Application of Imhof formula in the computation of the normalized detection threshold for the FFT summation detector

The FFT summation detector is a technique that has been widely used in civilian spectrum monitoring and military radio surveillance for detecting narrowband signals in noise. In practical implementations of the FFT summation detector, the detection threshold is set as the product of the channel noise power, which is often adaptively estimated from the FFT filter bank output, and the normalized detection threshold T_n, which is usually computed by solving a nonlinear equation. The golden section search algorithm is an ideal procedure for solving the nonlinear equation as it does not involve function derivative evaluations which can be a major source of numerical difficulties. However, for overlapped input data blocks, even the golden section search algorithm often breaks down, especially for relatively large values of P_fa. In this paper, an alternative technique based on the Imhof formula is investigated for the computation of T_n. It is shown that, for many cases of practical interest, the Imhof formula can be used to compute T_n reliably for overlapped input data blocks when the technique based on the numerical solution of an algebraic equation fails.