-\rho u} \mu(u) \, du $, where $ \mu(u) $ is non-negative and depends on the constellation geometry and minimum distance.
• Use the theory of completely monotone functions to characterize the SER's monotonicity properties via the Laplace transform representation.
• Derive a necessary and sufficient condition for SER complete monotonicity based on the rank of the constellation matrix: SER is completely monotone if the rank is 1 or 2.
• Generalize the SER representation to compound Gaussian noise, including non-Gaussian distributions like Middleton class-A and symmetric alpha-stable noise.
• Introduce a new stochastic order $\mathscr{G}_p$ defined by $ X_1 \leq_{\mathscr{G}_p} X_2 \Leftrightarrow \mathbb{E}[X_1^p e^{-\rho X_1}] \geq \mathbb{E}[X_2^p e^{-\rho X_2}] $ for all $ \rho \geq 0 $, enabling SER comparisons across fading distributions.
• Apply Bernstein's theorem to link the complete monotonicity of the SER to the positivity of its associated measure $ \mu(u) $, ensuring analytical tractability.

Experimental results

Research questions

• RQ1 Under what conditions is the SER of an arbitrary multi-dimensional constellation completely monotone with respect to the SNR?
• RQ2 Can the SER representation be generalized beyond AWGN to compound Gaussian noise models?
• RQ3 How does the rank of the constellation matrix determine the complete monotonicity of the SER?
• RQ4 What stochastic order can be used to compare average SERs of arbitrary constellations over quasi-static fading channels?
• RQ5 Can the SER be expressed as a positive mixture of decaying exponentials, and what are the implications for average SER analysis over fading channels?

Key findings

• The SER of any multi-dimensional constellation under AWGN can be represented as $ \rho^p \int_0^\infty e^{-\rho u} \mu(u) \, du $, where $ \mu(u) \geq 0 $, establishing a canonical form.
• The SER is completely monotone if and only if the rank of the constellation matrix is 1 or 2, providing a necessary and sufficient condition for complete monotonicity.
• For constellations with rank greater than 2, complete monotonicity depends on the specific constellation geometry and prior probabilities, not just the rank.
• The SER representation generalizes to compound Gaussian noise, including Middleton class-A and symmetric alpha-stable noise, broadening its applicability.
• A new stochastic order $\mathscr{G}_p$ is introduced, enabling comparison of average SERs over different fading distributions without closed-form SER expressions.
• The second derivative of the SER is non-negative for high SNR when $ \rho \geq 4(p + \sqrt{p}) / d_{\text{min}}^2 $, confirming convexity in the high-SNR regime for general constellations.