The Noise Folding Phenomenon In Compressed Sensing And The Recovery Of Noisy Signals

Compressed sensing fully considers the compressibility of sparse signals,which allows signals to be recovered exactly with fewer measurements.Most existing studies on CS assume that the signal itself is noiseless and that the measurements are contaminated by noise.However,the signal itself is often subject to random noise prior to measuremen-t.Signal noise will be amplified significantly during measurement when signals are contaminated by noise prior to measurement.Such a noise folding phenomenon can severely impact the recovery of signals.Typical noise suppression methods often con-sider noise during signal recovery by adding different penalty terms in the optimization target and applying proper stopping rules to control the sparsity and recovery error of signals simultaneously.These methods play an important role in de-noising but do not address the NF phenomenon.And in low signal-to-noise ratio environment,these methods are not adaptive and robust enough.Based on the multi-measurement model in compressed sensing,this paper intro-duces a 2-step approach to recovering the noisy signals with common sparse support.Firstly,the optimization of the compressed matrix is transformed into the estimation of the common sparse support of the signal set under the noisy model.And the se-lective compressive matrix is generated according to the support estimation based on the multiple-measurement vector model,which can be realized by the simultaneous anti-noise-folding orthogonal matching pursuit algorithm.The second step is adaptive recovery,where the random compressive matrix is replaced by the selective compressive matrix,and the recovery can be achieved by classical recovery algorithms based on the anti-noise-folding model.The selective compressive matrix can reduce the signal noise outside the sparse support during the compressive processing.This article contains a total of five chapters:The first chapter introduces the research background and significance of this paper by summarizing the recent development about compressed sensing and the existing research that focuss on the noise folding phenomenon in compressed sensing.The main contents and research methods of this paper are also briefly introduced.In the second chapter,the main theories of compressed sensing are introduced,and the noise-folding phenomenon in compressed sensing as well as the effect of this phenomenon on signal recovery are described in detail.And the existing methods to solve the problem of noise folding are studied.The third chapter mainly introduces the noisy signal recovery method based on the multi-measurement model proposed in this paper.Firstly,the multi-measurement model in compressed sensing is introduced,and the noise folding problem under the multi-measurement model is analyzed.Secondly,this chapter gives the design scheme of the compressive matrix and the sparse support estimation algorithm.Finally,a two-step recovery method for the noisy signals is introduced utilizing the selective compressive matrix.In the fourth chapter,the effect of the signal recovery method proposed in the third chapter is evaluated by numerical experiments,including the support estimation of noisy signal by the simultaneous anti-noise-folding orthogonal matching pursuit algorithm and the performance of 2-step approach to recovering the noisy signals.The fifth chapter summarizes the whole paper,and points out the improvement of the recovery method proposed in this paper,and the future research direction.