On Parameter Estimation of Two Dimensional Chirp Signal

Transcription

1 On Parameter Estimation of Two Dimensional Chirp Signal Ananya Lahiri & Debasis Kundu, & Amit Mitra Abstract Two dimensional (-D) chirp signals occur in different areas of image processing. In this paper, we consider the least squares estimation procedures for estimating the different unknown parameters of the -D chirp signal model in presence of stationary noise. The proposed least squares estimators are strongly consistent and we obtained the asymptotic distribution of the least squares estimators. It is observed that asymptotically the least squares estimators are normally distributed. We perform some simulation experiments to observe the behavior of the least squares estimators, and it is observed that the performances are quite satisfactory. Key Words and Phrases: Chirp signals; least squares estimators; strong consistency, asymptotic distribution; linear processes. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 0806, India. Corresponding author. Phone: , Fax:

2 Introduction In this paper we consider the two dimensional (-D) chirp signal model which has the following form, y(m,n) = A 0 cos(α 0 m + β 0 m + γ 0 n + δ 0 n ) + B 0 sin(α 0 m + β 0 m + γ 0 n + δ 0 n ) + X(m,n); m =,,M; n =,,N, () here X(m,n) is stationary error, A 0 and B 0 are distinct non zero amplitudes and α 0,γ 0,β 0,δ 0 s are distinct frequencies and frequency rates respectively, and lie strictly between 0 and π. The explicit assumptions on the errors X(m,n) will be provided later. The above -D chirp signal model () is a natural generalization of the -D chirp signal model considered by several authors, see for example the article by Kundu and Nandi (008) and the references therein. The model () has quite a bit of applications in signal processing, mainly in the areas of image processing. The model () and its different variants have been used quite extensively in modeling black and white grey images, mainly the finger printing images, see for example Friedlander and Francos (996). Several others, see for example Peleg and Porat(99), Friedlander and Francos (996), Francos and Friedlander (998,999), Cao, Wang and Wang (006), Zhang and Liu(006), Zhang, Wang and Cao (008) considered similar models and discussed different estimation techniques. But, surprisingly no where the least squares estimators of the -D chirp signal have been considered, nor their properties have been discussed. The reason might be that, although it is expected that the least squares estimators will be very efficient, but deriving the properties of the least squares estimators is not simple. The main aim of this paper is to consider the least square estimators of the unknown parameters of the model () and to study their statistical properties. It is expected due to the complexity of the model that deriving the exact distribution of

3 the least squares estimators may not be possible, and therefore we mainly rely on the asymptotic results. It may be mentioned that the properties of -D chirp signal model have been discussed by Kundu and Nandi (008). They established the strong consistency and asymptotic normality properties of the least squares estimators. Unfortunately, their results cannot be used directly to establish the asymptotic properties of the least squares estimators of the -D model (). It can be easily seen similar to the -D chirp signal model that this model also dose not satisfy the sufficient conditions of Jennrich (969) and Wu(98) for the least squares estimators to be consistent and asymptotically normal. Therefore, the results of Jennrich (969) or Wu (98) cannot be used directly to establish the asymptotic properties of the least squares estimators. In this paper we establish the strong consistency and asymptotic normality properties of the least squares estimators of the unknown parameters of the model (). Interestingly, it is observed that the least squares estimators of α 0 (γ 0 ) and β 0 (δ 0 ) have the convergence rates O p (M 3/ N / ) (O p (N 3/ M / )) and O p (M 5/ N / ) (O p (N 5/ M / )) respectively. Moreover, the least squares estimators of A 0 and B 0 have the convergence rate () /. Therefore, it is clear that the convergence rates of the least squares estimators of the linear parameters are much slower than the convergence rates of the least squares estimators corresponding to the non linear parameters. We perform some extensive simulation experiments to study the effectiveness of the least squares estimators for finite samples, and it is observed that the performances of the least squares estimators are quite satisfactory. The rest of the paper is organized as follows. In Section, we provide the necessary assumptions, preliminary results and the methodology for the least squares estimators. Strong consistency and asymptotic results are established in Section 3. Extensive numerical results are presented in Section 4. Finally we conclude the paper in Section 5. Most of the proofs are provided in the Appendix. 3

