Likelihood ratio tests for PSK modulation classification in unknown noise environment

We propose a computationally non-expensive and powerful modulation classifier (MC) in order to determine to the number of constellation points of a phase shift keying (PSK) signal in additive white Gaussian noise (AWGN). We consider the complex amplitude of the signal as well as the information sequence as the unknown parameters. In contrast of the existing literature, we assume that the variance of the noise is also unknown. A new MC algorithm is introduced based on the generalized likelihood ratio test (GLRT) assuming unknown deterministic parameters. We show that the GLRT solution does not outperform a random classifier, because the constellations under study are nested. To overcome this problem, we propose a classifier which uses the mixture of the concepts of GLRT and the average likelihood ratio test (ALRT). The ML estimates of the signal amplitude and the noise variance are employed and assuming the data as a sequence of independent uniformly distributed symbols, the likelihood function is averaged over the data symbols. This classifier is based on the fact that all the possible data symbols are equally likely to be transmitted whereas the GLRT based criterion does not use any indication of how probable a particular data symbol is. The implementation of these two classifiers requires either a search or an averaging over the possible information symbols, i.e., their computational complexities grow exponentially with the sequence length. Thus, we propose suboptimal implementations with almost no performance loss. The computational complexity of these implementations increase linearly with the sequence length. As we expect, simulation results confirm that the classification accuracy increases when the number of observed symbols or the signal-to-noise ratio (SNR) increases