Digital Processing of Random Oscillations 9783110627978, 9783110625004

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Viacheslav Karmalita Digital Processing of Random Oscillations

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Viacheslav Karmalita

Digital Processing of Random Oscillations

Author Dr. Viacheslav Karmalita 255 Boylan Avenue Dorval H9S 5J7 Canada [email protected]

ISBN 978-3-11-062500-4 e-ISBN (PDF) 978-3-11-062797-8 e-ISBN (EPUB) 978-3-11-062517-2 Library of Congress Control Number: 2019935308 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston and co-pub Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck Cover image: maxkabakov/iStock/Getty Images Plus www.degruyter.com

Preface Living in the era of information and communication, mankind is faced with the fastest evolution of technology to date. Sharing knowledge and technical information among the scientists all around the world makes it easier to develop applicable solutions for sophisticated engineering problems. Access to such large amounts of information presents a big challenge for scientists and engineers to determine which information is valid. In such circumstances, the experience and background of these people become more important. This book is a very good example of such sharing of invaluable knowledge and experience. More than four decades of activities in test and engineering projects make Dr. Karmalita a knowledgeable expert in the gas turbine engine field. The highest level of education and engineering experiences in the USSR and Russia provides Dr. Karmalita with an insight on the applied random vibration procedures and methods. This book has been written thanks to his second-to-none spectacular experience in real projects related R&D, manufacturing as well as repair, maintenance and overhaul (MRO) activities. During the past century, engineering science and technology has changed drastically. The scientific society worldwide did its best to share the latest findings to pave the way for the younger generation to push forward the boundaries of knowledge. There is a complex situation here through which the technical findings from the industrial application must be shared while considering the competitive nature of such activities and the core values for the companies. Dr. Karmalita makes this real by offering reliable and novel autoregressive methods for digital processing of random oscillations. Test engineering has a vital role in developing and monitoring of the existing air engines. Once a technological optimization of technical improvement has been made in such a system, test engineering approaches can endorse the validity of the whole process. Also, lifetime expansion and reduction of maintenance costs can be achieved by persistence and online monitoring of heavy-duty industrial machines with sensible results. Digital processing of random oscillations is a strong method to be utilized in such engineering projects. I have spent nearly three decades of my life in the development and test engineering of aircraft engines. While reviewing this book, I was completely impressed by the extensive approaches that had been followed and more importantly the applicable methods and considerations mentioned in the text. Not only can the professionals and engineers dealing with industrial projects make the best use of this book, but also it is highly recommended for the university students to read book and get in touch with very useful applicable methods. Dr. Hossein Pourfarzaneh August, 2018 Tehran, Iran

https://doi.org/10.1515/9783110627978-201

Contents Preface

V

Introduction 1

1

Discrete model of random oscillations 7 1.1 Random variable and function 7 1.2 Random process and linear series 17 1.3 Equivalence of the AR(2) process to random oscillations

2

Estimation of random oscillation characteristics 29 2.1 Criterion of maximum likelihood 30 2.2 Estimation of Yule model factors 33 2.3 Influence of additive broadband noise 38 2.4 Processing of multicomponent signals 41

3

Estimates accuracy 45 3.1 Optimal sampling of continuous realizations 3.2 Statistical error of estimates 49 3.3 Bias of decrement estimates 56 3.4 Identification of oscillations 61

4

46

Experimental validation of autoregressive method 65 4.1 Digitization of measurement signals 65 4.2 Metrology of autoregressive method 68 4.3 Analysis of combustion noise 70 4.4 Contactless measurements of blade oscillation characteristics 75 4.5 Application of tracking filter to vibration signals

References Index

89

87

80

23

Introduction An inalienable part of scientific and engineering activity is cognitive action. Cognitive action is realized on a basis of formulated cognitive tasks which allow to decompose a research problem into sequential steps. Cognitive tasks may be classified as empirical, theoretical, and logical. Empirical tasks consist of revealing, careful study and precise description of facts related to research phenomena. Solutions to empiric tasks are realized by means of specific cognitive methods: observation, experiment (test), and measurements. Observation is the cognitive method which does not involve any actions related to the researched object. Observation deals only with registering an object’s properties, performance, and so on. During testing there is a deliberate interference to operational routines of the researched object as well as to its operational conditions. Testing allows to find out an object’s properties (performance) in standard operational and unconventional conditions as well as examining, for instance, the results of changes to the object’s design. Measurement is the cognitive method of obtaining quantified data about research objects. Measurement includes two relatively independent procedures: a quantified estimation of measured physical values and an empirical verification of measurement reliability (impartiality). This book deals with results of measurement of oscillation processes (vibrations) observed in mechanical machines and structures, which are related to high cycle fatigue (HCF) phenomena. Prevention of component failure caused by the HCF phenomena is the main problem, which needs to be solved to guarantee secure operation of these machines and structures. Applications of statistical methods for the analysis of vibration measurements arise from the fact that the loads experienced by machineries and structures often cannot be described by the deterministic vibration theory. Dynamic airframe loads due to air turbulence, cinematic effects inside a car driving on rough roads, ship vibrations caused by sea waves, and so on have quite an irregular and random nature. Therefore, a sufficient description of real oscillatory processes (vibrations) calls for the use of random functions. As a rule, vibration theory problems are formulated in terms of differential equations describing the system state as follows: ΛXðtÞ = EðtÞ. The operator Λ relates to an external effect E(t) (system input) to the system state X(t). It is set by corresponding equations for which suitable initial and boundary

https://doi.org/10.1515/9783110627978-001

2

Introduction

conditions have been defined. E(t) is a stochastic process when oscillations are random. In engineering practice, the linear vibration theory (described by common linear differential equations) is generally used. This theory’s fundamental concepts such as natural frequency, oscillation decrement, resonance, and so on are credited for its wide use in various technical applications. A great development of linear vibration theory, the generality of its laws and their physical interpretation motivate the researchers’ intention to transfer the studied phenomena to linear models. To do this, different linearization techniques were developed to allow notions of the linear theory such as the logarithmic decrement and natural oscillation frequency to be applied to nonlinear oscillations with close-to-sine waveform [1]. In technical applications, two types of research tasks exist: direct and inverse. The former consists of determining stochastic characteristics of the system output X(t) resulting from a random process E(t). The operator Λ (object model) is assumed to be known. The direct task enables to evaluate the effect of an operational environment on the designed object and to predict its operation under various loads even at the design stage. Thus, using a corresponding operator Λ to describe such a design, solutions may be found that most closely meet specified requirements. The inverse task is aimed at evaluating the operator Λ based on known processes E(t) and X(t), that is, finding factors of differential equations. This task is usually met at the R&D tests of prototypes to check whether some chosen designs comply with the requirements and when the parameters of operator Λ must be identified (or verified) experimentally. This information is often obtained from the observations of an operated object under specified conditions, sometimes, from natural “noise” generated by the tested object. To characterize random processes a notion of a “shaping dynamic system” is commonly used. This concept allows to consider the observing process as the output of a hypothetical system with the input being stationary Gauss-distributed (“white”) noise. Therefore, the process may be exhaustively described in terms of characteristics of that system. In the case of random oscillations, the “shaping system” is an elastic system described by the common differential equation of the second order: € + 2hXðtÞ _ + ω2 XðtÞ = EðtÞ, XðtÞ 0 where ω0 = 2π/Т0 is the natural frequency, T0 is the oscillation period, and h is the damping factor. As a result, the process X(t) can be characterized in terms of the natural frequency and logarithmic oscillations decrement δ = hT0 = 2πh/ω0 as well pffiffiffiffiffiffi as the process variance Dx or its root-mean-square (rms) deviation σx = Dx . Evaluation of these parameters is subjected to experimental data processing based on frequency or time-domain representations of oscillations. During the last century, the

Introduction

3

approach to evaluate ω0 and δ values as well as corresponding data processing concepts did not change much at all. First of all, we will look at a representation of random oscillations in the frequency domain. The expression for the spectral density (at ω > 0) of a process generated by a linear elastic system under the effect of stationary noise E(t) is as follows [2]: SðωÞ =

4Dx hω0 . · π ðω2 − ω2 Þ2 − 4h2 ω2 0

In the case of spectral density utilization, evaluation of the decrement values is linked with bandwidth measurements Δω at the points of half-power (Fig. 1) of the observed oscillations.

S(ω)

Δω

0

ω0

ω

Fig. 1: Spectral image of random oscillations.

Knowing Δω allows to estimate a decrement value by means of the following formula [3]: δ=

π · Δω : ω0

A time-domain representative characteristic of the process X(t) is given by a covariance function in the following form: ΓðτÞ = Dx · e − hjτj ½cosðωh τÞ + ðh=ωh Þ · sin ðωh jτjÞ, where ωh2 = ω02 – h2, and ωh is a natural frequency corrected for damping [2]. A typical image of the covariance function is presented in Fig. 2. Decrement evaluation requires measuring two covariance values delayed by a time interval τ divisible by T0 , for instance, for the time interval nT0. In this case a value of δ may be evaluated by proceeding from a decrement definition:

4

Introduction

Γ(τ)

0

nT0

τ

Fig. 2: Covariance function of random oscillations.

δ=

1 Γðτ = 0Þ . · ln n Γðτ = nT0 Þ

Both estimation procedures are derived from a continuous description of research phenomena, so the accuracy of estimates is linked directly to the adequacy of discrete representation of random oscillations and their spectral/covariance functions. To render the discrete representation more accurate, it is imperative to increase the sampling rate and realization size and as a result, to meet the growing requirements for computing power. This approach is based on a concept of transforming differential equations to difference ones with derivative approximation by corresponding finite differences. The resulting discrete model, being an approximation, features a methodical error which can be decreased but never eliminated. The spectral density and covariance function estimates comprise a nonparametric (nonformal) approach. In principle, any nonformal approach is a kind of art, that is, the results depend on the performer’s skills. Because of the interference of subjective factors in spectral or covariance estimates of random signals, as well as a presence of a broadband instrumentation noise in measurement signals, the accuracy of results cannot be properly determined or justified. To avoid the abovementioned difficulties, the application of linear time series models [4] with well-developed procedures for parameter estimation is more advantageous. A method for the analysis of random oscillations that is based on a parametric model corresponding discretely (no approximation error) to a linear elastic system is presented in this book. As a result, a one-to-one transformation of the model’s numerical factors to logarithmic decrement and natural frequency of random oscillations is established. This transformation allows for the development of a formal processing procedure from experimental data to obtain the estimates of δ and ω0. A straightforward mathematical description of the procedure makes it possible to optimize the discrete representation of oscillations. The proposed

Introduction

5

approach allows researchers to replace traditional subjective techniques with a formal processing procedure providing efficient estimates with analytically defined statistical uncertainties.

1 Discrete model of random oscillations The experimental data resulting from any observations of process X(t) represent a time series. This time series can be a result of continuous observation or observation at discrete instances of time. In fact, even for continuous observation of a process, the subsequent digital processing has to do with data sampled at discrete times. The time series should be understood as a sequence of real numbers х(t1), . . ., х (ti), . . ., representing results of measurements made at equally spaced instances of time. Let us take as a time measure an interval Δt = ti − ti−1, and assign the series terms in the form of symbols х1, . . ., хi, . . . The values of process X(t) can be either random or deterministic. However, values хi are always to be considered random due to the existence of measurement errors. This chapter is devoted to the presentation of mathematical models related to the probability concept as well as to examine properties of these models.

1.1 Random variable and function Because of statistical variance retesting can give different results, that is, an observed variable may have values lying within an area that forms a sample space A. Elements (points) of this space may be grouped by different ways into subspaces A1, . . ., Ai, . . ., Ak referred to as events. Finding an experimental result inside any subspace implies the occurrence of a specific event. That is to say, the experiment always results in the following event: A = A1 + A 2 + . . . + A k . A certain event Ai may be given a quantitative characteristic through the frequency of this event’s occurrence in n experiments. If m(Ai) is the number of experiments in which the event Ai was observed, then the frequency of this event (event frequency) is determined by the following expression: νðAi Þ = mðAi Þ=n. It is evident that the event frequency can be calculated only at the experiment’s completion and, generally speaking, depends on the kind of experiments and their number. Therefore, in mathematics it is postulated an objective measure P(Ai) of the event frequency called the event probability, which is independent on the results in individual experiments. It is possible to state that: PðAi Þ = lim νðAi Þ. n!∞

https://doi.org/10.1515/9783110627978-002

8

1 Discrete model of random oscillations

If the experiment result is presented by a real number Ξ called a random variable, one may represent events in a form of conditions Ξ < ξ, where ξ is a certain number. In other words, an event may be determined as a multitude of possible outcomes satisfying the non-equality Ξ < ξ. The probability of such an event is a function of ξ and is called the cumulative distribution (or just distribution) function F(ξ) of the random variable Ξ: FðξÞ = PðΞ ≤ ξÞ. It is clear that if a ≤ b, then PðaÞ ≤ PðbÞ;

Pð − ∞Þ = 0;

Pð + ∞Þ = 1.

Any distribution function is monotonous and non-diminishing. An example of such a function is represented by Fig. 3. F (ξ ) 1

ξ

0

Fig. 3: A view of probability distribution function.

If the probability distribution function F(ξ) is continuous and differentiable, its first derivative of the form f ðξÞ =

dFðξÞ dξ

is termed as the probability density function (PDF) of the random variable Ξ. Note that: PðΞ ≤ aÞ = FðaÞ = Pða ≤ Ξ ≤ bÞ = Ð∞

Ðb

Ða −∞

f ðξÞdξ;

f ðξÞdξ = FðbÞ − FðaÞ;

a

−∞

f ðξÞdξ = 1.

In practice, these are the parameters of the distribution function that are often used rather than the function itself. One of them is a mathematical expectation of the random variable Ξ:

1.1 Random variable and function

9

∞ ð

μξ = M½Ξ =

ξ · f ðξÞdξ: −∞

The expectation of any real, single-valued, continuous function g(Ξ) may be expressed in a similar way: ∞ ð

gðξÞ · f ðξÞdξ:

M½ gðΞÞ = −∞

Note that mathematical expectations are not random but are deterministic values. Of particular interest are functions of the type: gl ðΞÞ = ðΞ − μξ Þl ; whose expectations are referred to as the lth-order central moments noted as follows: αl = M½ðΞ − μξ Þl : Specifically, the value α2 = Dξ = σξ2 is the lowest-order moment, which evaluates the mean deviation of a random variable around its expectation. This central moment is called variance and σξ is referred to as the root-mean-square (rms) deviation. As an example, let us examine the probability density of random variables referred to in this chapter. The first example is related to a probability scheme characterized by maximum uncertainty of the results. It is a case when all values of variable Ξ in the range a. . .b have the same probability. The corresponding probability density (called uniform) of such a random variable Ξ is as follows: ( 1 , a ≤ ξ ≤ b, f ðξÞ = b − a 0, ξ < 0, ξ > b. A view of the uniform density probability is represented in Fig. 4.

f (ξ )

1/(b – a)

0

a

b

ξ

Fig. 4: Uniform density probability.

