Detector type

Probability of detection

Probability of false alarm

Probability of miss-detection

Matched filtering

$Q\left(\sqrt{\frac{2E}{{\sigma }^2}},\sqrt{\frac{2{\lambda }^2}{E{\sigma }^2}}\right)$

$\exp \left(-\frac{{\lambda }^2}{E{\sigma }^2}\right)$

$1-Q\left(\sqrt{\frac{2E}{{\sigma }^2}},\sqrt{\frac{2{\lambda }^2}{E{\sigma }^2}}\right)$

Energy

$Q_{\frac{N}{2}}\left(\sqrt{\frac{\mu }{{\sigma }^2}},\sqrt{\frac{\lambda }{{\sigma }^2}}\right)$

$\frac{\Gamma \left(\frac{N}{2},\frac{\lambda }{2{\sigma }^2}\right)}{\Gamma \left(\frac{N}{2}\right)}$

$1-Q_{\frac{N}{2}}\left(\sqrt{\frac{\mu }{{\sigma }^2}},\sqrt{\frac{\lambda }{{\sigma }^2}}\right)$

cyclostationary

$Q\left(\frac{\sqrt{2\gamma }}{\sigma },\frac{\left(2N+1\right)\lambda }{{\sigma }^2}\right)$

$\exp \left(-\frac{\left(2N+1\right){\lambda }^2}{E{\sigma }^2}\right)$

$1-Q\left(\frac{\sqrt{2\gamma }}{\sigma },\frac{\left(2N+1\right)\lambda }{{\sigma }^2}\right)$