Periodic signals composed of periodic mixtures admit sparse representations in nested periodic dictionaries (NPDs). Therefore, their underlying hidden periods can be estimated by recovering the exact support of said representations. In this paper, support recovery guarantees of such signals are derived both in noise-free and noisy settings. While exact recovery conditions have long been studied in the theory of compressive sensing, existing conditions fall short of yielding meaningful achievability regions in the context of periodic signals with sparse representations in NPDs, in part since existing bounds do not capture structures intrinsic to these dictionaries. We leverage known properties of NPDs to derive several conditions for exact sparse recovery of periodic mixtures in the noise-free setting. These conditions rest on newly introduced notions of nested periodic coherence and restricted coherence, which can be efficiently computed and verified. In the presence of noise, we obtain improved conditions for recovering the exact support set of the sparse representation of the periodic mixture via orthogonal matching pursuit based on the introduced notions of coherence. The theoretical findings are corroborated using numerical experiments for different families of NPDs. Our results show significant improvement over generic recovery bounds as the conditions hold over a larger range of sparsity levels.
翻译:由定期混合物组成的周期性信号在嵌入周期性词典(NPDs)中反映的周期性信号很少。因此,可以通过恢复准确的上述表述来估计其潜在的隐藏期。在本文件中,支持此类信号的恢复保障来源于无噪音和吵闹的环境。虽然压缩感学理论早已研究过确切的回收条件,但现有条件不足以在NPDs中以稀少的定期信号产生有意义的可实现的区域,部分原因是现有的界限没有捕捉这些词典所固有的结构。我们利用NPDs已知的特性为无噪音环境中的周期性混合物的准确回收获得若干条件。这些条件以新引入的嵌入周期一致性和限制一致性概念为基础,这些概念可以高效地计算和核实。在噪音存在的情况下,我们根据引入的一致性概念,改善了恢复周期性混合物的稀少代表的确切支持条件。理论结论得到证实,对NPDs的不同家庭进行了数字实验。我们的结果显示,由于条件在更大范围内,普遍恢复的范围有了显著改善。