10

1 Discrete model of random oscillations

The uniformly distributed variable Ξ has the expectation ðb μξ =

ξ · f ðξÞdξ =

a+b ; 2

a

and variance ðb Dξ = ðξ − μξ Þ · f ðξÞdξ =

ðb − aÞ2 : 12

a

The uniform distribution has its merit when one is looking for maximum entropy of experimental results. Another type of probability density under examination is called the normal (Gaussian) distribution. The distribution of the normal value is described by the Gauss law: " # ðξ − μξ Þ2 1 exp − : f ðξÞ = pffiffiffiffiffi 2Dξ 2π · σξ Conditionally, a view of the PDF of the normal random value is presented in Fig. 5.

f (ξ )

–3σξ –2σξ

–σξ

0

σξ

2σξ

3σξ ξ

Fig. 5: PDF of Gauss distribution.

The Gauss distribution function is entirely defined by two moments: μξ and Dξ. In this case the expectation is a center of grouping of random variable values, the variance being a measure of their scatter around the expectation. When the variance is small, random variable values come grouped in the neighborhood of the expectation, and if σξ is large, generally speaking, the values will be more spread around the mathematical expectation μξ. An importance of Gaussian distribution in probability theory is grounded by the central limit theorem. Its engineering interpretation declares that summarizing (action) a big number of independent random values

1.1 Random variable and function

11

(factors) with similar distributions produces a random value (result) with a distribution tending to the normal distribution. In various applications, particularly in technical ones, there is often a need for the use of a set of random variables. In this case it is more practical to use random vector Ξ Т = (Ξ1, . . ., Ξn) instead of several random variables Ξ1, . . ., Ξn. Symbol “т” denotes the vector transposition: 0

Ξ1

1

B C B. C C Ξ=B B . C. @ A Ξn When a vector variable is used, one has to deal with multivariate distribution function of the kind: Fðξ T Þ = Fðξ 1 , ..., ξ n Þ = FðΞ1 ≤ ξ 1 , ..., Ξn ≤ ξ n Þ: If function Fðξ T Þ has partial derivatives with respect to ξi, the joint probability density of variables Ξ1, . . ., Ξn has the following form: f ðξ T Þ = f ðξ 1 , ..., ξ n Þ =

∂n Fðξ 1 , ..., ξ n Þ . ∂ξ 1 ...∂ξ n

Probability densities of the type: ∞ ð

f ðξ i Þ =

∞ ð

... −∞

f ðξ 1 , ..., ξ n Þdξ 1 ...dξ i − 1 dξ i + 1 ...dξ n −∞

are referred to as marginal. Random variables Ξ1,. . ., Ξn are called independent if f ðξ 1 , . . . , ξ n Þ = f ðξ 1 Þ . . . f ðξ n Þ. When variables are dependent, that is, the probability of Ξi depends on the remaining variables’ magnitude, then: f ðξ 1 , . . . , ξ n Þ = f ðξ i =ξ 1 , . . . , ξ i − 1 , ξ i + 1 , . . . , ξ n Þ × f ðξ 1 , . . . , ξ i − 1 , ξ i + 1 , . . . , ξ n Þ. Here f(ξi / ξ1,. . ., ξi-1, ξi+1,. . ., ξn) is a conditional probability density determining the probability of an event ξi < Ξi ≤ ξi + dξi when values of remaining (n−1) variables are known.

12

1 Discrete model of random oscillations

A statistical relationship between variables Ξi and Ξj is characterized by the second-order joint moment called the cross-covariance: γij = M½ðΞi − μi ÞðΞj − μj Þ = M½ðΞj − μj ÞðΞi − μi Þ. As it follows from its definition, the covariance is positive if values Ξi > μi (Ξi < μi) appear most often along with values Ξj > μj (Ξj < μj), otherwise the covariance is negative. It is more convenient to quantify the statistic relationship between variables through the use of the cross-correlation coefficient: ρij =

γij σ i σj

,

whose values are set to within −1. . .+1 range. The range limits (values ±1) correspond to a linear dependence of two variables; the cross-correlation coefficient and covariance are null when variables are independent. Statistical relationships between n random variables are given by a covariance matrix of the following form: 0 1 γ11 ... γ1n B C · A. Γ=@ · γn1 ... γnn From their definition, covariances γij = γij , so the covariance matrix is symmetric. For example, consider n random Gauss-distributed variables whose joint probability density is given as follows:   1 f ðξ T Þ = ð2πÞ-n=2 jHj1=2 exp − ðξ − μÞT Hðξ − μÞ ; 2 where (ξ − μ)Τ = (ξ1 − μ1, . . ., ξn − μn); H is a matrix inverse to the covariance matrix and is written as follows: n o H = Γ − 1 = ηij ,

i, j = 1, . . . , n; jHj = jΓj − 1 .

Therefore, the exponent index in the expression of the probability density f(ξT) may be represented in terms of matrix H elements as follows: ðξ − μÞT Hðξ − μÞ =

n X n X

ηij ðξ i − μi Þðξ j − μj Þ:

i=1 j=1

In particular, for two random variables Ξ1 and Ξ2 the covariance matrix Γ and inverse matrix Η are as follows:

1.1 Random variable and function

Γ=

σ21

γ12

γ21

σ22

!

1 H= 2 2 2 σ1 σ2 − γ12

;

σ22

− γ12

− γ21

σ21

13

! ,

while the probability density is written as follows: f ðξ 1 , ξ 2 Þ =  exp −

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × 2 2π σ1 σ22 − γ212 

σ22 ðξ 1 − μ1 Þ2 + σ21 ðξ 2 − μ2 Þ2 − 2γ12 ðξ 1 − μ1 Þðξ 2 − μ2 Þ 2ðσ21 σ2 − γ2 Þ 2 12

.

The joint probability density of bivariate Gaussian distribution is conditionally presented in Fig. 6. f (ξ1 ,ξ2 ) f (ξ1 )

f (ξ2) 0

μ1

μ2

ξ1

ξ2

Fig. 6: Bivariate Gauss distribution.

As rule, measurement results are a subject of processing procedure providing estimates for power, energy efficiency, economy, and so on that are characteristics of tested objects. Such characteristics may be represented as a function y of random variables ξi: y = λðξ 1 , ..., ξ i , ..., ξ n Þ. The function y may be linearized in the point (μ1, . . ., μi, . . ., μn), which allows to represent it in the form of the following equation: y=

n X

ai ξ i + b,

i=1

where b = λðμ1 , . . . , μn Þ,

ai =

∂y . ∂ξ i

The random function y as the variables ξi will have a probability density f(y) which is tied to f(ξi).

14

1 Discrete model of random oscillations

It is noteworthy to examine a sum of two independent variables ξ1 and ξ2: y = ξ 1 + ξ 2. The distribution function of y may be found in the following way: ðð FðyÞ = Pðξ 1 + ξ 2 < yÞ = f ðξ 1 , ξ 2 Þdξ 1 dξ 2 , A

with the integration region A shown in Fig. 7.

ξ1

A

ξ2 ξ1 + ξ2 = y

Fig. 7: The integration region A.

Because of independence of variables ξ1 and ξ2 their joint probability density will be just a product of f(ξi): f ðξ 1 , ξ 2 Þ = f ðξ 1 Þ · f ðξ 2 Þ. Hence the expression of F(y) may be transformed to the following expression: f ðξ 1 Þ · f ðξ 2 Þdξ 1 dξ 2 =

FðyÞ =

y −ðξ 1

∞ ð

ðð

f ðξ 1 Þdξ 1 −∞

A

f ðξ 2 Þdξ 2 . −∞

Corresponding probability density of the function y will be equal to: dFðyÞ f ðyÞ = = dy

∞ ð

f ðξ 1 Þ · f ðy − ξ 1 Þdξ 1 . −∞

Such a type of integral is called the convolution integral noted as follows: f ðyÞ = f ðξ 1 Þ*f ðξ 2 Þ. In particular, the probability density of a sum of two independent variables with Gauss law will be as follows: ∞ ð

f ðξ 1 Þ · f ðy − ξ 1 Þdξ 1 =

f ðyÞ = −∞

1.1 Random variable and function

1 2πσ1 σ2

∞ ð

−∞

15

"

# ðξ 1 − μ1 Þ2 ðy − ξ 1 − μ2 Þ2 exp − − dξ 1 = 2σ21 2σ22

1 2πσ1 σ2

∞ ð

expð − Aξ 21 + 2Bξ 1 − CÞdξ 1 , −∞

where σ21 + σ22 , 2σ1 σ2

A=

B=

μ1 y − μ2 + , 2 2σ1 2σ22

C=

μ21 ðy − μ2 Þ2 + . 2σ21 2σ22

After taking into account an existence of a solution to the integral of an exponential function in the following form: ∞ ð

rffiffiffiffi   π AC + B2 , expð − Ax ± 2Bx − CÞdx = exp − A A 2

−∞

the f(y) may be written as follows: ( ) 1 ½y − ðμ1 + μ2 Þ2 ffi × exp − f ðyÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2ðσ21 + σ22 Þ 2πðσ21 + σ22 Þ Therefore, the function y = ξ1 + ξ2 has a Gaussian distribution with the following expectation: μy = μ1 + μ2 , and the square root of the variance (rms): σy =

pffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dy = σ21 + σ22 .

Thus, the sum of two normal variables has the same distribution; its expectation and variance are the sum of the same characteristics of those variables. In the case of dependent variables, their sum still has the Gaussian law with the same expectation, but the variance will be determined by taking into account characteristics of their correlations: Dy = M½ðy − μy Þ2 y = M½ðξ 1 + ξ 2 − μ1 − μ2 Þ2  = M½ðξ 1 − μ1 Þ2 + ðξ 2 − μ2 Þ2 + 2ðξ 1 − μ1 Þðξ 2 − μ2 Þ = D1 + D2 + γ12 = σ21 + σ22 + 2ρ12 σ1 σ2 .

16

1 Discrete model of random oscillations

If the function y has a number of arguments n > 2, that is, n X

y=

ai ξ i + b,

i=1

the commutative and associative properties of the convolution integral: ½ f ðξ 1 Þ*f ðξ 2 Þ*....*f ðξ n Þ = f ðξ 1 Þ*...*½ f ðξ n − 1 Þ*f ðξ n Þ; f ðξ 1 Þ*f ðξ 2 Þ*....*f ðξ n Þ = f ðξ 2 Þ*f ðξ 1 Þ*...*f ðξ n Þ, allow to determine f(y) via the following step-by-step approach: y1 = a1 ξ 1 + a2 ξ 2 , y 2 = y 1 + a3 ξ 3 , . . yn = yn − 1 + an ξ n . In case of variables ξi with the normal distribution, the function y will have the same normal distribution with: μy =

n X

ai μi + b;

i=1

Dy =

n X

a2i Di + 2

i=1

X

ai aj γij .

i≠j

Very often, data processing may be presented as a system of linear equations: y1 =

n X

a1i ξ i + b1 ,

i=1

... yk =

n X

aki ξ i + bk ,

i=1

which may be written in the matrix form as follows: Y = A · Ξ + B. The mathematical expectation of Y is: M½Y = μY = AμΞ + B.

1.2 Random process and linear series

17

The elements of the covariance matrix of the vector Y: 1 0 D1 γ12 · γ1k C B B γ21 D2 · γ2k C C, B ΓY = B · · · C A @ · γk1 γk2 · Dk may be interpreted as the mathematical expectation of ij th element being a product of vector (Y − μΥ) and (Y – μΥ)T: 1 0 y1 − μ1 C B B y2 − μ2 C C. B ðY − μΥ Þ = B C A @. yk − μk Such an approach allows to represent the matrix ΓY as follows: ΓΥ = M½ðY − μΥ ÞðY − μΥ ÞT  = M½ðAΞ + B − AμΞ − BÞðAΞ + B − AμΞ − BÞT  = M½A · ðΞ − μΞ Þ · ðΞ − μΞ ÞT · AT  = A · M½ðΞ − μΞ ÞðΞ − μΞ ÞT  · AT = AΓΞ AT . Therefore, knowing the covariance matrix of variables allows to calculate the covariance matrix of results of these variables processing.

1.2 Random process and linear series By a random process, one implies a function X(t) whose instantaneous values are those of random variables. At discrete moments t1,. . ., ti,. . ., tn these values X(t1),. . ., X(ti),. . ., X(tn) may be viewed as a random vector XT=[X(t1),. . ., X(tn)]. Note that the order in the sequence X(t1),. . ., X(tn) is of importance for a random process, whereas the before mentioned random vector had indexes for notational convenience only. Introduced earlier, a random series x1, . . ., xi, . . . (resulting from any phenomena observation) represents a random process realization. The random series is referred to as stationary if the distribution function F(xi+1,. . ., xi+n) of any of its n values is independent of “i”. In this case any n consecutive variables have the same distribution regardless of their position in the series with the following expectation:

18

1 Discrete model of random oscillations

∞ ð

μx = M½Xðti Þ =

xi f ðxi Þdxi ; −∞

and variance Dx = σ2x



∞ ð

2

ðxi − μx Þ2 f ðxi Þdxi :

= M½ Xðti Þ − μx  = −∞

The relationship between two values xi and xi+k of the stationary series separated by a lag k is expressed by a covariance of the type:    γk = M½ Xðti Þ − μx Xðti + k Þ − μx  = γ − k ; as well as by a correlation coefficient: ρk =

γk = ρ − k: Dx

A set (in sequence) of process covariances is termed the covariance function, and a set of correlation coefficients is termed the correlation function. The Fourier transformation of the stationary process covariance function is called a one-side power spectral density: SðθÞ =

∞ 1 X γ cosðθ · iÞ, π i= −∞ i

0 ≤ θ ≤ π.

The expression for covariances represented in terms of S(θ) is as follows: ðπ γk =

cosðθ · kÞSðθÞdθ; 0

and variance, in particular, is: ðπ Dx = γ0 =

SðθÞdθ: 0

The power spectral density shows the distribution of the random process variance within a continuous frequency range 0. . .π. So a value S(θ)⋅dθ may be interpreted as an approximate portion of the process variance within the frequency range θ. . .(θ+dθ). As the power spectral density, which will simply be called the spectrum going forward, is the Fourier cosine transform of the covariance function, the knowledge of the latter is mathematically equivalent to the spectrum knowledge and vice versa. The use of random series models is based on an assumption that these series are generated by a sequence of independent impulses εi. These impulses are

1.2 Random process and linear series

19

realizations of random variables featuring a normal (Gaussian) distribution with zero expectation and variance Dε. In technical literature such a sequence of random impulses εi, εi+1, . . . is called “white” noise. It is considered that white noise transforms into the process X(ti) by means of a linear filter that computes a weighted sum of preceding values of E in the following way: xi = εi + c1 εi − 1 + c1 εi − 2 + . . . The introduction of a back-shift operator B set as Bxi = xi−1 allows to express the following: xi = CðBÞ · εi ,

(1)

where a linear filter operator C(B) called a filter transfer function is CðBÞ = 1 + c1 B + c2 B2 + . . . =

∞ X

cj Bj ,

ðc0 = 1Þ.

j=0

In such an approach B, which is taken as a dummy variable, has different values including complex ones, and whose j th power is a factor at cj. Theoretically, a series c1, c2, . . . can be finite or infinite. If this series converges at |Β |≤ 1, the filter is called stable and the series xi is stationary [4]. Let us review several types of filter models having practical applications and directly linked with a matter of this book. The autoregressive model represents a current value xi of a process in terms of a linear combination of preceding values xi−j (j > 0) and impulse εi as follows: xi = a1 xi − 1 + a2 xi − 2 + . . . + ap xi − p + εi .

