Asymptotic detection performance of discrete power and higher-order spectra estimates

Several detection statistics are compared in the frequency domain based on the asymptotic probability of detection (APD) criterion. They include second-order, fourth-order, normalized fourth-order, and kurtosis estimates. The results show that for randomly occurring signals which can be characterized as non-Gaussian, the fourth-order, normalized fourth-order, and kurtosis estimates can have higher asymptotic probability of detection levels compared with second-order estimates. But only for the normalized fourth-order and kurtosis estimates do the results seem significant. Moreover, if a second-order estimate of the noise is available to normalize a fourth-order estimate of signal and noise, the resultant normalized fourth-order estimate has higher asymptotic probability of detection levels even for Gaussian signals. This result holds only when there is a significant positive covariance between the numerator and the normalizing noise sample in the denominator. On the other hand, if an independent noise sample is used to normalize a second-order or fourth-order estimate, the overall performance based on the asymptotic probability of detection will be degraded compared with the unnormalized second-order or fourth-order estimates, respectively.