Comments on “A Representation for the Symbol Error Rate Using Completely Monotone Functions”

It was shown in the above-titled paper by Rajan and Tepedelenlioglu (see ibid., vol. 59, no. 6, p. 3922–31, June 2013) that the symbol error rate (SER) of an arbitrary multidimensional constellation subject to additive white Gaussian noise is characterized as the product of a completely monotone function with a nonnegative power of signal-to-noise ratio (SNR) under minimum distance detection. In this comment, it is proved that the probability of correct decision of an arbitrary constellation admits a similar representation as well. Based on this fact, it is shown that the stochastic ordering
$\{\leq_{{\cal G}_{\alpha}},\alpha\geq 0\}$
proposed by the authors as an extension of the existing Laplace transform order to compare the average SERs over two different fading channels actually predicts that the average SERs are equal for any constellation of dimensionality smaller than or equal to
$2\alpha$
. Furthermore, it is noted that there are no positive random variables
$X_{1}$
and
$X_{2}$
such that the proposed stochastic ordering is satisfied in the strict sense, i.e.,
$X_{1}<_{{\cal G}_{\alpha}}X_{2}$
, when
$\alpha=N/2$
for any positive integer
$N$
. Additional remarks are noted about the fading scenarios at low SNR and the generalization to additive compound Gaussian noise originally discussed in the subject paper.