(2)

The model given by eq. (2) describes a random process referred to as the p-order autoregressive process and is denoted AR( p). In the following text, a process generated by the p-order autoregressive model will be denoted as the “AR(p) process”. The term “autoregression” may be explained as follows. It is known that a linear model of the kind: X = a1 Y1 + a2 Y2 + . . . + ap Yp + E, relating to the dependent variable X with independent variables Y1,. . .,Yp plus the term E (error) is referred to as a regression model. A specific term “a regression of X on values Y1,. . ., Yp” is applicable in this case. As in eq. (2), xi is related to its preceding values and the term “autoregression” was chosen for this case. If the p-order autoregression operator is defined as follows: AðBÞ = 1 − a1 B − a2 B2 − ... − ap Bp ;

20

1 Discrete model of random oscillations

the AR(p) model may be written in a more compact form: AðBÞxi = εi . The autoregressive model is a particular form of the linear filter (1). In fact, acting in a formal way, from the above-presented expression: xi = CðBÞεi , where C(B) = A−1(B). The AR(p) process can be stationary or nonstationary. Conditions for the stationarity of the process may be defined from the above-specified condition of the convergence of series C(B). To do this let us make use of a notion of the process characteristic equation [5] that for the p-order autoregression has the form: zp − a1 zp − 1 − ... − ap =

p X

aj zp − j = 0; ða0 = 1Þ.

(3)

j=0

The eq. (3) has p roots that may be written as z1,. . ., zp, and the autoregression operator may be represented in terms of characteristic equation roots: AðBÞ =

p Y j=1

ð1 −

p Y B Þ= ð1 − zj BÞ; Bj j=1

because in the equation A(B) = 0 the roots Bj = 1/zj. Decomposing A(B) into common fractions allows us to write: xi = A − 1 ðBÞεi =

p X j=1

p ∞ X X Qj Qj ðzj BÞl εi : εi = ð1 − zj BÞ j=1 l=0

It follows that to make the series C(B) = A−1(B) converge at |B| < 1, |zj| must be less than 1; here j = 1,. . .,p. In other words, the roots of the characteristic eq. (3) have to be inside the unit circle. Let us consider the covariance function of the stationary autoregressive process. Multiplying eq. (2) by xi−j yields the expression: xi xi − j = a1 xi − 1 xi − j + a2 xi − 2 xi − j + . . . + ap xi − p xi − j + εi xi − j . A mathematical expectation of both parts of this equation provides the following expression for the covariance function (j ≥ 1): γj = a1 γj − 1 + a2 γj − 2 + . . . + ap γj − p , because preceding values xi−j are not correlated with following εi, that is, M[xi−jεi] = 0. In the case j = 0, the expression for the covariance function will be transformed to:

1.2 Random process and linear series

21

γ0 = Dx = a1 γ − 1 + a2 γ − 2 + . . . a − p γ − p + Dε , as М[xiεi] = M[εi2] = Dε. Taking into account γ−i = γi and moving covariance members to the left part of the expression yields the following expression for the AR(p) process variance: Dx =

Dε . 1 − a1 ρ1 − a2 ρ2 − ... − ap ρp

(4)

In the case j > 1, the autoregressive process covariances divided by γ0 may be related in the difference equation for process correlation coefficients (later on correlations): ρj = a1 ρj − 1 + . . . + ap ρj − p . Substituting values j =1, . . ., p in this equation will generate a system of linear equations for ai: ρ1 = a1 + a2 ρ1 + . . . + ap ρp − 1 ; ···

(5)

ρp = a1 ρp − 1 + a2 ρp − 2 + . . . + ap ; referred to as a system of Yule–Walker equations. The next type of reviewed models describes random series whose value xi is a linear function of a finite number of preceding εi−j and current εi. Such a process is described by a difference equation: xi = εi − d1 εi − 1 − d2 εi − 2 − . . . − dq εi − q

(6)

is called the q-order moving average process, MA(q). The term “moving average” can lead to a misunderstanding as the sum of weights 1, d1, . . ., dq is not necessarily equal to 1 (as in the case evaluating an average value). Nevertheless, this term has found use in a literature, and we shall keep it. The moving average operator may be defined as follows: DðBÞ = 1 − d1 B − d2 B2 − . . . − dq Bq , and the MA(q) model can be briefly written as follows: xi = DðBÞεi. It follows that the moving average process may be presented as the response of a linear filter with the transfer function D(B) to its input – noise E. Note, the MA(q) process is always stationary due to the fact that the series C(B) = D(B) is finite. Acting in the same way as in the case of the AR process, a covariance function of the MA process may be obtained from eq. (6):

22

1 Discrete model of random oscillations

( γj =

Dε ð − dj + d1 dj + 1 + . . . + dq − j dq Þ,

j = 1, . . . , q;

0,

j > q.

The corresponding correlation function has the following form: 8 < − dj + d1 dj + 1 + ... + dq − j dq , j = 1, ..., q; 1 + d21 + d22 + ... + d2q ρj = : j > q, 0, because the MA process variance is: Dx = γ0 = ð1 + d1 2 + . . . + dq 2 ÞDε . Therefore, the correlation function of MA(q) process stops with the lag q. To achieve a greater flexibility in the observed series description the AR and MA processes may be combined into one model. The combined process of the type: xi = a1 xi − 1 + . . . + ap xi − p + εi − d1 εi − 1 − . . . − dq εi − q

(7)

or AðBÞ xi = DðBÞ εi is referred to as the (p,q)-order autoregressive – moving average process and denoted ARMA(p,q). Equation (7) may be written as follows: xi =

DðBÞ 1 − d1 B − ... − dq Bq εi ; εi = 1 − a1 B − ... − ap Bp AðBÞ

and the ARMA process may be interpreted as a response to white noise E of a linear filter with a rational transfer function. As it was said previously, the MA process is always stationary. Therefore, the corresponding right-side terms in eq. (7) have no impact on conclusions defining conditions of autoregressive process stationarity. Hence, the model ARMA(p,q) describes a stationary process if the roots of the characteristic equation of AR(p) model are inside the unit circle. An expression for the ARMA process covariance function may be obtained in a way already used for the AR and MA processes: γj = a1 γj − 1 + . . . + ap γj − p + γxε ðjÞ − d1 γxε ðj − 1Þ − . . . − dq γxε ðj − qÞ. The cross-covariances of random processes X and E is given as: γxε ðjÞ = M½xi − j εi .

(8)

1.3 Equivalence of the AR(2) process to random oscillations

23

It follows from (7) that xi−j is a function of input noise impulses observed up to the instant (i − j). As such, the cross-covariance function γхε(j) = 0 for j > 0, whereas γхε (j) ≠ 0 for j ≤ 0. It means that for the ARMA(p,q) process there are q correlations ρq,. . ., ρ1 whose values are related to parameters d of the MA process and parameters a of the AR process. However, at j ≥ q+1 ρj = a1 ρj − 1 + . . . + ap ρj − p , that is, the correlation function of the ARMA process is entirely determined by the autoregression parameters. The reviewed models of random series allow to describe a large class of random processes. These models have found a practical application because they require a small number of parameters to represent a linear process. In general case these models are employed in scientific and engineering practice for approximations of experimental data. But there are cases, as it will be shown later, when these models can describe observed phenomena and model factors can have a physical interpretation.

1.3 Equivalence of the AR(2) process to random oscillations Let us review the properties of a process generated by the second-order autoregressive model of the form: xi − a1 xi − 1 − a2 xi − 2 = εi , which is called the Yule series. The stationary conditions of this model can be determined by examining its characteristic equation: z2 − a1 z − a2 = 0.

(9)

In the previous section, stationary conditions of the AR process were formulated as roots of the characteristic eq. (9) that must be inside the unit circle, that is, |zi| < 1. It is known that roots of the quadratic equation are related to factors a1 and a2 by expressions: z1 · z2 = − a2 , z1 + z2 = a1 ,

(10)

which allow to identify the stationary conditions of the Yule series with respect to the factors of the AR(2) model. From the first equation of expression (10) it follows that j − a2 j = j z1 j · j z2 j. Taking into account the requirements |z1| < 1 and |z2| 0;

− 1,

a1

< 0.

(13)

26

1 Discrete model of random oscillations

1=2

Equation (13) describes a damped sinusoid where − a2 is the damping factor, θ is the frequency, and φ is the phase. Parameters θ and φ may be expressed through autoregressive model factors in the following form [4]: a1 cos θ = pffiffiffiffiffiffiffiffiffi ; 2 − a2 tg’ =

1 − a2 tgθ. 1 + a2

Specifically, in region “4” (Fig. 8) where φ < π/2, the initial correlations are always positive and reverse the sign with growing lags. In region “3”, the correlation function always changes its sign on passage from ρo to ρ1; phase φ may have a value within π/2 . . . π range. For completeness of AR(2) model examination the spectrum of the Yule series has to be estimated. Let us take advantage of the fact that the spectrum of the output of the linear system described by the transfer function C(B) is related to the uniform white noise spectrum Dε/π through the |C(B)|2 with B = e−jθ [6]: SðθÞ =

Dε jC ðe − jθ Þj2 . π

CðBÞ =

1 ; 1 − a1 B − a2 B 2

The AR(2) model operator

therefore, the process spectrum is given as follows: SðθÞ =

Dε . π½1 + a21 + a22 − 2a1 ð1 − a2 Þ cos θ − 2a2 cosð2θÞ

(14)

The Yule series variance determined by eq. (4) takes the following form: Dx =

Dε . 1 − a1 ρ1 − a2 ρ2

The solution of the following Yule–Walker equations set for the AR(2) process as follows: ρ1 = a1 + a2 ρ1 ; ρ2 = a1 ρ1 + a2 , yields the following expressions: ρ1 =

a1 ; 1 − a2

1.3 Equivalence of the AR(2) process to random oscillations

ρ2 = a 2 +

27

a21 . 1 − a2

Their substitution into the Dx expression enables to complete variance description with model factors: Dx =

ð1 − a2 ÞDε ð1 + a2 Þ½ð1 − a2 Þ2 − a21 

.

(15)

Let us make an example materializing a view of the correlation function of the Yule series described by the following model: xi = 1.52xi − 1 − 0.9xi − 2 + εi . The theoretical correlation function of this process is calculated from eq. (11) with initial conditions ρo = 1 and ρ1= a1/(1 – a2) = 0.8. Its graphic representation is given in Fig. 9 from which it follows that a basic period of the correlation function equals 10. ρi 1

0

i

10

–1 Fig. 9: Correlation function of the Yule series.

This example shows that for model factor values from region “4” (0 ≤ а1 ≤ 2; −1 < а2 ≤ 0) the AR(2) process behavior is pseudo-periodic. Such a fact alludes to the physical interpretation of AR(2) model as a discrete analog of the linear differential equation: € + 2hXðtÞ _ + ω2 XðtÞ = EðtÞ. XðtÞ 0 This assumption is founded from a view of covariance function presented in Fig. 2 as well as from an expression of the correlation function of random oscillations in the form [6]: ρðτÞ =

e − hjτj sinðωh jτj + ’Þ , sin ’

(16)

28

1 Discrete model of random oscillations

where sinφ = ωh/ω0. Equation (16) corresponds to the expression (13) for a correlation function of the AR(2) process with factors values that lie within the region “4” (Fig. 8) ρi =

ð − a2 Þi=2 sinðθ · i + ’Þ . sin ’

(17)

So British statistician Udny Yule took advantage of the AR(2) process model in his attempt to describe a pendulum motion in the air as early as 1921. He assumed that air resistance is directly proportional to the motion rate of a pendulum, subjected to random shocks equally spaced in time. In fact, the AR(2) model factors may be linked to oscillation characteristics (decrement δ and natural frequency ω0) from the following condition: ρi = ρðτ = Δt · iÞ,

(18)

where Δt = ti – ti−1 is the time interval between neighboring members of the Yule series. Expression (18) sets the condition of statistical equivalency between sampling random oscillations X(t) and the AR(2) process. Substitution of eqs. (16) and (17) into the expression (18) gives: 2π ln ð − a2 Þ1=2 . θ pffiffiffiffiffiffiffiffiffi Taking into consideration that cosθ ¼ a1 =ð2 − a2 Þ, the relationships between factors of the AR(2) model and oscillations characteristics may be expressed as follows: ωh · Δt = θ;

ωh = where ωh = following:

δ=

1 a1 π lnð − a2 Þ, arccos pffiffiffiffiffiffiffiffiffi ; δ = 2 − a2 Δt ωh Δt

(19)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2o − h2 . The decrement definition δ = 2πh/ω0 allows to write the

ωh = ω0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi δ 1− . 2π

In particular, for values δ ≤ 0.3, the above-mentioned frequency ratio is limited by range 0,999 ≤ ωh/ω0 ≤ 1. Hence, in engineering practice a value of the natural frequency and its damped value may be assumed to be equal. As a result, processing of random oscillations is transformed into a linear task of estimating AR(2) model factors based on discrete readouts of oscillation realizations. In accordance with expression (19), the factor estimates always allow to evaluate the decrement and natural frequency of random oscillations realizations. In accordance with expression (19), the factor estimates always allow to evaluate the decrement and natural frequency of random oscillations.

2 Estimation of random oscillation characteristics Once the type of discrete model of random oscillation is identified, the methods for mathematical description of its observations by means of the AR(2) models may be implemented. This chapter deals with the elements of mathematical statistics related to adapting the autoregressive model to experimental data. Such model adaptation is realized by an estimation which yields the values of model factors. The values of factors obtained by processing measurement results are called estimates. In practice, experimental data are available only in a few process observations (realizations), sometimes only one realization. Examined statistical properties of random series, on the other hand, are determined for a multitude of realizations. Nevertheless, this fact will not limit the possibility of an evaluation procedure if the observed processes satisfy certain assumptions. Suppose there is a sequence of measurements x1, . . ., xn (realization) of a random process. Each value of xi features an expectation μx = M[Xi] and a variance Dx = M[(Xi − μx)2]. Let us describe a time average value x of a realization as follows: x = lim

n 1X xi . n i=1

n!∞ In accordance with the Birkhoff–Khinchin theorem [5], for a stationary series x1, …, xn with a finite mathematical expectation the probability of existence of variable x is P = 1. Additionally, if a stationary series meets the following condition: lim

n 1X ρ = 0, n i=1 i

(20)

n!∞ then such a series features an ergodic property which signifies an equality of the multitude and time averages: μx = x. Condition (20) is not too hard to be realized since in practice it is reduced to the requirement of a correlation function’s tendency to zero with an increase of time lag. Actually, the stationary second-order autoregressive process may be considered as ergodic because its correlation function is a damped sinusoid tending to zero. This fact enables the estimation of Yule model factors a1 and a2 based on a single realization of random oscillations.

https://doi.org/10.1515/9783110627978-003

30

2 Estimation of random oscillation characteristics

2.1 Criterion of maximum likelihood The task of model factor estimation may be stated as follows: there is a random variable X with the known probability density f(x/а) featuring parameters аТ = (а1, . . ., аk). Existing values of these parameters provided observations xT=(x1, . . ., xn) are unknown although their values are fixed. It is required to estimate parameters a to consider that some of their values “most likely” provide observations x. In 1925, R.A. Fisher, a British statistician and geneticist, has formulated the maximum likelihood criterion which holds the key position in the statistic estimation theory. Its essence may be described in the following manner. Before being observed, the possible values of random variable X are described by the probability density f(x/а). Once observations х are obtained, it is appropriate to proceed with considering the possible values of parameters a that provided those observations. To do this one constructs a likelihood function L(a/x) which is proportional to f(x/a) with observations x and unknown values of a. The maximum likelihood estimate (MLE) of parameters a provides the maximum value of the function L(a/x). In other words, the MLE correspond to the highest probability of appearance of observations x1, . . ., xn. The likelihood function plays a fundamental role in the estimation theory since it carries all information about model parameters obtainable from experimental data [7]. Often it is more convenient to use the logarithmic likelihood function lða=xÞ = lnLða=xÞ, which comprises additive constants instead of proportionality coefficients. Furthermore, l(a/x) takes on a simple view for exponential forms of distribution laws which are the most often case in technical applications. To illustrate the principle of maximum likelihood let us take an example of covariance estimation for a stationary series considered as a random vector xT = (x1,. . ., xn) with μх = 0 and Gauss probability density: ( ) n X n 1X − n=2 1=2 f ðx1 , ..., xn Þ = ð2πÞ jHj exp − η xi xj : 2 i = 1 j = 1 ij In this expression the matrix H is inverse to the covariance matrix Γ: n o H = fηij g = Γ − 1 ; Γ = γij , i, j = 1, . . . , n;jHj = jΓj − 1 : It is appropriate to remain that the element ηij of the inverse matrix equals the algebraic complement Γji to the element γji related to the determinant |Γ|. The Γji to the element γji is the determinant of a matrix derived from Γ by crossing out jth row and ith column as well as multiplying by (−1)j+i.

