A Criterion For Testing Hypothesis About Impulse Response Function

In this paper a time-invariant continuous linear system with a real-valued impulse response function is considered. A new method for the estimator construction of the impulse response function is proposed. Two criteria on the shape of the impulse response function are given.


Introduction
The problem of estimation of a stochastic linear system has been a matter of active research for the last five decades.One of the simplest models considers a black box?with some input and giving a certain output.The input may be single or multiple and there is the same choice for the output.This generates a great amount of models that can be considered.The sphere of applications of these models is vary extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models).For more details, see [15], [22] and [23].
We are interested in the estimation of the so-called impulse function from observations of responses of a SISO (single-input single-output) system to certain input signals.This problem can be considered both for linear and nonlinear systems.To solve this problem, different statistical approaches were used as well as various deterministic methods that are based on a perturbation of the system by stationary stochastic processes and the further analysis of some characteristics of both input and output processes.Let us mention two monographs on this problem by Bendat and Piersol [3] and Schetzen [22].Akaike [1] studied a MISO (multiple-input single-output) linear system and obtained estimates of the Fourier transform of the response function in each component.He considered later a scenario involving non-Gaussian processes [2].
Some methods for estimation of unknown impulse response function of linear system and the study of properties of corresponding estimators were considered in the works of Buldygin and his followers.These methods are based on constructing a sample cross-correlogram between the input stochastic process and the response of the system.

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The discrete-time sample inputoutput cross-correlogram as estimator of the response function was considered by Buldygin, Utzet, Kurochka and Zaiats [8], [11].Both asymptotic normality of finite-dimensional distributions of the estimates and their asymptotic normality in spaces of continuous functions were studied.An inequality of the distribution for supremum of estimation error in the space of continuous functions in the case of integral-type cross-correlogram estimator was obtained in [19].
In this paper a time-invariant continuous linear system with a real-valued impulse response function is considered.A new method for the construction of estimator of the impulse response function is proposed.
The paper consists of 7 sections.
In the second Section we describe the main definitions and general properties of the estimator.The input signal process is supposed to be a zero mean Gaussian stochastic process which is represented as a treamed sum with respect to orthonormal basis in L 2 (R).
In section 3 properties of the Hermite polynomials are described.Since the family of the Hermite functions forms an orthonormal basis in L 2 (R), then the input process of the system can be represented as a series with respect to the Hermite polynomials.Estimates of mathematical expectation, variance and variance of the increments for the estimator of impulse function are found.
Section 4 deals with square Gaussian random variables and processes.Inequalities for the C(T ) norm as well as for L p (T ) norm of a square Gaussian stochastic process are shown.
In the fifth section the convergence rate for the estimator of unknown impulse response function in the space of continuous functions and in the space L p ([0, A]) is investigated.
In the sixth section two criteria are developed on the shape of the impulse response function.

The estimator of an impulse response function and its properties
Consider a time-invariant continuous linear system with a real-valued square integrable impulse response function L(τ ), τ ∈ R.This means that the response of the system to an input signal X(t), t ∈ R, has the following form: and L ∈ L 2 (R).In practice the system is often supposed to be a causal linear system, that is, L(τ ) = 0 as τ < 0.
Hereinafter we will consider only such feasible system.Then the system (1) can be written as One of the problems arising in the theory of linear systems is to estimate the function L from observations of responses of the system to certain input signals.
Consider a real-valued Gaussian zero mean stochastic process X N = (X N,t (u), u ∈ R), that can be presented as where a fixed value t > 0, the system of functions {φ k (t), k = 1, ∞} is an orthonormal basis (ONB) in L 2 (R) and random variables ξ k , k ≥ 1, are independent with Eξ k = 0, Eξ k ξ l = δ kl , where δ kl is a Kronecker symbol.Let us denote If the system (1) is perturbed by the stochastic process X N , then for the output process we obtain Consider the sequence of independent copies {X N,i (u), i = 1, ..., n} of the Gaussian process (2), that perturb the system (1).That is, where ξ k,i are independent normal distributed random variables, Eξ k,i = 0 and Eξ k,i ξ l,j = δ kl δ ij , k, l = 1, N , i, j = 1, n.
By {Y N,i (t), i = 1, ..., n} denote the reactions of the system on input signals {X N,i (u)}.
An estimator for impulse function L at the point τ, τ > 0, is defined by Since L ∈ L 2 (R), then the following remarks hold true.
Remark 2.1.The integral in (1) is considered as the mean-square Riemann integral.The integral in (1) exists if and only if there exists the Riemann integral (see [14], p. 278) Since the covariance function of the process X N is , then the integral (7) exists.Therefore, there exists also the integral in (1).
Remark 2.2.The process X N,i (t − τ ) in ( 6) depends only on τ and doesn't depend on t.It follows from the definition of the process in (5).

