Open Access

On general filtering problem of stationary processes with fixed transformation

Journal of Inequalities and Applications20112011:68

https://doi.org/10.1186/1029-242X-2011-68

Received: 5 May 2011

Accepted: 23 September 2011

Published: 23 September 2011

Abstract

A fixed transformation are given for one-dimensional stationary processes in this paper. Based on this, we propose a general filtering problem of stationary processes with fixed transformation. Finally, on a stationary processes with no any additional conditions, we get the spectral characteristics of P H η ( t ) ξ in the space L2(F X ()), and then we calculate the value of the best predict quantity Q of the general filtering problem.

Keywords

stationary processes fixed transformation fillering

1. Introduction

The Prediction theory is an important part of stationary processes, also linear filter problems is an important part of Prediction theory. The linear filtering problem of multidimensional stationary sequence and processes for a linear system are firsted studied by Rosanov in [1], and then a series of general filter problem of stationary process for a linear system are studied in [29]. Theoretically, this problem is a extend of the classic prediction problem. But it has high practical value, it also widely applied in communication, exploration, space technology and automatic control, etc.

2. Propose the problem

Let X(t), t R be (simple) wide stationary process. Let
H X = L { X ( t ) , t R } H X ( t ) = L { X ( s ) , s t , s R }

Suppose the complex-value function b(t) statisfing the following conditions

1)
b ( t ) L [ 0 , + ) L 2 [ 0 , + )
(2-1)
2)
b ( t ) f o r t < 0
(2-2)
Let
B ( λ ) = 0 b ( t ) e - i λ t d t
(2-3)

then B(λ) is boundary values analytic function B(z) in the low half plane

We can obtain
B ( λ ) 0 , a . e . L e b
Let
E = { λ : B ( λ ) = 0 } , E ¯ = ( - , ) - E
(2-4)
then
L ( E ) = 0
Let
η ( t ) = 0 b ( s ) X ( t - s ) d s = 0 b ( s ) U - s X ( t ) d s
(2-5)

where U is the shift oprator of X(t) in the space H X .

Then, for each ξ H X
  1. 1)
    Find out the value of Q,
    Q = inf ξ ̃ H X ( t ) 0 b ( s ) U - s ξ ̃ d s - ξ ̃ 2
    (2-6)
     
  2. 2)
    Then, we will prove that
    Q = | | P H η ( t ) ξ - ξ | | 2
     

and solve the spectral characteristics of P H η ( t ) in the space L2(F X ()).

3. Main result

Let the random spectral measure of X(t) is Φ X (), and the spectral measure is F X (), the broad spectral measure which also named the spectral measure of absolutely continuous part of F is f X (λ).

The Lebesgue decomposition of F X () is
F X ( d λ ) = f X ( λ ) d λ + δ ( d λ )
(3-1)
where the Lebesgue measure of δ() is singular, namely δ(λ) = χΔ(λ)F X ()
Δ ( , ), L ( Δ ) = 0, Δ ¯ = ( , ) Δ
Let
A t = h : h = 0 b ( s ) U - s y d s , y H X ( t )
(3-2)

Obviously, At is linear set.

Lemma 1. Let X(t), t R is stationary processes, F () and Z() are spectral measure and random spectral measure respectively. f(λ), φ(λ) L2(F), and (λ)|M, M > 0, where M is real number, we have
- f ( λ ) φ ( λ ) Z ( d λ ) M f ( λ ) Z ( d λ )
Proof. According to the nature of the random integral, we have
f ( λ ) φ ( λ ) Z ( d λ ) = [ | f ( λ ) φ ( λ ) | 2 F ( d λ ) ] 1 2 = [ | f ( λ ) | 2 | φ ( λ ) | 2 F ( d λ ) ] 1 2 M [ | f ( λ ) | 2 F ( d λ ) ] 1 2 = M f ( λ ) Z ( d λ )
Lemma 2.
L { A t } = H η ( t )
(3-3)

where L ( A t ) is the linear closed manifold of A t .

Proof. η (τ0), τ0t, Let y = X(τ0), we have y H X (t), and
η ( τ 0 ) = 0 b ( s ) U - s y d s = 0 b ( s ) U - s X ( τ 0 ) d s
namely, η (τ0) A t , then
H η ( t ) L ( A t )
On the other hand, h A t , we have
h = 0 b ( s ) U - s y d s

where y H X (t).

