Computationally Efficient Signal Detection with Unknown Bandwidths

Signal detection in environments with unknown signal bandwidth and time intervals is a basic problem in adversarial and spectrum-sharing scenarios. This paper addresses the problem of detecting signals occupying unknown degrees of freedom from non-coherent power measurements where the signal is constrained to an interval in one dimension or hypercube in multiple dimensions. A Generalized Likelihood Ratio Test (GLRT) is derived, resulting in a straightforward metric involving normalized average signal energy on each candidate signal set. We present bounds on false alarm and missed detection probabilities, demonstrating their dependence on signal-to-noise ratios (SNR) and signal set sizes. To overcome the inherent computational complexity of exhaustive searches, we propose a computationally efficient binary search method, reducing the complexity from O(N2) to O(N) for one-dimensional cases. Simulations indicate that the method maintains performance near exhaustive searches and achieves asymptotic consistency, with interval-of-overlap converging to one under constant SNR as measurement size increases. The simulation studies also demonstrate superior performance and reduced complexity compared to contemporary neural network-based approaches, specifically outperforming custom-trained U-Net models in spectrum detection tasks.