Approximation of multidimensional stochastic processes from average sampling

Abstract

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.

1 Introduction

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 [8]. Soon after that, Balakrishnan [9] proved that a bandlimited random signal can be recovered from its sampled values in an $L^2$ (i.e., mean square) sense, and Belyaev [10] 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 $X(t,\omega )$ at times $t_k$. They are just local averages of $X(t,\omega )$ near $t_k$. These are integrals of the stochastic process $X(t,\omega )$ 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 $L^2$ sense. In 2009, Song, Wang and Xie [18] proved that second-order moment processes can be approximated by average sampling. Recently, He and Song [19] 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 [20], Djokovic and Vaidyanathan [21], Aldroubi [22], Sun and Zhou [23].

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.

Before stating the result of new work, we first introduce some notations. $L^p(\mathbb{R})$ is the space of all measurable functions on ℝ for which ${\parallel f\parallel }_p<+\mathrm{\infty }$, where

$B_{\pi \mathrm{\Omega },2}$ is the set of all entire functions f of exponential type with type at most π Ω that belong to $L^2(\mathbb{R})$ when restricted to the real line; see [5]. By the Paley-Wiener theorem, a square integrable function f is bandlimited to $[-\pi \mathrm{\Omega },\pi \mathrm{\Omega }]$ if and only if $f\in B_{\pi \mathrm{\Omega },2}$.

Given a probability space $(\mathcal{W},\mathcal{A},\mathcal{P})$, a real or complex valued stochastic process $X(t):=X(t,\omega )$ defined on $\mathbb{R}\times \mathcal{W}$ is said to be stationary in a weak sense if for all $t\in \mathbb{R}$, $E[X(t)\overline{X(t)}]<\mathrm{\infty }$, $\overline{X(t)}$ is the complex conjugate of $X(t)$, and the autocorrelation function

$$R_X(t,t+\tau ):={\int }_{\mathcal{W}}X(t,\omega )\overline{X(t+\tau ,\omega )}dP(\omega )$$

is independent of $t\in \mathbb{R}$, i.e., $R_X(t,t+\tau )$ depends only on $\tau \in \mathbb{R}$. For this reason, we will use $R_X(\tau )$ to denote $R_X(t,t+\tau )$.

The following results on a one-dimensional real- or complex-valued stochastic process are known. Recall that the function sinc is defined as

$$sinc(t)=\{\begin{array}{cc}\frac{sin\pi t}{\pi t},\hfill & \text{if }t\ne 0;\hfill \\ 1,\hfill & \text{if }t=0.\hfill \end{array}$$

(1.1)

Proposition A ([9], Theorem 1)

Let $X(t)$, $-\mathrm{\infty }<t<\mathrm{\infty }$, be a real or complex valued stochastic process, stationary in the ‘wide sense’ (or ‘second-order’ stationary) and with a spectral density vanishing outside the interval of angular frequency $[-\pi \mathrm{\Omega },\pi \mathrm{\Omega }]$. Then $X(t)$ has the representation

$$X(t)=lim\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}X\left(\frac{n}{\mathrm{\Omega }}\right)sinc(\mathrm{\Omega }t-n)$$

(1.2)

for every t, where lim stands for limit in the mean square sense, i.e.,

$$\underset{N\to \mathrm{\infty }}{lim}E\left\{\right|X(t)-\sum _{n=-N}^{N}X\left(\frac{n}{\mathrm{\Omega }}\right)sinc(\mathrm{\Omega }t-n){|}^2\}=0.$$

(1.3)

Proposition B ([10], Theorem 5)

Let $X(t)$, $-\mathrm{\infty }<t<+\mathrm{\infty }$ be a process with a bounded spectrum. If its covariance has the form

$$R_X(\tau )={\int }_{-\pi \mathrm{\Omega }}^{\pi \mathrm{\Omega }}{e}^{i\tau \lambda }dF(\lambda ),$$

