Truncation Error Estimate on Random Signals by Local Average

Truncation Error Estimate on Random Signals by Local Average Gaiyun He1 , Zhanjie Song2, , Deyun Yang3 , and Jianhua Zhu4 1

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China [email protected] 2 School of Science, Tianjin University, Tianjin 300072, China [email protected] 3 Department of Information Science, Taishan College, Taian 271000, China [email protected] 4 National Ocean Technique Center, Tianjin 300111, China [email protected]

Abstract. Since signals are often of random characters, random signals play an important role in signal processing. We show that the bandlimited wide sense stationary stochastic process can be approximated by Shannon sampling theorem on local averages. Explicit truncation error bounds are given. Keywords: stochastic process, random Signals, local averages, truncation error, Shannon sampling theorem.

1

Introduction and the Main Result

The Shannon sampling theorem plays an important role in signal analysis as it provides a foundation for digital signal processing. It says that any bandlimited function f , having its frequencies bounded by πW, can be recovered from its sampled values taken at instances k/W, i.e. f (t) =

+∞  k=−∞

 f

k W

 sinc(Wt − k),

(1)

where sinc(t) = sinπt/(πt), t = 0, and sinc(0) = 1. This equation requires values of a signal f that are measured on a discrete set. However, due to its physical limitation, say the inertia, a measuring apparatus may not be able to obtain exact values of f at epoch tk for k = 0, 1, 2, · · ·. Instead, what a measuring apparatus often gives us is a local averages of f near tk for each k. The sampled values defined as local averages may be formulated by the following equation 

Corresponding author. Supported by the Natural Science Foundation of China under Grant (60572113, 40606039) and the Liuhui Center for Applied Mathematics.

Y. Shi et al. (Eds.): ICCS 2007, Part II, LNCS 4488, pp. 1075–1082, 2007. c Springer-Verlag Berlin Heidelberg 2007 

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 f, uk  =

f (x)uk (x)dx

(2)

for some collection of averaging functions uk (x), k ∈ ZZ, which satisfy the following properties,  σ σ uk (x)dx = 1. (3) supp uk ⊂ [xk − , xk + ], uk (x) ≥ 0, and 2 2 Where uk for each k ∈ ZZ is a weight function characterizing the inertia of measuring apparatus. Particularly, in an ideal case, the function is given by Dirac δ-function, uk = δ(· − tk ), because f, uk  = f (tk ) is the exact value of tk . The local averaging method in sampling was studied by a number of papers [1]- [6] form 1994 to 2006. The associated truncation error of (1) is defined dy   +N  k f (k/W) sin πWt  f . (4) = f (t) − f (−1)k RN sinc(Wt − k) = W π Wt − k |k|>N

k=−N

But on the one hand, we can not finish a infinite number of terms in practise, we only approximated signal functions by a finite number of terms. Which is called truncation error deal with bounds by a number of papers [7]- [15]. On the other hand, since signals are often of random characters, random signals play an important role in signal processing, especially in the study of sampling theorems. For example, a signal of speech, where the random portion of the function may be white noise or some other distortion in the transmission channel, perhaps given via a probability distribution. So there are a lots of papers on this topic too. Such as [16]-[24]. Now we give truncation error bounds random signals by local averages. Before stating the results, let us introduce some notations. Lp (IR) is the space of all measurable functions on IR for which f p < +∞, where  f p :=

1/p

+∞

−∞

|f (u)|p du

f ∞ := ess sup |f (u)|,

,

1 ≤ p < ∞,

p = ∞.

u∈IR

BπW,p is the set of all entire functions f of exponential type with type at most πW that belong to L2 (IR) when restricted to the real line [25]. By the PaleyWiener Theorem, a square integrable function f is band-limited to [−πW, πW] if and only if f ∈ BπW,2 . Given a probability space (Ω, A, P) [26] , a real-valued stochastic process X(t) := X(t, ω) defined on IR × Ω is said to be stationary in weak sense if E[X(t)2 ] < ∞, ∀t ∈ IR, and the autocorrelation function  RX (t, t + τ ) := X(t, ω)X(t + τ, ω)dP (ω) Ω

is independent of t ∈ IR, i.e., RX (t, t + τ ) = RX (τ ).

