Abstract In this paper, we propose a method to construct uni-modular tight frames (UMTFs), which are tight frames with the additional constraint that every entry of the matrix has the same magnitude. UMTFs are useful in many applications, since multiplication of a UMTF by a vector can be implemented in polar coordinates using very low computational cost. Since normalized UMTFs are unit norm tight frames (UNTFs), and since a UNTF is a minimizer of the frame potential, we propose an algorithm to find UMTFs by minimizing the frame potential. We show that minimizing the frame potential is equivalent to minimizing the total coherence when the frame is unimodular. We use the majorization-minimization approach to propose a low complexity, iterative, fast-converging algorithm for minimizing the frame potential. We also extend our algorithm to the cases where the phase angles of the sensing matrix are required to belong to a given finite set of feasible angles, and to the case where the signal being sampled is sparse in an arbitrary, possibly non-canonical basis. We illustrate the utility of our proposed construction in the context of sparse signal recovery. Partial DFT matrices, obtained by randomly selected rows from the full DFT matrix, are UMTFs. However, they perform poorly when dealing with signals that admit a sparse representation in the wavelet, Fourier and discrete cosine transform domains. In such scenarios, we illustrate the superior performance of our construction compared to the partial DFT, complex Gaussian and Bernoulli random matrices through simulations. The proposed algorithm offers the same performance as the partial DFT matrix, and outperforms the complex Gaussian and Bernoulli random matrices, when the signal is sparse in the canonical basis.

中文翻译：

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使用压缩感知的多数化-最小化构建单模紧框架

摘要 在本文中，我们提出了一种构造单模紧框架（UMTF）的方法，它是具有附加约束的紧框架，即矩阵的每个条目都具有相同的幅度。UMTF 在许多应用中都很有用，因为可以使用非常低的计算成本在极坐标中实现 UMTF 与向量的乘法。由于归一化的 UMTF 是单位范数紧帧 (UNTF)，并且由于 UNTF 是帧电位的最小化器，因此我们提出了一种通过最小化帧电位来找到 UMTF 的算法。我们表明，当框架是单模时，最小化框架电位等效于最小化总相干性。我们使用majorization-minimization方法来提出一种低复杂度、迭代、快速收敛的算法来最小化帧潜力。我们还将我们的算法扩展到需要传感矩阵的相位角属于给定的有限可行角度集的情况，以及被采样的信号在任意的、可能是非规范的基础上稀疏的情况。我们说明了我们提出的结构在稀疏信号恢复的背景下的效用。通过从完整 DFT 矩阵中随机选择的行获得的部分 DFT 矩阵是 UMTF。然而，当处理在小波、傅立叶和离散余弦变换域中允许稀疏表示的信号时，它们的性能很差。在这种情况下，我们通过模拟说明了与部分 DFT、复杂高斯和伯努利随机矩阵相比，我们的构造的优越性能。所提出的算法提供与部分 DFT 矩阵相同的性能，