4 Model Assumptions, Preliminary Results and Methodology. Model Assumptions Assumption : The error X(m,n) has the following form; X(m,n) = a(j,k)ε(m j,n k) () j= k= with a(j,k) < (3) j= k= Here ε(m, n) is a double array sequences of independent and identically distributed (i.i.d.) random variables with zero mean, finite variance σ and with finite fourth moment. Assumption : If we denote the true parameters by θ 0 = (A 0,B 0,α 0,β 0,γ 0,δ 0 ) and the parameter space by Θ = [ K,K] [ K,K] [0,π] [0,π] [0,π] [0,π] then It is assumed that θ 0 is an interior point of Θ. Moreover, α 0,β 0,γ 0,δ 0 s are distinct.. Preliminary Results We need the following results for further development. Proposition. If (ω,θ) (0,π) (0,π), then except for countable number of points lim min{m,n} lim min{m,n} n= m= n= m= cos(ωm +θn ) = cos (ωm +θn ) = lim min{m,n} lim min{m,n} n= m= sin(ωm +θn ) = 0. n= m= (4) sin (ωm +θn ) =. (5) Proof: The proof can be obtained from Vinogradov (954). 4

5 Proposition. If (ω,ω,θ,θ ) (0,π) (0,π) (0,π) (0,π), then except for countable number of points, and for s, t = 0,,, lim min{m,n} lim min{m,n} lim min{m,n} lim min{m,n} M (s+) N (t+) M (s+) N (t+) n= m= n= m= n= m= n= m= cos(ω m + ω m + θ n + θ n ) = 0 sin(ω m + ω m + θ n + θ n ) = 0. m s n t cos (ω m + ω m + θ n + θ n ) = m s n t sin (ω m + ω m + θ n + θ n ) = (s + )(t + ) (s + )(t + ). Proof: They can be obtained from Proposition. Lemma. If X(m,n) satisfies the Assumptions &, then as min{m,n}, sup X(m,n)e i(αm+βm +γn+δn ) 0 a.s. α,β,γ,δ n= m= Proof: See in the Appendix. Lemma. If X(m, n) satisfies Assumptions &, then as min{m, N}, and for s,t = 0,,, sup M s+ N t+ α,β,γ,δ m s n t X(m,n)e i(αm+βm +γn+δn ) n= m= 0 a.s. Proof: It follows along the same line as Lemma..3 Methodology We will use the following notations; A φ = B 5

6 and cos(α + β + γ + δ) sin(α + β + γ + δ) cos(α + 4β + γ + δ) sin(α + 4β + γ + δ).. cos(mα + M β + γ + δ) sin(mα + M β + γ + δ) W(α,β,γ,δ) =.. cos(α + β + Nγ + N δ) sin(α + β + Nγ + N δ) cos(α + 4β + Nγ + N δ) sin(α + 4β + Nγ + N δ).. cos(mα + M β + Nγ + N δ) sin(mα + M β + Nγ + N δ), and Y is the data vector defined as follows: Y = (y(, ),,y(m, ),,y(,n),,y(m,n)) T. (6) For θ = (A,B,α,β,γ,δ), we want to minimize Q(θ) = (Y W(α,β,γ,δ)φ) T (Y W(α,β,γ,δ)φ) (7) with respect to θ. Now using separable regression technique of Richards (96), it can be seen that for fixed (α,β,γ,δ), the minimization of Q(θ) with respect to A and B can be obtained as ] [Â(α,β,γ,δ) φ(α,β,γ,δ) = B(α,β,γ,δ) = (W(α,β,γ,δ) T W(α,β,γ,δ)) W(α,β,γ,δ) T Y. Therefore, the minimization of Q(θ) can be obtained by minimizing R(α,β,γ,δ) = Y T (I P(α,β,γ,δ))Y with respect to (α,β,γ,δ), where P(α,β,γ,δ) = W(α,β,γ,δ)(W(α,β,γ,δ) T W(α,β,γ,δ)) W(α,β,γ,δ) T is the projection matrix on the column space of W(α,β,γ,δ). If ( α, β, γ, δ) minimizes R(α,β,γ,δ), the least squares estimates of A and B can be obtained as  = Â( α, β, γ, δ) and B = B( α, β, γ, δ). 6

