[Paper Review] Solutions to Integrals Involving the Marcum Q-Function and Applications

Paschalis C. Sofotasios, Sami Muhaidat+2 more|arXiv (Cornell University)|May 10, 2015

Wireless Communication Networks Research31 references3 citations

TL;DR

This paper derives novel closed-form solutions for integrals involving the generalized Marcum Q-function, exponential terms, and arbitrary powers, enabling tractable analysis in wireless communications. The key contribution is a generic, algebraically tractable framework that generalizes prior results and is applied to derive exact expressions for energy detection performance and channel capacity under correlated Nakagami-m fading with switch-and-stay combining.

ABSTRACT

Novel analytic solutions are derived for integrals that involve the generalized Marcum Q-function, exponential functions and arbitrary powers. Simple closed-form expressions are also derived for the specific cases of the generic integrals. The offered expressions are both convenient and versatile, which is particularly useful in applications relating to natural sciences and engineering, including wireless cpmmunications and signal processing. To this end, they are employed in the derivation of the channel capacity for fixed rate and channel inversion in the case of correlated multipath fading and switched diversity.

Motivation & Objective

• To address the lack of generic, closed-form solutions for integrals involving the Marcum Q-function, exponential functions, and arbitrary powers.
• To overcome limitations of prior works that restrict parameters to integer values or lack generality.
• To provide a unified, tractable mathematical framework applicable to diverse problems in wireless communications and signal processing.
• To enable exact performance analysis of energy detection and channel capacity in correlated Nakagami-m fading channels.
• To extend existing analyses beyond integer-order fading parameters (m) to arbitrary m values, enhancing practical applicability.

Proposed method

• Derives a closed-form expression for the integral $\mathcal{G}(k,m,a,b,p) = \int_0^\infty x^{k-1} Q_m(a, b\sqrt{x}) e^{-px} dx$ using the Humbert and Kummer confluent hypergeometric functions.
• Derives a corresponding closed-form solution for $\mathcal{F}(k,m,a,b,p) = \int_0^\infty x^{k-1} Q_m(a\sqrt{x}, b) e^{-px} dx$ via series expansion and special function identities.
• Applies the derived integrals to model average probability of detection in energy detection over Nakagami-m fading channels.
• Uses the same framework to compute channel capacity with fixed rate and channel inversion in correlated Nakagami-m fading using switch-and-stay combining.
• Validates results through extensive comparison with Monte Carlo simulations across various fading severity and correlation levels.
• Employs the generalized Marcum Q-function and its integral representations to model signal detection and fading statistics accurately.

Experimental results

Research questions

• RQ1 Can closed-form solutions be derived for integrals involving the Marcum Q-function, exponential decay, and arbitrary power terms, without restricting parameters to integers?
• RQ2 How can these generalized integrals be applied to compute the average probability of detection in energy detection over Nakagami-m fading channels with arbitrary m?
• RQ3 What is the exact expression for channel capacity under fixed-rate transmission and channel inversion in correlated Nakagami-m fading with switch-and-stay combining?
• RQ4 How does the fading severity (m) and correlation (ρ) affect the performance of energy detection and capacity in these systems?
• RQ5 Can the derived expressions be validated numerically and shown to outperform or generalize existing approximations?

Key findings

• A new closed-form expression is derived for $\mathcal{G}(k,m,a,b,p)$ that is valid for arbitrary real $k, p > 0$, integer $m$, and real $a, b$, using the Humbert and Kummer hypergeometric functions.
• The solution for $\mathcal{F}(k,m,a,b,p)$ is also derived in closed form, enabling exact analysis of systems with Marcum Q-function arguments involving square roots.
• The average probability of detection in energy detection over Nakagami-m fading is derived in closed form for arbitrary $m$, extending prior results limited to integer $m$.
• The channel capacity with fixed rate and channel inversion over correlated Nakagami-m fading using switch-and-stay combining is expressed in closed form via the derived integral $\mathcal{F}$.
• Numerical results confirm the accuracy of the derived expressions, showing perfect agreement with Monte Carlo simulations across various $m$, $\rho$, and $\gamma_T$ values.
• The impact of fading severity ($m$) and correlation ($\rho$) on system performance is clearly quantified, demonstrating significant sensitivity in both energy detection and capacity metrics.