R E S E A R C H
Open Access
Approximation of multidimensional
stochastic processes from average sampling
Zhanjie Song
**Correspondence: [email protected] School of Science & SKL of HESS, Tianjin University, Tianjin, 300072, China
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 Peruniˇci´c (Glas. Mat. 36(1):155-167, 2001) were improved too.
MSC: 42C15; 60G10; 94A20
Keywords: sampling theorems; mean square sampling reconstruction; almost sure sampling reconstruction; scalar and vectorial wide sense stationary processes
1 Introduction
Many mathematicians and engineers have discussed the Shannon sampling theorem or Whittaker-Kotel’nikov-Shannon sampling theorem (also called the WKS sampling theo-rem by some authors) for deterministic signals [–]. Kolmogorov, who is the most fa-mous mathematician in the last century, drew the information theorists’ attention that Kotel’nikov had published the deterministic sampling theorem years before Shannon and mentioned the stationary flow of new information investigation at Symposium on Information theory []. Soon after that, Balakrishnan [] proved that a bandlimited
ran-dom signal can be recovered from its sampled values in anL(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 [–].
In practice, signals and their sampled values are often not given in an ideal shape. For
ex-ample, 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 ofX(t,ω) at timestk.
They are just local averages ofX(t,ω) neartk. These are integrals of the stochastic process
X(t,ω) in small time intervals. In , Song, Sun, Yangetc.[, ] gave some
surpris-ing results on the average samplsurpris-ing theorems for univariate bandlimited processes in the
Lsense. In , 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 [–]. Following their idea and using some results on bandlimited random signals from [–], 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.Lp(R) is the
space of all measurable functions onRfor whichfp< +∞, where
fp:= +∞
–∞
f(u)pdu
/p
, ≤p<∞,
f∞:=ess sup
u∈R
f(u), p=∞.
Bπ ,is the set of all entire functionsf of exponential type with type at mostπ that
belong toL(R) when restricted to the real line; see []. By the Paley-Wiener theorem, a
square integrable functionf is bandlimited to [–π ,π ] if and only iff ∈Bπ ,.
Given a probability space (W,A,P), a real or complex valued stochastic processX(t) :=
X(t,ω) defined on R×W is said to be stationary in a weak sense if for all t∈ R,
E[X(t)X(t)] <∞,X(t) is the complex conjugate ofX(t), and the autocorrelation function
RX(t,t+τ) :=
WX(t,ω)X(t+τ,ω)dP(ω)
is independent oft∈R,i.e.,RX(t,t+τ) depends only onτ∈R. For this reason, we will use
RX(τ) to denoteRX(t,t+τ).
The following results on a one-dimensional real- or complex-valued stochastic process
are known. Recall that the functionsincis defined as
sinc(t) =
⎧ ⎨ ⎩
sinπt
πt , ift= ;
, ift= . (.)
Proposition A ([], Theorem ) Let X(t), –∞<t<∞, 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[–π ,π ].Then X(t) has the representation
X(t) =lim ∞
n=–∞
X
n
sinc(t–n) (.)
for every t,wherelimstands for limit in the mean square sense,i.e.,
lim
N→∞E
X(t) –
N
n=–N
X
n
sinc(t–n)
= . (.)
Proposition B([], Theorem ) Let X(t), –∞<t< +∞be a process with a bounded spectrum.If its covariance has the form
RX(τ) = π
–π
where F(λ)is a spectral function of X(t),then for any fixed number*>and almost all
sampling functions,the formula
X(t,ω) = ∞
k=–∞ X
k *,ω
·sinc*t–k (.)
is valid,i.e.,
P
X(t,ω) = lim
N→∞ N
k=–N
X
k *,ω
·sinc*t–k
= . (.)
Given a probability space (W,A,P), ad-dimensional real- or complex-valued stochastic
processX–––→(t) :=–––––X(t,→ω) = (X(t),X(t), . . . ,Xd(t)) defined onRd×Wis said to be stationary
in a weak sense if for allt∈Randi= , , . . . ,d,E[Xi(t)Xi(t)] <∞, and the autocorrelation
function
RXiXj(t,t+τ) :=
WXi(t,
ω)Xj(t+τ,ω)dP(ω)
is independent oft∈R,i.e., the functionsRXiXj(t,t+τ) depend onτ∈Ronly. We will use
RXiXj(τ) to denoteRXiXj(t,t+τ); see [].
Ad-dimensional weak sense stationary process–X––→(t) is said to be bandlimited to an
in-terval [–π ,π ] if and only ifR→–
X,→–X(τ) belongs toBπ ,. More precisely, this means that
RXjXk(τ),j,k= , , . . . ,dbelongs toBπ jk,,i.e.,
RXjXk(τ) = π jk
–π jk
eiτ λdF
jk(λ), (.)
and=max(jk,j,k= , , . . . ,d).
It is well known that all metrics onRdare equivalent. Thus, we will apply the max-norm
of–X––→(t):
–––→
X(t)= max
i=,,...,dE
Xi(t)Xi(t)
. (.)