7 We will use θ = (Â, B, α, β, γ, δ). Note that by using the separable regression technique, the least squares estimators of the unknown parameters of the model () can be obtained by solving a four dimensional optimization problem, rather than a six dimensional optimization problem. 3 Asymptotic Properties of the Least Squares Estimators 3. Consistency of the Least Squares Estimators In this section we provide the consistency results of the estimators. Theorem. If the Assumptions & are satisfied then θ, the least squares estimators of θ 0, is a strongly consistent estimator of θ 0. Proof: See in the Appendix. The following result might be useful for error analysis of the model, or it may have some independent interests also. Lemma 3. If the Assumptions & are satisfied, then M( α α 0 ) 0 a.s., M ( β β 0 ) 0 a.s., N( γ γ 0 ) 0 a.s., N ( δ δ 0 ) 0 a.s. Proof: See in the Appendix. Using Lemma 3, we immediately obtain  = A 0 + o() a.s., B = B 0 + o() a.s., α = α 0 + o(m) a.s., β = β 0 + o(m ) a.s., γ = γ 0 + o(n) a.s., δ = δ 0 + o(n ) a.s., 7

8 So we get, ŷ(m,n) = Â cos( αm + βm + γn + δn ) + B sin( αm + βm + γn + δn ) (8) = A 0 cos(α 0 n + β 0 n + γ 0 n + δ 0 n ) + B 0 sin(α 0 n + β 0 n + γ 0 n + δ 0 n ) + o() a.s. which gives, y(m,n) ŷ(m,n) = X(m,n) + o() a.s. (9) Therefore, clearly (9) can be used for checking the error assumptions. 3. Asymptotic normality of the estimators In this section we will provide the asymptotic normal distribution for the estimators. Theorem. If the Assumptions & are satisfied, then (ˆθ θ 0 )D N 6 (0, cσ Σ ) where the matrix D is a 6 6 diagonal matrix as follows: c = ( ) D = diag M N,M N,M 3 N,M 5 N,M N 3,M N 5, j= k= Σ = B A0 B 0 Σ = A0 3 B 0 A B 0 A 0 + B 0 A0 B 0 A0 3 3 a(j,k) and A 0 +B 0 A 0 +B 0 A 0 +B 0 4 A 0 +B B 0 3 A0 A 0 +B 0 4 A 0 +B 0 5 A 0 +B A 0 +B 0 5 B 0 A0 3 A 0 +B 0 4 A 0 +B A 0 +B 0 3 A 0 +B 0 4 B 0 3 A0 3 A 0 +B A 0 +B 0 5 A 0 +B 0 4 A 0 +B 0 5 A 0 +7B 0 8A 0 B 0 8B 0 5B 0 8B 0 5B 0 8A 0 B 0 7A0 +B 0 8A 0 5A 0 8A 0 5A 0 8B 0 8A B 0 5A B 0 8A B 0 5A (0) () Proof: See in the Appendix. 8

9 4 Simulations and Data Analysis 4. Simulations We perform some simulation experiments to see how the proposed least squares estimators behave for different sample sizes and for different error structures. We consider the model () with following parameter values: A 0 = 5.0, B 0 = 5.0, α 0 =.0, β 0 = 0.05, γ 0 =.5,δ 0 = 0.5. () We have taken two different error structures namely; Error-I: X(m, n) = ε(m, n); (3) Error-II: X(m,n) = ε(m,n) + 0.5ε(m,n) ε(m,n ); (4) where ε(m,n)s are i.i.d. Gaussian random variables with mean 0 and variance σ. We have considered different sets of sample sizes and different σ for our simulation experiments. We have used the random number generator RAN of Press et al. (99) for generating the uniform random numbers. In each case the least squares estimators of the unknown parameters are obtained by using the Downhill Simplex Algorithm, see for example Press et. al. (99), whereas, the initial guesses are obtained by using four dimensional grid search method. In each case we computed the least squares estimators, and obtained the average estimates, mean squared errors and variances over 000 replications. We report the true parameter values (PARA), the average estimates (MEAN), the associated mean square errors (MSE), variances(var). For comparison purposes we report the asymptotic variances (ASYV) also obtained using Theorem. All the results are reported in Tables -4. Some of the points are quite clear from these tables. First of all, it is observed that as the error variances decrease or the sample size increases the variances and the mean 9