Proof
The joint covariance function the processes X N,i and Y N,i equals From the equality above it follows that Therefore, relation ( 8) is proved.Since L ∈ L 2 (R), then the function L can be expanded into the series by orthonormal basis {φ k (t), k ≥ 1}.We obtain where a k is from (3).
Equalities (11) and ( 8) imply that We prove now the following auxiliary Lemma.

Lemma 2.2
The joint moments of LT,N is equal to where the coefficients a k are defined in (3).

Proof
By definition of estimator LT,N (6) and the values of the processes X N,i from (5) and Y N,i from (4) we have To prove the assertion it's enough to find Eξ k,i ξ l,i ξ u,j ξ v,j .Using the Isserlis?formulafor the centered Gaussian random variables [7] we obtain: By the definition of the processes X N,i and Y N,i , we have (12) will be completely proved if the equalities above and ( 14) will be substituted in (13).
The next Corollary follows from Lemma 2.2.

Corollary 2.1
The variance of the estimator LN,n is equal to Lemma 2.3 Suppose that the Lipschitz condition with a rate α ∈ (0, 1] for the functions φ k (t) holds on a segment [0, A].It means that there exist constants c k,φ such that In this case where Proof From relations ( 8) and ( 12) it follows that Therefore, by ( 15), ( 16) and (19), making elementary reduction we obtain The Lemma will be fully proved if in (20) the Lipschitz condition (17) for the function φ k (τ ), τ ∈ [0, A], will be used.

The use of Hermite polynomials for the presentation of the estimator
Consider the Hermite polynomials of the degree n ≥ 1: It is shown in the book [13] that In particular case k = 1 we have It is known that the system of the Hermite functions is an orthonormal basis (ONB) in L 2 (R) (see, for example, [13]).Suppose now that the input signal processes of the system (1) are zero mean Gaussian stochastic processes that are formed by the Hermite ONB (22).This means that the processes X N,i (u), i = 1, n, are represented in such way: It follows from (1) that the output processes Y N,i (t), t ∈ R, equal The estimator for impulse function L in the point τ, τ > 0, is defined by (6).

Consider the following conditions:
Condition A. There exists an integral where the function Z L (z) is equal to Denote Condition B. The function L(z) increases on z no faster than e z 2 /4 .It means that there exists a constant c ∈ (0, 1 4 ) that L(z)e −cz 2 → 0, z → +∞.
The following Lemma gives an estimate for |L(τ ) − E LN,n (τ )| and for the variance V ar( LT,N (τ )).

Lemma 3.1
Assume that the conditions A and B are satisfied.Then where the number K = 1, 086435.

