Study On Compressive Sampling And Recovery Algorithm Of Sparse Analog Signal

With the development of electronic information technology, as well as the rapidgrowth of data needs, as a guiding protocol for signal acquisition, the Shannon-Nyquistsampling theorem has brought great challenges to the signal processing capabilities andhardware equipment. However, to recover the original signal, the Shannon-Nyquistsampling theorem is sufficient but not necessary. In recent years, a new data acquisitiontheory called compressed sensing has been proposed, and it has the ability to breakthrough the restriction of the Shannon-Nyquist sampling theorem for sparse signal.According to the sparsity of signal, this dissertation presents some hardwareimplementable compressive sampling methods and signal reconstruction algorithmsbased on compressed sensing. The main work is as follows:1. Study on compressive sampling method based on random equivalent sampling.As an alternative approach to the traditional uniform sampling, for the periodic signal,the random equivalent sampling has the potential to obtain a high equivalent samplingwaveform form low speed samples. This dissertation has analyzed the principle of therandom phase and the impact of uneven sampling distribution on signal reconstruction.According to the requirement of prior information for the random equivalent samplingand the compressed sensing theory, the feasibility of appling compressed sensing torandom equivalent sampling signal reconstruction has been studied. Based on therelationship between the non-uniform random samples and the uniform signal to bereconstructed, we present a compressive sampling matrix in the context of the randomequivalent sampling and a stopping rule for sampling. Compared to the time alignmentrandom samples reconstruction method, the proposed compressed sensing based signalreconstruction method can not only improve the reconstruction precision but also reducethe random samples for reconstruction.2. Considering the input bandwidth barrier of analog-to-digital convertor (ADC),this dissertation has analyzed the random demodulation based compressive samplingmethod. First, we introduce the random demodulation based analog-to-informationconversion (AIC), and then, a parallel AIC and a segmented parallel AIC are proposed, and their compressed sensing matrixes are also suggested. Considering theimplementation and the sparse signal with multibands, we present a Hadamardstructured compressive sampling model based on random demodulation and prove that,a Rademacher random sequence can reduce the coherence between the Hadamardstructured measurement matrix and the sparse representation basis. Based on theanalysis of the proposed model in Fourier domain, a compressed sensing matrix ispresented. In the above sampling models, since the samples are captured after theintergrator or low pass filter, the models will not be subject to the input bandwidthbarrier of ADC.3. Study on the reconstruction for compressive samples with the mismatchedbasis. We introduce the principle of basis mismatching and its impact on the signalreconstruction. To address this problem, a reconstruction algorithm is suggested.Different from the classic reconstruction algorithms, the proposed algorithm estimatesthe frequencies contained in the signal using root-MUSIC approach. Since the proposedalgorithm does not employ the representation basis in the reconstruction, it will not besubject to the problem raised by the mismatched basis.4. Study on the signal reconstruction algorithm based on the statistic property ofwavelet coefficients. Based on the study of the multiscale analysis of wavelet coefficient,this dissertation makes a conclusion that the wavelet coefficients with significantamplitude will construct an upside down structured subtree. Base on the conclusion, animproved orthogonal matching pursuit (OMP) for wavelet domain compressive signalreconstruction is proposed. In the computation of the inner product between theresiduals and the column vectors of measurement matrix, the proposed algorithm willweight the columns on the subtree and enforce the solution to be sparse.