(1.4)

where $F(\lambda )$ is a spectral function of $X(t)$, then for any fixed number ${\mathrm{\Omega }}^{\ast }>\mathrm{\Omega }$ and almost all sampling functions, the formula

$$X(t,\omega )=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}X(\frac{k}{{\mathrm{\Omega }}^{\ast }},\omega )\cdot sinc({\mathrm{\Omega }}^{\ast }t-k)$$

(1.5)

is valid, i.e.,

$$P\{X(t,\omega )=\underset{N\to \mathrm{\infty }}{lim}\sum _{k=-N}^{N}X(\frac{k}{{\mathrm{\Omega }}^{\ast }},\omega )\cdot sinc({\mathrm{\Omega }}^{\ast }t-k)\}=1.$$

(1.6)

Given a probability space $(\mathcal{W},\mathcal{A},\mathcal{P})$, a d-dimensional real- or complex-valued stochastic process $\overrightarrow{X(t)}:=\overrightarrow{X(t,\omega )}=(X_1(t),X_2(t),\dots ,X_d(t))$ defined on ${\mathbb{R}}^d\times \mathcal{W}$ is said to be stationary in a weak sense if for all $t\in \mathbb{R}$ and $i=1,2,\dots ,d$, $E[X_i(t)\overline{X_i(t)}]<\mathrm{\infty }$, and the autocorrelation function

$$R_{X_i{X}_j}(t,t+\tau ):={\int }_{\mathcal{W}}{X}_i(t,\omega )\overline{X_j(t+\tau ,\omega )}dP(\omega )$$

is independent of $t\in \mathbb{R}$, i.e., the functions $R_{X_i{X}_j}(t,t+\tau )$ depend on $\tau \in R$ only. We will use $R_{X_i{X}_j}(\tau )$ to denote $R_{X_i{X}_j}(t,t+\tau )$; see [28].

A d-dimensional weak sense stationary process $\overrightarrow{X(t)}$ is said to be bandlimited to an interval $[-\pi \mathrm{\Omega },\pi \mathrm{\Omega }]$ if and only if $R_{\overrightarrow{X},\overrightarrow{X}}(\tau )$ belongs to $B_{\pi \mathrm{\Omega },2}$. More precisely, this means that $R_{X_j{X}_k}(\tau )$, $j,k=1,2,\dots ,d$ belongs to $B_{\pi {\mathrm{\Omega }}_{jk},2}$, i.e.,

$$R_{X_j{X}_k}(\tau )={\int }_{-\pi {\mathrm{\Omega }}_{jk}}^{\pi {\mathrm{\Omega }}_{jk}}{e}^{i\tau \lambda }d{F}_{jk}(\lambda ),$$

(1.7)

and $\mathrm{\Omega }=max({\mathrm{\Omega }}_{jk},j,k=1,2,\dots ,d)$.

It is well known that all metrics on ${\mathbb{R}}^d$ are equivalent. Thus, we will apply the max-norm of $\overrightarrow{X(t)}$:

$$\parallel \overrightarrow{X(t)}\parallel =\underset{i=1,2,\dots ,d}{max}E[X_i(t)\overline{X_i(t)}].$$

(1.8)

The measured sampled values for $\overrightarrow{X(t)}$ for $t_k$, $k\in \mathbb{Z}$ are

$$〈\overrightarrow{X(\cdot )},\overrightarrow{u_k}〉=\int \overrightarrow{X(t)}\cdot \overrightarrow{u_k(t)}dt$$

(1.9)

for some collection of averaging functions $\overrightarrow{u_k(t)}=(u_{1k}(t),u_{2k}(t),\dots ,u_{dk}(t))$, $k\in \mathbb{Z}$, which are convex functions satisfying the following properties:

$$\begin{array}{r}supp{u}_{ik}\subset [t_k-{\sigma }_{ik}^{\mathrm{\prime }},t_k+{\sigma }_{ik}^{\mathrm{\prime }\mathrm{\prime }}],\\ u_{ik}(t)\ge 0,i=1,2,\dots ,d,\text{and}\int u_{ik}(t)dt=1,\end{array}$$

(1.10)

where ${\delta }_i/2\le {\sigma }_{ik}^{\mathrm{\prime }}$, ${\sigma }_{ik}^{\mathrm{\prime }\mathrm{\prime }}\le {\delta }_i$, ${\delta }_i$ are some positive numbers. In this paper, we will use notations ${\delta }^{\ast }=max\{{\delta }_1,{\delta }_2,\dots ,{\delta }_d\}$ and ${\delta }_{\ast }=min\{{\delta }_1,{\delta }_2,\dots ,{\delta }_d\}$.

