A comparison of the normalized detection threshold for the overlapped and non-overlapped FFT summation detectors

A specified probability of false alarm can be obtained for the FFT summation detector by setting the detection threshold as the product of the channel noise floor and the normalized detection threshold T_n. The channel noise floor is often adaptively estimated from the FFT filter bank output and T_n is mostly computed by solving a non-linear equation, which can be reliably accomplished when, L, the number of input data blocks, and, N, the number of FFT bins assigned to a channel, are both relatively small. However,when L or N is large, numerical procedures often break down, yielding invalid results. This paper provides new theoretical results concerning the computation of T_n. Using two inequalities involving positively weighted sums of central χ² random variables, it is shown that the T_n for overlapped input data with an over- lap ratio not exceeding 50% is larger than the T_n for non-overlapped input data, assuming that the same number of input data blocks is processed by the FFT. This inequality can be used as a first level of sanity check when verifying results obtained by procedures such as the Newton-Ralphson and the golden section search algorithms.