The measured sampled values for–X––→(t) fortk,k∈Zare
––→ X(·), –→uk
=
–––→
X(t)·–––uk→(t)dt (.)
for some collection of averaging functions–––uk→(t) = (uk(t),uk(t), . . . ,udk(t)),k∈Z, which
are convex functions satisfying the following properties:
suppuik⊂
tk–σik,tk+σik
,
uik(t)≥, i= , , . . . ,d, and
uik(t)dt= ,
(.)
whereδi/≤σik,σik ≤δi,δiare some positive numbers. In this paper, we will use notations
δ*=max{δ
,δ, . . . ,δd}andδ*=min{δ,δ, . . . ,δd}.
Proposition C([], Theorem ) Let–X––→(t), –∞<t< +∞be a d-dimensional weak sense stationary process with a bounded spectrum and cross-correlation functions RXjXk(τ)
sat-isfying(.),then we have
lim
N→∞ –––→
X(t) –X**(t),X**(t), . . . ,Xd**(t)= , (.)
where
Xi**(t) =
Ni
ni=–Ni
X
ni
* ii
sinc*iit–ni
, i= , , . . . ,d,
N=min{N,N, . . . ,Nd}and*ii>iiare fixed numbers.
Proposition D([], Theorem ) Let–X––→(t),RXjXk(τ),X **
i (t),N and*iibe as in
Proposi-tionC,then we have
P–X––→(t) = lim
N→∞
X**(t),X**(t), . . . ,Xd**(t)= . (.)
2 Lemmas and the main results
Let us introduce some preliminary results first.
Lemma .([], Lemma .) For any> and p,q > satisfying/p + /q = and
> ,we have
∞
k=–∞
sinc(t–kπ)q ≤ +
π
q
q
q – <p. (.)
Lemma . Suppose that X(t,ω)is a weak sense stationary stochastic process with its au-tocorrelation function RXX belonging to Bπ ,and satisfying R XX(t)∈C(R).For all j∈Z+,
and|σ*| ≤δ,|σ**| ≤δ,let
D(j/;δ) :=supRXX(j/) –RXX
j/–σ**–RXX
j/+σ*+RXX
j/+σ*–σ**
=sup
–σ**
σ*R
XX(j/+u+v)du dv .
Then for all r,M,N∈Z+,we have
N
j=–M
D(j/;δ)r≤(M+N+ )δrR XX(t)r∞. (.)
Proof SinceRXXis even andR XX(t)∈C(R), we have
N
j=–M
D(j/;δ)r
=D(;δ)r+
M
j=
D(j/;δ)r+
N
j=
=sup
–σ**
σ*R
XX(u+v)du dv r+
M
j=
sup
–σ**
σ* R
XX(j/+u+v)du dv r
+
N
j=
sup
–σ**
σ*R
XX(j/+u+v)du dv r
≤δR XX(t)∞r+ (M+N)δR XX(t)∞r ≤(M+N+ )δrR XX(t)r∞.
The proof is now completed.
Following Belyaev’s traces in [], we easily conclude the following results.
Lemma . Suppose that X(t,ω)is a weak sense stationary stochastic process with its au-tocorrelation function RXXbelonging to Bπ ,.For any*>,we have
E
X(t,ω) – N
k=–N
X
k *,ω
sinc*t–k
≤E|X(t)| πN
*|t|+
( –/*)
. (.)
The following results are the main results in this paper. To state these results, we
intro-duce the following notations. For any integerk,
Xi*(k/,ω) =
k/+σik
k/–σik
uik(t)Xi(t,ω)dt (.)
and
–––––––––→
X*(k/,ω) =X*(k/,ω),X*(k/,ω), . . . ,Xd*(k/,ω), (.)
where{uk(t)}is a sequence of continuous functions defined by (.).
ForMi,Ni∈Z+,i= , , . . . ,d, we define M*=max{M,M, . . . ,Md},N*=max{N,N,
. . . ,Nd}, andM*=min{M,M, . . . ,Md},N*=min{N,N, . . . ,Nd},
Xi*(t,Mi,Ni,ω) = Ni
k=–Mi
k/+σik
k/–σik
uik(t)Xi(t,ω)dt (.)
and
–––––––––––––––––––––––→
X*t,M*,N*,M*,N*,ω
=X*(t,M,N,ω),X*(t,M,N,ω), . . . ,Xd*(t,Md,Nd,ω)
. (.)
Theorem . Suppose that–X––→(t)is a weak sense stationary stochastic process with its cor-relation function R→–
/δ≥min{M
iNi} ≥,the following is valid:
E
Ni
k=–Mi
Xi
k/*,ω–Xi*k/*,ωsinc*t–k
≤.R X
iXi(t)∞
ln(MiNi)
MiNi
. (.)
Consequently,
lim
M*,N*→∞ E
N*
k=–M*
––––––––––→
Xk/*,ω––––––––––––X*k/*,→ωsinc*t–k
= , (.)
where M*=min{M,M, . . . ,Md},N*=min{N,N, . . . ,Nd}.
