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An Efficient and Fast Algorithm for Estimating the Parameters of Sinusoidal Signals

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An Efficient and Fast Algorithm for Estimating the Parameters of Sinusoidal Signals An Efficient and Fast Algorithm for Estimating the Parameters of Sinusoidal Signals Swagata Nandi1 Debasis Kundu2 Abstract A computationally e±cient algorithm is proposed for estimating the parameters of sinusoidal signals in presence of stationary errors. The proposed estimators are con- sistent and they are asymptotically equivalent to the least squares estimators. Monte Carlo simulations are performed to compare the proposed one with the other exist- ing comparable methods. It is observed that the proposed estimator works quite well in terms of biases and mean squared errors. The main advantage of the proposed method of estimation is that the estimators can be obtained using only ¯xed number of iterations. Some real data sets have been analyzed for illustration purposes. Key Words and Phrases: Sinusoidal frequency model; least squares estimators; approx- imate least squares estimators; consistent estimators; asymptotic distributions; iterations; convergence. Short Running Title: Sinusoidal Frequency. MS Classi¯cations: 62F10; 60K40. Address of correspondence: Debasis Kundu, e-mail: [email protected], Phone no. 91- 512-2597141, Fax no. 91-512-2597500. 1 Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7, S.J.S. Sansan- wal Marg, New Delhi - 110016. 2 Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India. 1 1 Introduction We consider the following model; p y(t) = [Aj cos(!jt) + Bj sin(!jt)] + X(t); t = 1; : : : ; N: (1) jX=1 Here Ajs and Bjs are unknown amplitudes and none of them is identically equal to zero. The !js are unknown frequencies lying strictly between 0 and ¼ and they are distinct. The error random variables X(t)s are stationary linear processes and they satisfy the following assumption. Assumption 1: The error random variables X(t)s have the following structure. 1 X(t) = a(j)e(t ¡ j); (2) ¡1 j=X where e(t)s are independent and identically distributed (i.i.d.) random variables with mean zero and ¯nite variance σ2 and a(j)s are real numbers such that 1 ja(j)j < 1: (3) ¡1 j=X The number of components p is assumed to be known. The problem is to estimate the unknown parameters Ajs, Bjs and !js, given a sample of size N, namely y(1); : : : ; y(N). In this paper, we mainly consider e±cient estimation of !js, even though the estimation of the linear parameters is also very important. Estimation of frequencies in presence of additive noise is a very important problem in the time series analysis and in the area of statistical signal processing. Starting with the work of Fisher (1929), this problem has received considerable attention. Brillinger (1987) discussed some of the very important real life applications of this particular problem, see Kay and Marple (1981) also. Stoica (1993) provided an extensive list of references up to that time and see Kundu (2002) for some of the recent references. 2 The optimum rate of convergence can be obtained by the least squares estimators and 3 ¡ ¡± ± ¡± their convergence rate is Op(N 2 ). Here Op(N ) means N Op(N ) is bounded in probabil- ity. The periodogram estimators (without the constraint of Fourier frequency) also provide the best possible rate and they are asymptotically equivalent to the least squares estimators. Finding the least squares estimators tends to be computationally intensive as the functions to be optimized are highly non-linear in parameters. Thus, very good (close enough to the true value) initial estimates are needed. Several techniques are available in the literature, for example Pisarenko (1973), Chan, Lavoie and Plant (1981), which attempt to ¯nd computa- tionally e±cient frequency estimators. However, these procedures produce estimators having ¡ 1 convergence rate Op(N 2 ). The periodogram maximizer over Fourier frequencies does not ¡1 generally provide good initial estimates as the convergence rate is Op(N ), see for example ¡1¡± Rice and Rosenblatt (1988), whereas an initial estimate of the convergence rate Op(N ) (± > 0) is needed for most of the iterative techniques to work. In this paper, we propose a new iterative procedure similar to the procedure of Bai, et al. (2002). The method uses a correction term based on PN (j) and QN (j) to be de¯ned in theorem 1, which are functions of the data vector as well as the available frequency estimator corresponding to j-th component. The forms of the functions PN (j) and QN (j) are motivated by the least squares method. It is observed that if the initial guess is accurate up to the order N ¡1, then our three step iterative procedure produces fully e±cient frequency estimator which has the same rate of convergence as the least squares estimators. In the proposed method we do not use the ¯xed sample size available for estimation at each step. At ¯rst step we use a fraction of it and at the last step we use the whole data set by gradually increasing the e®ective sample sizes. The rest of the paper is organized as follows. Two di®erent estimators and their properties are discussed in Section 2. The proposed algorithm is presented in Section 3. Simulation 3 results and analysis based on some real data are presented in Section 4 and ¯nally the conclusion appears in Section 5. All the proofs are provided in the appendix. 