10 Table : The MEAN, MSE, VAR and ASYV of the least squares estimators when σ = 0.05 and for Error-I M=N=50 PARA MEAN MSE ( 0.38E-0) ( E-0) ( 0.43E-06) ( E-0) ( 0.385E-06) ( 0.305E-09) VAR ( 0.365E-0) ( E-0) ( 0.4E-06) ( E-0) ( E-06) ( 0.97E-09) ASYV ( E-03) ( E-03) ( E-07) ( E-0) ( E-07) ( E-0) M=N=75 PARA MEAN MSE ( 0.434E-03) ( E-03) ( E-07) ( E-) ( E-07) ( E-) VAR ( 0.435E-03) ( E-03) ( E-07) ( 0.64E-) ( 0.377E-07) ( E-) ASYV ( E-03) ( E-03) ( 0.36E-07) ( 0.07E-) ( 0.36E-07) ( 0.0E-) Table : The MEAN, MSE, VAR and ASYV of the least squares estimators when σ = 0.5 and for Error-I M=N=50 PARA MEAN MSE ( 0.486E-0) ( E-0) ( 0.333E-05) ( 0.406E-09) ( E-05) ( 0.873E-08) VAR ( 0.485E-0) ( E-0) ( 0.80E-05) ( 0.406E-09) ( E-05) ( 0.87E-08) ASYV ( E-0) ( E-0) ( E-06) ( E-09) ( E-06) ( E-09) M=N=75 PARA MEAN MSE ( E-0) ( E-0) ( 0.757E-06) ( E-0) ( E-06) ( E-0) VAR ( 0.305E-0) ( E-0) ( E-06) ( 0.468E-0) ( E-06) ( E-0) ASYV ( E-0) ( E-0) ( 0.36E-06) ( 0.07E-0) ( 0.36E-06) ( 0.07E-0) Table 3: The MEAN, MSE, VAR and ASYV of the least squares estimators when σ = 0.05 and for Error-II M=N=50 PARA MEAN MSE ( E-03) ( 0.686E-03) ( 0.59E-06) ( E-0) ( 0.5E-06) ( E-0) VAR ( E-03) ( E-03) ( 0.565E-06) ( E-0) ( 0.50E-06) ( E-0) ASYV ( E-03) ( E-03) ( E-07) ( E-0) ( E-07) ( E-0) M=N=75 PARA MEAN MSE ( E-03) ( E-03) ( 0.444E-07) ( E-) ( E-07) ( E-) VAR ( E-03) ( E-03) ( E-07) ( E-) ( E-07) ( E-) ASYV ( E-03) ( E-03) ( 0.649E-07) ( E-) ( 0.649E-07) ( E-) 0

11 Table 4: The MEAN, MSE, VAR and ASYV of the least squares estimators when σ = 0.5 and for Error-II M=N=50 PARA MEAN MSE ( E-0) ( E-0) ( 0.444E-05) ( E-09) ( 0.94E-05) ( E-09) VAR ( E-0) ( E-0) ( E-05) ( E-09) ( 0.900E-05) ( E-09) ASYV ( E-0) ( E-0) ( E-06) ( E-09) ( E-06) ( E-09) M=N=75 PARA MEAN MSE ( 0.446E-0) ( 0.446E-0) ( E-06) ( 0.636E-0) ( E-06) ( E-0) VAR ( E-0) ( 0.446E-0) ( E-06) ( 0.63E-0) ( E-06) ( 0.535E-0) ASYV ( E-0) ( E-0) ( 0.649E-06) ( E-0) ( 0.649E-06) ( E-0) squared errors decrease in all the cases considered, as expected. The simulation results show that the maximum likelihood estimates are quite close to the true parameter values. For both the error structures it is observed that the mean squared errors of the least squares estimators match quite well with the corresponding asymptotic variances. It indicates that the asymptotic results work quite well even for moderate sample sizes. 4. Data Analysis For illustrative purposes, mainly to show how the proposed method can be implemented in practice we have analyzed one simulated data set obtained from the model (). We have used the following parameter values: A 0 = 5.0, B 0 =.0, α 0 =.55, β 0 = 0.05, γ 0 =.5, δ 0 = X(n)s are as follows, X(m,n) = ǫ(m,n) + 0.5ǫ(m,n) ǫ(m,n ) + 0.ǫ(m,n ) where ǫ s are assumed to be i.i.d. Gaussian random variables with mean 0 and variance σ =.5. We have generated the data y(m,n) for M = N = 00, and the data are plotted in Figure. Figure represents the -D image plot of a simulated noise corrupted y(m,n), whose gray level at (m,n) is proportional to the value of y(m,n).