The following results on a d-dimensional real stochastic process were known in 2001.

Proposition C ([28], Theorem 2)

Let $\overrightarrow{X(t)}$, $-\mathrm{\infty }<t<+\mathrm{\infty }$ be a d-dimensional weak sense stationary process with a bounded spectrum and cross-correlation functions $R_{X_j{X}_k}(\tau )$ satisfying (1.7), then we have

$$\underset{N\to \mathrm{\infty }}{lim}\parallel \overrightarrow{X(t)}-(X_1^{\ast \ast }(t),X_2^{\ast \ast }(t),\dots ,X_d^{\ast \ast }(t))\parallel =0,$$

(1.11)

where

$$X_i^{\ast \ast }(t)=\sum _{n_i=-N_i}^{N_i}X\left(\frac{n_i}{{\mathrm{\Omega }}_{ii}^{\ast }}\right)sinc({\mathrm{\Omega }}_{ii}^{\ast }t-n_i),i=1,2,\dots ,d,$$

$N=min\{N_1,N_2,\dots ,N_d\}$ and ${\mathrm{\Omega }}_{ii}^{\ast }>{\mathrm{\Omega }}_{ii}$ are fixed numbers.

Proposition D ([28], Theorem 3)

Let $\overrightarrow{X(t)}$, $R_{X_j{X}_k}(\tau )$, $X_i^{\ast \ast }(t)$, N and ${\mathrm{\Omega }}_{ii}^{\ast }$ be as in Proposition  C, then we have

$$P\{\overrightarrow{X(t)}=\underset{N\to \mathrm{\infty }}{lim}(X_1^{\ast \ast }(t),X_2^{\ast \ast }(t),\dots ,X_d^{\ast \ast }(t))\}=1.$$

(1.12)

2 Lemmas and the main results

Let us introduce some preliminary results first.

Lemma 2.1 ([16], Lemma 2.1)

For any $\mathrm{\Omega }>0$ and $p^{\mathrm{\prime }},q^{\mathrm{\prime }}>1$ satisfying $1/p^{\mathrm{\prime }}+1/q^{\mathrm{\prime }}=1$ and $\mathrm{\Omega }>0$, we have

$$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{|sinc(\mathrm{\Omega }t-k\pi )|}^{q^{\mathrm{\prime }}}\le 1+{\left(\frac{2}{\pi }\right)}^{q^{\mathrm{\prime }}}\frac{q^{\mathrm{\prime }}}{q^{\mathrm{\prime }}-1}<p^{\mathrm{\prime }}.$$

(2.1)

Lemma 2.2 Suppose that $X(t,\omega )$ is a weak sense stationary stochastic process with its autocorrelation function $R_{XX}$ belonging to $B_{\pi \mathrm{\Omega },2}$ and satisfying $R_{XX}^{\mathrm{\prime }\mathrm{\prime }}(t)\in C(\mathbb{R})$. For all $j\in {\mathbb{Z}}^{+}$, and $|{\sigma }^{\ast }|\le \delta$, $|{\sigma }^{\ast \ast }|\le \delta$, let

$$\begin{array}{rcl}D(j/\mathrm{\Omega };\delta )& :=& sup|R_{XX}(j/\mathrm{\Omega })-R_{XX}(j/\mathrm{\Omega }-{\sigma }^{\ast \ast })-R_{XX}(j/\mathrm{\Omega }+{\sigma }^{\ast })+R_{XX}(j/\mathrm{\Omega }+{\sigma }^{\ast }-{\sigma }^{\ast \ast })|\\ =& sup|{\int }_{-{\sigma }^{\ast \ast }}^0{\int }_{{\sigma }^{\ast }}^0{R}_{XX}^{\mathrm{\prime }\mathrm{\prime }}(j/\mathrm{\Omega }+u+v)dudv|.\end{array}$$