2.1 Criterion of maximum likelihood

31

As it was mentioned previously, the statistical relationship between stationary series terms is determined by their relative positions within the realization x1, . . ., xn, that is, for stationary series 1 0 γ−1 . γn − 1 γ0 C B γ0 . γn − 2 C   B γ−1 C= γ , Γ=B k B . . . . C A @ γ − ðn − 1Þ γ − ðn − 2Þ . γ0 where k = i−j = −(n−1), . . ., (n−1), and ηk = Γ−k / |Γ|. Accordingly, the index of the exponential function in the expression of f(x1, . . ., xn) transforms into the following: n X n X i=1 j=1

ηij xi xj =

n−1 X

ηk

nX − jk j

k = − ðn − 1Þ

xj xj + jkj .

j=1

The likelihood function is derived from the probability density providing fixed x and variable parameters ηk. In particular, the logarithmic likelihood function will have the following form: lðηk =xÞ = −

nX − jkj n−1 n 1 1 X ηk xj xj + jkj . lnð2πÞ + lnjHj − 2 2 2 k = − ðn − 1Þ j=1

The conditions of a maximum of the logarithmic likelihood function with respect to parameters ηk will be: ∂lðηk =x1 , ..., xn Þ = 0, ∂ηk which will result in the following equation: n − jk j 1 ∂jHj X − xj xj + jkj = 0. · jHj ∂ηk j=1

(21)

Now, let us represent a derivative of the determinant |Η|. It may be done through the algebraic complement Hk to elements ηk lying on the same diagonal as follows: n − jkj ∂jHj X = Hkj ; ∂ηk j=1

where j is an ordinal number attributed to diagonal elements ηk. Finally, expression (21) takes the form:

32

2 Estimation of random oscillation characteristics

nX − jk j j=1

n − jk j Hkj X xj xj + jkj . = j Hj j=1

As Η−1 = Γ, the algebraic complement Hk divided by the determinant |H| is a corresponding element of matrix Γ: Hk =jHj = γ − k . Thus, the expression for calculations of the covariance MLE of stationary series will be determined as follows: ~γk = ~γ − k =

n−k 1 X xj xj + k . n − k j=1

(22)

In fact, this expression is obtained from the condition of the maximum of the likelihood function with respect to the parameter ηk but not to γk. Therefore, a question arises: Can covariance estimates calculated with the help of eq. (22) be considered the MLE? To answer this question, let us examine an injective (one-to-one) function g(a) instead a parameter a. A derivative of this function may be presented in the following form: ∂l ∂l ∂gðaÞ = · : ∂a ∂gðaÞ ∂a So the condition ∂l =0 ∂gðaÞ corresponds to the condition: ∂l = 0; ∂a

∂gðaÞ ≠0. ∂a ~ is the MLE of the parameter a, then g This is a property of MLE invariance: if a ~) will be the MLE of any function of parameter a (not necessarily a one-to-one (a function) [7]. The invariance property allows to define the covariance estimates calculated with the help of eq. (22) as the MLE. Accordingly, the MLE of stationary series correlations may be obtained from the following: if

nP −k

~k = ρ

~γk n = · ~γ0 n − k

i=1

xi xi + k

n P

i=1

. xi2

(23)

2.2 Estimation of Yule model factors

33

The invariance property of the MLE is particularly important for estimations of random series model factors. It substantially simplifies obtaining their MLE when the relationship between the model factors and the statistical characteristics of the series are known. In this case, it suffices only to obtain the MLE of these characteristics. Then, the knowledge of a functional link between those statistical characteristics and model factors allows calculating their MLE.

2.2 Estimation of Yule model factors Besides illustrative purposes of the MLE, expressions (22) and (23) become important for a practical application due to the invariance property of the MLE. As it was introduced in Section 1.3, correlations of the Yule series may be composed in a system of Yule–Walker equations of the kind: ρ1 = a1 + a2 ρ1 ; ρ2 = a1 ρ1 + a2 : This system may be written in the matrix form as follows: BA = P,

(24)

where A = ða1 , a2 Þ; P = ðρ1 , ρ2 Þ; B = T

T

1

ρ1

ρ1

1

! .

Implementation of the Cramer’s rule for solving this system yields a functional linkage of Yule model factors with series correlations: a1 =

ρ1 ð1 − ρ2 Þ ; 1 − ρ21

a2 =

ρ2 − ρ21 . 1 − ρ21

These expressions allow to determine the MLE of factors a1 and a2, if the MLE of correlations ρ1 and ρ2 are already known: ~1 = a

~2 Þ ~1 ð1 − ρ ρ ; ~21 1−ρ

~2 = a

~21 ~2 − ρ ρ . ~21 1−ρ

34

2 Estimation of random oscillation characteristics

Using eq. (19) which relate а1 and а2 to random oscillation characteristics, it is not complicated to derive suitable expressions for the MLE of the decrement and natural frequency: ~0 = ω

~1 1 a arccos pffiffiffiffiffiffiffiffiffi ; Δt ~2 2 −a

δ~ = −

π ~2 Þ. lnð − a ~ 0 Δt ω

Thus, due to the invariance property of the MLE the processing algorithms can be developed without considering the likelihood function for model factors. It is necessary to note that the method of the least squares may be used for model factor estimation as well. This statement is grounded by a fact that MLE and least squares estimates (LSE) are asymptotically consistent, that is, they become the same values when the realization size tends to infinity. To prove this statement, let us examine the realization x1, . . ., xn featuring the mathematical expectation μx = 0. If a process is generated by the AR(2) model, one may pass from the xT = (х1, . . ., хn) to the complex (х1, х2, ε3, . . ., εn) using εi = xi − a1 xi − 1 − a2 xi − 2 , i = 3, . . . , n. The probability density of that complex can be presented in the form of the following product: f ðx1 , x2 , ε3 , . . . , εn Þ = f ðε3 , . . . , εn =x1 , x2 Þf ðx1 , x2 Þ. The white noise E has a Gauss distribution with με = 0 and Dε = 1; therefore, f ðε3 , ..., εn =x1 , x2 Þ = ð2πÞ − ðn − 2Þ=2 expð − ð2πÞ − ðn − 2Þ=2 exp½ −

n 1X ε2 Þ = 2 i=3 i n 1X ðxi − a1 xi − 1 − a2 xi − 2 Þ2 . 2 i=3

As the AR(2) model is a linear, x1 and x2 have the Gauss distribution too. Statistical relationships between these values are represented by the following covariance matrix: ! ! γ0 γ1 1 ρ1 1 = Γ= . Dx ρ1 1 γ1 γ0 In view of matrix Γ the probability density of terms x1 and x2 may be written as follows:

2.2 Estimation of Yule model factors

−1

35

! x1 1 exp½ − ðx1 , x2 ÞH . 2 x2

1=2

f ðx1 , x2 Þ = ð2πÞ jHj

Let us recall that matrix H is the inverse of the covariance matrix, therefore its expression is as follows: ! 1 − ρ1 1 −1 . H=Γ = Dx ð1 − ρ21 Þ − ρ1 1 In order to construct the likelihood function, the probability density has to be represented as an explicit function of the model factors. In Section 1.3, an expression for ρ1 was derived in the following form: ρ1 =

a1 : 1 − a2

Additionally, the variance of the Yule series was determined via factors a1 and a2 by expression (15): Dx =

ð1 − a2 Þ ð1 + a2 Þ½ð1 − a2 Þ2 − a21 

,

assuming Dε = 1. Making use of these two latter expressions as well as the expression for matrix H, the logarithmic likelihood function may be constructed by setting values of x1, . . ., xn as fixed and factors a1, a2 as variable: 1 lða1 , a2 Þ = const + lnð1 − a22 Þ + x1 x2 a1 ð1 + a2 Þ − 2 n 1 1X ðxi − a1 xi − 1 − a2 xi − 2 Þ2 . ð1 − a22 Þðx12 + x22 Þ − 2 2 i=3 In this expression only the following term depends on n: Sða1 , a2 Þ =

n 1X ðxi − a1 xi − 1 − a2 xi − 2 Þ2 . 2 i=3

That is why asymptotically (n → ∞), the maximum of the likelihood function corresponds to the minimum of the sum S(a1,a2): max lða1 , a2 Þ ⁓ min Sða1 , a2 Þ a

a

An extreme value of S(a1,a2) is determined by the conditions:

36

2 Estimation of random oscillation characteristics

∂Sða1 , a2 Þ = 0; j = 1, 2, ∂aj which lead to the following system of equations: n X

ðxi − a1 xi − 1 − a2 xi − 2 Þ · ð − xi − 1 Þ = 0;

i=3 n X ðxi − a1 xi − 1 − a2 xi − 2 Þ · ð − xi − 2 Þ = 0. i=3

The second derivatives of S(a1,a2) are n ∂2 Sða1 , a2 Þ X = xi2− j > 0; ∂2 aj i=3

j = 1, 2.

Therefore, an extreme value of S(a1,a2) is a minimum. The system of equations derived above transforms to: a1

n X i=3

a1

n X

xi2− 1 + a2

n X

xi − 1 xi − 2 =

i=3

x i − 1 x i − 2 + a2

i=3

n X

xi xi − 1 ;

i=3 n X i=3

xi2− 2 =

n X

xi xi − 2 ;

i=3

which has the following matrix form ΒΑ = Ρ. Here B=

b11

b12

b21

b22

bij =

n X

! ; A=

a1 a2

! ; P=

b01 b02

! ;

xl − i xl − j ; i = 0, 1, 2; j = 1, 2.

l=3

The form of this matrix equation is similar to eq. (24). As it was done earlier, implementation of the Cramer’s formula leads to the LSE of the factors a1 and a2 in the following forms: a1 =

b01 b22 − b02 b12 ; b11 b22 − b212

a2 =

b11 b02 − b01 b12 . b11 b22 − b212

(25)

37

2.2 Estimation of Yule model factors

_

_

Let us mark ρ 1 and ρ 2 as the LSE: nP −2

xi xi + 1 i=1 nP −2 x2i i=1

_

_

ρ1 =

γ1 _ = γ0

nP −2

xi xi + 2 i=1 nP −2 x2i i=1

_

_

ρ2 =

;

γ2 = γ0

_

.

As result, expressions (25) may be written in the form: _

_

a1 =

_

ρ 1 ð1 − ρ 2 Þ _2

1 − ρ1 _

_

a2 =

;

(26)

_

ρ 2 − ρ 21 _

1 − ρ 21

:

The expressions (26) correspond to expressions for the MLE of factors a1 and a2. Let us recall that the expression of MLE of correlations is given by the expression: nP −k

~k = ρ

n n−k

i=1

xi xi + k

n P

i=1

, xi2 _

~k = ρ k when that is, the LSE and MLE are differing by the factor n/(n−k). Hence, ρ n → ∞; both types of estimates will be consistent asymptotically. In practice, the difference between MLE and LSE is insignificant for n > 100. To utilize the LSM for an evaluation of factors а1 and а2, the overdetermined system of equations is composed from values х1, . . ., хn x3 = a1 x2 + a2 x1 ; x4 = a1 x3 + a2 x2 ; · xn = a1 xn − 1 + a2 xn − 2 . This system has the matrix form as well: ΒΑ = Ρ,

38

2 Estimation of random oscillation characteristics

where

A=

a1 a2

!

0

x2

x1

xn − 1

xn − 2

B ; B=@ ·

1

0

x3

1

C B C · A; P = @ · A. xn

In this case factors of AR(2) model are determined directly from observations х1, . . ., хn. Because the least squares approach has been implemented as a standard computer procedure, it has frequent applications for experimental data processing.

2.3 Influence of additive broadband noise The estimation procedure described in the previous section is applicable when experimental data contains only values of the process X(t). This is an idealized situation since in reality a measurement signal Z(t) may include broadband instrumental noise Ξ(t) with a variance Dξ. This noise presence will transform observations of the process X into a series of the form: zi = xi + ξ i . The presence of noise Ξ in the measurement signal Z means that correlations of Z will not be equal to correlations of the process X. This fact is evident after comparing expressions for correlations of the random oscillations and the measurement signal: ρxi = ρzi =

γxi ; Dx

γzi γxi + γξi = : Dz Dx + Dξ

In the case of using the least squares method, a substitution of zi for xi into the sum: Sx ða1 , a2 Þ =

n n 1X 1X ðxi − a1 xi − 1 − a2 xi − 2 Þ2 = ½AðBÞxi 2 2 i=3 2 i=3

yields the expression of the kind: Sz ða1 , a2 Þ =

n n 1X 1X ½AðBÞzi 2 = ½AðBÞxi + AðBÞξ i 2 = 2 i=3 2 i=3

Sx ða1 , a2 Þ + ΔS . Therefore, a minimum of the sum Sz(a1,a2) will differ from a minimum of Sx(a1,a2) due to ΔS. As result, values of estimates of factors a1 and a2 calculated from readouts zi will not be equal to the estimates calculated from xi. The corresponding

2.3 Influence of additive broadband noise

39

estimates of oscillation characteristics will be different as well. The changes of oscillation characteristic estimates is illustrated in Fig. 10 which gives a conventional representation of process X, Ξ, and their sum Z spectra.

S(ω) Sz(ω)

Δωz Δωx Sx(ω) Sξ (ω) ω0

0

ω

Fig. 10: Spectrum of X, Ξ, and Z processes.

