Approximation of multidimensional stochastic processes from average sampling
© Song; licensee Springer 2012
Received: 30 April 2012
Accepted: 8 October 2012
Published: 24 October 2012
The convergence property of sampling series, the estimate of truncation error in the mean square sense and the almost sure results on sampling theorem for multidimensional stochastic processes from average sampling are analyzed. These results are generalization of the classical results which were given by Balakrishnan (IRE Trans. Inf. Theory 3(2):143-146, 1957) and Belyaev (Theory Probab. Appl. 4(4):437-444, 1959) for random signals. Using inequalities in the mean square sense, the results of Pogány and Peruničić (Glas. Mat. 36(1):155-167, 2001) were improved too.
MSC:42C15, 60G10, 94A20.
Many mathematicians and engineers have discussed the Shannon sampling theorem or Whittaker-Kotel’nikov-Shannon sampling theorem (also called the WKS sampling theorem by some authors) for deterministic signals [1–7]. Kolmogorov, who is the most famous mathematician in the last century, drew the information theorists’ attention that Kotel’nikov had published the deterministic sampling theorem 16 years before Shannon and mentioned the stationary flow of new information investigation at 1956 Symposium on Information theory . Soon after that, Balakrishnan  proved that a bandlimited random signal can be recovered from its sampled values in an (i.e., mean square) sense, and Belyaev  proved that a bandlimited random signal can be recovered from its sampled values in the almost sure (with probability one) sense. For other results on bandlimited random signals, see [11–15].
In practice, signals and their sampled values are often not given in an ideal shape. For example, due to physical reasons, e.g., the inertia of the measurement apparatus, the sampled values of a signal obtained in practice may not be the exact values of at times . They are just local averages of near . These are integrals of the stochastic process in small time intervals. In 2007, Song, Sun, Yang etc. [16, 17] gave some surprising results on the average sampling theorems for univariate bandlimited processes in the sense. In 2009, Song, Wang and Xie  proved that second-order moment processes can be approximated by average sampling. Recently, He and Song  proved that a real-valued weak stationary process can be approximated by its local average in the almost sure sense. For reference to the results on average sampling for deterministic signals, see Gröchenig , Djokovic and Vaidyanathan , Aldroubi , Sun and Zhou .
Recently, quite a few researchers have started to discuss the Shannon sampling theorem for multi-band deterministic signals; see [24–26]. Following their idea and using some results on bandlimited random signals from [27–29], we give some results on the average sampling theorems for multi-band processes in this paper.
is the set of all entire functions f of exponential type with type at most π Ω that belong to when restricted to the real line; see . By the Paley-Wiener theorem, a square integrable function f is bandlimited to if and only if .
is independent of , i.e., depends only on . For this reason, we will use to denote .
Proposition A (, Theorem 1)
Proposition B (, Theorem 5)
is independent of , i.e., the functions depend on only. We will use to denote ; see .
where , , are some positive numbers. In this paper, we will use notations and .
The following results on a d-dimensional real stochastic process were known in 2001.
Proposition C (, Theorem 2)
and are fixed numbers.
Proposition D (, Theorem 3)
2 Lemmas and the main results
Let us introduce some preliminary results first.
Lemma 2.1 (, Lemma 2.1)
The proof is now completed. □
Following Belyaev’s traces in , we easily conclude the following results.
where is a sequence of continuous functions defined by (1.10).
where , .
Obviously, when , , where δ stands for the Dirac delta-function, Theorem 2.4 and Theorem 2.5 reduce to Proposition C and Proposition D, respectively. When , we recover Proposition A and Proposition B, respectively.
3 Proof of the main results
Thus (2.8) is valid. The second assertion of the theorem follows immediately. This completes the proof. □
The proof is completed. □
In this paper, we have analyzed the convergence property of sampling series, the estimate of truncation error in the mean square sense and the almost sure results on sampling theorem for multidimensional random signals from average sampling. The proposed results significaently improve the classical Shannon sampling theorem.
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 60872161, 60932007). The author would like to express his sincere gratitude to anonymous referees and professors Yanxia Ren and Renming Song for many valuable suggestions and comments which helped him to improve the paper.
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