Computationally Efficient Signal Detection with Unknown Bandwidths

TL;DR

This work tackles signal detection when both the bandwidth and time interval are unknown, formulating a GLRT-based detector that reduces to a normalized average energy metric over candidate signal sets. To address the prohibitive search over all possible intervals, it introduces a dyadic, binary-search approach that achieves O(N) complexity in one dimension while preserving near-exhaustive detection performance. The authors provide false-alarm and missed-detection bounds, derive asymptotic consistency guaranteeing interval recovery as data grows, and demonstrate favorable comparisons against U-Net baselines with substantially lower computational cost. The results indicate practical, scalable spectrum sensing performance for adversarial and spectrum-sharing environments, with clear pathways for extension to multiple concurrent signals and more realistic signal models.

Abstract

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.

Figures (11)

• Figure 1: Detection problem examples: Top: $d=1$ example for finding an interval $[a,b)$ of signal energy in $N$ frequency bins; Bottom: $d=2$ example of finding a bounding box in time-frequency.
• Figure 2: Threshold values $u_{\ell}$ from equation \ref{['eq:uisolve']} as a function of signal interval size, showing higher thresholds for shorter intervals due to increased noise power variance.
• Figure 3: The diagram illustrates the computationally efficient maximum likelihood method using binary search over possible intervals. The figure demonstrates the method for $N=16$, where $N$ represents the number of power measurements. The blue and purple lines respectively depict the trajectory of the detected start and end of the optimal interval across different stages. The signal, detected within the interval $X_{[7,9]}$, is highlighted by green circles.
• Figure 4: The architecture of the baseline U-Net used for signal detection in this paper. This architecture is designed for one-dimensional data. However, the same principles and methods can be extended to handle higher-dimensional data. For each feature map, dimensions are annotated at the bottom left, while the number of channels is indicated at the top. As illustrated in the figure, the process initiates with the input signal, upon which a convolutional block comprising two stages of $1\times3$ convolution, Batch Normalization, and ReLU activation is applied. This is followed by a Max Pooling operation, which is iteratively repeated until achieving a $1\times 1$ feature map with 2048 channels. The decoding phase commences with up-convolutions, wherein feature maps are concatenated with the corresponding encoder maps, applying the same convolutional block utilized during the encoding stage. Finally, a Conv$1\times1$ layer is employed to generate the output segmentation map.
• Figure 5: The detection IOU error rate of the exhaustive and binary search maximum likelihood estimations and U-Net for 1D and 2D signals with SNR ranging from -3 to 20 dB. Each plot represents the performance for a specific fixed signal size. As expected, increasing the SNR improves the performance of both methods. The figures indicate that the performance of the two methods is nearly identical for larger signal sizes. For smaller signal sizes, the binary search method still performs similarly to the exhaustive ML at low and high SNR, with only a slight, non-significant performance drop observed at intermediate SNR. In most cases, the binary search ML method surpasses the U-Net in the IOU metric while maintaining significantly lower computational complexity. The performance is generally better for 2D data compared to 1D data.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 5
• proof