As it follows from comparison of spectra Sξ(ω) and Sx(ω), their maximums correspond to the natural frequency, so estimates of ω0 will not have changed, if they are calculated from values of process Z. Unlike that spectrum widths for half of their power levels of processes X and Z are different. As Δωz > Δωx the decrement estimates calculated from the process Z will be overestimated. The term ΔS of the sum Sz ða1 , a2 Þ can be determined as the following mathematical expectation: M½ΔS  =

n X

M½AðBÞxi · AðBÞξ i  +

i=3

n n o 1X M ½AðBÞξ i 2 . 2 i=3

Assuming that the instrumental noise ξi is white and is not correlated with xi, the expression for M½ΔS  may be transformed to the following form: M½ΔS =

n 1X n−2 ð1 + a21 + a22 ÞDξ = ð1 + a21 + a22 ÞDξ . 2 i=3 2

Therefore, a value of Sx(a1,a2) may be represented as follows: Sx ða1 , a2 Þ = Sz ða1 , a2 Þ −

n−2 ð1 + a21 + a22 ÞDξ . 2

A specific manner of estimate changes, when a value of the natural frequency is not changed, can be utilized to correct the estimates of model factors. Taking into account the formula:

40

2 Estimation of random oscillation characteristics

ω0 =

1 a1 arccos pffiffiffiffiffiffiffiffiffi , 2 − a2 Δt _

_

and the above-mentioned specific of a 1 and a 2 estimates, the following relationship may be declared: a1 pffiffiffiffiffiffiffiffiffi = const. 2 − a2 This fact allows to modify the estimation procedure to a single-parameter task. _ _ If initial estimate received utilizing values zi are marked as a 10 and a 20 , it allows taking into consideration only the factor a2 since a value of a1 can always expressed in terms of a2: rffiffiffiffiffiffiffi a2 _ a1 = a 10 _ : a 20 Therefore, the autoregressive model factors can be estimated during the iterative procedure at the jth step of which one obtains the functional estimate _

_

Sx ða1j , a2j Þ = Sz ða 10 , a 20 Þ −

n−2 ð1 + a21j + a22j ÞDξ , 2

where rffiffiffiffiffiffiffi a2j . a2j = a2ðj − 1Þ + Δa2 , a1j = a10 a20 Values of factors a1j and a2j providing the least magnitude of Sx(a1j,a2j) are to be taken as LSE. A value of the increment Δa2 defines the accuracy of calculations of factor a2 as well as the estimate of decrement δ. As far as the MLE estimates are concerned, a procedure for the estimation of the model factors can be set up as described below. A broadband spectrum of noise Ξ means that its correlation function will decrease faster than a correlation function of the process X being insignificant (worthless) from some lag q. Therefore, a covariance function of the process Z will actually correspond to the covariances of process X from the lag q. This fact allows to assume that for covariances of process Z featuring lags i ≥ q the following equality is true: γzi = γxi . The equation for covariances γi of the process AR(p) presented in Section 1.2 will have the following form for the Yule series: γi = a1 γi − 1 + a2 γi − 2 .

2.4 Processing of multicomponent signals

41

Therefore, four covariances of the process Z with lags i > q may be combined in a system of equations of the type: γzðq + 2Þ = a1 γzðq + 1Þ + a2 γzq ; γzðq + 3Þ = a1 γzðq + 2Þ + a2 γzðq + 1Þ . This system solution yields expressions for factors of the AR(2) model: a1 =

γzðq + 1Þ γzðq + 2Þ − γzq γzðq + 3Þ γ2zðq + 1Þ − γzq γzðq + 2Þ

a2 =

γzðq + 1Þ γzðq + 3Þ − γ2zðq + 2Þ γ2zðq + 1Þ − γzq γzðq + 2Þ

;

(27)

;

which may be used for calculations of estimates of model factors on a basis of known estimates ~γzi . Therefore, to obtain the MLE of the AR(2) model factors one should calculate the MLE of covariance γq, γq+1, γq+2, γq+3 using the realization of process Z and substitute the obtained values into expressions in (27). The initial lag q may be identified on a basis of a priori data of the noise frequency bandwidth determined by fmax. A value of fmax determines a correlation interval τcor being a minimum interval after which the statistical relationship between instantaneous noise values may be neglected. Rough estimating of τcor may be done by assuming the measurement signal as an output of a low band filter. In this case the fmax may be interpreted as a cutoff frequency and an estimate of a correlation interval is as follows: 1 . τcor = 2πfmax When any quantitative characteristics are not available at all, estimates of factors a1 and a2 should be computed sequentially for q = 1, 2,. . . until stable estimates are obtained, that is, until a moment when the increase of q no longer results in a sufficient variation of factors estimates.

2.4 Processing of multicomponent signals Besides the instrumental noise, the measurement signal may feature oscillations of different natural frequencies (Fig. 11).

S( f )

0

f

Fig. 11: Spectrum of multicomponent measurement signal.

42

2 Estimation of random oscillation characteristics

Such a signal realization enables the estimates of oscillation parameters to be computed in two ways. The first way provides for the development of a detailed multicomponent signal model and for estimation of its parameters. The detailed signal model parameters will obviously characterize a component of interest as well. This approach, however, requires a large amount of a priori data on all the signal components and forces the development of a more complicated model. The second way consists of a multicomponent signal decomposition. If signal components differ by energy distribution over a frequency range then the multicomponent signal decomposition is performed by filtering. In particular, one makes use of band-pass filters to extract random oscillations with the main part of their power concentrated around f0. The filters of this type allow signals to pass unaltered within the frequency range f1 ≤ f ≤ f2 while suppressing all the others. A spectrum Sy of the filtered signal Y is coupled to a spectrum Sx of the initial signal X: Sy = Sx Af , where Af is the frequency response (FR) of a filter. Since the pass-band filter suppresses signal energy outside its bandwidth Δf = f2 – f1, Dξ value will be diminished as well. The level of suppression is correlated with the bandwidth – the narrower Δf, the smaller a residual value of Dξ. Determining the bandwidth value starts with determining the decrement values range under consideration. As it was presented earlier, the value of δ is coupled with the bandwidth of random oscillations at the level of half its power δ=

Δω Δf = . ω0 f0

For example, a value of the oscillation bandwidth for δ ≤ 0.2 will be Δf ≤ 0.016 (f0∙Δt = 0.25). In the case Δf =10 Δf, the relationship between the oscillations spectrum and the filter FR is demonstrated by Fig. 12.

Af Sx

f0

Fig. 12: The filter FR and spectrum of random oscillations.

43

2.4 Processing of multicomponent signals

It appears that the main part of oscillation energy is inside the filter bandwidth. In each situation, the filter bandwidth is conditioned by set of factors being under considerations (experiment specific, source/nature of noise, etc.). The subject of this work is digital data processing, so only digital filters will be taken into consideration. The theory of digital filtering is explicitly presented, for instance, in [8]. We shall consider its aspects that deal with the processing of random oscillations. Filters with infinite and finite impulse response, respectively called IIR and FIR filters, may be employed for signal processing. The choice of filter type may involve a large variety of factors; there are no common recommendations to be given on its specific application. For estimation of oscillation parameters it is useful to employ FIR filters (non-recursive). What is critical is that these filters are stable and their feasibility is not in question. The FIR filter input xi and output yi are related by the expression: yi =

q−1 X

cj xi − j = CðBÞxi ;

i=0

that is, the FIR filter implements the MA(q) operator, therefore it is always stable. In the case when the FIR filter input is white noise ε, its output process Φ has a variance Dϕ = Dε

q−1 X

c2j .

j=0

Obviously, after bandpass filtering the signal energy Dϕ < Dε, so a value of

qP −1 j=0

c2j < 1.

It means that an energy the broadband instrumental noise Ξ(t) will be decreased by the bandpass filter. Another even more important feature of the FIR filter is the fact that the AR(2) process correlation function experiences no influence of filtering beyond the qth lag. Now consider the filter output when the AR(2) process is its input: yi = CðBÞa1 xi − 1 + CðBÞa2 xi − 2 + CðBÞεi = a1 yi − 1 + a2 yi − 2 + CðBÞεi . Thus, the filtering result is the ARMA(2,q) process whose covariances are described by expression (8): γj = a1 γ1 + a2 γj − 2 + c0 γyε ðjÞ + c1 γyε ðj − 1Þ − ... − cq − 1 γyε ðj − q + 1Þ, where γyε ðjÞ = M½yi − j · εi  is a cross-covariance of Y and E processes. As it was demonstrated in Section 1.2, after the (q+1)th lag the covariances of ARMA(2,q) process will correspond to AR(2) process covariances. It means they may be used to construct the Yule–Walker system of equations already presented in Section 2.3:

44

2 Estimation of random oscillation characteristics

γq + 3 = a1 γq + 2 + a2 γq + 1 ; γq + 4 = a1 γq + 3 + a2 γq + 2 . This system solution (27) allows to find estimates а1 and а2, and as a result, to calculate the estimates for the decrement and natural frequency of oscillations.

3 Estimates accuracy Experimental evaluation of random process parameters is inevitably complemented by errors committed during the measurements as well as subsequent processing of measurement results. Assuming that a continuous realization of a process under study is available, a measurement procedure linked with specifics of dedicated experiments is outside our consideration. This means that only the accuracy of data processing conditioned by used algorithms and their implementation will be examined in this chapter. Concurrently, processing approaches that increase the accuracy of estimates will be worked out as well. The parameter estimates in themselves cannot be correct or incorrect because they are somewhat arbitrary. Nevertheless, some estimates could be considered as “better” than others. To compare them, one makes use of the mean square value of ~: the error of estimate a ~ − Mða ~Þ2 g + Mf½Mða ~Þ − a2 g. ~ − aÞ2  = Mf½a M½ða The first term in the right-hand side of this expression is the estimate variance which is a measure of a “random” fraction of the error: ~ − Mða ~Þ2 g. Da~ = Mf½a The second term – square of the estimate bias – gives a systematic deviation: ~Þ − a2 g. b2a~ = Mf½Mða Depending on properties of components of the error, estimates come subdivided into several categories. First, if the estimate expectation equals the parameter which is to be estimated, such as: ~Þ = a; Mða that is, ba~ = 0, then the estimate is called unbiased. ^ is less than that of any other estimate a ~, Second, if the variance of the estimate a that is: Da^ < Da~ , ^ is referred to as an efficient estimate. then the estimate a And finally, if with the increase of the realization size n the estimate draws near the parameter a with a probability tending to 1, in other words at any c > 0

https://doi.org/10.1515/9783110627978-004

46

3 Estimates accuracy

~ − aj ≥ cÞ = 0, lim Pðja n!∞ then the estimate is called consistent. From the Chebyshev inequality of the form: ~ − aj ≥ cÞ = Pðja

Da~ , c2

it follows that a sufficient (but not required) condition of consistency is: lim Da~ = 0. n!∞ In other words, estimate accuracy must increase with an unlimited increase of the realization size. Both conditions of estimate’s consistency are, in fact, requirements for the convergence in probability and the root-mean-square.

3.1 Optimal sampling of continuous realizations A digital processing of experimental data starts with a transformation of a continuous-time realization x(t) into a time series. Terms xi of this time series are readouts (samples) of the x(t) at a point in time ti = Δt . i. The interval Δt is referred to as a sampling interval. Let us examine such a sampling (discretization) procedure from the standpoint of its influence on the accuracy of estimates. The theoretical base of the discretization procedure is the Nyquist–Shannon– Kotelnikov (sampling) theorem. Particularly, Shannon has formulated it in the following terms: if a function x(t) contains no frequencies higher than fm, then this function is completely determined by giving its ordinates at a series of points spaced Δt = 1/(2fm) apart. The value 2fm is called the Nyquist frequency (rate). The theorem sets up a maximum value of interval Δt, however, for practical purposes its value is obtained from the expression of the form: Δt =

1 , 2κ · fm

where κ > 1 is a margin coefficient whose value depends on how much a real signal is approximated by one featuring a spectrum limited by fm as well as on the means of subsequent processing. It is known that energy of random oscillations is concentrated within a narrow bandwidth around the natural frequency f0. Random oscillations X(t) are interpreted as a reaction of a linear elastic system on white noise. This means that the

3.1 Optimal sampling of continuous realizations

47

shape of random oscillation spectrum will correspond to square values of the FR of the elastic system presented in Fig. 13 (in the logarithmic scale).

A( f ) 40 10

󰛿 = 0.06

󰛿 = 0.3

1

0.2 0.1

1 21/2 4

f|f0 Fig. 13: The FR of elastic system.

Theoretically, the spectrum of random oscillation, like the spectrum of any physical process, is infinite. At the same time, the fact that its energy is concentrated around f0 (especially for δ ≤ 0.3) allows to consider random oscillations as a process with a pseudo-limited spectrum. The infinite spectrum of any signal theoretically implies its under-sampling leading to low-frequency aliases. In Section 2.4, it was proposed to start digital processing with bandpass filtering of measurement signals. The main purpose of filtering is to extract an oscillation mode of interest from the multicomponent signal. Taking into account the above-mentioned statements, the high cutoff frequency of the bandpass filter has to be f2 ≤ (κ2f0). Such measure will allow to avoid the aliasing effect during measurement signal discretization. Because of the absence of a strictly determined value of the frequency fm, the definition fm = f0 is more characteristic for the oscillations spectrum. In this case, as it follows from Fig. 13 the value of the margin coefficient must be intentionally κ > 21/2 ≈ 1.4. We will determine its value with a reference to particularities of estimation algorithms. Formally, in all cases considered in Chapter 2 the estimation of oscillation characteristics is reduced to the solution of the matrix equation for vector of factors АТ =(а1, а2): BA = P,

(28)

where elements of vector P and matrix B are composed of the observations xi. It is a known fact that the solution of eq. (28) provides stable estimates of the vector A if the

48

3 Estimates accuracy

matrix B is well-conditioned [9]. In this case small alterations of values of vector P elements will cause no large deviations of estimates of factors a1 and a2. Therefore, a criterion of choice for κ may be settled as having a requirement for matrix B to be well-conditioned. A matrix is not well-conditioned if its properties approach the properties of a singular matrix whose determinant is zero. Thus, formally, the criterion of choice for value κ may be written as follows: jBj ! max . κ

It must be noted that an absolute value of the determinant may be increased simply through multiplying eq. (28) with a certain number. Following that, a value of κ which provides an absolute maximum of the matrix B determinant needs to be found. Utilizing the Yule–Walker system for estimating factors а1 and а2, the matrix B has the following form: ! 1 ρ1 B= . ρ1 1 Its determinant |B| = 1 – ρ12 obviously reaches a maximum with value ρ1 = 0. Taking into account expression (17) this condition can be rewritten as follows: ρ1 =

ð − a2 Þ1=2 sinðθ + ’Þ = sin ’

ð − a2 Þ1=2 sinð2πf0 · Δt + ’Þ ð − a2 Þ1=2 sinðπ=κ + ’Þ = = 0, sin ’ sin ’ due to θ =ω0Δt and Δt = 1=ð2κ · f0 Þ. Therefore, a fulfillment of the condition ρ1 = 0 may be reached when sinðπ=κ + ’Þ = 0. As sinφ = ωh/ω0 [6], for small decrement values (δ ≤ 0.3) the argument φ will be equal to: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ’ = arcsinðωh =ω0 Þ = 1 − ðδ=2πÞ2 ≈ π=2. This means that the condition of ρ1 = 0 is fulfilled with κ = 2. Simultaneously with the ρ1, all of the following odd correlations are equal to zero as well. Thus, the optimal (κ = 2) discretization of random oscillations is provided when the sampling rate equals four times the value of natural frequency. Changing κ in either direction causes ρ1 ≠ 0 and leads the determinant of matrix B to decrease (Fig. 14).

3.2 Statistical error of estimates

49

|B|

󰛿 = 0.02

0,5

󰛿 = 0.01

0

2

4

󰜅

Fig. 14: Optimal sampling selection.