Truncation Error Estimate on Random Signals by Local Average

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A weak sense stationary process X(t) is said to be bandlimited to an interval [−πW, πW] if RX belongs to BπW,p for some 1 ≤ p ≤ ∞. Now we assume that uk which are given by (3) satisfy the following properties. i)

supp uk ⊂ [ constants;

ii)

uk (t) ≥ 0,

k k − σk , + σk ], where σ/4 ≤ σk , σk ≤ σ/2, σ are positive W W

 uk (t)dt = 1; 

iii) m = inf {mk }, where mk := k∈Z

k/ W +σ/4

k/ W −σ/4

uk (t)dt.

(5)

In this cases, The associated truncation error of random signals X(t, ω) is defined dy +N  X RN = X(t) − X, uk  sinc(Wt − k). (6) k=−N

where the autocorrelation function of the weak sense stationary stochastic process X(t, ω) belongs to BπW,2 , and W > W > 0. The following results is proved by Belyaev and Splettst¨ osser in 1959 and 1981, respectively. Proposition A. [16, Theorem 5]) If the autocorrelation function of the weak sense stationary stochastic process X(t, ω) belongs to BπW,2 , for W > W > 0, we have ⎛

2 ⎞   N



  X∗ 2  k



X E |RN | = E ⎝ X(t, ω) − , ω sinc(W t − k) ⎠



W k=−N

≤

16RX (0)(2 + |t| W)2 . π 2 (1 − W/ W)2 N 2

(7)

Proposition B. [17, Theorem 2.2]) If the autocorrelation function of the weak sense stationary stochastic process X(t, ω) belongs to Bπ W,p for some 1 ≤ p ≤ 2 and Ω > 0, then ⎛

2 ⎞   N



 k



lim E ⎝ X(t, ω) − (8) X , ω sinc(W t − k) ⎠ = 0. N →∞



W k=−N

For this case, we have the following result. Theorem C. If the autocorrelation function RX of a weak sense stationary stochastic process X(t, ω) belongs to BW,2 , for W > W > 0 and 2/δ ≥ N ≥ 100, we have   2  X 2 ln N 32RX (0)(2 + |t| W)2  , (9) E |RN | ≤ 14.80R ∞ + π 2 (1 − W/ W)2 N where {uk (t)} is a sequence of continuous functions defined by (5).

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Or, in other words E

2



X 2 | |RN



 =O

ln N N

2 N → ∞.

,

(10)

Proof of the Main Result

Let us introduce some preliminary results first. Lemma D. [27] One has for q  > 1, 1/p + 1/q  = 1, and W > 0,  q ∞   2 q < p . |sinc(W t − k)|q ≤ 1 +  π q −1

(2.1)

k=−∞

Lemma E. [26] If a stationary stochastic process X(t, ω), t ∈ [a, b] is continuous in mean square, f (t), g(t) are continuous function on [a, b], then

     b b b b E f (s)X(s)ds · g(t)X(t)dt = f (s)g(t)R(s − t)dsdt. (2.2) a

a

a

a

Lemma F. Suppose that the autocorrelation function RX of the weak sense stationary stochastic process X(t, ω) belongs to Bπ W,2 and W > 0, and satisfies  RX (t) ∈ C(IR). Let   j δ ; D W 2

       



j j j j := sup

RX −RX −δ ∗∗ − RX +δ ∗ + RX +δ ∗ −δ ∗∗

W W W W |δ∗ |≤ δ 2 |δ ∗∗ |≤ δ2

=



sup

δ

|δ∗ |≤

2

0



−δ ∗∗

0 δ∗

 RX





j + u + v dudv

. W

|δ ∗∗ |≤ δ2

Then we have for r, N ≥ 1,  2r +2N    jπ δ r δ  ; ≤ (4N + 1)(RX (t)∞ )r . D Ω 2 2 j=−2N

 Proof. Since RX is even and RX (t) ∈ C(IR), we have      r r +2N 2N   r   jπ δ jπ δ δ ; ; = D 0; +2 D D Ω 2 2 Ω 2 j=1 j=−2N  2r δ  r ≤ (4N + 1)(RX (t)∞ ) . 2

which completes the proof.

