Two-Dimensional Interpolation Criterion Using DFT Coefficients

: In this paper, we address the frequency estimator for 2-dimensional (2-D) complex sinusoids in the presence of white Gaussian noise. With the use of the sinc function model of the discrete Fourier transform (DFT) coefficients on the input data, a fast and accurate frequency estimator is devised, where only the DFT coefficient with the highest magnitude and its four neighbors are required. Variance analysis is also included to investigate the accuracy of the proposed algorithm. Simulation results are conducted to demonstrate the superiority of the developed scheme, in terms of the estimation performance and computational complexity.

al. [So, Chan, Lau et al. (2010)] suggests principal-singular-vector utilization for modal analysis (PUMA), which utilizes linear prediction of 2-D multiple exponentials to achieve optimal estimation performance in only high signal-to-noise ratio (SNR). Although all these methods can provide the optimal performance, they suffer from the high complexity, especially in the case of big data [Wu (2018)]. Then the class of interpolating on discrete Fourier transform (DFT) coefficients, is devised, which is shown efficient by means of estimation performance and computational requirements. Among this class of methodologies, a coarse estimation and then a finetune step are required, where the former one is usually realized by finding the index of the peak magnitude of DFT spectrum, and the latter one refers to interpolation on DFT peak to improve the estimation accuracy. In Quinn et al. [Quinn (1997); Provencher (2010); Candan (2013Candan ( , 2015], different interpolation schemes are developed with the use of DFT peak and its neighbor bins. Although these methods can achieve the optimal or nearly optimum estimation performance, they can apply only for the one-dimensional single complex-tone. In the paper, a 2-D non-iterative and accurate frequency estimator (2-D NIA) is proposed. Here, a new interpolation criterion is devised by utilizing the relationship of ratios of midway magnitudes to the largest one. For the estimation of the first dimensional frequency, the left-and-right neighbors of DFT peak is employed, while the up-and-low neighbors of the peak is used to obtain the second dimensional frequency estimate. The variance analysis is also provided, which indicates the high performance of our method. The rest of this paper is organized as follows. The NIA method is derived in Section 2, whose variance analysis is also provided in Section 3. Computer simulations in Section 4 are carried out to show that the developed methods can attain Cramer-Rao lower bound (CRLB). Finally, conclusions are drawn in Section 5.
In the following, we first discuss the variance analysis for the first dimension frequency From (12) and (17), we can devise the relationship between δ and 1 δ , which is Therefore, the variance analysis of δ can be devised easily form that of 1 δ . As N → ∞ , we have from (13) With the use of the discussion in Aboutanios et al. [Aboutanios and Mulgrew (2005) Expanding and simplifying (21) Kay (1993)], the variance of δ , 1 var( ) δ is The variance of ω , var( ) ω , has the form of ( ) 2 2 2 2 2 2 2 (1 2 ) (1 2 ) (1 4 ) sin sin cos 2 2 var( ) 4 cos cos 2 Similarly, the variance of ν , var( ) ν , has the form of ( )  ν . The Cramer-Rao lower bound (CRLB) ] is included as the benchmark while comparisons with PUMA [So, Chan, Lau et al. (2010)] and ESPRIT [Sun and So (2004)] methods are also provided. First of all, we investigate the performance of the proposed methods in different noise conditions. The MSEs and biases of ω and ν versus SNR are plotted in Figs. 1-4. It is observed in Fig. 1 and Fig. 2 that the proposed method is superior to the other two estimators since it can attain CRLB fastest. Fig. 3 and Fig. 4 also verifies this result since our approach can provide stable estimates when SNR>-5 dB, but those of the other methods are SNR>0 dB.
Then the estimation performance and the computational cost versus the data length M are studied. Here all parameters are set to as the same with the previous experiment. The stopwatch timer is utilized to measure the operation times of all methods. It is indicated in Fig. 5 and Fig. 6 that our method can still provide a high estimation accuracy. Furthermore, it can be seen in Fig. 7 that in the case of nearly optimal estimation performance, the complexity of our approach is significantly lower than those of the PUMA and ESPRIT methods. It is worth to point out that in the case of varying N , the results are similar.
Third, the estimation performance for different δ and µ is examined with 1 3 L = , 2 8 L = and the SNR is 10 dB. We vary δ when µ is fixed to -0.1, while in the case of varying , 0.08 µ δ = is selected. It is shown in Fig. 8 and Fig. 9 that the gap between the MSE of the proposed method and CRLB is smallest than the other two estimators, in all values of δ and µ .

Conclusion
In this paper, an accurate frequency estimators of 2-D measurements using Fourier coefficients interpolation are proposed, which can attain lower complexity than that of existing methods. Computer simulations show that the proposed algorithms perform superior to PUMA and ESPRIT methods in terms of high accuracy and low complexity. Moreover, it is indicated that with the increasing observation data set, the computational complexity of our methods has the smaller rate of complexity than that of other methods, which can be applied in big data.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.