Study On Frequency Estimators Under Noise Environment

Estimating one or more frequencies from the sample data distorted by noise is one of the most valuable technologies in modern signal processing. It has wide applications in various fields such as radar, wireless communications, sonar, vibration measurement, seismic signal processing and electronic monitoring.The mean square error is often used as a measure of system inaccuracy. Estimation bias is of secondary importance. The computational complexity, the signal-to-noise ratio (SNR) and the size of the sample data are also problems have to be considered. According to the probability theory, the mean square error of estimated frequency can achieve the Cramer Rao Bound (CRB). Since Gaussian white noise is the most common noise in the nature, how to extract frequencies from the sample data distorted by Gaussian white noise has inspired a lot of work. The existing estimators achieve the CRB when SNR is high. However, large iterations are necessary, which bring in computational complexity. Furthermore, the existing estimators still have performance gap compared with the CRB when SNR is low. Thus, efficient estimators which approach the CRB when SNR is low are the key point in the study of frequency estimation. Sinusoidal signals are narrow-banded, whereas white noise spreads statistically equally in the whole spectrum. Hence the effects of noise can be reduced when estimating frequency in the frequency domain and the performance of the estimators can be improved in low SNR cases.This dissertation has made deep research on the estimators in frequency domain. Several new estimators are proposed. DFT is performed as the coarse estimation followed by various fine estimators.1) The author proposes a new estimator considering the fact that the derivative of a sinusoidal signal contains the frequency information of the sinusoidal signal. We also deduce the effects of narrow-band approximation and noise on the performance. The signal and its derivative are used to estimate frequency and they are both approximated by the spectral lines inside the narrow-band around the actual tone. The theoretical analyses not only show that the proposed estimator approaches the CRB, but also give criterions for the choice of parameters. The choice of parameters and the SNR thresholds are further discussed in the dissertation. The fine estimation of the proposed estimator is more efficient compared with the other exsiting estimators. Furthermore, the proposed estimator has a lower SNR threshold and is fit for the cases of long samples and low SNRs. The drawback of the proposed estimator is that 3 times of zeroes have to be padded in the coarse estimation. 2) A new estimator based on the narrow-band autocorrelation function is proposed. The autocorrelation of the signal has the same frequency as the signal, thus researchers proposed a number of estimators by estimating the phase of the autocorrelation function. Most of these estimators can only be used for estimating single-frequency signal, and the frequency is ambiguous. The theoretical performances of the existing autocorrelation based estimators are analyzed. We propose a new estimator based on the narrow-band autocorrelation function, which can be applied in multiple-frequency estimation and is unambiguous. The theoretical analyses and simulations show that the proposed estimator outperforms the existing autocorrelation-based estimators and approaches the CRB. Finally, an improved narrow-band autocorrelation estimator is proposed bearing the advantages of M&M estimator in mind. The improved estimator can be used in short samples and only one time zeroes have to be padded in the coarse estimation.3) Based on the least square approximation in frequency domain, two fast estimators approaching the CRB are proposed. We firstly expand the power density spectrum of sinusoidal signal into first order Talor series around the frequency which is the peak of the periodgram achieved in coarse estimation. And then the sample data is used to approximate the sinusoidal signal in frequency domain. The approximation is applied inside the narrow-band, which decreases the effects of noise and reduces the computational complexity. Modifying the coefficients, we can further deduce a frequency estimator by rational combination of spectrum lines. Theoretical bound is obtained to show that the proposed estimator approaches the CRB. The combinational weights can be pre-calculated, thus the proposed estimator is more computational efficient than all the existing estimators, especially for short sample cases. Additionally, the estimator has a lower SNR threshold.4) We apply the proposed estimators to real multi-frequency signals and deduce several new sinusoidal analyses methods. The speech sinusoidal model is a new speech model, which is popular in low-bitrate speech code. We deduce several new sinusoidal analyses methods based on the narrow-band spectrum, which are used for extracting frequency and amplitude from the speech/audio signal samples. Simulations show that the proposed sinusoidal analyses methods are effective.