Damped sinusoidal signals parameter estimation in frequency domain☆
Highlights
► Parameter estimation of noisy damped sinusoidal signals in the frequency domain is presented. ► A low complexity frequency domain three-sample estimation algorithm (TSEA) is proposed. ► A high performance frequency domain nonlinear least squares (FD-NLS) estimation algorithm is proposed. ► The Cramer–Rao lower bound (CRLB) of the frequency domain estimation algorithms is derived. ► The theoretical analysis shows that the FD-NLS can yield a near-optimal performance with few energy-concentrated samples.
Introduction
The problem of estimating the parameters of damped sinusoidal signals, especially buried in noise, is very important in different areas of applications. There exist many algorithms to estimate the sinusoidal parameters with time domain data, namely, the widely used Prony's method [1], the KT algorithm [2] and its modified versions [3], [4], the matrix pencil method [5], and the expanded linear prediction (ELP) method [6].
The time domain estimation algorithms usually have good estimation performance while their computational complexities are relatively high. The authors in [7] proposed a simple frequency domain (FD) algorithm, which is called as “Alg 1” hereafter, to perform sinusoidal parameter estimation based on two DFT samples. Although the “Alg 1” is computationally efficient, its performance is not good enough for different signal conditions.
In this paper, least squares (LS) estimation algorithms based on the frequency domain samples of damped sinusoidal signal are discussed. It is shown that the LS error criterion is nonlinear for some parameters that makes it be a nonlinear LS (NLS) problem. Firstly, a three-sample-based estimation algorithm (TSEA) is proposed. The basic idea of the algorithm is to form two pairs of DFT samples from the three samples, each pair yields a frequency estimate and these two estimates are combined to give the final estimate in an LS criterion. Evaluation shows that the TSEA performs much better than the “Alg 1” at the cost of moderate increase in computation.
The noise effect in the TSEA causes the mean square error performance cannot approach the CRLB particularly for damped sinusoidal signals. To improve estimation performance for damped sinusoidal signals, iterative NLS estimation algorithms are proposed next. Time domain and frequency domain NLS approaches dealing with the undamped sinusoidal signals parameter estimation can be found respectively in [8], [9], [10], [11], [12], [13]. In this paper, FD-NLS and TD-NLS estimation algorithms for damped sinusoidal signals, using the TSEA to give the initial estimate and the Gauss–Newton method to refine the estimate, are developed. Simulation results show that the new NLS algorithms provide the Cramer–Rao lower bound (CRLB) performance in Gaussian noises.
The CRLB analysis of frequency domain estimation algorithms for finite data set is carried out. The theoretical analysis shows that a small number of samples are enough for the frequency domain algorithm to achieve estimation performance close to the CRLB. Computer simulation verifies that the new FD-NLS algorithm can make use of a few significant DFT samples to achieve a near CRLB performance more effective than the TD-NLS algorithm. This property can significantly reduce the computational complexity of the FD-NLS algorithm.
The rest of the paper is organized as follows. In Section 2, the TSEA is described. Then, the FD-NLS estimation algorithm is proposed in Section 3, where a TD-NLS is also discussed for comparison and their computational complexities are analyzed. In Section 4, the CRLB of frequency domain estimation algorithms is analyzed. Some simulation results and discussions are presented in Section 5. Finally, conclusions are drawn in Section 6.
Notation: In the following, all boldface lower case and upper case letters denote vectors and matrices respectively. The , , denote matrix transpose, conjugate transpose and inverse. The functions and denote the natural logarithm and the angle respectively. The denotes the discrete Fourier transform.
