为了通过一组线性测量值重建原始信号, 并以低于奈奎斯特采样频率的速率采样, 开展本研究。对于线性调频信号, 分数阶傅里叶变换是在线性调频信号基上稀疏分解信号, 因此采用离散分数阶傅里叶变换的变换矩阵构造正交基字典矩阵, 通过调整调制参数(调制法)寻找最佳稀疏基, 可以较好地稀疏表示线性调频信号。同理, 采用调制法寻找正切形非线性调频信号的最佳稀疏基, 对其进行稀疏表示与重构的压缩感知研究。仿真结果表明: 该方法能够找到最佳稀疏基, 实现正切形非线性调频信号的稀疏表示与信号重构, 信号波形恢复效果较好。

The study is intended to reconstruct the original signal through a linear measurement, and sampling at a speed lower than the Nyquist sampling frequency. Fractional Fourier Transform(FrFT) is applied to sparse represent signal on the Linear Frequency Modulation(LFM) basis to the LFM signal, so Ψ is substituted to the transforming matrix of Digital Fractional Fourier Transform(DFrFT), and the optimal sparse basis is gained by modulation method, then the sparse representation and reconstruction of signal is researched. Similarly, modulation method is utilized to find the optimal sparse basis of tangent-shape Nonlinear Frequency Modulation(NLFM) signals and complete the compressive sensing research of sparse representation and reconstruction. Simulation results show that the optimal sparse basis can be found with this method; and the sparse representation and reconstruction of tangent-shape NLFM signals can be accomplished with good recovery results.