12 The problem is to extract the regular texture, see Figure, from the contaminated one Figure : Noisy signal We use the least squares technique and estimate the unknown parameters, and they are as follow: Â = , B = , α =.5498, β = , γ =.5085, δ = The estimated y(m,n), namely ŷ(m,n) as in (8) are plotted in Figure 3. It is clear that the original and the estimated plots match quite well. 5 Conclusion In this paper we consider the -D chirp signal model and study the properties of the least squares estimators of the unknown parameters. It has been proved that the least squares estimators are strongly consistent and asymptotically normally distributed. It is observed that the least squares estimators can be obtained as a four-dimensional optimization problem. Our simulation results suggest that the asymptotic properties

13 Figure : True signal of the least squares estimators can be used quite effectively even for moderate sample sizes. There are several open issues and generalizations which are of interests for future work. For example, it is observed that the least squares estimators can be obtained using a four dimensional optimization problem. It will be interesting to develop some numerically efficient algorithm to find a solution of this optimization problem. One possible generalization can be a multiple chirp signal model, of course finding the least squares estimators and deriving their properties will be a challenging problem. 6 Acknowledgement Part of the work of the first author is financially supported by the Center for Scientific and Industrial Research (CSIR). The Department of Science and Technology, Government of India has partially supported the work of the second and third authors. 3

14 Figure 3: Estimated signal Appendix Proof of Lemma- We will use the following results to prove Lemma : Result : For fixed j,k =,,,N, and for fixed l,t =,,,N, using Holder s inequality we have N t+k M l+j E ε(m,n)ε(m j,n k)ε(m + l,n + t)ε(m j + l,n k + t) n= m= ( N t+k E = [ E n= N t+k n= = O() M l+j m= M l+j m= ε(m,n)ε(m j,n k)ε(m + l,n + t)ε(m j + l,n k + t) ) (ε(m,n)ε(m j,n k)ε(m + l,n + t)ε(m j + l,n k + t)) ] The equality at the third step of the above expression holds as contribution over cross product terms is zero. 4

15 Result : If we denote q(θ,m,n) = αm + βm + γn + δn, then for a given t, again using Holder s inequality, we have; E sup ε(m,n)ε(m j,n k)e itq(θ,m,n) θ n= m= ( ) E sup ε(m,n)ε(m j,n k)e itq(θ,m,n) θ = [ [ E sup E θ n= m= ( N n= m= n= m= ) ( N ε(m,n)ε(m j,n k)e itq(θ,m,n) ε(m,n) ε(m j,n k) n= m= N + E ε(m,n)ε(m,n + )ε(m j,n k)ε(m j,n k + ) n= m= M + E ε(m,n)ε(m +,n)ε(m j,n k)ε(m j +,n k) n= m= + + E ε(m, )ε(m,n)ε(m j, k)ε(m j,n k) m= ] +E ε(,n)ε(m,n)ε( j,n k)ε(m j,n k) n= = O( +.() ) = O(() 3 4 ) ε(m,n)ε(m j,n k)e itq(θ,m,n) )] 5

16 Let m j = m,n k = n. Then E sup X(m,n)e iq(θ,m,n) θ n= m= = E sup a(j,k)ε(m j,n k)e iq(θ,m,n) θ n= m= k= j= = E sup a(j, k) ε(m,n )e iq(θ,m,n) θ k= j= n= m= E sup θ a(j,k) ε(m,n )e iq(θ,m,n) k= j= n= m= = E sup a(j, k) ε(m,n )e iq(θ,m,n) θ k= j= n= m= [ ] a(j, k) E sup ε(m,n )e iq(θ,m,n) θ k= j= n= m= a(j, k) E sup ε(m,n )e iq(θ,m,n) θ = k= j= k= j= [ E sup θ k= j= a(j,k) ( N n= m= n= m= ) ( N ε(m,n )e iq(θ,m,n) a(j,k) [ E n= m= n= m= ε(m,n ) ε(m,n )e iq(θ,m,n) )] N +E ε(m,n )ε(m,n + ) + E M ε(m,n )ε(m +,n ) n= m= n= m= ] + + E ε(m, )ε(m,n) +E ε(,n )ε(m,n ) m= n= = O( +.()3 4 ) = O(() 8 ) Therefore, E sup α,β,γ,δ () 9 N 9 M 9 n= m= X(m,n)e iq(θ,m,n) O(() 9 8 ) (5) 6