Let z l = k = 1 m l a k l X ( t k l ) , t k l t , k = 1 , 2 , , m l , a k l , is complex number. Let
| | y - z l | | 0 ( l )
while
0 b ( s ) U - s z l d s = k = 1 m l a k l 0 b ( s ) U - s X ( t k l ) d s H η ( t ) , l = 1 , 2 ,
Let, the spectral characteristics of y and z t are ψ y (λ) and ψ z l ( λ ) in the space L 2 (F X ()) respectively. According to the equation (2-3)
| B ( λ ) | = 0 b ( s ) e - i λ s d s 0 | b ( s ) | d s = M
where M > 0 is constant. According to the lemma 1, we get
h 0 b ( s ) U s z l d s = 0 b ( s ) U s ( y z l ) d s = 0 b ( s ) e i s λ ( ψ y ( λ ) ψ z l ( λ ) ) Φ X ( d λ ) d s = ( ψ y ( λ ) ψ z l ( λ ) ) 0 b ( s ) e i s λ d s Φ X ( d λ ) = ( ψ y ( λ ) ψ z l ( λ ) ) B ( λ ) Φ X ( d λ ) M ( ψ y ( λ ) ψ z l ( λ ) ) Φ X ( d λ ) = M [ | ψ y ( λ ) ψ z l ( λ ) | 2 F ( d λ ) ] 1 2 = M y z l 0 ( l )
so, h H η (t), namely
L { A t } H η ( t )

Then, it shows the equation (3-3) is correct.

According to the lemma 2, we have
Q = inf ξ ˜ H X ( t ) 0 b ( s ) U s ξ ˜ d s ξ 2 = inf h A t h ξ 2 = inf h L { A t } h ξ 2 = inf h H η ( t ) h ξ 2 = P H η ( t ) ξ ξ 2
(3-4)
According to the equation (2-5) and (2-1), we get
η ( t ) = 0 b ( s ) X ( t - s ) d s = 0 b ( s ) - e i ( t - s ) λ Φ X ( d λ ) d s = - e i t λ 0 b ( s ) e - i s λ d s Φ X ( d λ ) = - e i t λ B ( λ ) Φ X ( d λ )
on the other hand
η ( t ) = - e i t λ Φ η ( d λ )
According to the stochastic process spectral theorem and the Relevant function spectral theorem, we have
Φ η ( d λ ) = B ( λ ) Φ X ( d λ )
(3-5)
F η ( d λ ) = | B ( λ ) | 2 F X ( d λ )
(3-6)
f η ( d λ ) = | B ( λ ) | 2 f X ( λ )
(3-7)

where Φη (), F η (), f η (λ) representative the random spectral measure, spectral measure and broad spectral measure of η(t).

Lemma 3. Let ξ H X , ξ = ξ ^ + ξ ̃ , where ξ ^ H η , ξ ̃ H η , then
Q = | | ξ ̃ | | 2 + | | P H η ( t ) ξ ^ - ξ ^ | | 2
(3-8)
If, the wold decomposition of η (t)is
η ( t ) = η r ( t ) + η s ( t )
and
ξ = ξ r + ξ s
where η r is regular process, η s (t) is singular process, and ξ ^ r S η · ξ ^ s S η , S η = t H η ( t ) , then
Q = | | ξ ̃ | | 2 + | | P H η r ( t ) ξ ^ r - ξ ^ r | | 2
(3-9)
Proof. According to ξ = ξ ^ + ξ ̃ , ξ ^ H η , ξ ̃ H η , So
P H η ( t ) ξ = P H η ( t ) ξ ^ + P H η ( t ) ξ ̃ = P H η ( t ) ξ ^
According to the equation (3-4),
Q = | | P H η ( t ) ξ - ξ | | 2 = | | P H η ( t ) ξ - ( ξ ^ + ξ ̃ ) | | 2 = | | ( P H η ( t ) ξ ^ - ξ ^ ) - ξ ̃ | | 2
also, according to ξ ̃ ( P H η ( t ) ξ ^ - ξ ^ ) , So
Q = | | ξ ̃ | | 2 + | | P H η ( t ) ξ ^ - ξ ^ | | 2

namely, the equation (3-8) is correct.

