Elsevier

Signal Processing

Volume 92, Issue 2, February 2012, Pages 381-391
Signal Processing

Damped sinusoidal signals parameter estimation in frequency domain

https://doi.org/10.1016/j.sigpro.2011.08.003Get rights and content

Abstract

Parameter estimation of noisy damped sinusoidal signals in the frequency domain is presented in this paper. The advantage of the frequency domain approach is having the spectral energy concentrated in frequency domain samples. However, the least squares criterion for frequency estimation using frequency domain samples is nonlinear. A low complexity three-sample estimation algorithm (TSEA) for solving the nonlinear problem is proposed. Using the TSEA for initialization, a frequency domain nonlinear least squares (FD-NLS) estimation algorithm is then proposed. In the case of white Gaussian noise, it yields maximum likelihood estimates, verified by simulation results. A time domain NLS (TD-NLS) estimation algorithm is also proposed for comparison.

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. On the other hand, the TD-NLS does not have the energy concentration property and requires more time domain samples to perform satisfactory estimation. Simulation results verify that the frequency domain estimation algorithms provide better tradeoff between computational complexity and estimation accuracy than time domain algorithms.

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 (·)T, (·), (·)1 denote matrix transpose, conjugate transpose and inverse. The functions ln(·) and arg(·)[0,2π) denote the natural logarithm and the angle respectively. The F[·] denotes the discrete Fourier transform.

Section snippets

Three-sample-based estimation algorithm—TSEA

The observed time domain data, {y(n),n=0,1,,N1}, of a sum of M damped sinusoidal signals in complex white Gaussian noise, w(n), can be expressed asy(n)=x(n)+w(n)=i=1MAie(αi+j2πfi)n+w(n)forn=0,1,,N1where w(n) has zero mean and variance of σ2 per dimension. For the ith damped sinusoidal signal of x(n), Ai=ηiejφi is the complex amplitude, αi<0 is the damping factor and fi[0,1) is the pole frequency. To estimate the parameters Ai and θi=eαi+j2πfi, it is usually required that the data sequence

FD-NLS estimation algorithm

The LS criterion in (5) can be expressed in vector form asJ=[YX(A,θ)][YX(A,θ)]where the vector Y=[Y(0),Y(1),,Y(N1)]T, the vector X(A,θ)=[X(0),X(1),,X(N1)]T, the parameter vectors A=[A1,A2,,AM]T and θ=[θ1,θ2,,θM]T. From (3), the vector X(A,θ) can be expressed asX(A,θ)=H(θ)Awhere the N×M matrix H(θ) isH(θ)=1θ1N1θ11θ2N1θ21θMN1θM1θ1N1θ1ej2πk/N1θ2N1θ2ej2πk/N1θMN1θMej2πk/N1θ1N1θ1ej2π(N1)/N1θ2N1θ2ej2π(N1)/N1θMN1θMej2π(N1)/NSubstituting (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 ϕ=[v1,,vM,α1,,αM,η1,,ηM,φ1,,φM]T, where vi=2πfi. Assume the variance of time domain noise ω(n) is σ2 per dimension, the frequency domain noise W(k) is also white Gaussian distributed with variance σF2=Nσ2 per dimension. If the DFT vector Y=[Y(N1),Y(N1+1),,Y(N2)] with NY=N2N1+1 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 asSNR=10log1012σ2dBOn each SNR, 5000 trial runs are performed. In Procedure II, Procedure III, tmax and τ are set equal to 20 and 10−4 respectively. For the GN method, three values are assigned to μ in (23): [μ1=1/3,μ2=2/3,μ3=1].

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 N/4×3N/4

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)

  • N. Kannan et al.

    Estimating parameters in the damped exponential model

    Signal Process.

    (2001)
  • S.M. Kay

    Modern Spectral Estimation

    (1988)
  • R. Kumaresan et al.

    Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise

    IEEE Trans. Acoust. Speech Signal Process.

    (1982)
  • B. Porat et al.

    A modification of the Kumaresan–Tufts method for estimating rational impulse responses

    IEEE Trans. Acoust. Speech Signal Process.

    (1986)
  • K.J.R.L.Y. Li et al.

    A parameter estimation scheme for damped sinusoidal signals based on low-rank Hankel approximation

    IEEE Trans. Signal Process.

    (1997)
  • Y. Hua et al.

    Matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise

    IEEE Trans. Acoust. Speech Signal Process.

    (1990)
  • C.O.M. Bertocco et al.

    Analysis of damped sinusoidal signals via a frequency-domain interpolation algorithm

    IEEE Trans. Instrum. Meas.

    (1994)
  • A.J.P. Stoica 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)
There are more references available in the full text version of this article.

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The work described in this paper was substantially supported by a grant from CityU (7002459).

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