Theorem . Suppose that–X––→(t)is a weak sense stationary stochastic process with its
corre-lation function R→–
X→–Xbelonging to Bπ ,and satisfying R Xi,Xi(t)∈C(R).Then for
*>≥
andδ≤/N,we have
P
–––––→
X(t,ω) = lim
N→∞
N
k=–N
–––––––––––→
X*k/*,ω·sinc*t–k
= . (.)
Obviously, when uik(t) =δ(·– kn), i= , , . . . ,d, where δ stands for the Dirac
delta-function, Theorem . and Theorem . reduce to Proposition C and Proposition D,
re-spectively. Whend= , we recover Proposition A and Proposition B, respectively.
3 Proof of the main results
Proof of Theorem. Using Proposition A and following the methods in [], we have
E
Ni
k=–Mi
Xi
k *,ω
–Xi*
k *,ω
sinc*t–k
=E Ni
k=–Mi σk
–σk
Xi
k *,ω
uik
k *+s
ds
–
σk
–σk
uk
k/*+tXi
k/*+t,ωdt
sinc*t–k
= Ni
k=–Mi Ni
j=–Mi σik
–σik
σij
–σij
RXiXi
k–j *
–RXiXi
k–j * –v
–RXiXi
k–j * +u
+RXiXi
k–j * +u–v
uk
k *+u
uj
j *+v
du dv
×sinc*t–ksinc*t–j
≤
Ni
k=–Mi Ni
j=–Mi σik
–σik
σij
–σij
D
k–j * ;δi
uik
k *+u
uij
j *+v
×sinc*t–ksinc*t–j
=
Ni
k=–Mi Ni
j=–Mi
D
k–j * ;δi
sinc*t–ksinc*t–j.
Applying Hölder’s inequality, we get
Ni
k=–Mi Ni
j=–Mi
D
k–j * ;δi
sinc*t–k·sinc*t–j
≤p*
N i
k=–Mi
Ni
j=–Mi
D
k–j * ;δi
sinc*t–j
p*/p* ,
where /p*+ /q*= . By the Hausdorff-Young inequality (see [], p.), we have
Ni
k=–Mi
Ni
j=–Mi
D
k–j * ;δ
sinc*t–j
p*/p*
≤(Ni+ Mi+ )/r
* R X
iXi(t)∞
MiNi
∞
j=–∞
sinc*t–jπs*
/s*
,
where ≤/s*+/r*– = /p*. Letr*=ln(M
iNi)/. Notice thatMi,Ni≥ andMiNi≥,
we have
(Ni+ Mi+ )/r
*
≤.e.
Lets*= r*/(r*– ) ands = r*. Then /s*+ /s = andp*= r*=ln(MiNi)/. We get
∞
j=–∞
sinc*t–jπs *
/s*
≤ + π
s*
s*– /s*
=
p*+
π
p*/p *
p*≤ln(MiNi)
π .
Hence, it holds
E
Ni
k=–Mi
Xi
k *,ω
–
k/*+σik
k/*–σ
ik
uik(t)Xi(t,ω)dt
sinc*t–k
≤.R X(t)∞ln
(M iNi)
MiNi
.
Thus (.) is valid. The second assertion of the theorem follows immediately. This
com-pletes the proof.
Proof of Theorem. Define
YXNi(t,ω) :=Xi(t,ω) –Xi*(t,N,N,N,N,ω), (.)
ZNXi(t,ω) :=Xi(t,ω) – N
k=–N
Xi
k *,ω
WXNi(t,ω) :=
N
k=–N
Xi
k *,ω
sinc*t–k–Xi*(t,N,N,N,N,ω). (.)
Using Theorem ., we have
EYXNi(t,ω)
≤.R XiXi(t)∞
ln
N N
+E|X
i(t)|
πN
*|t|+
( –/*)
. (.)
Thus, for any fixedt, we have
∞
n=N
EYXNi(t,ω)<∞. (.)
Thus, the series converges uniformly iftlies in any bounded interval. Using the Chebyshev
inequality, for allε> ,tlies in any bounded interval, we have
P
max
i
Xi(t,ω) –Xi*(t,N,N,N,N,ω)≥ε,i= , , . . . ,d
< ∞
n=N
On–<∞. (.)
Using the famous Borel-Cantelli lemma, we have
P
max
i
Xi(t,ω) –Xi*(t,N,N,N,N,ω)≥ε,i= , , . . . ,d
= . (.)
In other words,
P
–––––→
X(t,ω) = lim
N→∞ N
k=–N
–––––––––––→
X*k/*,ω·sinc*t–k
= . (.)
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 the-orem for multidimensional random signals from average sampling. The proposed results significaently improve the classical Shannon sampling theorem.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
ZS is the unique author of the paper, who proposed the paper idea, carried out the theory derivation, and composed the whole paper.
Acknowledgements
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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doi:10.1186/1029-242X-2012-246
