Frequency Estimation Of Sinusoid Based On Interpolation Of FFT And DTFT

The frequency estimation of sinusoidal signals in white noise background is a classic subject in the field of digital signal processing,which can be widely used in many fields.The factors that measure the performance of a frequency estimator include frequency estimation accuracy,algorithm complexity,SNR threshold at low SNR,performance at high SNR and so on.The existing algorithms can not always make all the above factors satisfactory,so the sinusoidal frequency estimation technology has been continuously concerned by researchers.Although the number of the FFT spectral lines used in the existing frequency estimators based on FFT interpolation is different,the location of the spectral lines used in all the existing estimators is fixed.No one has done a study of the variation rule of the algorithm performance when the spectral lines locate arbitrary locations near the main lobe.No one has studied the existence of a general algorithm that can cover all the different situations when the spectral lines locate arbitrary locations near the main lobe.Based on the above considerations,the main research work done in this paper is as follows:1.The calculation formulas for the correlation coefficients of the DTFT spectrum of complex white noise,and the correlation coefficients of the DTFT spectrum of complex sinusoidal signal plus complex white noise(including rectangular window and non rectangular window two cases)are derived.The problem of calculating the correlation coefficients in the MSE formula derivation of the frequency estimators based on arbitrary interval frequency interpolation is solved.2.Several commonly used frequency estimators are improved.(1)An improved estimator based on phase of autocorrelation function is proposed.The SNR threshold of the proposed estimator is lower,and there is no frequency estimation range limit.(2)The possible problems of segmented-FFT phase difference algorithm in measuring phase difference are analyzed,and an improved estimator based on segmented-FFT phase difference and a synthetic frequency estimator are proposed.The proposed synthetic estimator has higher accuracy in frequency estimation and is suitable for wider SNR range.(3)An improved unbiased frequency estimator based on three spectral line interpolation of FFT is proposed.The estimation accuracy is higher than the typical estimators,and it performs better than the estimators based on N-point FFT interpolation at low SNR.(4)An improved unbiased frequency estimator based on two spectral line interpolation of FFT is proposed.The estimation accuracy is higher than the similar estimators,and it performs better than the estimators based on N-point FFT interpolation at low SNR.3.For white noise background,a frequency estimator base on three-spectral-line(in any position)interpolation of FFT and DTFT is proposed.The coarse estimation is made by searching the location of the maximum FFT spectral line.In the fine estimation step,two auxiliary spectral lines located at arbitrary position(symmetrically distributed beside the maximum spectral line)within the main lobe or the first side lobe of the frequency spectrum of the sinusoid can be used,and these two spectral lines are obtained by calculating the DTFT of the sampled sinusoid signal.Formula for calculating the MSE of frequency estimation in additive white noise is derived.Unlike the existing estimators,the frequency interval between the spectral lines can be smaller than the interval of the adjacent spectral lines of N-point FFT,and the noise on different spectral lines are correlated.Therefore,the derivation of the formula for calculating the MSE is more complicated,and the correlation coefficients of the DTFT spectrum of complex sinusoidal signal plus complex white noise(in chapter 2)are needed.The influence of the interval between the auxiliary spectral lines and the maximum spectral line on the MSE is analyzed through theoretical analysis and simulation experiments.The proposed estimator is more general than the existing estimators,and contains all the results when the interval between the auxiliary spectrum line and the maximum spectral line is not larger than the interval of the adjacent spectral lines of N-point FFT.The proposed estimator also contains the cases of some typical algorithms.Performance analysis shows that the proposed estimator is more accurate than the typical estimators,and has less computation than the estimators based on zero-padding and 2N-point FFT interpolation.The performance of the proposed estimator is better than that of the typical estimators.4.For the presence of interference signals,the frequency estimator base on three-spectral-line(in any position)interpolation of FFT and DTFT is generalized,and the frequency estimation formula of the estimator with window functions is constructed.Formula for calculating the MSE in additive white noise is derived.As the frequency interval between the spectral lines can be smaller than the interval of the adjacent spectral lines of N-point FFT,the correlation coefficients in chapter 2 of the DTFT spectrum of complex sinusoidal signal plus complex white noise are needed.Theoretical analysis and simulation experiments show that the proposed estimator is more general than the existing estimators,and contains all the results when the interval between the auxiliary spectrum line and the maximum spectral line is not larger than the interval of the adjacent spectral lines of N-point FFT.The proposed estimator also contains the cases of some typical estimators with window functions.Performance analysis shows that the proposed estimator can effectively reduce the impact of interference signals,and its performance is better than that of the typical estimators with window functions.