Shnidman’s Equation

It is notoriously difficult to evaluate receiver operating characteristic curves exactly. This is due to the complexity of these expressions in terms of incomplete Gamma, incomplete Toronto, confluent hypergeometric functions or other special functions. Albersheim fit a set of parametric curves to the exact expression of an ROC curve for a receive chain with a filter, linear detector and independent sample integration. His fit applied to within 0.2 dB of the exact curves for a wide range of detection and false alarm probabilities, non-coherent independent sample averaging and signal to noise ratios. However, his fit modeled only non-fluctuating statistical signals.

Shnidman provides an empirical fit applicable to non-fluctuating and Swerling 1 – 4 fluctuating signals and square law detection. Square law detection is usually implemented given today’s floating point signal processing.  The trade-off is that the equations are accurate to only 0.5 dB over a smaller range (1 – 100 versus 1 – 8096) of independent samples. The expressions are valid for a wider range of detection probabilities (0.99 versus 0.9 at the upper end of the range) and a wider range of false alarm probabilities (10^-9 versus 10^-7 at the lower end of the range). Richards provides a complete summary with algorithms for both Albersheim’s and Shnidman’s curves. Either is easy to program into an Excel spreadsheet using that program’s Visual Basic for Applications or into Matlab.

One can use this receiver operating characteristic curve to relate correct and false classification probabilities through detection and false alert probabilities (using auxiliary expressions) to detector output signal to noise ratio and, via a range equation (or propagation curve), to a detectable range.