In compressive sensing theory (CS), the restricted isometry property (RIP) is commonly used for the measurement matrix to guarantee the reliable recovery of sparse signals from linear measurements. Although many works have indicated that random matrices with excellent recovery performance satisfy the RIP with high probability, Toeplitz-structured matrices arise naturally in real scenarios, such as applications of linear time-invariant systems. Thus, the corresponding measurement matrix can be modeled as a Toeplitz (partial) structured matrix instead of a completely random matrix. The structure characteristics introduce coherence and cause the performance degradation of the measurement matrix. To enhance the recovery performance of the Toeplitz structured measurement matrix in multichannel convolution source separation, an efficient construction of measurement matrix is presented, referred to as sparse random block-banded Toeplitz matrix (SRBT). The sparse signal is pre-randomized by locally scrambling its sample locations. Then, the signal is subsampled using the sparse random banded matrix. Finally, the mixing measurements are obtained. Based on the analysis of eigenvalues, the theoretical results indicate that the SRBT matrix satisfies the RIP with high probability. Simulation results show that the SRBT matrix almost matches the recovery performance of random matrices. Compared with the existing banded block Toeplitz matrix, SRBT significantly improves the probability of successful recovery. Additionally, SRBT has the advantages of low storage requirements and fast computation in reconstruction.

To efficiently collect sensor readings in cluster-based wireless sensor networks, we propose a structural compressed network coding (SCNC) scheme that jointly considers structural compressed sensing (SCS) and network coding (NC). The proposed scheme exploits the structural compressibility of sensor readings for data compression and reconstruction. Random linear network coding (RLNC) is used to re-project the measurements and thus enhance network reliability. Furthermore, we calculate the energy consumption of intra- and inter-cluster transmission and analyze the effect of the cluster size on the total transmission energy consumption. To that end, we introduce an iterative reweighed sparsity recovery algorithm to address the all-or-nothing effect of RLNC and decrease the recovery error. Experiments show that the SCNC scheme can decrease the number of measurements required for decoding and improve the network's robustness, particularly when the loss rate is high. Moreover, the proposed recovery algorithm has better reconstruction performance than several other state-of-the-art recovery algorithms.