Published online: 23 Aug 2011.

Transcription

1 This article was downloaded by: [Professor Tibor K. Pogány] On: 07 November 2013, At: 09:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: Average Sampling Restoration of Harmonizable Processes Andriy Olenko a & Tibor Pogány b a Department of Mathematics and Statistics, La Trobe University, Melbourne, Australia b Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia Published online: 23 Aug To cite this article: Andriy Olenko & Tibor Pogány (2011) Average Sampling Restoration of Harmonizable Processes, Communications in Statistics - Theory and Methods, 40:19-20, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Communications in Statistics Theory and Methods, 40: , 2011 Copyright Taylor & Francis Group, LLC ISSN: print/ x online DOI: / Average Sampling Restoration of Harmonizable Processes ANDRIY OLENKO 1 AND TIBOR POGÁNY 2 1 Department of Mathematics and Statistics, La Trobe University, Melbourne, Australia 2 Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia 1. Introduction The harmonizable Piranashvili type stochastic processes are approximated by finite time shifted average sampling sums. Explicit truncation error upper bounds are established. Various corollaries and special cases are discussed. Keywords Average sampling reconstruction; Harmonizable stochastic process; Local averages; Sampling theorem; Time shifted sampling. Mathematics Subject Classification 60G12; 94A20; 42C15. Recovering a stochastic signal from discrete samples or assessing the information lost in the sampling process are the fundamental problems in sampling and interpolation theory. An essential question of the theory of approximation of random functions is the problem of determining the classes of approximant. We start with the Paley-Wiener class PW 2 of all non random complex-valued L 2 -functions whose Fourier spectrum is bandlimited to ; see Higgins (1996). The classical Whittaker Shannon Kotel nikov sampling theorem states that any f PW 2 n can be reconstructed at arbitrary point x by its values at points w fx = n= f ( n w ) sinwx n (1) wx n Although this formula yields an explicit reconstruction of fx, it is usually considered to be of theoretical interest only, because it requires computations of infinite sums. In practice, we truncate the series in (1). Typically, to estimate fx we should only use the values of the function at points around x for example n n x N. The sampling size N is determined by the relative error accepted in the reconstruction. Hence the reconstruction possesses time adapted sampling Address correspondence to Tibor Pogány, Faculty of Maritime Studies, University of Rijeka, Studentska 2, Rijeka HR-51000, Croatia; poganj@pfri.hr 3587

3 3588 Olenko and Pogány size. Thus, the error analysis plays a crucial role in setting up the interpolation formula. The time shifted truncation version of (1) Y N f x = n x N f ( n ) sinwx n w wx n plays an important role in applications, when the appropriate remainder is conveniently bounded by a term which vanishes with the growing sampling size parameter N. The scheme is called a time shifted sampling, since it depends on the location x of the reconstruction. For a study of time shifted sampling schemes and numerous references; see Flornes et al. (1999); Micchelli et al. (2009); and Olenko and Pogány (2006, 2007);?. Then, the second problem arises. By physical and applications reasons the measured samples in practice may not be the values of the measured function f precisely at the sample time n, but only the local average of the function f around n w. So, the measured sample values are f u n Un = w U n fxu n xdx U n = suppu n [ t n n t n + n for some sequence u = u n x n of non negative, normalized, that is 1u n 1, averaging functions. The local averaging models and methods were introduced by Gröchenig (1992), and developed by Butzer and Lei (1998, 2000). Recently, Sun and Zhou (2002, 2003) gave some results in this direction, while the stochastic counterpart of the average sampling was intensively studied by He et al. (2007) and Song et al. (2006a,b) also see their references inside. The interested reader can consult, for example, the exhaustive list of references in (He et al., 2007) too. However, the mentioned stochastic average sampling results were restricted to weakly stationary stochastic processes, while the approximation average sampling sums were used around the origin. The obtained error reconstruction bounds were derived under various decay assumptions on the function f for the deterministic case or covariance functions for the stochastic case. The error reconstruction bounds were not sharp in the sense that they are constant and do not depend on the time variable x. It has to be mentioned as well that no extremal functions were given in the listed works for the average approximation problem. The third problem is that the stationarity assumption is unacceptable for many practical problems in signal processing. In recent years, there has been growing interest in various specific models of nonstationarity. The concept of harmonizability is a natural generalization of stationarity that includes a large class of nonstationary processes, while retaining some spectral representations of stationary processes; see Sec. 2. A wide class of real systems has the output which is a harmonizable process; see Bochner (1956). Our intention is to extend the mentioned above stochastic sampling results to the time shifted average sampling, considered for harmonizable processes. New techniques are required to obtain the desired results. ]

