关于噪声信号恢复的一些理论与实际结果
Some theoretical and practical results on noisy signals recovery
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中文总结 AI 辅助
本文针对无线通信中噪声稀疏信号恢复问题，在字典满足RIP型条件下为OMP和WOMP算法建立Lebesgue型不等式，并应用于过采样傅里叶字典的稀疏信道恢复，证明了OMP算法能恢复大系数元素并估计精度。
中文摘要 AI 辅助
我们讨论了一些理论结果及其在无线通信中特定实际问题中的应用。我们假设已知信号在有限个点处的噪声版本，该信号相对于给定元素系统（字典）是稀疏的，并且我们希望近似恢复它。这个噪声信号恢复问题与为相应算法建立Lebesgue型不等式的问题密切相关，这促使我们去证明此类不等式。在字典的某些条件下（RIP型条件、相干性条件），我们获得了OMP及其变体WOMP算法的不同类型的Lebesgue型不等式。我们新理论结果的最重要特征是假设字典具有RIP型性质，而不是先前结果中使用的Riesz基假设。我们的方法允许我们处理冗余（过完备）系统，这在应用中很重要。我们考虑了无线通信OFDM设置中出现的高相干字典的稀疏恢复设置。我们的主要例子是过采样傅里叶字典和稀疏信道频率响应的恢复。我们开发了一种解决此类问题的通用方法，并证明了在特定条件下OMP算法能恢复所有具有大系数的字典元素，并根据信噪比估计其精度（以NMSE度量）。
英文摘要
We discuss some theoretical results and their applications to specific practical problems from wireless communications. We assume that we know the noisy version of the signal, which is sparse with respect to a given system of elements (dictionary), at a finite number of points and we want to approximately recover it. This problem of recovery of a noisy signal is closely related to the problem of establishing the Lebesgue-type inequalities for the corresponding algorithms and it motivates us to prove such inequalities. Under certain conditions on a dictionary (RIP-type condition, coherence condition) we obtain different kinds of the Lebesgue-type inequalities for the OMP and its version WOMP algorithms. The most important feature of our new theoretical result is the assumption that the dictionary has the RIP-type property instead of the assumption that it is the Riesz basis, which was used in the previous results. Our approach allows us to treat redundant (overcomplete) systems, which is important in applications. We consider the setting of sparse recovery for highly-coherent dictionaries that appear in OFDM setting for wireless communication. Our main example is the oversampled Fourier dictionary and the recovery of the frequency response of a sparse channel. We develop a general approach to such problems and we prove that under certain conditions the OMP algorithm recovers all dictionary elements with large coefficients, and estimate its accuracy (in NMSE metric) in terms of signal-to-noise ratio.
发表机构
- Moscow RTT Laboratory（莫斯科RTT实验室）
- Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院马丘克数值数学研究所）
- Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
- Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）
- Moscow Center of Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）
机构由 AI 辅助整理，请以论文原文为准。
