Editing Receiver operating characteristic (section)

=== Radar in detail ===
As mentioned ROC curves are critical to [[radar]] operation and theory. The signals received at a receiver station, as reflected by a target, are often of very low energy, in comparison to the [[noise floor]]. The ratio of [[Signal-to-noise ratio|signal to noise]] is an important metric when determining if a target will be detected. This signal to noise ratio is directly correlated to the receiver operating characteristics of the whole radar system, which is used to quantify the ability of a radar system.

Consider the development of a radar system. A specification for the abilities of the system may be provided in terms of probability of detect, <math>P_{D}</math>, with a certain tolerance for false alarms, <math>P_{FA}</math>. A simplified approximation of the required signal to noise ratio at the receiver station can be calculated by solving<ref>{{Citation |title=Fundamentals of Radar |date=2008-01-29 |url=http://dx.doi.org/10.1002/9780470377765.ch4 |work=Digital Signal Processing Techniques and Applications in Radar Image Processing |pages=93–115 |access-date=2023-05-20 |place=Hoboken, NJ, USA |publisher=John Wiley & Sons, Inc.|doi=10.1002/9780470377765.ch4 |isbn=9780470377765 }}</ref>

: <math>P_{D}=\frac{1}{2}\operatorname{erfc}\left(\operatorname{erfc}^{-1}\left(2P_{FA}\right)-\sqrt{\mathcal{X}}\right)</math>

for the signal to noise ratio <math>\mathcal{X}</math>. Here, <math>\mathcal{X}</math> is not in [[decibel]]s, as is common in many radar applications. Conversion to decibels is through <math>\mathcal{X}_{dB}=10\log_{10}\mathcal{X}</math>. From this figure, the common entries in the radar range equation (with noise factors) may be solved, to estimate the required [[effective radiated power]].