When η (t) have the wold decomposition, notice
ξ ^ S = P H η ξ ^ , H η ( t ) = H η r ( t ) S η
we get
P H η ( t ) ξ ^ s = P H η ( t ) ( P S η ξ ^ ) = P S η ξ ^ = ξ ^ s
(3-10)
P H η ( t ) ξ ^ r = P H η r ( t ) ξ ^ r + P S η ξ ^ r = P H η r ( t ) ξ ^ r
(3-11)
So
P H η ( t ) ξ ^ ξ ^ 2 = ( P H η ( t ) ξ ^ r + P H η ( t ) ξ ^ s ( ξ ^ r + ξ ^ s ) 2 = P H η r ( t ) ξ ^ r ξ ^ r 2

Then, according to the equation (3-8), we get that the equation (3-9) is correct.

Lemma 4. Let ξ H X , ξ = ξ ^ + ξ ̃ , where ξ ^ H η , ξ ̃ H η , ψ(λ) is spectral characteristics of ξ in the space of L2(F X ()), ψ ^ ( λ ) is spectral characteristics of ξ ^ in the space of L2(F η ()), then
  1. 1)
    when λ E ¯ ,
    ψ ( λ ) = ψ ^ ( λ ) B ( λ ) , a . e . F X ( d λ )
    (3-12)
     
2)
| | ξ ̃ | | 2 = E | ψ ( λ ) | 2 F X ( d λ )
(3-13)
Proof. 1) According to the given conditions, we have
ξ = - ψ ( λ ) Φ X ( d λ ) ξ ^ = - ψ ^ ( λ ) Φ η ( d λ ) = - ψ ^ ( λ ) B ( λ ) Φ X ( d λ ) η ( t ) = - e i t λ Φ η ( d λ ) = - e i t λ B ( λ ) Φ X ( d λ )
So
( ξ , η ( t ) ) = e i t λ ψ ( λ ) B ( λ ) ¯ F X ( d λ )
on the other hand
( ξ , η ( - t ) ) = ( ξ ^ + ξ ̃ , η ( - t ) ) = ( ξ ^ , η ( - t ) ) = - e i t λ ψ ^ ( λ ) | B ( λ ) | 2 F X ( d λ )
According to the Fourier transformation, we have
ψ ( λ ) B ( λ ) ¯ = ψ ^ ( λ ) | B ( λ ) | 2 , a . e . F X ( d λ )
So, when λ E ¯ , we have
ψ ( λ ) = ψ ^ ( λ ) B ( λ ) , a . e . F X ( d λ )
  1. 2)
    According to ξ = ξ ^ + ξ ̃ , and ξ ^ ξ ̃ , Thus
    | | ξ | | 2 = | | ξ ^ + ξ ̃ | | 2 = | | ξ ^ | | 2 + | | ξ ̃ | | 2
    | | ξ ̃ | | 2 = | | ξ | | 2 - | | ξ ^ | | 2 = - | ψ ( λ ) | 2 F X ( d λ ) - - | ψ ^ ( λ ) | 2 F η ( d λ ) = - | ψ ( λ ) | 2 F X ( d λ ) - - | ψ ^ ( λ ) | | B ( λ ) | 2 F X ( d λ ) = - χ E ( λ ) | ψ ( λ ) | 2 F X ( d λ ) + - χ E ¯ ( λ ) | ψ ( λ ) | 2 F X ( d λ ) - - χ E ¯ ( λ ) | ψ ^ ( λ ) | 2 | B ( λ ) | 2 F X ( d λ ) = - χ E ( λ ) | ψ ( λ ) | 2 F X ( d λ ) = E | ψ ( λ ) | 2 F X ( d λ ) (6) 
     

namely, the equation (3-13) is correct.

Theorem. Let ξ H X , ξ = ξ ^ + ξ ̃ , where ξ ^ H η , ξ ̃ H η , ψ(λ) representative the spectral characteristics of ξ in the space of L2(F X ()), ψ ^ ( λ ) representative the spectral characteristics of ξ ^ in the space of L2(F η ()). Then
  1. 1)
    When log f X ( λ ) 1 + λ 2 L 1 ( - , )
    Q = E | ψ ( λ ) | 2 F X ( d λ )
    (3-14)
     