17 Take and for ǫ > 0 Z(M,N) = sup α,β,γ,δ () 9 N 9 M 9 n= m= X(m,n)e iq(θ,m,n) N= M= P(Z(M,N) > ǫ) N= M= EZ(M,N) ǫ N= M= O(() 9 8) ǫ < Therefore, by Borel Cantelli lemma we have N 9 M 9 sup X(m,n)e iq(θ,m,n) () 9 α,β,γ,δ n= m= 0 a.s. Denote, {(J,K) : N 9 < K (N + ) 9, M 9 < J (M + ) 9 } = S JK Define, U(M,N) = sup sup α,β,γ,δ S JK () 9 = sup sup α,β,γ,δ S JK () 9 + K J () 9 n= m= sup N 9 α,β,γ,δ () 9 N 9 M 9 n= m= N 9 M 9 n= m= n= m= X(m,n)e iq(θ,m,n) JK X(m,n)e iq(θ,m,n) X(m,n)e iq(θ,m,n) JK M 9 X(m,n)e iq(θ,m,n) ( + () 9 (M + ) 9 (N + ) 9) V (M,N) + W(M,N), K () 9 K n= m= (N+) 9 n= (N+) 9 n= n= m= J X(m,n)e iq(θ,m,n) K J n= m= J X(m,n)e iq(θ,m,n) (M+) 9 m= (M+) 9 m= X(m,n)e iq(θ,m,n) X(m,n)e iq(θ,m,n) X(m,n)e iq(θ,m,n) where V (M,N) = sup α,β,γ,δ (N+) 9 M 9 n=n 9 + m= + sup α,β,γ,δ X(m,n)e iq(θ,m,n) (N+) 9 (M+) 9 n=n 9 + m=m sup N 9 α,β,γ,δ X(m,n)e iq(θ,m,n) (M+) 9 n= m=m 9 +, X(m,n)e iq(θ,m,n) W(M,N) = ( ) () 9 (M + ) 9 (N + ) 9 7 sup α,β,γ,δ (N+) 9 n= (M+) 9 m= X(m,n)e iq(θ,m,n).

18 We have and Therefore, E sup We also have α,β,γ,δ n=n 0 + m=m 0 + P(V (M,N) > ǫ) N= M= X(m,n)e iq(θ,m,n) O((M m 0 )(N n 0 )) 7 8 (6) EV (M,N) ǫ P(V (M,N) > ǫ) 7 O((M9N8) 8 +(M 8 N 9 ) 8 7 ) () 9 N= M= ǫ O(() 9 8) ǫ <.. (7) P(W(M,N) > ǫ) EW(M,N) ǫ O(( + 9 )(M + ) 8 (N + ) 9 8 ) M N ǫ Therefore, N= M= P(W(M,N) > ǫ) N= M= O(() 9 8) ǫ < Now by Borel Cantelli lemma we get U(M,N) 0 a.s.. Proof of Lemma : It can be obtained along the same line. To prove the Theorem- we need the following Lemma. Lemma 4. Let θ be the least squares estimator of θ 0, and consider the set S c = {θ : θ Θ; A A 0 c, B B 0 c, α α 0 c, β β 0 c, γ γ 0 c, δ δ 0 c}. If for any c > 0, lim inf inf θ S c (Q(θ) Q(θ0 )) > 0 a.s. then ˆθ θ 0 a.s.. Here the function Q(θ) is same as defined in (7). Proof of Lemma 4: The proof can be obtained along the same line as in Wu (98), so it is avoided. Proof of Theorem : To prove Theorem, it is enough to prove that lim inf inf θ S c (Q(θ) Q(θ0 )) > 0 a.s. 8

