This paper proposes to unify fading distributions by modeling the magnitude-squared of the instantaneous channel gain using generalized gamma convolutions (GGC). A random variable is said to be GGC, if it can be written as the limit of a sum of independent gamma variables potentially with different parameters. We show that GGC random variables have many useful mathematical properties, that are applied in the performance analysis of wireless systems. The proposed unification subsumes several classes of multipath and shadowing fading distributions previously proposed in the wireless communications literature.

A new stochastic order between two fading distributions is introduced. A fading channel dominates another in the ergodic capacity ordering sense, if the Shannon transform of the first is greater than that of the second at all values of average signal-to-noise ratio. It is shown that some parametric fading models, such as the Nakagami-m, Rician, and Hoyt are distributions that are monotonic in their line of sight parameters with respect to the ergodic capacity order. Some operations under which the ergodic capacity order is preserved are also discussed. Through these properties of the ergodic capacity order, it is possible to compare under two different fading scenarios, the ergodic capacity of a composite system involving multiple fading links with coding/decoding capabilities only at the transmitter/receiver. Such comparisons can be made even in cases when a closed form expression for the ergodic capacity of the composite system is not analytically tractable. Applications to multiple access channels, and extensions to multiple-input multiple-output systems are also discussed.

The symbol error rate of an arbitrary multidimensional constellation in the absence of coding impaired by additive white Gaussian noise is characterized as the product of a completely monotone function with a nonnegative power of the signal-to-noise ratio, when the minimum distance detector is used. 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.

In this paper, a new stochastic order between two fading distributions is introduced. A fading channel dominates another in the capacity ordering sense, if the ergodic capacity of the first is greater than that of the second at all values of average signal to noise ratio. We show that many parametric fading models such as the Nakagami-m, Rician and Hoyt fading satisfy the capacity order, in the sense that the distribution with a larger line of sight parameter is larger in the ergodic capacity sense. Further, we obtain closure properties of the capacity order for the first time, because such a stochastic order has not been considered in either stochastic ordering literature, or information theory literature. Through these properties, we develop sufficient conditions for comparing the ergodic capacity of a composite system involving multiple capacity ordered fading links with coding/decoding capabilities only at the transmitter/receiver, when operated in two different fading scenarios. Such comparisons can be made even in cases when a closed form expression for the ergodic capacity of the composite system is not analytically tractable. We also show that capacity ordering of point-to-point links has applications to multiple access channels.

In this paper, it is shown that the symbol error rate of an arbitrary multi-dimensional constellation impaired by independent and identically distributed additive white Gaussian noise under maximum likelihood detection is completely monotonic in the average SNR, if the rank of the constellation matrix is either one or two. Further, under minimum distance detection, this result generalizes to the case when the noise follows a dependent and identically distributed compound Gaussian distribution. Classes of constellations with dimensionality greater than two, which have completely monotone and convex error rates are also identified herein. The applications of the complete monotonicity of the symbol error rate in obtaining generic comparisons of average error rates over pairs of fading channels are also discussed.

Stochastic orders are binary relations defined on probability distributions which capture intuitive notions like being larger or being more variable. This paper introduces stochastic ordering of instantaneous SNRs of fading channels as a tool to compare the performance of communication systems over different channels. Stochastic orders unify existing performance metrics such as ergodic capacity, and metrics based on error rate functions for commonly used modulation schemes through their relation with convex and completely monotonic (c.m.) functions. Toward this goal, performance metrics such as instantaneous error rates of M-QAM and M-PSK modulations are shown to be c.m. functions of the instantaneous SNR, while metrics such as the instantaneous capacity are seen to have a completely monotonic derivative (c.m.d.). It is shown that the frequently used parametric fading distributions for modeling line of sight (LoS) exhibit a monotonicity in the LoS parameter with respect to the stochastic Laplace transform order. Using stochastic orders, average performance of systems involving multiple random variables are compared over different channels, even when closed form expressions for such averages are not tractable. These include diversity combining schemes, relay networks, and signal detection over fading channels with non-Gaussian additive noise, which are investigated herein. Simulations are also provided to corroborate our results.

This paper addresses the bit error rate performance of diversity combining schemes for a single-user communication system operating over Rayleigh flat fading channels subject to impulsive alpha-stable noise, which is known to arise due to network interference from a Poisson field of interference sources. Under this setting, a theoretical benchmark receiver is analyzed, the maximum possible diversity order attainable is derived to be Lα/2, where L is the number of antennas and α is the characteristic exponent of the noise. Moreover, trade-offs between the diversity gain and the array gain have been identified. The conventional linear matched filter receiver (also known as maximum ratio combining in the Gaussian case) has been shown to have a diversity order of α/2, thereby not benefiting from spatial diversity. Closed form expressions for the diversity gain and array gain of post-detection combining have been derived. As an improvement to conventional combining schemes, the optimum maximum likelihood detector is derived. A simpler high SNR approximation of the detector is shown to not depend on the noise parameters, and is compared to the generalized Cauchy detector under stable noise. Monte-Carlo simulations are used to supplement our analytical results and compare the performance of the detectors.