(11)

Truncation Error Estimate on Random Signals by Local Average

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Proof of Theorem C. From Proposition A, Proposition B and Lemma E we have ⎡

2 ⎤  k/ W +δk N



  X 2



uk (t)X(t, ω)dtsinc(W t − k) ⎦ E |RN | = E ⎣ X(t, ω) −



 k=−N k/ W −δk

  N

 k

= E X(t, ω) − X , ω sinc(W t − k)

W k=−N   N  k X + , ω sinc(W t − k) W k=−N

2 ⎤  k/ W +δk N



uk (t)X(t, ω)dtsinc(W t − k) ⎦ −

 k=−N k/ W −δk  X∗ 2  = 2E |RN | ⎡

2 ⎤ 

   kπ/Ω+δk N



k



+2E ⎣

uk (t)X (t, ω) dt sinc(W t − k) ⎦ X ,ω −



 W k/ W −δk k=−N

N      

 δk  X∗ 2  k k

X , ω uk + s ds = 2E |RN | + 2E

 W W −δk k=−N

2 ⎤   δk



uk (k/ W +t)X (kπ/Ω + t, ω) dt sinc(W t − k) ⎦ −

 −δ k

N   X∗ 2  | +2 = 2E |RN



N 

k=−N j=−N

       k j j k E X ,ω X , ω uk ( + u)uj + v dudv   W W W W −δk −δk         δk  δk   k j k j E X − ,ω X + v, ω uk + u uj + v dudv   W W W W −δk −δk         δk  δk   k j k j E X − + u, ω X , ω uk + u uj + v dudv   W W W W −δk −δk          δk  δk   k j k j E X +u, ω X +v, ω uk +u uj +v dudv +   W W W W −δk −δk  δk



 δk

·|sinc(W t − k)||sinc(W t − j)|   N  δk N    X∗ 2  = 2E |RN | +2

   (k − j) (k − j) − RX −v   W W k=−N j=−N −δk −δk         (k − j) (k − j) k j −RX +u +RX +u−v uk +u uj +v dudv W W W W  δk





RX

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G. He et al.

·|sinc(W t − k)||sinc(W t − j)|   N  δk N    X∗ 2  ≤ 2E |RN | + 2 k=−N j=−N



 −δk



 δk

 −δk

D

(k − j) δ ; 2 W



   k j + u uj + v dudv · |sinc(W t − k)||sinc(W t − j)| uk W W   N N    X∗ 2  (k − j) δ ; D |sinc(W t − k)||sinc(W t − j)| = 2E |RN | + 2 2 W k=−N j=−N Using H¨older’s inequality and Lemma D, we have N 

N 

 D

k=−N j=−N

(k − j) δ ; 2 W

 |sinc(W t − k)| · |sinc(W t − j)|



p∗ ⎞1/p∗



  N N 

⎟ (k − j) δ ⎜

D ; ≤⎝ |sinc(W t − j)|

⎠

2 W

k=−N j=−N ⎛



1/q∗

N 

·

|sinc(W t − k)|q

∗

k=−N

⎛



p∗ ⎞1/p∗



  N N 

⎟ (k − j) δ ∗ 1/q∗ ⎜

≤ (p ) D ; |sinc(W t − j)|

⎠ ⎝

2 W

k=−N j=−N ⎛



p∗ ⎞1/p∗

N

 



⎟ (k − j) δ ∗⎜

≤p ⎝ D , ; |sinc(W t − j)|

⎠

2 W

k=−N j=−N N 

where 1/p∗ + 1/q ∗ = 1. By Hausdorff-Young inequality [28, page176] and Lemma F, we have

p∗ ⎞1/p∗



N  

⎟

 (k − j) δ ⎜

|sinc(W t − j)|

⎠ D ; ⎝

2 W

k=−N j=−N ⎛

N 

⎛ ≤⎝

2N  

 D

j=−2N

≤ (4N + 1)1/r

j δ ; W 2

r∗

⎞1/r∗ ⎛ ⎠

⎝ ⎛

∗

2N  j=−2N

⎞1/s∗ ∗

|sinc(W t − j)|s ⎠

⎞1/s∗  2 2N  ∗ δ  ⎝ RX (t)∞ |sinc(Ωt − jπ)|s ⎠ 2 j=−2N

Truncation Error Estimate on Random Signals by Local Average

∗

 ≤ (4N + 1)1/r RX (t)∞



1 N

2

⎛ ⎝

∞ 

1081

⎞1/s∗ ∗

|sinc(Ωt − jπ)|s ⎠

,

j=−∞

where 0 ≤ 1/s∗ + 1/r∗ − 1 = 1/p∗ . Let r∗ = ln N/2. Notice that N ≥ 100, we have

∗

(4N + 1)1/r