The Okamoto lower bound for the normalized detection threshold of the FFT filter bank-based summation detector

The FFT filter bank-based summation detector is well suited for detecting narrowband signals embedded in wideband noise and has numerous applications in areas such as civilian spectrum monitoring, electronic warfare, and spectrum sensing in cognitive radio networks. In practical implementations, the detection threshold is obtained as the product of the channel noise power, which is adaptively estimated from the FFT filter bank output, and the theoretical normalized detection threshold T_n corresponding to the desired probability of false alarm. Since the probability of false alarm P_fa is a strictly decreasing function of T_n, T_n is usually computed by finding the root of a non-linear equation using the Newton-Ralphson or golden section search algorithm. The initialization of the numerical procedures, however, hinges on good a priori knowledge of lower and upper bounds for T_n. In this article, a new and easily computable lower bound for T_n, the Okamoto lower bound, is derived. This lower bound is larger than the lower bounds derived earlier by the authors and hence should be preferred when initializing the numerical procedures in the computation of T_n.