19 Observe that where [Q(θ) Q(θ0 )] = f(θ) + g(θ), f(θ) = g(θ) = n= m= [ A cos(αm + βm + γn + δn ) + B sin(αm + βm + γn + δn ) A 0 cos(α 0 m + β 0 m + γ 0 n + δ 0 n ) B 0 sin(α 0 m + β 0 m + γ 0 n + δ 0 n ) ] X(n) [ A 0 cos(α 0 m + β 0 m + γ 0 n + δ 0 n ) + B 0 sin(α 0 m + β 0 m + γ 0 n + δ 0 n ) n= A cos(αm + βm + γn + δn ) B sin(αm + βm + γn + δn ) ] Now g(θ) is going to zero a.s. because of Lemma-. We now observe that S c S A c S B c S α c S β c S γ c S δ c, where S A c = {θ;θ Θ, A A 0 c}, and the other sets are also similarly defined. It can be shown along the same line as in Kundu (997) that lim inf inf S ξ c and hence the result is proved. f(θ) > 0, a.s. for ξ = A,B,α,β,γ,δ Proof of Lemma 3: Let us denote Q (θ) as the 6 first derivative matrix and Q (θ) as the 6 6 second derivative matrix. Now using multivariate Taylor series expansion of Q (ˆθ) around θ 0 and we get Q ( θ) Q (θ 0 ) = ( θ θ 0 )Q ( θ) (8) where θ is a point on line joining ˆθ and θ 0. Since, Q (ˆθ) = 0, then for the diagonal matrix D, same as defined in Theorem, we obtain Q (θ 0 )D = (ˆθ θ 0 )D [DQ ( θ)d] (9) which gives, (ˆθ θ 0 )D = [ Q (θ 0 )D][DQ ( θ)d] (0) 9

20 Dividing by the expression becomes (ˆθ θ 0 )( D) = [ Q (θ 0 )D][DQ ( θ)d] () Since, θ θ 0 a.s., θ θ 0 a.s.. Therefore, [ DQ ( θ)d ] [ DQ (θ 0 )D ] Moreover, using Lemma and Proposition, So, Hence using (3) we get Q (θ 0 )D 0 a.s. () ( θ θ 0 )( D) 0 a.s. (3) M(ˆα α 0 ) 0 a.s., M (ˆβ β 0 ) 0 a.s., N(ˆγ γ 0 ) 0 a.s., N (ˆδ δ 0 ) 0 a.s. Proof of Theorem : Note that [ Q (θ 0 )D ] T = N M n= N M n= N N n= N N n= N M n= m= cos(q(θ0,m,n))x(m,n) N M n= m= sin(q(θ0,m,n))x(m,n) M m= m[a0 sin(q(θ 0,m,n)) B 0 cos(q(θ 0,m,n))] X(m,n) M m= m [A 0 sin(q(θ 0,m,n)) B 0 cos(q(θ 0,m,n))] X(m,n) M m= n[a0 sin(q(θ 0,m,n)) B 0 cos(q(θ 0,m,n))] X(m,n) M m= n [A 0 sin(q(θ 0,m,n)) B 0 cos(q(θ 0,m,n))] X(m,n). Now using Central Limit Theorem of linear processes, see Fuller (996,), page 39, it follows that [ Q (θ 0 )D ] T N6 (0, cσ Σ). (4) Since [ DQ (θ 0 )D ] Σ we immediately get ( θ θ 0 )D N 6 (0, cσ Σ ). 0

21 References [] Cao, F.,Wang, S., Wang, F. (006), Cross-Spectral Method Based on -D Cross Polynomial Transform for -D Chirp Signal Parameter Estimation ICSP006 Proceedings, DOI:0.09/ICOSP [] Fuller, W. A. (996), Introduction to Statistical Time Series, second Edition, John Wiley and Sons, New York. [3] Francos, J.M., Friedlander, B. (998), Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds Multidimensional Systems and Signal Processing, 9, [4] Francos, J.M., Friedlander, B. (999), Parameter Estimation of -D Random Amplitude Polynomial-Phase Signals, IEEE Transactions on Signal Processing, 47, [5] Friedlander, B.,Francos, J.M. (996), An Estimation Algorithm for -D Polynomial Phase Signals, IEEE Transactions on Image Processing, 5, [6] Jennrich, R.I. (969), Asymptotic properties of non-linear least square estimators, The Annals of Mathematical Statistics, 40, [7] Kundu, D. (997), Asymptotic properties of the least squares estimators of sinusoidal signals, Statistics, 30, [8] Kundu, D. and Nandi, S. (008), Parameter estimation of chirp signals in presence of stationary noise, Statistica Sinica, 8, [9] Nandi, S. and Kundu, D. (004), Asymptotic properties of the least squares estimators of the parameters of the chirp signals, Annals of the Institute of Statistical Mathematics, 56,

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