We present an efficient algorithm based on the robust Chinese remainder theorem (CRT) to perform single frequency determination from multiple undersampled waveforms. The optimal estimate of common remainder in robust CRT, which plays an important role in the final frequency estimation, is first discussed. To avoid the exhausted searching in the optimal estimation, we then provide an improved algorithm with the same performance but less computation. Besides, the sufficient and necessary condition of the robust estimation was proposed. Numerical examples are also provided to verify the effectiveness of the proposed algorithm and related conclusions.

Frequency estimation from noisy sinusoid is a hot research topic in engineering, which is applied in almost all engineering fields, such as communications, radar, astronomy, medicine, measurements, and geophysical exploration [

The problem of frequency determination in undersampled condition is summarized as follows.

Assume that a single frequency signal is modelled as a complex sinusoid form:

Let the sampling rates be

The

This problem has been studied in [

The remaining of this paper is organized as follows. In Section

In this section, we first briefly describe the extended CRT where all the moduli have a common factor larger than 1. We then briefly describe the error sensitivity of CRT.

For convenience, we denote by

Let

Let

Now, we consider the case when the remainders have errors.

Let the

According to (

Now, we consider condition (

Note that the fact that

In this section, we give the optimal estimate of the common remainder firstly; then we propose the efficient frequency estimation algorithm. Before introducing our method, we introduce a kind of circular distance function.

For real numbers

Putting

The optimal estimate

According to the definition of the circular distance, we have

Let

For

For

Let

Theorem

Based on the proposed algorithm, we have a sufficient and necessary condition of the robust estimate as shown in Algorithm

Let

We proof the sufficiency.

From (**), we have

According to (

According to (

According to (

We then proof the necessity.

Combination of (

By (

By (

By (

Therefore, we have

In the simulations, the coprime numbers from

We compare the proposed method with the ICRT algorithm which is the most effective one of the existing methods. To assess the performance, we take the trail-fail rate (TFR) as the measurement. In each trial, if the estimate of the frequency is

Figure

TFR versus SNR.

Figure

Total running time versus remainders number.

Figure

Robust estimation probability of common remainder versus SNR.

In this paper, we have proposed an efficient algorithm based on the robust CRT to perform single frequency estimation from multiple undersampled waveforms. We transformed the problem into the robust CRT firstly. Then we presented the optimal estimate of the common remainder and the original frequency. Based on the proposed algorithm, we presented the sufficient and necessary condition of the robust estimation. We finally applied the efficient algorithm to estimate the original frequency from multiple undersampled waveforms. Compared with the improved robust CRT algorithm, it has nearly the same performance but less computation. After using the undersampled waveforms, the sampling rate reduced sharply.

The authors would like to thank the referee for his (her) valuable comments and suggestions which improved this work to a great extent. This work was supported by the National Natural Science Foundation of China (Grants nos. 61172092, 61302069) and the Research Fund for the Doctoral Programs of Higher Education of China (no. 20130201110014).