Then for all $r,M,N\in {\mathbb{Z}}^{+}$, we have

$$\sum _{j=-M}^N{[D(j/\mathrm{\Omega };\delta )]}^r\le (M+N+1){\delta }^{2r}{\parallel R_{XX}^{\mathrm{\prime }\mathrm{\prime }}(t)\parallel }_{\mathrm{\infty }}^r.$$

(2.2)

Proof Since $R_{XX}$ is even and $R_{XX}^{\mathrm{\prime }\mathrm{\prime }}(t)\in C(\mathbb{R})$, we have

The proof is now completed. □

Following Belyaev’s traces in [14], we easily conclude the following results.

Lemma 2.3 Suppose that $X(t,\omega )$ is a weak sense stationary stochastic process with its autocorrelation function $R_{XX}$ belonging to $B_{\pi \mathrm{\Omega },2}$. For any ${\mathrm{\Omega }}^{\ast }>\mathrm{\Omega }$, we have

(2.3)

The following results are the main results in this paper. To state these results, we introduce the following notations. For any integer k,

$$X_i^{\ast }(k/\mathrm{\Omega },\omega )={\int }_{k/\mathrm{\Omega }-{\sigma }_{ik}^{\mathrm{\prime }}}^{k/\mathrm{\Omega }+{\sigma }_{ik}^{\mathrm{\prime }\mathrm{\prime }}}{u}_{ik}(t)X_i(t,\omega )dt$$

(2.4)

and

$$\overrightarrow{X^{\ast }(k/\mathrm{\Omega },\omega )}=[X_1^{\ast }(k/\mathrm{\Omega },\omega ),X_2^{\ast }(k/\mathrm{\Omega },\omega ),\dots ,X_d^{\ast }(k/\mathrm{\Omega },\omega )],$$

(2.5)

where $\{u_k(t)\}$ is a sequence of continuous functions defined by (1.10).

For $M_i,N_i\in {\mathbb{Z}}^{+}$, $i=1,2,\dots ,d$, we define $M^{\ast }=max\{M_1,M_2,\dots ,M_d\}$, $N^{\ast }=max\{N_1,N_2,\dots ,N_d\}$, and $M_{\ast }=min\{M_1,M_2,\dots ,M_d\}$, $N_{\ast }=min\{N_1,N_2,\dots ,N_d\}$,

$$X_i^{\ast }(t,M_i,N_i,\omega )=\sum _{k=-M_i}^{N_i}{\int }_{k/\mathrm{\Omega }-{\sigma }_{ik}^{\mathrm{\prime }}}^{k/\mathrm{\Omega }+{\sigma }_{ik}^{\mathrm{\prime }\mathrm{\prime }}}{u}_{ik}(t)X_i(t,\omega )dt$$

(2.6)

and

(2.7)

Theorem 2.4 Suppose that $\overrightarrow{X(t)}$ is a weak sense stationary stochastic process with its correlation function $R_{\overrightarrow{X}\overrightarrow{X}}$ belonging to $B_{\pi \mathrm{\Omega },2}$. For ${\mathrm{\Omega }}^{\ast }>\mathrm{\Omega }>2$, $M_i,N_i\ge 10$, $i=1,2,\dots ,d$, and $1/{\delta }^2\ge min\{M_i{N}_i\}\ge {10}^4$, the following is valid:

(2.8)

Consequently,

$$\underset{M_{\ast },N_{\ast }\to \mathrm{\infty }}{lim}E\left[\right|\sum _{k=-M_{\ast }}^{N_{\ast }}[\overrightarrow{X(k/{\mathrm{\Omega }}^{\ast },\omega )}-\overrightarrow{X^{\ast }(k/{\mathrm{\Omega }}^{\ast },\omega )}]sinc({\mathrm{\Omega }}^{\ast }t-k){|}^2]=0,$$

(2.9)

where $M_{\ast }=min\{M_1,M_2,\dots ,M_d\}$, $N_{\ast }=min\{N_1,N_2,\dots ,N_d\}$.