The value κ = 2 corresponds to the diagonal type of matrix B confirming a known fact that orthogonal and diagonal matrices are always well-conditioned. In addition, Fig. 14 illustrates that the range 1.6 ≤ κ ≤ 2.5 provides a value of the decrement |B| ≥ 0.9. This means that this particular range of κ is characterized by a degradation of accuracy of a solution to the eq. (28) of less than 10%. This fact allows to declare that the condition κ = 2 is not too strict as the sampliling interval T0 T0 ≤ Δt < 5 3 may still be considered as pseudo-optimal.

3.2 Statistical error of estimates In the discussion of estimates’ properties it was stated that their statistical error features a certain variance: n o ~ − Mða ~Þ2 . Da~ = M ½a In the previous chapter it was established that estimates of Yule model factors are linked with correlation estimates calculated on a basis of xi values: nP −k

~k = ρ

n · n−k

xi xi + k P 2 . xi

i = 1n

i=1

~2 is In the case of optimal sampling (a1 = 0) the correlation ρ2 = a2 so the estimate a a function of xi and xi+2:

50

3 Estimates accuracy

nP −2

n ~2 = a λðxi , xi + 2 Þ ≈ n−2

i=1

xi xi + 2

n P i=1

. xi2

It was demonstrated in Section 1.1 that a random function may be linearized that ~2 variance via the Yule series variance: allows to present the a 82 32 2 32 9 n2 n n2 n n2 P P P P P > > > 2 > >  2  2 xiþ2 · xi  xi xiþ2 · 2 xi 7 6 xi 7 > = < 6 ∂λ ∂λ 6 i¼1 i¼1 i ¼ 1 7 6i ¼ 1 7 Da~2 ≈ Dx þ D x ¼ 6i ¼ 1 7 þ6 7 Dx : > 4 5 4P 5> n n ∂xi ∂xiþ2 2 P > > > ; : ð xi2 Þ xi2 > i¼1

i¼1

Both parenthesized terms have the power of summing members of the denominator larger than the power of summing members of the numerator. In addition, the denominators have members in even powers. Because xi change their signs, the denominator sums will grow much faster than the ones in the numerators. As a result, the value of the expression in parentheses will tend to zero with unlimited increase of the realization size (n ! ∞). Hence, the MLE of the factors as well as the decrement and natural frequency are asymptotically consistent. In practice, however, estimates are always computed from a realization of limited size, therefore asymptotic properties of estimates are not practically valuable. They enable the researcher to neither quantify errors in estimates obtained from a realization of a specified size, nor to identify the realization size which is required to obtain certain estimate accuracy. To overcome these problems, it is necessary to know the quantitative relationships between Da~ and the size of realization (n). As oscillation characteristics are derived from the Yule model factors, it is logical to start with examination of statistical errors of vector A set by covariance matrix: ! Da~1 γa~1 a~2 (29) Γa~ = γa~2 a~1 Da~2 ~2 . ~1 and a which characterizes errors of estimates a In Section 1.1 it was demonstrated that in the case of a linear linkage of vectors Y and Ξ, Y = AΞ, covariance matrices of these vectors are linked by the following expression: ΓY = AΓΞ AT . Therefore, in the case of Yule–Walker system of equations BA = P,

(30)

3.2 Statistical error of estimates

51

the expression (30) allows to write: Γa~ = B − 1 Γρ~ B − T = ðBT WBÞ − 1 = N − 1 where Γρ~ =

D~ρ1

γρ~1 ρ~2

γ~ρ2 ρ~1

Dρ~2

(31)

! .

Matrix W ¼ Γ~ρ1 is called the weighting matrix. The quadratic form of matrices B and W as well as a symmetric form of matrix B allows to write: jNj = jBj2 · jWj. Convexity of the curve in Fig. 14 representing a dependence of |B| upon κ grounds the conclusion that the maximum of |N| corresponds to a maximum of |B|2 provided by value κ = 2. Because of the equality jΓa~ j = jNj − 1 , a maximum of |N| corresponds to a minimum of jΓa~ j. A value of the determinant jΓa~ j is called the generalized variance of estimates of the vector A. This variance characterizes errors of estimates of factors ai, under consideration they are an indivisible set of variables. In other words, jΓa~ j deals with errors of the model factors, in particular the AR(2). A minimum of jΓa~ j provided by the optimal discretization implies the best congruence of the model to experimental data. Earlier in Section 3.1, it was established that the optimal sampling transforms matrix B to the unit form, that is, |B| = 1. This fact allows to write: Γa~ = Γρ~ , that is, Da~1 = Dρ~1 , Da~2 = Dρ~2 , and γa~1 a~2 = γ~ρ1 ~ρ2 . Elements of the matrix Γρ~ as well as Γa~ may be determined by Bartlett’s formulas [5]: Dρ~k ≈ γ~ρ ~ρ

∞ 1 X ðρ2 + ρi + k ρi − k − 4ρk ρi ρi − k + 2ρ2i ρ2k Þ; n i= −∞ i

j j+k

≈

(32)

∞ 1 X ðρ ρ + ρ ρ + 4ρj ρj + k ρ2i − n i = − ∞ i i + j i i + 2j + k

2ρj ρi ρi + j + k − 2ρj + k ρi ρi + k Þ. In the case of optimal sampling, all odd correlations may be considered null. Therefore, γa~1 a~2 = γ~ρ1 ρ~2 = 0 because every term in the second eq. (32) includes ρ1 = 0 or a ~1 and product of odd and even correlations (j = k = 1). This implies that estimates a ~1 and a ~2 will be equal ~2 are not correlated, and the generalized variance a a

52

3 Estimates accuracy

jΓa~ j = Da~1 · Da~2 . As the optimal sampling provides minimum value of jΓa~ j, the variances of model factors have minimum values as well. In other words, the value κ = 2 provides effi^i . cient estimates of Yule model factors which will be marked as a The first equation of (32) provides an expression for the variance of the estimate ^1 in the form: a Da^1 = Dρ^1 ≈

∞ 1 X ðρ2 + ρi + 1 ρi − 1 Þ, n i= −∞ i

as two other terms of the sum include ρ1 = 0. Because of the fact that all odd correlations equal zero, only even correlations in the sum have to be taken into account. They may be represented in the form ρ2i = a22i transforming the first sum into: ∞ X

∞ X

ρ2i 2 =

i= −∞

ð − a2 Þ2i cos2 ðπ · iÞ = 2

i= −∞

∞ X

i a2 2 − 1.

i=0

2 i

The even correlations ða2 Þ composes the infinite convergent geometric series zi whose sum is ∞ X

zi = 1=ð1 − zÞ, if j zj < 1.

i=0

Therefore, the first sum of the Da~1 expression will be equal to ∞ X

ρ2i 2 =

i= −∞

2 1 + a22 − 1 = . 1 − a22 1 − a22

As ρ2i − 1 = 0 and ρ − i = ρi the second sum of Da~1 expression may be written in the following form: ∞ X i= −∞

ρi + 1 ρi − 1 = ... + ρ − 2 ρ − 4 + ρ0 ρ − 2 + ρ0 ρ2 + ρ2 ρ4 + ... = 2

∞ X i=0

ρ2i ρ2i + 2 .

Taking into account expression (17), the summing term can be evaluated in the following way: ρ2i ρ2i + 2 = ð − a2 Þi cosðπ · iÞð − a2 Þi + 1 cos½π + ðπ · iÞ = − ð − a2 Þ2i + 1 , due to the product of such cosine functions always being equal to −1. As a result, the evaluated sum is reduced to an infinite convergent geometric series as well. Hence, its sum can be obtained by using the above-mentioned rule:

3.2 Statistical error of estimates

−2

∞ X

ð − a2 Þ2i + 1 = 2a2

i=0

∞ X

ð − a2 Þ2i = 2a2

i=0

∞ X

ð − a22 Þi =

i=0

53

2a2 . 1 − a22

^1 variance becomes: Finally, the expression of estimate a   1 1 + a22 2a22 1 1 + a2 = · + . Da^1 ≈ n 1 − a22 1 − a22 n 1 − a2 ^2 variance can proceed utilizing the following approach. The corEvaluation of the a relation equation of the AR(2) process at the optimum sampling rate (a1 = 0) takes the form: ρi = a 2 ρi − 2 , and looks similar to the first-order autoregressive process AR(1) correlations: rl = α rl − 1 . The expression for the variance of the rl estimates is known as [4]:   1 ð1 + α2 Þð1 − α2l Þ 2l D~rl ≈ · . − 2l · α n 1 − α2 In case l=1 the variance expression will be reduced to: D~r 1 ≈

1 − α2 . n

^2 variance because l = i/2 = 1 and The formula for D~r1 may be applicable as a ρ2 = a2 = r1 = α: Da^ 2 = D~r1 ≈

1 − a22 . n

Knowledge of the estimates of Yule model factors allows to determine the covariances of the estimates of oscillation characteristics. In the case of the optimal sampling (Δt = T0/4), the decrement value depends only on the factor a2 : δ=−

π π · 4f0 lnð − a2 Þ ≈ 2ð1 + a2 Þ. lnð − a2 Þ = − 2πf0 ω0 Δt

Therefore, the rms error is a doubled value of σa^2 : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 + a2 Þð1 − a2 Þ 4δð4 − δÞ δð4 − δÞ = = , σδ = 2 n 2n2 n and its relative value will be:

(33)

54

3 Estimates accuracy

σδ = δ

rffiffiffiffiffiffiffiffiffiffi 4−δ . δ·n

(34)

The natural frequency of random oscillations may be calculated from factors a1 and a2 via the following expression: a1 cos θ = cos 2πf0 · Δt = pffiffiffiffiffiffiffiffiffi . 2 − a2 This equation may be linearized for a1 = 0 and f0 . Δt = 0.25 providing an expression for frequency deviations in the following form: Δf0 =

4πΔt

1 f0 pffiffiffiffiffiffiffiffiffi Δa1 = pffiffiffiffiffiffiffiffiffi Δa1 , π − a2 − a2

which allows to obtain an expression for the frequency rms error: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f0 1 + a2 f0 δ·2·2 = = σ f0 = π ð − a2 Þð1 − a2 Þn π 2ð2 − δÞð4 − δÞn pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 · f0 δ . π ð2 − δÞð4 − δÞn

(35)

It follows that an appropriate relative error of the oscillation frequency estimates may be represented as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ f0 δ . (36) ≈ 0, 5 f0 ð2 − δÞð4 − δÞn Expressions (34) and (36) represent errors of estimates of oscillations characteristics as a function of the decrement value and realization size. The latter can be replaced by a parameter NT determined a realization size in oscillations period: NT =

n n = . 2κ 4

The values of relative rms errors are plotted in Fig. 15 against the values of decrement δ and realization size NT. σ𝛿/𝛿, %

σf /f, %

75 50

𝛿 = 0.01

25

𝛿 = 0.2

0

250

500

750

𝛿 = 0.2

0.2

𝛿 = 0.01

NT

Fig. 15: Relative errors of estimates.

0

250

500

750

NT

3.2 Statistical error of estimates

55

These plots indicate that the stability of estimates is achieved when the realization size is at least 500 periods. The realization size NT ≥ 500 must be chosen in view of the requirement for ergodic properties as well. As pointed out earlier, the ergodic property assumes correlation attenuation over an interval within the realization size. For δ = 0.01, the first and last terms of the series are not virtually correlated only when NT > 400. Hence, estimation of oscillation parameters based on a single realization may be carried out if its length is at least 500 periods. As it is known, the least squares method is an effective approach to increase the accuracy of estimates by using a big number of initial data. It looks likely that the accuracy of estimates ^δ and ^f0 may be increased by using, for instance, k correlation ρ1, . . ., ρk. In this case, those correlations allow to compose an overdetermined system of equations which may be solved by utilizing the least squares method. In the case of optimal sampling, the Yule series reduces to AR(1) process: rl = αrl − 1 , where l = i/2, α = а2. It was pointed out that a variance of ρl estimates is determined by the expression:   1 ð1 + a22 Þð1 − a2l2 Þ 2l − 2l · a2 . Drl ≈ · n 1 − a22 Therefore, for δ = 0.01 and δ = 0.2 corresponding variances of r1 estimates are as follows: D^r1 ðδ = 0.01Þ ≈

0.01 , n

D^r1 ðδ = 0.2Þ ≈

0.2 , n

D^r10 ðδ = 0.01Þ ≈

0.94 , n

D^r10 ðδ = 0.2Þ ≈

and for ^r10 5.9 . n

As i = 2l, the 10th correlation of the AR(1) process corresponds to the 20th correlation of the AR(2) process. If 20 covariance estimates have variances equal to D^r , 1 utilizing the least squares method may decrease the estimate variance of decrement up to 20 times. Actually, variances of correlation estimates with lag k = 20 are larger by 94 (δ = 0.01) and 29 (δ = 0.2) times compared to D^r making the use of the least 1 squares method insufficient. This conclusion agrees with the numeric simulation of this approach. The decrement estimates obtained from 25 correlations and identified for 20 realizations of the AR(2) process are denoted by “ο” sign in Fig. 15. Involvement of a large number of correlations for estimate calculations does not

56

3 Estimates accuracy

reduce the error compared to that in decrement estimates obtained by using only ρ1 and ρ2 estimates. Results of accuracy analysis of estimates determine the data processing procedure in the following form: – the sampling interval must be close to a quarter of oscillations’ period; – the length of the processing realization must be larger than 500 periods of oscillations; – the number of correlations utilized for evaluation of oscillations characteristics is not significant from the standpoint of statistical accuracy of the estimates.

3.3 Bias of decrement estimates It was mentioned in Section 2.3 that the measurement signal Z may include broadband instrumental noise Ξ(t) with variance Dξ. The presence of this noise will lead to measurement signal readouts of the following form: zi = xi + ξ i . Straightforward use of zi for estimation of Yule model factors will lead to a bias of decrement estimates which was illustrated in Fig.10. The value of this bias may be evaluated in the following manner. The optimal sampling (κ = 2, а1 = 0) transforms the equations for the AR(2) process and its covariance to: xi = a2 xi − 2 + εi ; γx2 = a2 γx0 . Utilizing covariances of the process Z produces the expression: ^2 = a

γz2 M½zi zi − 2  M½ðxi + ξ i Þðxi − 2 + ξ i − 2 Þ = = = γz0 M½zi2  M½ðxi + ξ i Þ2  M½xi xi − 2 + xi − 2 ξ i + xi ξ i − 2 + ξ i ξ i − 2  . Dx + Dξ + 2γxξ

Because of independence of X and Ξ this expression may be written as follows: ^2 = a

γx2 + γξ 2 Dx + Dξ

=

a2 γx0 + γξ2 Dx + Dξ

=

a2 Dx + γξ2 Dx + Dξ

.

