On Solving Ambiguity Resolution With Robust Chinese Remainder Theorem for Multiple Numbers

Xiao, Hanguang; Xiao, Guoqiang

doi:10.1109/tvt.2019.2905240

Cited by 15 publications

(13 citation statements)

References 17 publications

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“…In this section, the simulation is carried in MATLAB R2016b, with an Intel Core i5 2.60 GHz. To emphasize, the existing CRT-based multi-tone frequency determination approaches [ 19 , 20 , 21 ] cannot apply to the case that the reconstruction path number L is smaller than the component number

. Therefore, this section will first verify the feasibility of the case

, meaning that there are more frequency components to be estimated than the data acquisition paths.…”

Section: Simulation Results and Discussionmentioning

confidence: 99%

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Multi-Tone Frequency Estimation Based on the All-Phase Discrete Fourier Transform and Chinese Remainder Theorem

Huang

Cao

Lü

2020

Sensors

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The closed-form robust Chinese Remainder Theorem (CRT) is a powerful approach to achieve single-frequency estimation from noisy undersampled waveforms. However, the difficulty of CRT-based methods’ extension into the multi-tone case lies in the fact it is complicated to explore the mapping relationship between an individual tone and its corresponding remainders. This work deals with this intractable issue by means of decomposing the desired multi-tone estimator into several single-tone estimators. Firstly, high-accuracy harmonic remainders are calculated by applying all-phase Discrete Fourier Transform (apDFT) and spectrum correction operations on the undersampled waveforms. Secondly, the aforementioned mapping relationship is built up by a novel frequency classifier which fully captures the amplitude and phase features of remainders. Finally, the frequencies are estimated one by one through directly applying the closed-form robust CRT into these remainder groups. Due to all the components (including closed-form CRT, the apDFT, the spectrum corrector and the remainder classifier) only involving slight computation complexity, the proposed scheme is of high efficiency and consumes low hardware cost. Moreover, numeral results also show that the proposed method possesses high accuracy.

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. Therefore, this section will first verify the feasibility of the case

, meaning that there are more frequency components to be estimated than the data acquisition paths.…”

Section: Simulation Results and Discussionmentioning

confidence: 99%

“…In recent years, many endeavors have been made to generalize CRT-based estimators to multi-tone undersampled waveform cases [ 19 , 20 , 21 ]. Generally, these estimators solve this problem through the remainder redundancy coding, which actually pays the cost of heavy computation burden and sacrificing the dynamic estimation range.…”

Section: Introductionmentioning

confidence: 99%

Multi-Tone Frequency Estimation Based on the All-Phase Discrete Fourier Transform and Chinese Remainder Theorem

Huang

Cao

Lü

2020

Sensors

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“…Such tradeoff between the dynamic range and N is studied in [26], [27], [28], whereas the tight bound is only known when N = 2 [29]. To tackle both robustness and correspondence ambiguity simultaneously, the polynomial-time (statistical) robust multi-objective estimation is only known recently in [30], [31], [32], [33]. Co-prime Sampling (Array): Different from CRT method, which handles the spectrum directly, co-prime sampling es-timates the auto-correlation on the temporal domain.…”

Section: Introductionmentioning

confidence: 99%

On the Foundation of Sparse Sensing (Part I): Necessary and Sufficient Sampling Theory and Robust Remaindering Problem

Xiao,

Zhang,

Xiao

2021

Preprint

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In the first part of the series papers, we set out to answer the following question: given specific restrictions on a set of samplers, what kind of signal can be uniquely represented by the corresponding samples attained, as the foundation of sparse sensing. It is different from compressed sensing, which exploits the sparse representation of a signal to reduce sample complexity (compressed sampling or acquisition). We use sparse sensing to denote a board concept of methods whose main focus is to improve the efficiency and cost of sampling implementation itself. The "sparse" here is referred to sampling at a low temporal or spatial rate (sparsity constrained sampling or acquisition), which in practice models cheaper hardware such as lower power, less memory and throughput.We take frequency and direction of arrival (DoA) estimation as concrete examples and give the necessary and sufficient requirements of the sampling strategy. Interestingly, we prove that these problems can be reduced to some (multiple) remainder model. As a straightforward corollary, we supplement and complete the theory of co-prime sampling, which receives considerable attention over last decade.On the other hand, we advance the understanding of the robust multiple remainder problem, which models the case when sampling with noise. A sharpened tradeoff between the parameter dynamic range and the error bound is derived. We prove that, for N -frequency estimation in either complex or real waveforms, once the least common multiple (lcm) of the sampling rates selected is sufficiently large, one may approach an error tolerance bound independent of N .

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“…In the first part of the series papers, we have theoretically studied the necessary and sufficient (robust) condition for sparse sensing theory, characterized by (multiple) remaindering encoding problem. We connected and specified the relationship between Chinese Remainder Theorem method [1], [2], [3], [4], [5], [6], [7], [8] and co-prime sensing [9], [10]. On the other hand, we also pointed out that the deterministic necessary sampling constraint can be significantly relaxed at a cost of negligible failure, where co-prime sensing provides a concrete example.…”

Section: Introductionmentioning

confidence: 99%

On the Foundation of Sparse Sensing (Part II): Diophantine Sampling and Array Configuration

Xiao,

Zhou,

Xiao

2021

Preprint

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In the second part of the series papers, we set out to study the algorithmic efficiency of sparse sensing. Stemmed from co-prime sensing, we propose a generalized framework, termed Diophantine sensing, which utilizes generic Diophantine equation theory and higher-order sparse ruler to strengthen the sampling time, the degree of freedom (DoF), and the sampling sparsity, simultaneously.With a careful design of sampling spacings either in the temporal or spatial domain, co-prime sensing can reconstruct the autocorrelation of a sequence with significantly denser lags based on Bézout theorem. However, Bézout theorem also puts two practical constraints in the co-prime sensing framework. For frequency estimation, co-prime sampling needs Θ((M1 + M2)L) samples, which results in Θ(M1M2L) sampling time, where M1 and M2 are co-prime down-sampling rates and L is the number of snapshots required. As for Direction-of-arrival (DoA) estimation, the sensors cannot be arbitrarily sparse in co-prime arrays, where the least spacing has to be less than a half of wavelength.Resorting to higher-moment statistics, the proposed Diophantine framework presents two fundamental improvements. First, on frequency estimation, we prove that given arbitrarily large down-sampling rates, there exist sampling schemes where the number of samples needed is only proportional to the sum of DoF and the number of snapshots required, which implies a linear sampling time. Second, on Direction-of-arrival (DoA) estimation, we propose two generic array constructions such that given N sensors, the minimal distance between sensors can be as large as a polynomial of N , Θ(N q ), which indicates that an arbitrarily sparse array (with arbitrarily small mutual coupling) exists given sufficiently many sensors. In addition, asymptotically, the proposed array configurations produce the best known DoF bound compared to existing coarray designs.

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On Solving Ambiguity Resolution With Robust Chinese Remainder Theorem for Multiple Numbers

Cited by 15 publications

References 17 publications

Multi-Tone Frequency Estimation Based on the All-Phase Discrete Fourier Transform and Chinese Remainder Theorem

Multi-Tone Frequency Estimation Based on the All-Phase Discrete Fourier Transform and Chinese Remainder Theorem

On the Foundation of Sparse Sensing (Part I): Necessary and Sufficient Sampling Theory and Robust Remaindering Problem

On the Foundation of Sparse Sensing (Part II): Diophantine Sampling and Array Configuration

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