2 Estimation Procedures There are mainly two di®erent methods of estimation of the unknown parameters. We will briefly discuss both of them below. Least Squares Estimators: The least squares estimators (LSEs) of the unknown parameters can be obtained by minimizing the residual sum of squares, namely; 2 N p R(A; B; !) = y(t) ¡ [A cos(! t) + B sin(! t)] ; (4) 0 j j j j 1 Xt=1 jX=1 @ A with respect to A = (A1; : : : ; Ap), B = (B1; : : : ; Bp) and ! = (!1; : : : ; !p). Note that R(A; B; !) can be written as follows; R(A; B; !) = R(®; !) = [Y ¡ X(!)®]T [Y ¡ X(!)®] ; (5) T T T where Y = [y(1); : : : ; y(N)] , ® = [A1; : : : ; Ap; B1; : : : ; Bp] , ! = (!1; : : : ; !p) and X(!) is an N £ 2p matrix of the form cos(!1) : : : cos(!p) sin(!1) : : : sin(!p) 2 3 cos(2!1) : : : cos(2!p) sin(2!1) : : : sin(2!p) 6 7 X(!) = 6 7 : (6) 6 7 6 . 7 6 . : : : . : : : . 7 6 7 6 7 6 7 6 7 6 cos(N!1) : : : cos(N!p) sin(N!1) : : : sin(N!p) 7 4 5 From (5), it is clear that ® can be separated from !. Therefore, by using the separable regression techniques of Richards (1961), the LSE of ® can be obtained in terms of !. For ¯xed !, the LSE of ® can be obtained as ¡1 ®^ (!) = XT (!)X(!) XT (!)Y: (7) h i 4 If we replace ®^ (!) in (5), we obtain T Q(!) = R(®^ (!); !) = Y I ¡ PX(!) Y; (8) h i where ¡ T 1 T PX(!) = X(!) X (!)X(!) X (!) h i is the projection matrix on the column space spanned by the matrix X(!) for a given !. Therefore, the LSE of ! can be obtained by minimizing (8) with respect to !. If !^ minimizes (8), then the corresponding estimator of the linear parameter ® can be obtained as; ¡1 ®^ (!^ ) = XT (!^ )X(!^ ) XT (!^ )Y: (9) h i Most of the special purpose algorithms, for example the methods proposed by Bresler and Macovski (1988), Kumaresan, Scharf and Shaw (1986), Kundu and Kannan (1994) and Smyth (2000) attempt to minimize (8), which naturally saves computational time. Approximate Least Squares Estimators: Alternative way to estimate the frequencies is to maximize the periodogram function. The periodogram function I(!) can be de¯ned as follows; 2 1 N I(!) = y(t)e¡i!t : (10) ¯N ¯ ¯ t=1 ¯ ¯ X ¯ ¯ ¯ The periodogram estimators obtained under¯ the condition¯ that frequencies are Fourier fre- ¡1 quencies, provide estimators with convergence rate Op(N ). When this condition is dropped, the estimators obtained by ¯nding p local maxima of I(!) achieve the best possible rate and are asymptotically equivalent to the LSEs. So it is called approximate least squares esti- mators (ALSEs) in the literature. A frequency ¸ is a Fourier frequency if it is of the form 2¼k N ¸ = N , for some integer 1 · k · 2 . Let us denote the ALSE of ! = (!1; : : : ; !p) as ^ ^ !^ = (!^1; : : : ; !^p). It is known that !^ and !^ are both consistent estimators of ! with the 5 following asymptotic distribution. 3 2 2 N (!^ ¡ !)!Np 0; 24σ § ; (11) 3 ³ ´ 2 ^ 2 N (!^ ¡ !)!Np 0; 24σ § ; (12) ³ ´ where § is p £ p diagonal matrix as follows; c c § diag 1 ; : : : ; p : = 2 2 (13) "½1 ½p # Here 2 1 c = a(k)eik!j and ½2 = A2 + B2: j ¯ ¯ j j j ¯k=¡1 ¯ ¯ X ¯ ¯ ¯ ¯ ¯ ^ ^ The LSEs of Aj and Bj are¯estimated using¯ (9) and ALSEs, say A^j and B^j are obtained as ^ 2 N ^ 2 N A^ = y(t) cos(!^ t); B^ = y(t) sin(!^ t): (14) j N j j N j Xt=1 Xt=1 These expressions are used in section 4.2 to estimate the linear parameters. Thus, according to (11) and (12) both the LSEs and ALSEs produce frequency estimators which have conver- ¡ 3 gence rate Op(N 2 ). In this paper, we have considered the e±cient estimation of non-linear frequency parameters, so we do not discuss anything related to the rate of convergence of the linear parameters. In the next section, we describe a method which produces frequency estimators and which have the same convergence rate as the LSEs or the ALSEs. 3 Proposed Algorithm Given a consistent estimator !~j, we compute !^j using (15) for j = 1; : : : ; p as follows; 12 P (j) ! ! N ; ^j = ~j + 2 Im (15) N "QN (j)# where N N P (j) = y(t) t ¡ e¡i!~j t; (16) N 2 Xt=1 µ ¶ N ¡i!~j t QN (j) = y(t)e (17) Xt=1 6 and Im[.] denotes the imaginary part of a complex number.
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