4 Average Sampling of Harmonizable Processes 3589 The article addresses these three problems. We study the time shifted finite average sampling sums un t = n t N u n n t sinwt n wt n in approximating the initial stochastic signal t by the weighted averages of over n t = n/w n t n/w + n t 0 nt nt /2w. The aim of the article is to derive some reasonably simple efficient truncation error upper bounds appearing in the approximation t un t. The mean square sampling truncation error un t = E t un t 2 is investigated. We do not assume any decay conditions and construct sharp upper bounds on un t adapted to the time variable t. In comparison with all previous approaches, our method is superior in obtaining very simple, minimal, time shifted truncation error upper bounds in interpolating the harmonizable class process with Paley Wiener class kernel function in its spectral representation. The organization of this article is the following. In Sec. 2, we introduce the necessary background from the theory of harmonizable stochastic processes. In Sec. 3, some Piranshvili s results are adapted to the time shifted sampling procedures. In Sec. 4, we formulate and prove some auxiliary results on stochastic shifted average sampling. This section also contains the main theorem and its corollaries. Conclusions are made in Sec A Brief Review of Piranashvili Processes In this section, we give some key results concerning harmonizable processes and their spectral representations with respect to bimeasures. Let t t be a centered second order random process defined on certain fixed probability space P. Further, let the process t have a covariance function (associated to some domain with some sigma algebra ) inthe form: Bt s = ft f s F d d (2) where, for each, f can be extended to the complex plane as an complex analytic exponentially bounded kernel function, that is, for some M>0 ft Me t while F is a positive definite measure on 2 The total variation F of the spectral distribution function F satisfies F = F d d <

5 3590 Olenko and Pogány Notice that the sample function t t 0 and ft possess the same exponential types; see Belyaev (1959, Theorem 4), and Piranashvili (1967, Theorem 3). Then, by the Karhunen Cramér theorem the process t has the spectral representation t = ft Z d (3) where Z is a stochastic measure and F S 1 S 2 = EZ S 1 Z S 2 S 1 S 2 Such a process will be called Piranashvili process in the sequel; see Piranashvili (1967), and Pogány (1999). Being ft entire, it possesses the Maclaurin expansion Let = sup ft = f n 0 t n /n! n=0 c = sup lim n n f n 0 < Since the exponential type of ft is equal to, then for all w>there holds t = n ( n w ) sinwt n (4) wt n and the series converges uniformly in the mean square and almost surely (Piranashvili, 1967, Theorem 1). This result we call Whittaker Kotel nikov Shannon (WKS) stochastic sampling theorem (Pogány, 1999). The class of Piranashvili processes includes various well known subclasses of stochastic processes. Some particular cases of Piranashvili processes are listen below. Specifying F x y = xy F x in (2) one easily concludes the Karhunen representation of the covariance function Bt s = ft f s F d Also, putting ft = e it in (2) one gets the Loève-representation: Bt s = e it s F d d Here, c = and, therefore, WKS formula (4) holds for all w>= sup. Note that the Karhunen process with the Fourier kernel ft = e it is the weakly stationary stochastic process having the covariance B = e i F d = t s

6 Average Sampling of Harmonizable Processes 3591 A deeper insight into different kinds of harmonizabilities is presented in Kakihara (1997); Priestley (1988); Rao (1982), and the related references therein. Finally, using = w w for some finite w in this consideration, we get the band limited variants of processes of the same kind. 3. Truncation Errors in Sampling of Piranashvili Processes In this section, we adapt some Piranshvili s results on stochastic sampling to the time shifted sampling procedures. First of all we introduce few auxiliary results. Let N x stand for the integer nearest to xw/ x, and let us denote In what follows, the series N x = n xw/ n N { N x = z z N x < q = stands for the Dirichlet lambda function. n=1 ( N n 1 q ) } w N Theorem 3.1. Let fz be an entire bounded on the real axis function of exponential type <w. Denote L f = sup fx L 0 z = 4wL f sinwz w ( 1 e ) Then for all z N x and large enough N it holds ( ) n sinwz n f w wz n < L 0z (5) N \ N x Proof. The estimate can be obtained by the contour integration method. We adapt Piranashvili s results; see Piranashvili (1967, p. 648) to the set N x in the following way. We use the notation N x for the boundary of the set N x By Cauchy s residue theorem for the contour integral 1 f d 2i N x sinw x we obtain the left side of the inequality (5). Then we derive the right hand side upper bound by using the classical upper estimate for entire functions of exponential type <w; see Achieser (1992). Here, and in what follows, we denote by Y N t = ( n ) sinwt n w wt n N t (6) the time shifted truncated WKS restoration sum for the stochastic process.

