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On general filtering problem of stationary processes with fixed transformation

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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.

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 L2(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 ηr is regular process, ηs(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 ) dλ, 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.

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Acknowledgements

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

Author information

Correspondence to Long suo Li.

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5. Competing interests

The author declares that they have no competing interests.

6. Authors' contributions

The studies and manuscript of this paper was written by Longsuo Li independently.

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Keywords

  • stationary processes
  • fixed transformation
  • fillering