Sub-Nyquist Sampling of Pulse Streams Based on the Real Part of Fourier Coefficients
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摘要: 有限新息率(FRI)采样理论可以远低于信号Nyquist频率的采样速率实现对脉冲流信号的欠采样。经典的FRI重构算法大多基于傅里叶系数进行运算，其中存在大量的对复数矩阵的奇异值分解，降低了算法的执行效率。针对该问题，该文提出基于傅里叶系数实部的脉冲流信号FRI采样及重构方法。首先利用离散余弦变换从脉冲流信号的低速采样值中获取其傅里叶系数实部信息，并在重构算法中使用实部的Toeplitz矩阵以提高奇异值分解(SVD)的效率；其次，为了提升经典的零化滤波器算法的鲁棒性，该文从傅里叶系数实部协方差矩阵的旋转不变特性以及零空间特性出发，提出基于离散余弦变换的协方差矩阵分解算法以及基于离散余弦变换的零空间搜索算法来估计脉冲流信号的特征参数，并针对出现的共轭根问题，提出基于交替方向乘子法的去共轭算法。仿真结果表明：在信号新息率较高的情况下，使用傅里叶系数实部信息会极大提高算法的执行效率，同时保证参数估计的准确性。
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关键词:
- 欠Nyquist采样 /
- 有限新息率 /
- 脉冲流信号 /
- 离散余弦变换
Abstract: The Finite Rate of Innovation (FRI) theory can realize the sub-Nyquist sampling of pulse streams signal by a sampling rate much lower than its Nyquist frequency. Most classical FRI reconstruction algorithms operate on the basis of Fourier coefficients, and there is a lot of singular value decomposition of complex matrices, which reduces the efficiency of the algorithm. To solve this problem, an FRI sampling and reconstruction method based on the real part of Fourier coefficients is proposed in this paper. Firstly, the discrete cosine transform is used to obtain the real part of Fourier coefficients information from the low-speed sampling value of the pulse flow signal, and the Toeplitz matrix of the real part is used in the reconstruction algorithm to improve the efficiency of the Singular Value Decomposition (SVD). Secondly, in order to improve the robustness of the classical annihilating filter algorithm, a covariance matrix decomposition algorithm and a null space searching algorithm are proposed from the rotation invariant feature and the null space property of the real covariance matrix. The two methods are based on the discrete cosine transform to estimate characteristic parameters of the pulse stream signal. For the conjugate root problem, a new method of deconjugation based on the alternating direction multiplier is proposed in this paper. The simulation results show that using the real part information of Fourier coefficients can greatly improve the efficiency of the algorithm and ensure the accuracy of parameter estimation when the rate of innovation of the signal is high. -
算法1 时间延迟参数去共轭算法伪代码 输入：观测矩阵${\boldsymbol{\varPhi}}$，采样值向量${\boldsymbol{c}}$，参数$ \lambda $,$ \rho $；脉冲个数$ K $；算法
的迭代次数$ {I_{\max }} $；输出：$ K $稀疏向量${\boldsymbol{a}}$，其$ K $个最大元素的位置集合$ \Im $。 (1) 初始化$ {a^0} = c $，$ {u^0} = 0 $且$ {z^0} = 0 $； (2) While $ k < {I_{\max }} $ do, ${a^{k + 1} } = {({ {\boldsymbol{\varPhi} } ^{\text{T} } } \cdot {\boldsymbol{\varPhi} } + \rho {\boldsymbol{I} })^{ - 1} }({ {\boldsymbol{\varPhi} } ^{\text{T} } } \cdot {\boldsymbol{c}} + \rho ({z^k} - {u^k}))$, $ {z^{k + 1}} = {S_{\lambda /\rho }}({a^{k + 1}} + {u^k}) $, $ {u^{k + 1}} = {a^{k + 1}} - {z^{k + 1}} + {u^k} $, $ k = k + 1 $, End While; (3) 返回${\boldsymbol{a}}$以及其最大的$ K $个元素构成的集合$ \Im $。 
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算法2 CDoD算法伪代码 输入：傅里叶系数实部向量${{\boldsymbol{c}}^{ {\text{DCT} } } }$，脉冲参数$ K $，周期$ \tau $； 输出：Dirac脉冲幅度参数$ \{ {a_k}\} _{k = 1}^K $，时间延迟参数$ \{ {t_k}\} _{k = 1}^K $。 (1) 根据式(25)及定理1构建矩阵${ {\boldsymbol{R} }_{\boldsymbol{c} } }$以及$ {\boldsymbol{E}} $, $ {{\boldsymbol{E}}_1} $和$ {{\boldsymbol{E}}_2} $； (2) 根据定理1计算矩阵$ {\boldsymbol{W}}{\text{ = }}{\boldsymbol{E}}_1^\dagger \cdot {{\boldsymbol{E}}_2} $的$ 2K $个特征值； (3) 根据式(16)求解含有共轭的时间延迟参数$ \{ {\tilde t_r}\} _{r = 1}^{2K} $； (4) 根据去共轭算法1以及最小二乘算法计算原始信号的时间延
迟参数$ \{ {t_k}\} _{k = 1}^K $以及幅度参数$ \{ {a_k}\} _{k = 1}^K $。
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算法3 NSoD方法伪代码 输入：向量${{\boldsymbol{c}}^{ {\text{DCT} } } }$，脉冲参数$ K $，周期$ \tau $，矩阵束参数$ M $； 输出：脉冲幅度参数$ \{ {a_k}\} _{k = 1}^K $，时间延迟参数$ \{ {t_k}\} _{k = 1}^K $。 (1) 利用$ {{\boldsymbol{c}}^{{\text{DCT}}}} $构建矩阵${\boldsymbol{C}}$，并通过奇异值分解获取其零空间向量
$ \{ {v_i}|1 \le i \le M + 1 - 2K\} $；(2) 根据定理2构建向量$ {\boldsymbol{e}}(\omega ) $,通过搜索式(35)的伪谱峰值，获得
信号的时间延迟参数；(3) 根据式(16)求解含有共轭的时间延迟参数$ \{ {\tilde t_r}\} _{r = 1}^{2K} $； (4) 根据算法1计算信号的参数$ \{ {t_k}\} _{k = 1}^K $以及$ \{ {a_k}\} _{k = 1}^K $。
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