Section snippets
Three-sample-based estimation algorithm—TSEA
The observed time domain data, , of a sum of M damped sinusoidal signals in complex white Gaussian noise, w(n), can be expressed aswhere w(n) has zero mean and variance of per dimension. For the ith damped sinusoidal signal of x(n), is the complex amplitude, is the damping factor and is the pole frequency. To estimate the parameters Ai and , it is usually required that the data sequence
FD-NLS estimation algorithm
The LS criterion in (5) can be expressed in vector form aswhere the vector , the vector , the parameter vectors and . From (3), the vector can be expressed aswhere the N×M matrix isSubstituting (18) into the LS
CRLB analysis of frequency domain estimation algorithm
The frequency domain expression of the damped sinusoidal signals is shown in (3). We represent all the parameters of the M sinusoids as a 4M×1 vector , where . Assume the variance of time domain noise is per dimension, the frequency domain noise W(k) is also white Gaussian distributed with variance per dimension. If the DFT vector with samples is used in the FD-NLS estimation algorithm, the joint
Simulation results and discussions
In the following simulation, the estimation performance is expressed in terms of the mean square error (MSE). The SNR is defined asOn each SNR, 5000 trial runs are performed. In Procedure II, Procedure III, tmax and are set equal to 20 and 10−4 respectively. For the GN method, three values are assigned to in (23): .
The CRLB [2, Appendix], the MSEs of the “Alg 1” [7], the TSEA (Procedure I or II), the time domain KT algorithm [2] (with an
Conclusions
The TSEA, which is based on three DFT samples to minimize the LS error criterion, is presented. Its performance is much better than that of the “Alg 1” with moderate increase in computation and comparable to the well-known KT algorithm. With leakage cancellation, the TSEA is able to provide robust estimation for multiple sinusoids case as if performing for single sinusoid case, for which the “Alg 1” fails to perform.
Based on the GN method with the initial estimate generated by the TSEA, the
Acknowledgments
The authors would like to thank the anonymous reviewers for their careful review. It helps us to improve the presentation of this paper.
References (16)
- et al.
Estimating parameters in the damped exponential model
Signal Process.
(2001) Modern Spectral Estimation
(1988)- et al.
Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise
IEEE Trans. Acoust. Speech Signal Process.
(1982) - et al.
A modification of the Kumaresan–Tufts method for estimating rational impulse responses
IEEE Trans. Acoust. Speech Signal Process.
(1986) - et al.
A parameter estimation scheme for damped sinusoidal signals based on low-rank Hankel approximation
IEEE Trans. Signal Process.
(1997) - et al.
Matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise
IEEE Trans. Acoust. Speech Signal Process.
(1990) - et al.
Analysis of damped sinusoidal signals via a frequency-domain interpolation algorithm
IEEE Trans. Instrum. Meas.
(1994) - et al.
Cisoid parameter estimation in the colored noise case: asymptotic Cramer–Rao bound, maximum likelihood, and nonlinear least-squares
IEEE Trans. Signal Process.
(1997)
Cited by (20)
Damping parameter estimation using topological signal processing
2022, Mechanical Systems and Signal ProcessingCitation Excerpt :An alternative to this option is to analyze the frequency response of a damped oscillation through the Fourier spectrum [31]. This method has been shown to be robust to some degree of additive Gaussian noise [32]. However, it requires a least-squares estimation algorithm applied to the frequency domain of the signal, which is an additional computational expense.
Phase correction autocorrelation-based frequency estimation method for sinusoidal signal
2017, Signal ProcessingCitation Excerpt :Frequency estimation of sinusoidal signal has received much attention in the literature because of its wide application in numerous engineering applications, such as radar, sonar, communication, power systems, measurement and instrumentation [1–5].
Analytical solutions for frequency estimators by interpolation of DFT coefficients
2014, Signal ProcessingPhase correction of discrete Fourier transform coefficients to reduce frequency estimation bias of single tone complex sinusoid
2014, Signal ProcessingCitation Excerpt :The second iteratively refines a single solution toward the final solution. This can be done by using either generic numerical solutions such as Newton method [2,7] or specific solutions tailored for the problem [8–14]. Among the iterative estimators, the one proposed by Aboutanios and Mulgrew [9] is considered the best because it can approach CRLB asymptotically in two iterations.
A Parameter Estimation Algorithm for Damped Real-value Sinusoid in Noise
2023, Measurement Science ReviewCombined coulomb and viscous damping estimation using topological signal processing
2021, Proceedings of the ASME Design Engineering Technical Conference
- ☆
The work described in this paper was substantially supported by a grant from CityU (7002459).