Theorem 2.5 Suppose that $\overrightarrow{X(t)}$ is a weak sense stationary stochastic process with its correlation function $R_{\overrightarrow{X}\overrightarrow{X}}$ belonging to $B_{\pi \mathrm{\Omega },2}$ and satisfying $R_{X_i,X_i}^{\mathrm{\prime }\mathrm{\prime }}(t)\in C(\mathbb{R})$. Then for ${\mathrm{\Omega }}^{\ast }>\mathrm{\Omega }\ge 2$ and $\delta \le 1/N$, we have

(2.10)

Obviously, when $u_{ik}(t)=\delta (\cdot -\frac{k}{n})$, $i=1,2,\dots ,d$, where δ stands for the Dirac delta-function, Theorem 2.4 and Theorem 2.5 reduce to Proposition C and Proposition D, respectively. When $d=1$, we recover Proposition A and Proposition B, respectively.

3 Proof of the main results

Proof of Theorem 2.4 Using Proposition A and following the methods in [16], we have

Applying Hölder’s inequality, we get

where $1/p^{\ast }+1/q^{\ast }=1$. By the Hausdorff-Young inequality (see [30], p.120), we have

where $0\le 1/s^{\ast }+1/r^{\ast }-1=1/p^{\ast }$. Let $r^{\ast }=ln(M_i{N}_i)/4$. Notice that $M_i,N_i\ge 10$ and $M_i{N}_i\ge {10}^4$, we have

$${(2N_i+2M_i+1)}^{1/r^{\ast }}\le 1.8260e^4.$$

Let $s^{\ast }=2r^{\ast }/(2r^{\ast }-1)$ and $s^{\mathrm{\prime }}=2r^{\ast }$. Then $1/s^{\ast }+1/s^{\mathrm{\prime }}=1$ and $p^{\ast }=2r^{\ast }=ln(M_i{N}_i)/2$. We get

$${\left(\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|sinc({\mathrm{\Omega }}^{\ast }t-j\pi )|}^{s^{\ast }}\right)}^{1/s^{\ast }}\le 1+\frac{2}{\pi }{\left(\frac{s^{\ast }}{s^{\ast }-1}\right)}^{1/s^{\ast }}=(\frac{1}{p^{\ast }}+\frac{2}{\pi }{\left(p^{\ast }\right)}^{1/p^{\ast }})p^{\ast }\le \frac{ln(M_i{N}_i)}{\pi }.$$

Hence, it holds

Thus (2.8) is valid. The second assertion of the theorem follows immediately. This completes the proof. □

Proof of Theorem 2.5 Define

(3.1)

(3.2)

(3.3)

Using Theorem 2.4, we have

(3.4)

Thus, for any fixed t, we have

$$\sum _{n=N}^{\mathrm{\infty }}E{|Y_{X_i}^N(t,\omega )|}^2<\mathrm{\infty }.$$

(3.5)

Thus, the series converges uniformly if t lies in any bounded interval. Using the Chebyshev inequality, for all $\epsilon >0$, t lies in any bounded interval, we have

$$P\{\underset{i}{max}|X_i(t,\omega )-X_i^{\ast }(t,N,N,N,N,\omega )|\ge \epsilon ,i=1,2,\dots ,d\}<\sum _{n=N}^{\mathrm{\infty }}O\left(n^{-2}\right)<\mathrm{\infty }.$$

(3.6)

Using the famous Borel-Cantelli lemma, we have

$$P\left\{\underset{i}{max}\right|X_i(t,\omega )-X_i^{\ast }(t,N,N,N,N,\omega )|\ge \epsilon ,i=1,2,\dots ,d\}=0.$$

(3.7)

In other words,

$$\mathbf{P}\{\overrightarrow{X(t,\omega )}=\underset{N\to \mathrm{\infty }}{lim}\sum _{k=-N}^N\overrightarrow{X^{\ast }(k/{\mathrm{\Omega }}^{\ast },\omega )}\cdot sinc({\mathrm{\Omega }}^{\ast }t-k)\}=1.$$

(3.8)

The proof is completed. □

Conclusions

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.