Computationally Efficient Signal Detection with Unknown Bandwidths

Signal detection in environments with unknown signal bandwidth and time intervals is a fundamental 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 a hyper-cube in multiple dimensions. A GLRT is derived, resulting in a straightforward metric involving normalized average signal energy for each candidate signal set. We present bounds on false alarm and missed detection probabilities, demonstrating their dependence on 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(N^2) 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.