(37)

Considering that δ ≈ 2(1+a2), the bias of the estimate may be evaluated as follows:

3.3 Bias of decrement estimates

 ^2 − a2 Þ = 2 bδ = 2ða

a2 Dx + γ2ξ Dx + Dξ

57

 2ðγξ2 − a2 Dξ Þ − a2 = = Dx + Dξ

2ðγξ2 =Dξ − a2 Þ

2ð1 − a2 Þ 4−δ 4 < ≈ < . 1 + Dx =Dξ 1 + Dx =Dξ Dx =Dξ Dx =Dξ pffiffiffiffiffiffiffiffiffiffiffiffiffi The value σx/σξ = Dx =Dξ is called the signal-to-noise ratio which characterizes a level of noise in the measurement signal. If one defines a threshold value bδthr as an insignificant decrement bias, the corresponding value of the ratio σx/σξ providing such a bias may be determined as follows: 2 σx =σξ > pffiffiffiffiffiffiffiffiffi . bδthr For example, if the value bδthr = 0.001, the bias of decrement estimates may be ignored for σx/σξ > 63. However, a fixed value bδthr has a different influence on the real value of the decrement. For example, the importance of the above-mentioned value bδthr = 0.001 is different for decrements δ = 0.01 (10%) and δ = 0.2 (0,5%). Therefore, a definition of the value bδthr being in congruence with statistical uncertainty of decrement estimates rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffi δð4 − δÞ δ σδ = ≈ n NT looks more reasonable. If the threshold value bδthr = β⋅σδ (β < 1), it means that an insignificant bias is defined as a value that is a few times less than the rms error of a decrement value. For instance, if the value β = 0,33 is defined, the threshold value of bias will be three times less than σδ. If NT = 1000, an estimate of the value δ = 0.01 will have the insignificant bias with σx/σξ > 61, the value δ = 0.2 – with σx/σξ > 28. Such an approach looks more rational because it ties the threshold value bδthr with the statistical error depending on decrement values as well as realization size. In Section 2.4 it was demonstrated that bandpass filtering of measurement signals is an effective means to improve the signal-to-noise ratio. However, filtering will affect the process X as well. In particular, the FIR filter transforms the input process AR(2) into the ARMA (2,q) process: yi =

q−1 X j=0

cj xi − j = a1 yi − 1 + a2 yi − 2 +

q−1 X

cj εi − j = a1 yi − 1 + a2 yi − 2 + ϕi ,

j=0

where q is the number of filter weighing coefficients cj. The bandpass filter with acceptable performance has, as a rule, more than a hundred weighing coefficients. Actually, the correlation function of the ARMA (2,q) process will correspond to the correlation function of the AR(2) process starting from the (q+1) lag. It was demonstrated in Section 3.2 that a statistical error of correlation

58

3 Estimates accuracy

estimates with big lags increases drastically. For instance, a variance of the ρ20 estimate grows in 94 (δ = 0.01) or 29 (δ = 0.2) times. Besides that, trustworthy estimates may be obtained only for decrement values δ ≤ 0.04. For instance, in the case of δ = 0.2 we have values ρi ≥ 0.05 up to the 60th lag. For this reason, to enlarge the range of estimated decrement values one must use correlations featuring shorter lags. However, this will lead to summing correlations of Y and Φ processes up. It means that the use of ARMA(2,q) process for estimation of oscillation parameters results in a decrement estimate bias. Let us study the bias mechanism to establish measures which allow us if not to eliminate, then at least to reduce the decrement bias. Covariances of the ARMA(2,q) process (8) after the optimal sampling will take the following form: γj = a2 γj − 2 + c0 γyε ðjÞ + c1 γyε ðj − 1Þ − . . . − cq − 1 γyε ðj − q + 1Þ,

(38)

where γyε ðjÞ = M½yi − j · εi  is the cross-covariance function of Y (filter output) and E. Values γyε ðjÞ for j=0,1,2, . . . may be evaluated taking into account the fact that the preceding values yi do not depend on subsequent values εi and values of the process E are not correlated: γyε ð0Þ = M½yi · εi  = M½ða2 yi − 2 + εi Þ · εi  = Dε ; γyε ð − 1Þ = M½yi + 1 · εi  = M½ða2 yi − 1 + εi + 1 Þ · εi  = 0; γyε ð − 2Þ = M½yi + 2 · εi  = M½ða2 yi + εi + 2 Þ · εi  = a2 Dε ; · γyε ð − lÞ = al2 Dε , where l=j/2. The magnitude of Dε is related to the process AR(2) variance (15) which at κ = 2 becomes: Dε = Dy ð1 − a22 Þ. This expression and the expression (38) allow us to write the covariance γ2 as follows: γ2 = a2 γ0 +

qX − 1=2 l=2

c2l al2− 2 Dε = a2 γ0 + Dy ð1 − a22 Þ

Corresponding correlation will have the following form: ρ2 = a2 + ð1 − a22 Þ

qX − 1=2 l=2

c2l al2 .

qX − 1=2 l=2

c2l al2− 2 .

3.3 Bias of decrement estimates

59

During the optimal sampling the second correlation of the AR(2) ρ2 = a2 so the MA (q) process Φ will generate a bias Δρ of this correlation: Δρ = ð1 − a22 Þ

qX − 1=2

c2l al2 .

l=2

Its value may be evaluated by the following approach. In Section 2.3, it was established that the coefficients of the pass band FIR filter satisfy the following inequality: q−1 X

c2j < 1.

j=0

As noted previously, an acceptable quality of filtering demands utilizing more than a 100 coefficients. One can determine q = 100. Assuming an equality of coefficients, its values may be evaluated as follows: rffiffiffiffiffiffiffiffi 1 = 0.1. jcl j < 100 Therefore, the highest value of Δρ may be estimated as follows: Δρ < ð1 − a22 Þ

49 X l=2

0.1 ·

0.1 · al2 < 0.1 · ð1 − a22 Þ

∞ X

al2 =

l=2

ð1 − a22 Þ = 0.1 · ð1 + a2 Þ = 0.05 · δ. 1 − a2

Particularly for δ = 0.3 the value Δρ < 0.015. In Section 3.2, the rms error of the esti^2 was determined in the form: mate ρ rffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 1 − a22 . σρ^2 = Dρ^2 ≈ n In the case of NT > 500 (n > 2000) and δ ≤ 0.3 (a2 ≥ −0.85) the value of σ^ρ2 < 0.012 ^2. Hence, that is received value of Δρ = 0.015 is compared to a statistical error of ρ the ρ2 bias cannot be ignored, though it is possible to speculate that assuming |cl| < 0.1 was too inaccurate. The real values of coefficients cl may be demonstrated by an example of a passband filter (q = 128) with the FR is shown in Fig. 16. Its weighting coefficients have values 0.00035 < |cl| < 0.04016 (significantly less than the above-assumed value of 0.1) and different signs. Therefore, a corresponding value of Δρ calculated for a2 = −0.85 equals Δρ ≈ 0.0016, which is ten times less than the above-evaluated value Δρ = 0.015.

60

3 Estimates accuracy

Af , dB 0

–20 –40 –60

0

0.04

0.08 |f – f0|Δt

Fig. 16: The FR of a filter with 128 weighting coefficients.

In the case of δ = 0.3 (a2 = −0.85), the rms error σρ^2 < 0.0016 is associated with a big length of realization (NT>27000), that is, in engineering practice the influence of Δρ may be ignored. In other words, an application of the examined FIR filter does not provide a significant bias of decrement estimates evaluated via readouts of filtered measurement signals. Actually, bandpass filtering may decrease the power of the process X cutting out parts of its spectrum from the filter band width, that is, Dy ≤ Dx . As far as the broadband noise Ξ is concerned, filtering decreases its energy at a more significant rate and the variance of filtered signals Dξf is determined as follows: Dξf ¼

Xq1 j¼0

c2j Dξ :

For instance, the filter with the FR represented in Fig. 16 has the value

qP −1 j=0

c2j = 0.074.

This means that filtering decreases the noise variance by more than 13 times. At the same time, being weaker, the noise Ξ will still be present in filtered signals. Therefore, the expression (37) has to be written in the following form: ^2 = a

a2 Dy + γξ2f Dy + Dξf

.

This expression allows to realize a procedure which eliminates biases of decrement estimates in the following way. First, values of the variance Dξf and covariance γξ2f have to be evaluated. For this purpose, the pass band filter is relocated in a spectrum area where oscillations X are not present. The measurement signal filtered in such a manner is used for ~ ξf and ~γ . evaluations of D ξ2f

3.4 Identification of oscillations

61

~ Σ of the value DΣ = Dy + Dξf and ba2 of the factor а2 have to be calcuEstimates D lated via readouts of the signal Z filtered by the central frequency fo. After that, the signal-to-noise ratio of the signal Z may be evaluated: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ξf ~Σ − D ~Σ D D = − 1. σy =σξf = ~ ~ Dξf Dξf Then, choosing a value β the threshold bias bδthr is calculated. This value together ^2 (^ δ) and NT allows to determine a threshold value of the signal-towith known a ^2 noise ratio. If this value is less than the above-evaluated ratio σy/σξf, the estimate a has to be corrected: ~2 = a

~ Σ − ~γ ^2 D a ξ 2f . ~ ξf ~Σ − D D

~2). A decrement estimate without a bias will be calculated as ~δ = 2(1+a

3.4 Identification of oscillations In Section 3.3, it was shown that the additive broadband noise makes a decrement estimate bias (increase). As such, there is a possibility to get nonzero values of decrement estimates from processing of a signal which is, for instance, a sum of harmonic oscillations (constant amplitude of correlations) and broadband noise (exponentially diminished correlations). In addition, the discretization procedure is realized in a non-perfect way. For instance, the discretization interval Δt may fluctuate. A statistical simulation demonstrates that if a value of Δt is randomized with σΔt/Δt = 0.1%, processing the corresponding discrete realization of the harmonic signal (δ=0) yields a decrement estimate ^δ = 0.01. When σΔt/Δt = 0.2%, the decrement estimate equals δ^ = 0.07. Therefore, when decrement estimates are small (close to zero) a process type must be identified because the two above-mentioned processes are the product of drastically different phenomena. Random oscillations are a response of an elastic system to broadband noise, whereas harmonic oscillations can be provoked, for example, by resonance or auto-oscillation phenomenon. It is expedient to build-up the identification of oscillations proceeding with the difference of the random and harmonic oscillations distributions presented in Fig. 17. It is the obvious fact that the most probable values of these two oscillations do not coincide. The most probable values of random oscillations are grouped in the vicinity of zero (curve 1), whereas those of harmonic oscillations are in the vicinity of the oscillations amplitude ≈1,4σх (curve 2). One may state that unlikely values of one

62

3 Estimates accuracy

f(x) 1

2

x

0

Fig. 17: Oscillations probability density: 1 – random oscillations; 2 – harmonic oscillations.

kind of oscillation are the most likely of the other. In other words, random and harmonic oscillations differ in frequency of occurrences of the same values. Starting from what was mentioned earlier, a probability of exceeding a certain signal threshold level xthr may be taken as the identification criterion. The threshold level must provide a maximal difference between the respective probabilities: P1 for random and P2 for harmonic oscillations: ΔP = P1 − P2 = max . Calculations yielded the maximum of ΔP = 0.195 for xthr = 0.85σx (Fig. 18).

ΔP 0.2

0.1

0

0.5

1.0

x/σx

Fig. 18: Dependence of ΔP value from threshold level.

The corresponding values of probabilities are P1 = 0.395 and P2 = 0.59. Therefore, in the case of harmonic oscillations an occurrence of values greater than 0.85σx is almost 1.5 times higher than in the case of random oscillations. In Section 1.1, an empirical estimate of probability of the occurrence of event A was introduced in the form of frequency ν:

3.4 Identification of oscillations

63

ν = mðAÞ=n, where n is the length of a realization (number of samples), and m(A) is the number of samples in which the event A occurred. Let us write the frequency of occurrence of the event |xi| ≤ 0.85σx as ν(A1) = m(A1)/n, and that of the event |xi| > 0.85σx as ν (A2) = m(A2)/n while n = mðA1 Þ + mðA2 Þ. The empirical estimate of probability is a random variable. Therefore, in order to identify the kind of oscillations one should use approaches of the theory of statistical inferences, that is, the kind of oscillation has to be identified by testing the hypothesis of the ν(A2) correspondence to the probability P1 or P2. Because of the statistical nature of the ν estimates, errors of two kinds are possible [9]: the nonacceptance of the hypothesis even though it is true results in error of the first kind, whereas the acceptance of the hypothesis even though it is false results in error of the second kind. Assessment of the probability of these errors necessitates a quantified distribution of estimates of ν to be available. Formally the estimation of ν reduces to the following. The verification of the xi value leads to one of two incompatible events: either A1 ( |xi| ≤ 0.85σx) or A2 ( |xi| > 0.85σx). Probabilities of these events are P(A1) and P(A2) with P(A1) + P(A2) = 1, that is, A1 and A2 are mutually exclusive. Probability of occurrence of A2 event m(A2) times in n samples may be described by the binomial distribution with the expectation P(A2)n and the variance P(A1)P(A2)n [9]. Transformation from m(A2) to frequency ν is realized by dividing m(A2) by n, therefore ν gets the following characteristics: μν2 = PðA2 Þ; σ2ν2 =

PðA2 Þ½1 − PðA2 Þ . n

Consequently, the distribution of estimate of ν for two examined kinds of oscillations is quantified as follows: μvran = P1 = 0.395; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 ð1 − P1 Þ 0.49 ≈ pffiffiffi ; σνran = n n

μνhar = P2 = 0.59; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 ð1 − P2 Þ 0.49 σνhar = ≈ pffiffiffi . n n

For values of P > 0.3 when n > 30, the binomial distribution can be approximated by the Gauss law with the same quantitative characteristics. Thus, the boundaries of confidence intervals (P = 0.999) for estimates of ν2 (top νran and bottom νhar) are as follows:

64

3 Estimates accuracy

1.47 νran = 0.395 + pffiffiffi ; n

1.47 νhar = 0.59 − pffiffiffi . n

Since for n > 250 margins νran and νhar do not cross (Fig. 19), the kinds of oscillations are statistically identifiable.

0.6

ν νreg

0.5

0.4 0

νran 250

500

750

n

Fig. 19: Margins of alternative hypotheses acceptance ð − σ x =σ ξ = 1;  − σ x =σ ξ = 2; + − σ x =σ ξ = 3Þ.

It is noteworthy to mention that the nature of statistical inference is such that if the hypothesis is not accepted then it is possible to predetermine the probability of the error of the first kind. So, when ν > νran and oscillations are random the probability of error of the first kind is Pα = 1 – 0.999 = 0.001. If ν < νran, that is, the hypothesis is accepted this does not mean that it has been verified with a given probability. In this case one can determine the probability of error of the second kind Pβ (acceptance of wrong hypothesis) for the alternative hypothesis. The probability (1 – Pβ) is called criterion power. Since for n > 250 margins νran and νhar of alternative hypotheses do not cross, this means that in the case of either hypothesis acceptance Pβ < 0.001 and criterion power exceeds 0.999. The obtained results reflect an ideal case but a presence of noise Ξ will randomize harmonic oscillations, that is, the additive broadband noise will alter the signal distribution. Statistical simulation of harmonic signal mixed with white noise of different signal-to-noise ratios allows to state that the proposed criterion enables harmonic signal identification for σx/σξ ≥2 (symbols “” and “+” on Fig. 19).

4 Experimental validation of autoregressive method This chapter contains results of the metrology validation of autoregressive method as well as examples of the method’s implementations for random oscillations in the air path and structure of gas turbine engines (GTEs). Considered processes with various physical nature are subjects of diverse scientific disciplines (aerodynamic, combustion, and structure durability). Therefore, these applications confirm the general meaning of the theoretical concept of the autoregressive method. In addition, the presented examples serve to develop the method implementation skill for different kinds of tasks (data analysis, development of measurement method, design of data processing device).