7 3592 Olenko and Pogány Theorem 3.2. Let t be a Piranashvili process with exponentially bounded kernel function ft and let L f = sup sup Then for all t and large enough N we have ft L 0 t = 4 L f w sinwt w ( 1 e ) (7) E t Y N t 2 < L 2 0 t N 2 F (8) Proof. If t is a Piranashvili process, then by the spectral representation formula (3) we obtain that the truncated sampling sum (6) can be expressed (in the mean square sense) in the form ( n ) sinwt n Y N t = f w Z wt n d = Y N f t Z d where N t Y N f t = ( n ) sinwt n f w wt n N t Now, combining the above formulae with (3) yields that E t Y N t 2 ( = E ft YN f t ) 2 Z d ( = ft YN f t ) (f t Y N f t) EZ dz d ( = ft YN f t ) (f t Y N f t) F d d (9) Denote s N t = sup ft Y N f t Applying Theorem 3.1 twice to (9), we deduce that E t Y N t 2 s 2 N t F d d ) 2 ( L 0 t F N d d completing the proof. Remark 3.1. Let us point out that the straightforward consequence of (8) is not only the exact L 2 -restoration of the initial Piranashvili type harmonizable process by the sequence of approximants Y N t when N, but the a.s. reconstruction as well. It follows immediately from and the Borel Cantelli lemma. E t Y N t 2 = ( N 2)

8 Average Sampling of Harmonizable Processes Time Shifted Average Sampling In this section, we derive some simple efficient mean square truncation error upper bounds appearing in the approximation t un t. Instead of the approach used by Song et al. (2007), we take time shifted finite average sampling sums (consult (Olenko and Pogány, 2006, 2007)) in approximating the initial stochastic signal. First, we consider the weighted averages of t over n t = n/w n t n/w + n t, 0 nt, nt /2w instead of the measured values at n/w, n N t. Suppose that the following assumption holds true: = sup max n N t max ( nt n t) < Let us define the time shifted average sampling approximation sum in the form and its truncated variant as u t = u n n t un t = u n n t N t sinwt n wt n sinwt n wt n We will study the mean square, time shifted average sampling truncation error un t = E t un t 2 Lemma 4.1. Let t be a Piranashvili process with the covariance Bt s C 2, satisfying sup B t t <. Let p q be a conjugated Hölder pair of exponents: Then, 1 p + 1 q = 1 p > 1 E Y N t un t 2 2 C q t sup B t t 2N + 1 2/p (10) where C q t = (1 + 2q+1 sinwt q q ) 2/q q (11) Proof. Let us note that the first order difference xy B see Habib and Cambanis (1981) of Bt s on the plane satisfies ( xy B ) t s = Bt + x s + y Bt + x s Bt s + y + Bt s x y 2 = 0 0 uv B( t + u s + v ) dv du (12)

9 3594 Olenko and Pogány Having in mind (2), the properties of the sequence u of averaging functions and (12), we obtain E Y N t un t 2 = E = sinwt n wt n nm 2 N t x y 0 2 N t sup 0 xy n N t sinwt m wt m ( ) n u w n n t n t n t m t m t u n ( 2 uv B u + n ) w v+ m dv du dx dy v sinwt n wt n x y 0 0 sinwt m wt m ( 2 uv B u + n ) w v+ m v dvdu sinwt n wt n ( x + n w 2 )u m ( y + m w being u normalized. For the sake of brevity, let us denote by H n m the sup term in the last display. Then, by the Hölder inequality with conjugate exponents p>1 and q, weget E Y N t un t { } 1/p { 2 H p n m nm 2 N t It is not hard to see that for all n m N t there holds H n m 2 sup 2 2 Bt s ts n N t sinwt n wt n 2/q q} ) (13) Applying the Cauchy Bunyakovsky Schwarz inequality to 2 B, we deduce sup 2 2 Bt s ts sup 2 Bt t t 2 = sup B t t It remains to evaluate the sum of the qth power of the sinc functions. Since sinwt N t wt N t 1 we conclude sinwt n wt n n N t q 1 + Ct sup < 1 + 2Ct n=1 { } N 1 n + 1 q n + q n=1 1 n 1/2 q = 1 + 2q+1 q Ct

