On Novel RIP of Windowed Compressed Spectrum Sensing

Compressed spectrum sensing (CSS) offers great advantages in spectral analysis through sub-Nyquist sampling. However, conventional CSS approaches have not sufficiently addressed the effect of spectral leakage (SL) on sensing performance, a problem that fundamentally alters signal sparsity. Although WFs (WFs) are widely employed in traditional spectral analysis to mitigate SL, their systematic application within CSS lacks comprehensive analysis, and they have not been theoretically investigated. This study develops a rigorous theoretical framework for analyzing windowed measurement matrices in CSS by introducing two novel metrics: the Edge Zeroing Coefficient (EZC), which quantifies the boundary behavior of WFs and directly correlates with SL suppression, and the Window Scaling Coefficient (WSC), which characterizes how WFs affect the Restricted Isometry Property (RIP) of measurement matrices. By integrating these metrics, we formulate a dual-effects framework that quantitatively characterizes the inherent trade-offs between the accuracy of spectral analysis and the fidelity of signal reconstruction. We extend conventional block-sparsity concepts by explicitly incorporating WF characteristics, providing the first mathematical framework that directly links window properties to resulting signal sparsity patterns. By leveraging subspace counting theory, we establish sampling bounds for windowed CSS under various conditions, including block-sparse signal structures, ultra-sparse frequency distributions, and noisy environments. Our findings demonstrate that although WFs effectively reduce SL, excessively low EZC and WSC values can degrade RIP quality, potentially causing numerical instability during signal reconstruction. This research provides theoretical foundations and practical guidelines for selecting appropriate WFs based on application-specific requirements, enabling more accurate and efficient spectrum sensing across diverse domains.