Proof
From ( 9) it follows that Calculate the required values a k by using equality (21) and partial integration: Since H k (z)e −z 2 /4 tends to zero as z → ±∞, and by condition A the function L(z) increases on z no faster than e z 2 /4 , then Let's apply the integration by parts two times more.We obtain where g k+3 (z) is the Hermite function of and the function Z L (z) is from (25).By using ( 28) and ( 29) we obtain that From the Cramér inequality [13] it follows that for all z ∈ R |g k (z)| < K, where K = 1, 086435.Therefore, the coefficients a k can be estimated as follows From relations (30) and (31) it follows that Evaluate now the sum in (32).
Since |g k (τ )| < K and using inequality (31) for a k , we obtain that Similarly to (33) the sums above can be evaluated in the following way If substitute ( 35), ( 36) into (34), then we obtain (27).
The following Lemma gives an estimate for the difference of the Hermite functions be the Hermite functions from (22).Assume that the conditions of Lemma 3.1 are satisfied.Then for any x, y ∈ [0, A], where A > 0 is some constant, the inequality holds true, where the value c k,g is equal to where Γ(r) = ∫ ∞ 0 t r−1 e −t dt is an Euler gamma function.Proof Without limiting the generality assume that x > y ≥ 0. By the definition of the Hermite function: 4 − e y 2 4 To estimate the values I 1 and I 2 let's use the next formula that is, actually, a characteristic function for a standard Gaussian random variable.By (40) we have From (41) follows that the the quantity I 1 in the case of x, y ∈ [0, A] is bounded in a such way: then For u ≥ 0 and v > 0 the inequality holds true (see, for example, [18]) If (43) will be used for the value (42), then we obtain that Remind that Γ(r) = ∫ ∞ 0 t r−1 e −t dt is an Euler gamma function.From (41) follows that Estimate now the value I 2 from (39).By (45) we have then for x > y ≥ 0, x, y ∈ [0, A], we obtain If we substitute (44) and ( 47) into (39), we will have that (37) completely proved.

Corollary 3.1
From conditions of Lemma 3.1 it follows that where

Square Gaussian random variables and processes
In this section the definition and some properties of square Gaussian random variables and processes are presented.Let (Ω, L, P ) be a probability space and let (T, ρ) be a compact metric space with metric ρ.
For more information about properties of square Gaussian random processes see [16], [17], [18], [7], [20].Denote by N (u) the metric massiveness, that is the least number of closed balls of radius u, covering the set T with respect to the metric ρ.Let ξ(t) = {ξ(t), t ∈ T} be a square Gaussian stochastic process.Assume that there exists a monotonically increasing continuous function σ(h), h > 0, such that σ(h) → 0 as h → 0, and the inequality sup ρ(t,s)≤h Define now the following values: Let C be a maximum of t 0 and γ 0 , C = max{t 0 , γ 0 }.The next theorem gives an estimate for the large deviation probability of square Gaussian process in the norm of continuous function.The proof of the Theorem can be found in the article [19].
Theorem 4.1 Let ξ(t) = {ξ(t), t ∈ T} be a separable square Gaussian stochastic process.Suppose that there exists an increasing function r(u) ≥ 0, u ≥ 1, with the properties: r(u) → ∞ and u → ∞ and let the function r(exp{t}) be convex.Assume that the following integral

An estimate for the L p (T ) norm of a square Gaussian stochastic process
The following theorem can be found in article [21].

On the rate of convergence of the estimator of impulse response function
This section is devoted to the investigation of the rate of convergence of estimators of unknown impulse response function in the space of continuous functions and in the space L 2 ([0, A]).From Theorem 5.2 it follows that to test the hypothesis H 0 , we can use the following criterion.Criterion 2. For a given level of confidence 1 − δ, δ ∈ (0, 1), the hypothesis H 0 is rejected if

It is clear that
otherwise the hypothesis H 0 is accepted, where ε * 2,δ is from (62).

Conclusions
In this paper we considere time-invariant continuous linear system in which the impulse response function was estimated applying a new proposed method.The input signal process is supposed to be a zero mean Gaussian stochastic process which is represented as a series with respect to an orthonormal basis in L 2 (R).A particular case where the orthonormal basis is given by the Hermite functions is studied in details.Some characteristics of the estimator of impulse function such as mathematical expectation, variance and variance of the increments are described.We also investigated the convergence rate for the estimator of unknown impulse response function in the space of continuous functions and in the space L p ([0, A]).For this reason the theory of square Gaussian random variables and processes is applied, namely we use inequalities for the C(T ) and L p (T ) norms of square Gaussian stochastic process.It gives us an opportunity to construct two criteria of testing hypothesis on the shape of the impulse response function.