10 Average Sampling of Harmonizable Processes 3595 where sinwtq Ct = q Collecting all these estimates, we deduce (10). We are ready to formulate our main upper bound result for the mean square, time shifted average sampling truncation error un t. The almost sure sense restoration procedure will be treated too. Since we use the average sampling sum un t instead of Y N t to obtain asymptotically vanishing un t it is not enough letting N as in Remark 3.1. For the average sampling reconstruction we need some additional conditions upon w or to guarantee smaller average intervals for larger/denser sampling grids. Theorem 4.1. Let the assumptions of Theorem 3.2 and Lemma 4.1 hold true. Then, un t 2 L 2 0 t N 2 F C q t sup B t t 2N + 1 2/p where L 0 and C q t are given, respectively, by (7) and (11). If = on 1/p then lim N un t = t in mean square Moreover, if = N 1/2 1/p, >0, then { } P lim un t = t = 1 (14) N for all t. Proof. By direct calculation we deduce un t = E t un t 2 = E t Y N t + Y N t un t 2 2E t Y N t 2 + 2E Y N t un t 2 Thus, we get the asserted upper bound by (8) and (10). Hence, for = on 1/p we obtain convergence in mean square. To derive (14), we apply the Chebyshov inequality: P N = P { t un t } 2 un t Since L 0 t = 1 as N, we obtain P N K ( ) 1 N N + 1 2/p < where K is a suitable absolute constant.

11 3596 Olenko and Pogány An application of the Borel Cantelli lemma results in the a.s. convergence (14), which completes the proof. Remark 4.1. Theorem 4.1 gives new truncation error upper bounds in time shifted average sampling restorations for wide classes of harmonizable processes. The estimate (13) and the upper bound for nm 2 N t H p n m in Lemma 4.1 can be specified for particular classes of stochastic processes which correlation or kernel functions decay rates are known. Thus, one can use the suggested approach and obtained upper bounds to sharpen the results, mentioned in the introduction. Remark 4.2. The upper bounds in Lemma 4.1 and Theorem 4.1 are sharp. It is easy to check choosing t = n and trivial stochastic process t 0 where 0 is a random variable with finite variance. Remark 4.3. By obvious reasons, in many applications they restrict the study to. In this case, we can replace the estimate (13) by 2w H n m 2 4w sup 2 Bt s 2 ts 2 Hence, instead of the estimate (10) in Lemma 4.1 we obtain E Y N t un t 2 2 C q t 4w 2 sup B t t 2N + 1 2/p In this case, Theorem 4.1 ensures the perfect time shifted average sampling restoration in the mean square sense when w = N 1/p+, >0 The a.s. sense restoration (14) requires a stronger assumption. It holds when w = N 1/2+1/p+. In this case, Theorem 4.1 gives the rate of mean square convergence when the observations get dense in the whole region (infill asymptotics). Remark 4.4. In both cases, we use the so-called approximate sampling procedure, when in the restoration procedure 0 or w in some fashion. The consequence of this is that we have to restrict ourselves to the case = Therefore we deal with the non bandlimited Piranashvili type harmonizable process case. The importance of the approximate sampling procedures for investigations of aliasing errors in sampling restorations and various conditions on the joint asymptotic behaviour of N and w were discussed in detail in (Olenko and Pogány, 2006). 5. Conclusions We analyzed truncation error upper bounds in time-shifted average sampling restorations for the stochastic initial signal case. The general class of harmonizable stochastic processes was studied. The convergence of the truncation error to zero was discussed. The results are obtained under simple conditions. The conditions are weaker than those in the former literature: there are no any assumptions on decay rates of correlation or kernel functions. The analysis is new and provides

