A unified MGF based approach to the analysis of area under the ROC curve of energy detection over generalized fading channels

Abstract

The performance of energy detection, in terms of the area under receiver operating characteristic (AUC), is investigated. By using its contour integral representation, the generalized Marcum Q-function is reformulated as an analytically attractive exponential integral, which in turn allows for a unified moment generating function (MGF) based approach to the AUC analysis of energy detection in an arbitrarily fading channel. The proposed MGF approach is then modified to include square-law combining (SLC) diversity and is further extended to address partial AUC with SLC.Index Terms-Generalized Marcum Q-function, moment generating function, energy detection I. INTRODUCTIONAs a classic application of noncoherent detection, energy detection [1] has attracted renewed interest recently, due primarily to its application in spectrum sensing for cognitive radio and dynamic spectrum access. Usually, the performance of energy detection is characterized by its probability of detection and probability of false alarm, and thereby fully described by the receiver operating characteristic (ROC) curve [2]-[7]. It is, however, inconvenient sometimes to compare different schemes using their ROC curves that may cross and hence cannot provide a simple definitive conclusion. In view of this situation, the area under the receiver operating characteristic (AUC) has been proposed in [8] as a single figure of merit to measure the performance of energy detection. The main difference between ROC and AUC is that ROC depends on the decision threshold whereas the effect of the threshold on AUC has been averaged out.Performance analysis of energy detection in terms of AUC has received relatively little attention so far. In [9], the moment generating function (MGF) of the received SNR, with the help of contour integration, is used to derive exact closedform expressions for the average AUC of energy detection in Nakagami-m fading with integer m and in η − µ fading with integer µ. But it is difficult to extend this approach to more general cases because the residual calculation involved can be challenging. In [10], the average AUC is given in an exact closed-form expression, for which high-order derivatives of the MGF of the received SNR are essential. The number of high-order derivatives of the MGF is determined by the timebandwidth product of energy detection. In [11], the average

On the performance of energy detection for spectrum sensing in cognitive radio over Nakagami-lognormal composite channels

“…In [20], an interesting exponential integral representation is derived for the generalized Marcum Q-function, which is reproduced without proof in the following…”