Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Designs and System Identification

Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Designs and System Identification
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发表时间：
2008-06
期刊：
ArXiv
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通讯作者：
Venkatesh Saligrama
Venkatesh Saligrama
中科院分区：
其他
文献类型：
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作者：
Venkatesh Saligrama

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在本文中，我们提出了一个新的家庭的离散序列具有“随机”一致衰减的自相关特性。这类新的无限长序列是用无理数构造的高阶线性调频脉冲。利用连续分数和丢番图近似理论的结果，我们表明，这样形成的序列类具有的属性，最坏情况下的自相关系数为每个有限长度的序列以多项式的速度衰减。这些序列也显示多普勒免疫。我们还表明，Toeplitz矩阵从这样的序列形成满足限制等距属性（RIP），最近在压缩感知应用中发挥了核心作用的概念。压缩感测传统上处理具有任意分量的感测矩阵。然而，这种任意的传感矩阵是不适合线性系统的识别，必须采用Toeplitz结构的传感矩阵。线性系统辨识在诸如多径无线系统的信道估计以及控制系统应用等各种各样的应用中起着核心作用。Toeplitz矩阵由于其滤波结构也是期望的，这允许快速实现以及降低的存储要求。
In this paper we present a new family of discrete sequences having "random like" uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from such sequences satisfy restricted-isometry-property (RIP), a concept that has played a central role recently in Compressed Sensing applications. Compressed sensing has conventionally dealt with sensing matrices with arbitrary components. Nevertheless, such arbitrary sensing matrices are not appropriate for linear system identification and one must employ Toeplitz structured sensing matrices. Linear system identification plays a central role in a wide variety of applications such as channel estimation for multipath wireless systems as well as control system applications. Toeplitz matrices are also desirable on account of their filtering structure, which allows for fast implementation together with reduced storage requirements.