12 Average Sampling of Harmonizable Processes 3597 a constructive algorithm for determining the number of terms in the sampling expansions to ensure the approximation of stochastic processes with given accuracy. However, certain new questions immediately arise: (i) to apply the developed techniques to L p processes using recent deterministic findings in Olenko and Pogány (2010a,b); (ii) to obtain similar results for irregular/non uniform sampling restorations using the methods exposed in Olenko and Pogány (2003, 2010b). Acknowledgments This work was partly supported by La Trobe University Research Grant Sampling, wavelets and optimal stochastic modelling. References Achieser, N. I. (1992). Theory of Approximation. New York: Dover Publications, Inc. Belyaev, Yu. K. (1959). Analytical random processes. Theor. Probab. Applic. IV(4): Bochner, S. (1956). Stationarity, boundedness, almost periodicity of random-valued functions. Proc. Third Berkeley Symp. Math. Statist. Prob. Vol. 2. Berkeley, CA: University of California Press, pp Butzer, P. L., Lei, J. (1998). Errors in truncated sampling series with measured sampled values for non-necessarily bandlimited functions. Funct. Approx. Comment. Math. 26: Butzer, P. L., Lei, J. (2000). Approximation of signals using measured sampled values and error analysis. Commun. Appl. Anal. 4: Gröchenig, K. (1992). Reconstruction algorithms in irregular sampling. Math. Comp. 59: Flornes, K. M., Lyubarskii, Yu., Seip, K. (1999). A direct interpolation method for irregular sampling. Appl. Comput. Harmon. Anal. 7(3): Habib, M. K., Cambanis, S. (1981). Sampling approximation for non band limited harmonizable random signals. Inform. Sci. 23: He, G., Song, Zh., Yang, D., Zhu, J. (2007). Truncation error estimate on random signals by local average. In: Shi, Y., et al., eds., ICCS 2007, Part II, Lecture Notes in Computer Sciences Berlin: Springer-Verlag, pp Higgins, J. R. (1996). Sampling in Fourier and Signal Analysis: Foundations. Oxford: Clarendon Press. Kakihara, Y. (1997). Multidimensional Second Order Stochastic Processes. Singapore: World Scientific. Micchelli, C. A., Xu, Yu., Zhang, H. (2009). Optimal learning of bandlimited functions from localized sampling. J. Complex. 25(2): Olenko, A., Pogány, T. (2003). Direct Lagrange Yen type interpolation of random fields. Theor. Stoch. Process. 9(25)(3 4): Olenko, A., Pogány, T. (2006). Time shifted aliasing error upper bounds for truncated sampling cardinal series. J. Math. Anal. Appl. 324(1): Olenko, A., Pogány, T. (2007). On sharp bounds for remainders in multidimensional sampling theorem. Sampl. Theor. Signal Image Proc. 6(3): Olenko, A., Pogány, T. (2010a). Universal truncation error upper bounds in sampling restoration. Georg. Math. J. 17(4): Olenko, A., Pogány, T. (2010b). Universal truncation error upper bounds in irregular sampling restoration. Appl. Anal. 9(3 4): Piranashvili, Z. (1967). On the problem of interpolation of random processes. Theor. Probab. Applic. XII(4):

13 3598 Olenko and Pogány Pogány, T. (1999). Almost sure sampling restoration of bandlimited stochastic signals. In: Higgins, J. R., Stens, R. L., eds. Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Oxford: Oxford University Press, pp , Priestley, M. (1988). Non linear and Non stationary Time Series. New York: Academic Press. Rao, M. (1982). Harmonizable processes: structure theory. Enseign. Math.(2) 28(3 4): Song, Zh., Zhu, Z., He, G. (2006a). Error estimate on non bandlimited random signals by local averages. In: Aleksandrov, V. N., et al., eds. ICCS 2006, Part I, Lecture Notes in Computer Sciences Berlin: Springer-Verlag, pp Song, Zh., Yang, Sh., Zhou, X. (2006b). Approximation of signals from local averages. Appl. Math. Lett. 19: Song, Zh., Sun, W., Yang, Sh., Zhu, G. (2007). Approximation of weak sense stationary stochastic processes from local averages. Sci. China Ser. A. 50(4): Sun, W., Zhou, X. (2002). Reconstruction of bandlimited signals from local averages. IEEE Trans. Inform. Theor. 48: Sun, W., Zhou, X. (2003). Reconstruction of functions in spline subspaces from local averages. Proc. Amer. Math. Soc. 131: