A New Upper Bound for the Normalized Detection Threshold of the FFT-Based Summation Detector

The FFT summation detector, a detection scheme extensively used in applications such as civilian spectrum monitoring and military radio surveillance, is also applicable to cognitive radio spectrum sensing applications. The practical implementation of the FFT summation detector depends on the reliable numerical computation of the normalized detection threshold, T
_n
,. Although this problem has been the subject of considerable study, the computation of T
_n
still presents difficulties. Existing approaches depend on numerical procedures such as the Newton-Raphson and the golden section search algorithms, which often fail to converge when the number of input data blocks, L, or the number of FFT bins, N, used for channel power estimation is large. To circumvent such numerical difficulties, easily computable lower and upper bounds for Tom, are of great interest as they are essential for verifying the accuracy of approximations to T
_n
when the true value of T
_n
is not available. Although several lower bounds for T
_n
have been proposed, few good upper bounds for T
_n
have been derived. This paper proposes a new upper bound for T
_n
, which is easily computed for larger values of the probability of false alarm, P
_fa
, using the Imhof integral formula.