now the spectral characteristics of P H η ( t ) ξ in the space of L2(F X ()) is
ψ ̃ ( λ ) = 0 , λ E , a . e . F X ( d λ ) ψ ( λ ) , λ E ¯ .
(3-15)
  1. 2)
    When log f X ( λ ) 1 + λ 2 L 1 ( - , )
    Q = E | ψ ( λ ) | 2 F X ( d λ ) + t | φ ( - s ) | 2 d s
    (3-16)
     
now the spectral characteristics of P H η ( t ) ξ in the space of L2(F X ()) is
ψ ̃ ( λ ) = 0 , λ E , ψ ( λ ) , λ E ¯ Δ , B ( λ ) - t e - i s λ φ ( s ) d s Γ ( λ ) , λ E Δ ¯
(3-17)

where b(t), B(λ), η(t), E and Δ is decided by the equation (2-1), (2-2), (2-3), (2-5), (2-4) and (3-1) respectively. Where Γ(λ) is the maximal factor boundary values of the spectral density f η (λ), φ(λ) is the Fourier transformation of ψ ^ ( λ ) Γ ( λ ) , where ψ ^ ( λ ) is determined by equation (3-12) and ψ ^ ( λ ) = ψ ( λ ) B ( λ ) , a.e. L.

Proof. 1) When log f X ( λ ) 1 + λ 2 L 1 ( - , ) . According to
f η ( λ ) = | B ( λ ) | 2 f X ( λ )
(3-18)
log | B ( λ ) | L 1 ( - , )
(3-19)
we know that
log f η ( λ ) 1 + λ 2 L 1 ( - , )
Thus, η(t) is singular process. So
H η = S η P H η ( t ) ξ ^ = P H η ξ ^ = P S η ξ ^ = ξ ^
According to (3-8)
Q = | | ξ ̃ | | 2 = E | ψ ( λ ) | 2 F X ( d λ ) .
On the other hand
P H η ( t ) ξ = P H η ( ξ ^ + ξ ̃ ) = P H η ξ ^ = P S η ξ ^ = ξ ^
The spectral characteristics of P H η ( t ) ξ in the space L2(F X ()) is
ψ ̃ ( λ ) = ψ ^ ( λ ) B ( λ ) , a . e . F X ( d λ )
According to the equation (3-12), we get
ψ ̃ ( λ ) = 0 , λ E , a . e . F X ( d λ ) ψ ( λ ) , λ E ¯ ,
  1. 2)

    When log f X ( λ ) 1 + λ 2 L 1 ( - , ) , according to (3-18) and (3-19), we get log f η ( λ ) 1 + λ 2 L 1 ( - , ) .

