Abstract In this paper, we investigate the spark of a family of transforms that are frequently used in signal processing applications such as Multicarrier Communication systems: we consider the even Discrete Cosine Transform matrices (DCTs). On one hand, we provide several theorems that prove that the submatrices obtained from its first rows present maximum spark; on the other hand, we give counterexamples of other sets of rows that do not present this good property. Besides the mathematical interest of our contributions, we also show that our results have important novel applications in compressed sensing problems such as sparse signal reconstruction and compressed channel estimation. We also demonstrate that a channel filter can be perfectly estimated by means of a small amount of its DCT coefficients, which can be furthermore arbitrarily selected. Thus, random compressive sampling schemes are valid for solving channel estimation problems when using even DCTs. Finally, numerical simulations show the good performance of the DCT transforms for practical sparse signal recovery in both noisy and noise-free scenarios.

中文翻译：

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偶数离散余弦变换的完整火花

摘要 在本文中，我们研究了在信号处理应用（如多载波通信系统）中经常使用的一系列变换的火花：我们考虑偶数离散余弦变换矩阵 (DCT)。一方面，我们提供了几个定理来证明从其第一行获得的子矩阵存在最大火花；另一方面，我们给出了不呈现这种良好特性的其他行集的反例。除了我们贡献的数学兴趣之外，我们还表明我们的结果在压缩感知问题中具有重要的新应用，例如稀疏信号重建和压缩信道估计。我们还证明了可以通过少量的 DCT 系数完美地估计信道滤波器，还可以任意选择。因此，当使用偶数 DCT 时，随机压缩采样方案对于解决信道估计问题是有效的。最后，数值模拟表明 DCT 变换在有噪声和无噪声情况下用于实际稀疏信号恢复的良好性能。