تجزیه و تحلیل عملکرد از نمونه برداری فشرده شده سگمنتال جزئی

Recently, a segmented AIC (S-AIC) structure that measures the analog signal by K parallel branches of mixers and integrators (BMIs) was proposed by Taheri and Vorobyov (2011). Each branch is characterized by a random sampling waveform and implements integration in several continuous and non-overlapping time segments. By permuting the subsamples collected by each segment at different BMIs, more than K samples can be generated. To reduce the complexity of the S-AIC, in this paper we propose a partial segmented AIC (PS-AIC) structure, where K branches are divided into J groups and each group, acting as an independent S-AIC, only works within a partial period that is non-overlapping in time. Our structure is inspired by the recent validation that block diagonal matrices satisfy the restricted isometry property (RIP). Using this fact, we prove that the equivalent measurement matrix of the PS-AIC satisfies the RIP when the number of samples exceeds a certain threshold. Furthermore, the recovery performance of the proposed scheme is developed, where the analytical results show its performance gain when compared with the conventional AIC. Simulations verify the effectiveness of the PS-AIC and the validity of our theoretical results.

During recent years, a new theory of compressed sampling (CS) has emerged, which exploits the sparse prior to recover the signals from fewer samples than the number stated in the Nyquist theorem [1], [2], [3] and [4]. According to CS, these few samples are obtained by multiplying signals with a sampling matrix, which should satisfy the restricted isometry property (RIP) with overwhelming probability to guarantee reliable reconstruction [1] and [2]. To obtain compressed samples directly from an analog signal, analog-to-information conversion (AIC) has been proposed as a practical scheme [5], [6] and [7]. The AIC structure consists of K parallel branches of mixers and integrators (BMIs), where each BMI multiplies the signal with a sampling waveform and the result is integrated over the sampling period T. In conventional AIC, the number of the collected samples equals to the number of BMIs. To get more samples, Taheri and Vorobyov [8] proposed the segmented AIC (S-AIC) scheme. The integration period T is divided into M equal segments such that K BMIs generate KM subsamples. By using the permuted results of these subsamples, this scheme can collect at most K2 samples. The authors showed that the equivalent measurement matrix (EMM) of this scheme satisfies the RIP with overwhelming probability if the original matrix of BMI sampling waveforms satisfies it. Note that the EMM of the S-AIC is in fact a dense matrix, and the resulted AIC suffers from a higher hardware complexity when compared with the conventional AIC. In this paper, we introduce a partial segmented AIC (PS-AIC) scheme, where each BMI only works within a partial time period and thus enjoys a reduced complexity. This scheme has been simply described in our recent short paper [9], where some preliminary results have been presented. PS-AIC is inspired by the favorable RIP feature of the block diagonal matrix (BDM). In this structure, K BMIs are divided into J groups that work in non-overlapping integration time (J<KJ<K), that is, the integration period in each BMI is reduced to T/JT/J instead of T in S-AIC. Each group implements the same operations as those in S-AIC, and KM/JKM/J subsamples are collected from J groups. By using these subsamples in their original form and their permuted form, a larger number of samples (at most K2/JK2/J samples) than the number of BMIs can be generated. We give the detailed proof for the RIP of the EMM of the PS-AIC, 1 which shows that the EMM satisfies the RIP when the number of samples is larger than a threshold. In addition, when the sparsity basis of the signal is the Fourier basis and the number of measurements satisfies a specific condition, EMM of the PS-AIC has the identical RIP condition as that of the S-AIC. Furthermore, we derive the mean squared error (MSE) of the PS-AIC scheme when empirical risk minimization method is used for recovery. The motivation for using this method is to make fair comparison with [8], which also develops MSE results when empirical risk minimization is adopted. Our analytical results show that PS-AIC enjoys better performance than the conventional AIC with K BMIs. Actually, PS-AIC can be implemented by using only K/JK/J BMIs in the conventional AIC. Therefore, the proposed PS-AIC is a promising candidate for CS. The rest of this paper is organized as follows. Section 2 presents the background of CS and S-AIC. The structure of the PS-AIC is described in Section 3, where we prove that the corresponding EMM satisfies the RIP. Section 4 provides the recovery performance analysis of PS-AIC. In Section 5, some simulation results are shown. Finally, Section 6 concludes this paper. Throughout the paper, we denote vectors and matrices by boldface lowercase letters and boldface uppercase letters, respectively, e.g., xx and AA. The l th element of xx is denoted as x l. The Euclidean norm of xx is View the MathML source∥x∥2=xHx, ∥x∥1=∑l|xl|∥x∥1=∑l|xl| is the l 1-norm, ∥x∥∞=maxl|xl|∥x∥∞=maxl|xl| is the l∞l∞-norm, and ∥x∥0∥x∥0 denotes the number of nonzero entries in xx. For a matrix AA, View the MathML source(A)ab denotes a submatrix containing the a th row to the b th row of AA, and Ai,jAi,j (or (A)i,j(A)i,j) means the entry in AA. ATAT stands for the transpose of AA. Furthermore, View the MathML sourceA=diag(A^1,A^2,…,A^J) is a block diagonal matrix with the j th block in the diagonal being View the MathML sourceA^j. We use Pr(ξ)Pr(ξ) to denote the probability of event ξξ.

In this paper, we propose a partial segmented AIC to reduce the implementation complexity of the segmented AIC in [8]. K BMIs are divided into J groups, and each group works in a shorter period, i.e., T/JT/J, when compared with the period T in segmented AIC. The subsamples collected on K BMIs are used to generate the first K samples in their original form and the other additional samples in their permuted form. The permutation is performed within subsamples obtained in the same time instant of each BMI group. In practical implementation, PS-AIC can be realized by using only K/JK/J BMIs working all the time, thus enjoys a reduced complexity when compared with the conventional AIC and S-AIC. We prove that the EMM of the PS-AIC satisfies the RIP. The result shows that PS-AIC has the same RIP condition as S-AIC when signals are sparse on the Fourier basis and K+KnpK+Knp is an integral multiple of 2J2J. We also provide the MSE performance analysis of the signal recovery based on empirical risk minimization, and the results show its performance gain over the conventional AIC. Simulation results confirm the analytical results and show the favorable performance of PS-AIC.