     
It shows that η(t) is non-singular process, so η(t) has regular singular decomposition, and it consistent with Lebesgue-Gramer decomposition. Let the decomposition equation is
η ( t ) = η r ( t ) + η s ( t )
where η r (t) is regular process, η s (t) is singular process
η r ( t ) = - e i t λ Φ η r ( d λ ) = - e i t λ χ Δ ¯ ( λ ) Φ η ( d λ ) η s ( t ) = - e i t λ Φ η s ( d λ ) = - e i t λ χ Δ ( λ ) Φ η ( d λ )
Let V (ds) is basic cross stochastic measure, namely
V ( Δ 1 ) = - e i λ t 2 - e i λ t 1 i λ Λ ( d λ )
where Δ1 = (t1, t2], stochastic measure Λ ( d λ ) = 1 Γ ( λ ) Φ η r ( d λ ) , Thus
η r ( t ) = - t C ( t - s ) V ( d s )
when C ( s ) = 1 2 π - e i s λ Γ ( λ ) d λ , and Γ(λ) satisfate the follow conditions
f ( λ ) = 1 2 π | Γ ( λ ) | 2
also Γ(λ) is the boundary values of the lower half plane maximum analytic functions Γ(z), notice
ξ ^ r = - ψ ^ ( λ ) χ Δ ¯ ( λ ) Φ η ( d λ ) = - ψ ^ ( λ ) Φ η r ( d λ ) = - ψ ^ ( λ ) Γ ( λ ) Λ ( d λ ) ξ ^ s = - ψ ^ ( λ ) χ Δ ( λ ) Φ η ( d λ ) = - ψ ^ ( λ ) Φ η s ( d λ )
Let φ(s) is the Fourier transforation of ψ ^ ( λ ) Γ ( λ ) , namely
φ ( s ) = 1 2 π - e i s λ ψ ^ ( λ ) Γ ( λ ) d λ
According to the Fourier transformation of random measure
ξ ^ r = - ψ ^ ( λ ) Γ ( λ ) Λ ( d λ ) = - - e - i λ s φ ( s ) d s Λ ( d λ ) = - - e i λ s φ ( - s ) d s Λ ( d λ ) = - φ ( - s ) V ( d s )
Thus
P H η r ( t ) ξ ^ r = t φ ( s ) V ( d s ) P H η r ( t ) ξ ^ r ξ ^ r 2 = t φ ( s ) V ( d s ) 2 = t | φ ( s ) | 2 d s
According to the equation (3-9), we get
Q = E | ψ ( λ ) | 2 F X ( d λ ) + t | φ ( - s ) | 2 d s
According to the Lemma 3
P H η ( t ) ξ = P H η ( t ) ξ ^ = P H η ( t ) ξ ^ r + P H η ( t ) ξ ^ s = P H η r ( t ) ξ ^ r + ξ ^ s
notice
P H η r ( t ) ξ ^ r = - t φ ( - s ) V ( d s ) = - - t e i λ s φ ( - s ) d s Λ ( d λ ) = - - t e - i λ s φ ( s ) d s 1 Γ ( λ ) Φ η r ( d λ ) = - - t e - i λ s φ ( s ) d s 1 Γ ( λ ) χ Δ ¯ ( λ ) B ( λ ) Φ X ( d λ ) ξ ^ s = - ψ ^ ( λ ) χ Δ ( λ ) Φ η ( d λ ) = - ψ ^ ( λ ) χ Δ ( λ ) B ( λ ) Φ X ( d λ )
Thus
P H η ( t ) ξ = - - t e - i λ s φ ( s ) d s B ( λ ) Γ ( λ ) χ Δ ¯ ( λ ) + ψ ^ ( λ ) B ( λ ) χ Δ ( λ ) Φ X ( d λ )
So, the spectral characteristics of P H η ( t ) ξ in the space of L2(F X ()) is
ψ ̃ ( λ ) = 0 , λ E ψ ( λ ) , λ E ¯ Δ B ( λ ) - t e - i λ s φ ( s ) d s Γ ( λ ) , λ E Δ ¯

4. Conclusions

A fixed transform is given which based on one-dimensional a stationary processes in this paper. Also, we propose a general filtering problem. Then, in the space of L2(F X ()), we get the spectral characteristics of P H η ( t ) ξ with no any additional conditions. Finally, we calculate the value of the best predict quantity of the general filtering problem.

Declarations

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant no. 10771047).

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology

References

  1. Rosanov A Iv: The spectral theory of multidimensional stationary random processes with discrete time. Uspekhi Mat. Nauk. Z3 1958, 93–142. (in Russian)Google Scholar
  2. Gong ZR, Zhu YS, Cheng QS: Optimum filter problem of random signal for a linear system. Journal of Harbin University of Science and Technology 1981., (1):Google Scholar
  3. Gong ZR, Tong GR, Cheng QS: On filtering problem of multidimensional stationary sequence for a linear system. Journal of Wuhan University (Natural Science 1982., (2):Google Scholar
  4. Zhu YS, Gong ZR: General filter problem of multidimensional stationary sequence for a linear system. Mathematical Theory and Applied Probability 1990.,5(1):Google Scholar
  5. Zhang DC: Solution of linear prediction problems with fixed transformation in stationary stochastic processes. Journal of Huazhong University of Science and Technology(Nature Science Edition) 1981.,9(2):Google Scholar
  6. Zhang XY, Xie ZJ: On the extrapolation of multivariate stationary time series for a linear system. Journal of Henan Normal University (Natural Science) 1983., (3):Google Scholar
  7. Zhang XY, Xie ZJ: On the extrapolation of multivariate stationary time series for a linear system. Journal of Henan Normal University(Natural Science) 1983., (4):Google Scholar
  8. Xie ZJ, Cheng QS: On the extrapolation of stationary time series for a linear system. Fifth European Meating on Cybesneties and Systems Research(edited by austrian society for cybernetie studies) 1980.Google Scholar
  9. Zhang XY: A few problems in the extrapolation of multivariate stationary time series for a linear system. Journal of Henan Normal University(Natural Science) 1984., (4):Google Scholar

Copyright

© Li; licensee Springer. 2011

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