基于双极性混沌序列的托普利兹块状感知矩阵

Toeplitz-block sensing matrix based on bipolar chaotic sequence

感知矩阵的构造是压缩感知从理论走向工程应用的关键技术之一.由于托普利兹感知矩阵能够支持快速算法且与离散卷积运算相对应,因此具有重要的研究意义.然而常用的随机托普利兹感知矩阵因其元素的不确定性,使得它在实际应用中受到了诸多约束,例如内存消耗较高和不易于硬件加载.基于此,本文结合双极性混沌序列的内在确定性和托普利兹矩阵的优点,提出了基于双极性混沌序列的托普利兹块状感知矩阵.具体地,首先介绍了双极性混沌序列的产生并分析了它的统计特性.其次,构造了双极性托普利兹块状混沌感知矩阵,从相关性方面证明了新建的感知矩阵具有近乎最优的理论保证,并同时证实了它满足约束等距条件.最后,研究了该感知矩阵针对一维信号和图像的压缩测量效果,并与典型感知矩阵进行了对比.结果表明,提出的感知矩阵对这些测试信号具有更好的测量效果,而且它在内存开销、计算复杂度和硬件实现等方面均具有明显的优势.特别地,该感知矩阵非常适用于多输入-单输出线性时不变系统的压缩感知测量问题.

Compressed sensing is a revolutionary signal processing technique,which allows the signals of interest to be acquired at a sub-Nyquist rate,meanwhile still permitting the signals from highly incomplete measurements to be reconstructed perfectly.As is well known,the construction of sensing matrix is one of the key technologies to promote compressed sensing from theory to application.Because the Toeplitz sensing matrix can support fast algorithm and corresponds to discrete convolution operation,it has essential research significance.However,the conventional random Toeplitz sensing matrix,due to the uncertainty of its elements,is subject to many limitations in practical applications,such as high memory consumption and difficulty of hardware implementation.To avoid these limitations,we propose a bipolar Toeplitz block-based chaotic sensing matrix(Bi-TpCM) by combining the intrinsic advantages of Toeplitz matrix and bipolar chaotic sequence.Firstly,the generation of bipolar chaotic sequence is introduced and its statistical characteristics are analyzed,showing that the generated bipolar chaotic sequence is an independent and identically distributed Rademacher sequence,which makes it possible to construct the sensing matrix.Secondly,the proposed Bi-TpCM is constructed,and it is proved that Bi-TpCM has almost optimal theoretical guarantees in terms of the coherence,and also satisfies the restricted isometry condition.Finally,the measurement performances on one-dimensional signals and images by using the proposed Bi-TpCM are investigated and compared with those of its counterparts,including random matrix,random Toeplitz matrix,real-valued chaotic matrix,and chaotic circulant sensing matrix.The results show that Bi-TpCM not only has better performance for these testing signals,but also possesses considerable advantages in terms of the memory cost,computational complexity,and hardware realization.In particular,the proposed Bi-TpCM is extremely suitable for the compressed sensing measurement of linear timeinvariant(LTI) systems with multiple inputs and single output,such as the joint parameter and time-delay estimation for finite impulse response.Moreover,the construction framework of the proposed Bi-TpCM can be extended to different chaotic systems,such as Logistic or Cat chaotic systems,and it is also possible for the proposed Bi-TpCM to derive the Hankel blocks,additional stacking of blocks,partial circulant blocks sensing matrices.With these block-based sensing architectures,we can more easily implement compressed sensing for various compressed measurement problems of LTI systems.