Research On Frequency Estimation Algorithm Based On Chinese Remainder Theorem

The frequency estimation of the signal is an important research content in radar,sonar,seismic monitoring,wireless communication,etc.In order to recover the frequency without distortion,the Nyquist sampling theorem states that the sampling rate should not be less than twice the highest frequency of the signal.However,when the frequency of the signal is very high,sampling the signal according to the Nyquist sampling theorem will bring extremely high requirements and challenges to the hardware performance and cost.Although these issues can be addressed by under-sampling,under-sampling will lead to spectrum aliasing.The Chinese remainder theorem provides a feasible scheme to estimate the frequency from under-sampled signals.In the scheme,multiple samples are obtained by multi-channel parallel under-sampling.Then,the frequency is reconstructed by the multiple samples according to the Chinese remainder theorem,achieving low-cost and fast frequency estimation for the high-frequency signal.The Chinese remainder theorem(CRT)based frequency estimation has been widely concerned in recent years.However,the current studies have two shortcomings: on the one hand,the error-tolerance bound of the current CRT-based multiple frequencies estimation algorithm decreases with the increase of the frequency component,and the robustness of the algorithm needs to be improved.On the other hand,most of the existing studies focus on complex sinusoidal signals,and few studies focus on the CRT-based frequency estimation for real sinusoidal signals.To solve these issues,this thesis makes the following two contributions by generalizing and improving the current algorithms:(1)Research on the CRT-based multiple frequencies estimation for the undersampled complex sinusoidal signals.In order to improve the error-tolerance bound of the current algorithm,this thesis presents a CRT-based multiple frequencies estimation algorithm for the under-sampled complex sinusoidal signals under a given error-tolerance bound.First,to solve the issue of lack of correspondence between frequencies(numbers to be recovered)and the locations of the spectrum peaks(residues)under the given errortolerance bound,this thesis proposes a scheme by randomly assigning residues into different clustering,where each clustering corresponds to a number to be recovered.Then,based on the pigeonhole principle,the dynamic range of the number to be recovered is derived,which verifies the validity of the random clustering.This thesis proves that once the random clustering is wrong,the number reconstructed by the random clustering will exceed the dynamic range.Second,to improve the error-tolerance bound of the algorithm,the error caused by the modulo operation is corrected by assigning random coefficients to each residue with errors.Besides,the criterion for verifying the validity of the random coefficients is given.(2)Research on the CRT-based frequency estimation for the under-sampled real sinusoidal signals.In order to deepen the study of CRT-based frequency estimation,this thesis takes the single-tone real sinusoidal signal as the research object,proposing a polynomial-time CRT-based frequency estimation algorithm for the single-tone real sinusoidal signal under a given error-tolerance bound.The real sinusoidal signal will have an additional fake peak in the spectrum compared to the complex sinusoidal signal.This thesis makes full use of the prior knowledge that the two peaks are symmetric with each other to solve the following problems: first,to address the difficulty in distinguishing the fake peak,this thesis constructs a quadratic congruence equation based on the prior knowledge,transforming the problem of distinguishing the fake peak into solving the quadratic congruence equation.Second,to ensure the robustness of the algorithm,the relationship between the residues is derived by using the above prior knowledge,and the error caused by the modulo operation by shifting the residues with errors based on the relationship.Finally,to evaluate the performance of the above algorithms,this thesis carries out experiments on the proposed algorithms compared with the mainstream CRT-based frequency estimation algorithms.Experimental results demonstrate that the proposed algorithms effectively improve the error-tolerance bound.