We consider the change detection problem where the pre-change observation vectors are purely noise and the post-change observation vectors are noise-corrupted compressive measurements of sparse signals with a common support, measured using a sensing matrix. In general, post-change probability density function (pdf) of the observations depends on parameters such as the support and variances of the sparse signal. When these parameters are unknown, we propose two approaches. In the first approach, we approximate the post-change pdf based on the known parameters such as mutual coherence of the sensing matrix and bounds on the signal variances. In the second approach, we parameterize the post-change pdf with an unknown parameter and try to estimate this parameter using two different methods, stochastic gradient descent and generalized likelihood ratio. In both these approaches, we employ the conventional cumulative sum (CUSUM) algorithm with various decision statistics such as the energy of the observations, correlation values with columns of the sensing matrix and the maximum value of such correlations. We analytically characterize the worst case detection delay and average run length to false alarm performance of pdf-approximation based approach. We also numerically study the performance and offer insights on the relevance of various decision statistics in different signal to noise ratio (SNR) regimes. We also address the problem of designing sensing matrices with small mutual coherence by using designs from quantum information theory. One such design using equi-angular lines has an additional structure which allows exact characterization of the post-change pdf of the correlation values, even when the support set of the sparse signal is unknown. We apply our detection algorithms with sensing matrix designed from equi-angular lines to a massive random access problem and show their superior performance over conventional Gold codes.

中文翻译：

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稀疏信号变化检测算法

我们考虑变化检测问题，其中变化前观察向量是纯噪声，变化后观察向量是具有公共支持的稀疏信号的噪声破坏压缩测量，使用传感矩阵测量。通常，观测值的变化后概率密度函数 (pdf) 取决于参数，例如稀疏信号的支持度和方差。当这些参数未知时，我们提出了两种方法。在第一种方法中，我们根据已知参数（例如传感矩阵的相互相干性和信号方差的界限）来近似更改后的 pdf。在第二种方法中，我们用一个未知参数参数化改变后的 pdf，并尝试使用两种不同的方法来估计这个参数，随机梯度下降和广义似然比。在这两种方法中，我们采用具有各种决策统计数据的传统累积和 (CUSUM) 算法，例如观测能量、与传感矩阵列的相关值以及此类相关的最大值。我们分析了基于 pdf 近似的方法的最坏情况检测延迟和平均运行长度对误报性能的表征。我们还对性能进行了数值研究，并提供了有关不同信噪比 (SNR) 机制中各种决策统计数据的相关性的见解。我们还通过使用量子信息理论的设计来解决设计相互相干性较小的传感矩阵的问题。使用等角线的这样一种设计具有附加结构，可以精确表征相关值的变化后 pdf，即使稀疏信号的支持集未知。我们将具有从等角线设计的传感矩阵的检测算法应用于大规模随机访问问题，并展示了其优于传统 Gold 代码的性能。