4.1 Digitization of measurement signals The conversion of a continuous signal x(t) into a sequence of numbers is carried out by the Analog-to-Digital Converter (ADC). The ADC realizes two consistent operations: signal discretization in time and quantization of xi readouts in value. These operations may influence data processing results – estimates of random oscillations characteristics. It was already mentioned in Section 3.4 that interval Δt randomization causes a decrement bias. In particular, data presented in that section allows to state that the value σΔt/Δt < 0.05% makes this bias negligible. The next step is the evaluation of quantization influence on data processing results. From a mathematical standpoint, digitization deals with value xi round off according to specified rules. Digitization may be exhibited in the following possible ways. Let values of xi be inside a range ±Хmax. Within this range, one can fix M values x(1), x(2), . . ., x(M) called quantization levels. The digitization procedure consists of pairing the value of xi with one of the quantization levels. Two algorithms: identification with either the nearest or nearest low quantization levels can be implemented. If the measured values are transduced into electrical analog signals initially (commonly used practice), then the second rule is implemented. A presentation of the measurement range via a set of discretization levels allows to assign just numbers of quantization levels to the values of xi and later on to deal with them only. The true value of a level is identifiable at any time with the knowledge of level scale Δx = x(j) – x(j−1). As it is known, for digital techniques the binary system is usually used and, as a result, the number of quantization levels corresponds to the base power 2, that is, M = 2m – 1, and accordingly the ADC output (result of digitizing) is the m-bit binary code (word).

https://doi.org/10.1515/9783110627978-005

66

4 Experimental validation of autoregressive method

As noted previously, from the mathematical standpoint digitization is the round-off operation, therefore the digitized value of xi will have an error. For instance, when values of xi are identified with a nearest low level, this error εq = xi – x(j−1) will have a value within the 0 ÷ Δx range. Over a small interval x(j−1) < x ≤ x(j) (M > 100) the probability density f(x) may be considered constant. It means that all values of εq have a virtually equal probability, that is, f(εq) has the form of a uniform distribution (Fig. 20).

f (εq)

1/Δ x

0

εq

Δx

Fig. 20: Probability density of quantization error.

The uniform distribution assumption allows to determine the εq mathematical expectation as follows: Δx ð

μq =

εq f ðεq Þdεq =

Δx . 2

0

The value of Δx is always known a priori, therefore a systematic error μq = Δx/2 can be assumed in subsequent processing. Because of the uniform distribution of the quantization error, its variance will be equal to Δx ð

εq −

Dq =

Δx 2

2 f ðεq Þdεq =

ðΔxÞ2 . 12

0

If values of digitized process change by several quantization levels when moving from xi to xi+1, one may truly assume the value of εqi to be statistically independent of εq(i+1). This is very applicable for the optimal sampling which provides four readouts for each period of oscillations. Statistical independence of εqi and εq(i+1) allows to consider the quantization error as non-correlated (“white”) noise, a sufficient feature of which is given by the variance Dq. In other words, the quantization error may be considered as an external additive

4.1 Digitization of measurement signals

67

noise independent of the process X(t). In Section 3.3 it was found that additive broadband noise results in oscillation decrement overestimate but its influence is insignificant when the “signal-to-noise” ratio >63. Let us now relate this ratio value to quantization parameters. Because of the Gauss distribution of random oscillations, their maximum values may be presented as Xmax = 3σx (Р = 0.997). Therefore, the discretization interval will be Δх = Xmax/(М − 1), and the quantization noise variance is related to the process variance and to the number of quantization levels by: Δx 3σx . σq = pffiffiffiffiffi = 12 3, 46ðM − 1Þ Hence, the condition σx/σq>63 will have the following form: σx =σy =

σx · 3, 46ðM − 1Þ ≈ 1.15ðM − 1Þ ≥ 63. 3σx

As a result, the effect of the quantization operation can be ignored when M > 56, that is, m ≥ 7. Incidentally, modern signal registration and processing devises (Fig. 21), such as those manufactured by the company MERA (www.nppmera.ru), use ADCs with 16-bit output words.

Fig. 21: Signal registration and processing device MIC-200.

68

4 Experimental validation of autoregressive method

Besides the 16-bits ADC, the MIC-200 device shown above has six ranges from ±0.02 V to ±8.5 V, ensured a multi-bit representation of measurement signals.

4.2 Metrology of autoregressive method The autoregressive method was validated by its application for electrical signal oscillations simulating outputs of any physical value sensor. To generate random oscillation realizations with different decrements, two precision oscillatory devices featuring natural frequencies f1 = 2334 Hz and f2 = 4856 Hz, respectively have been used. These devices were built using input and output amplifiers around the RLC passive circuit. Decrement magnitudes known up to the third decimal place had been set within 0.01 to 0.1 range by a varying the value of the feedback resistance. The FR of these devices is similar to the ones presented in Fig. 13. The broadband noise was supplied to the input amplifier, as the random oscillations were observed on the capacitor of the RLC circuit. The random oscillations were discretized by using an ADC with 12-bit output words. Its discretization error was characterized by the rms deviation σΔt < 0.01%. As it was mentioned in Section 4.2, such facts make the digitization impact negligible. Sampling of continuous realizations was done with f 1 = 0.28 ðκ = 1.78Þ and f 2 = 0.29 ðκ = 1.72Þ, that is, sampling rates were near to the optimum value κ = 2. For each value of the decrement in the series 0.01, 0.015, 0.05, 0.1 up to 16 realizations have been processed; the realizations size was equal to 250, 500, and 1000 oscillation periods. Values of σδ and σf were calculated from the obtained estimates of δ and f0. Processing results and curves of theoretical errors corresponding to expressions (34) and (36) are shown in Fig. 22. σ𝛿/𝛿, %

σf/f, % 0,2

80 60 40 20 0

𝛿 = 0,01

𝛿 = 0.1

0,15

0,015

0,1 0,05

0,05

𝛿 = 0,1

250

500

750

NT

0

𝛿 = 0.01

250

500

750

NT

Fig. 22: Comparison of theoretical and experimental errors ðo − f1 = 2334 Hz;  − f2 = 4856 HzÞ.

Taking into account that the confidence interval (P = 0.9) of rms errors derived from 16 estimates is ±50%, one can confirm that the experimental and theoretical values are in good agreement.

4.2 Metrology of autoregressive method

69

A more complete characteristic of the error is a confidence interval Δ = λσ which contains the estimates with a probability of P (usually P = 0.95). The value of λ is a function of P as well as an estimate distribution law. Therefore, to find Δδ and Δf it was necessary to identify the distribution of estimates. Particularly, Fig. 23 demonstrates a histogram (line 1) of decrement estimate deviations from the smallest value in the realization and the Rayleigh distribution (line 2).

l/1000 1

0,4

0,2 2

0

σ

(󰛿i – 󰛿min) 2σ 3σ 4σ 5σ

Fig. 23: Histograms of δ = 0.01 estimates.

Observing the similarity of experimental and Rayleigh distributions is confirmed by quantified distribution parameters. As it is known, the expectation of a random value R with the Rayleigh distribution may be represented as follows:  = 1, 86σR , R  An average value of the deviation estiwhere σR is the rms deviation of R from R. ^ ^ mate was ðδi − δmin Þ ≈ 1, 75σδ^ . The Rayleigh confidence interval for the probability P = 0.95 is  + 2σR . ΔR = 3.73σR ≈ R For the experimental distribution represented in Fig. 23, about 94% of decrement estimates have fallen inside the confidence interval 0. . .ð^δi − δ^min Þ + 2σ^δ . It is noteworthy to add that assuming the Gauss distribution of the decrement and frequency estimates would result in nearly the same confidence interval because its value (P = 0.95) is ΔP = ±1.96σ. Taking into account (33) and (35), the confidence intervals for the estimates may be evaluated via the following expressions:

70

4 Experimental validation of autoregressive method

Δ^δ ≈ ± 2σδ^ < ± 2ðδ=NT Þ 1=2 ; Δ^f ≈ ± 2σ^f < ± 0.2f0 ðδ=NT Þ 1=2 = ± 0.1f0 Δδ . 0

0

(39)

These interval values must be presented by just one significant figure.

4.3 Analysis of combustion noise Modern GTEs apply turbulent combustion which is characterized by a presence of acoustic (combustion) noise E(t). Having a broadband spectrum with an almost constant value, this noise is a stationary random process with the normal distribution. A spectrum sample of the combustion noise in a real chamber is presented in Fig. 24.

S( f )

f,Hz 0

250

500

750

Fig. 24: A spectrum of combustion noise.

Sometimes, such a noise may provoke an instable operating regime of a combustion chamber (“vibrating combustion”). This regime features oscillations X(t) of a chamber’s gas pressure with frequency f0. Vibrating combustion is accompanied by reduced combustion efficiency and may also lead to damage of chamber components. In practice of experimental research of an influence of chamber regime parameters as well as chamber design on its stable operation, a linear approach may be applied to describe vibrating combustion [10]. This means that the gas pressure oscillations X(t) will be interpreted as a response of a hypothetical linear elastic system for an input in the form of white noise E(t). In this case, the value of the decrement δ of gas oscillations will be a quantitative characteristic of the combustion chamber’s stability. Measurements of gas pressure oscillation has a specific which are conditioned by the presence of the processes X (oscillation mode) and E (combustion noise) in the same measurement signal Z. As it was demonstrated in Section 2.3, decrement values calculated from the process Z will always be overestimated, that is, estimates e δ will include the systematic bias bδ > 0. A value of the decrement bias can be evaluated using the approach proposed in [11]. Let us assume that the measurement signal includes just one oscillation mode X(t). In this case the measurement signal Z(t) is represented by readouts in the following form:

4.3 Analysis of combustion noise

71

zi = xi + εi . Another assumption will specify a signal discretization which has to be done with the optimal sampling (Δt = T0/4). Such sampling yields values a1 = γ1 = 0 and the relative frequency equals to f0 = f0 Δt = 0.25. These assumptions transform equations describing the Yule model and its covariances to the following forms: xi = a2 xi − 2 + εi ; γx2 = a2 γx0 . From the above-written equation, the estimate of the factor a2 may be calculated via covariances of process Z: ~2 = a

γz2 M½zi zi − 2  = . γz0 M½zi2 

~2 as The use of the expression for readouts zi allows to determine a ~2 = a

M½ðxi + εi Þðxi − 2 + εi − 2 Þ M½ðxi + εi Þ2 

=

M½xi xi − 2 + xi − 2 εi + xi εi − 2 + εi εi − 2  . Dx + Dε + 2γxε

(40)

Because the samples of “white” noise E are uncorrelated, and preceding values xi−j (j > 0) are independent from values εi, the numerator of eq. (40) will be transformed to M½xi xi − 2 + xi − 2 εi + xi εi − 2 + εi εi − 2  = M½xi xi − 2 + xi εi − 2  = M½ða2 xi − 2 + εi Þxi − 2  + M½ða2 xi − 2 + εi Þεi − 2  = a2 Dx + a2 M½xi − 2 εi − 2  = a2 Dx + a2 M½ða2 xi − 4 + εi − 2 Þεi − 2  = a2 ðDx + Dε Þ. The cross-covariance of X and E is as follows: γxε = M½xi εi  = M½ða2 xi − 2 + εi Þεi  = Dε . Finally, the expression for the estimate e a2 will be ~ 2 = a2 a

Dx + Dε . Dx + 3Dε

a2 j < ja2 j. The stationary Yule series has −1 < a2 < 0 which results in je

72

4 Experimental validation of autoregressive method

It means that the decrement value of the process Z is larger than the decrement of oscillations X which was illustrated in Fig. 10. As it was mentioned earlier, the variances Dε and Dx are linked by the expresa2 expression will become sion Dε = Dx ð1 − a22 Þ when a1 = 0. Because of this fact, the e ~ 2 = a2 a

Dx ð2 − a22 Þ 2 − a22 : = a2 2 4 − 3a22 Dx + 3Dx ð1 − a2 Þ

In the case of optimal sampling δ ≈ 2(1+a2), therefore the decrement bias may be evaluated as follows: ~ 2 − a2 Þ = bδ = δ~ − δ = 2ða   2 − a22 1 − a22 2a2 − 1 = − a2 . 2 4 − 3a2 1 − 0.75a22 For instance, if δ = 0.2 (a2 = −0.9), the value of the decrement bias equals bδ ¼ 0.44. Such a big bias (220%) leads to overestimated margin of chamber stability. In the Section 3.3, it was proposed to use bandpass filtering to extract the oscillation mode out of the measurement signal Z. Because the energy of oscillations is concentrated in the neighborhood of f0, signal X will be almost intact. As far as energy of the broadband noise is concerned, it will be reduced by bandpass filtering. So instead the variance Dε, a variance of filtered noise Dϕ equals Dϕ = Dε

q−1 X

c2j

j=0

~2 will be transformed must be taken into account. In particular, the expression for a to the following form: D + Dϕ ~ 2 = a2 x = a2 a Dx + 3Dϕ

Dx + Dε

Dx + 3Dε qP −1

1 + ð1 − a22 Þ

j=0

a2 1 + 3ð1 − a22 Þ

The FIR filter presented in Section 3.3 has

qP −1

qP −1 j=0

127 P j=0

c2j

j=0 qP −1 j=0

= c2j

c2j .

(41)

c2j

c2j = 0.074. Thus, the application of

this filter will provide the following expression for bδ:

4.3 Analysis of combustion noise

73

 bδ = 2a2

 1 + 0.074ð1 − a22 Þ 1 − a22 . − 1 = − a 2 4.128 − 0.75a22 1 + 3 · 0.074ð1 − a22 Þ

For the above-examined case with δ = 0.2, an implementation of the FIR filter reduces the decrement bias to bδ ¼ 0.048, that is, the filtering procedure decreases the value of bδ by almost ten times. The actual values of the decrement bias bδ may be illustrated by a test result for the combustion chamber of a real GTE. The purpose of this test was the determination of the operational margins related to the vibrating combustion. During one of the operational regimes of this chamber, gas pressure oscillations with f0 = 393 Hz was registered. Their spectrum is presented in Fig. 25. S( f )

f,Hz 0

393

750

Fig. 25: Spectrum of gas pressure oscillations.

First of all, filtering around f0 = 393 Hz of the measurement signal was done by applying the above-mentioned FIR filter. Readouts (after filtering) of the realization Z with the length NT = 1000 were utilized to calculate covariance estimates γzi used to build the system of Yule–Walker equations. Solving this system yields ~2 = −0.93 and the corresponding value ~δ = 0.14 ± 0.02. Substituting the estimate a 127 P ~2 as well as the value c2j = 0.074 in expression (41) the calculated value of a j=0 produces the following cubic equation: a32 + 2.79a22 − 14.51a2 − 15.36 = 0. The solution to this equation leads to a value of a2 ¼ −0.945 which allows to determine the decrement estimate e δ = 0.11 ± 0.02. As the value of the decrement bias bδ ¼ 0.03 is compared with the confidence interval Δδ = ±0,02 (Р = 0,95), it must be taken into account. During the test of the chamber, another regime (Fig. 26) was characterized by the value ~δ = 0.05 ða2 = − 0.975Þ. As it was mentioned in Section 3.4, a small value of the decrement may correspond to harmonic oscillations accompanied by broadband noise with a high level

74

4 Experimental validation of autoregressive method

S( f )

f, Hz 0

393

750

Fig. 26: Spectrum of gas pressure oscillations.

of energy. In Section 3.4, the corresponding criterion for identification of oscillations was proposed. Its value characterizes the frequency ν of the occurrence of event |xi| > 0.85σx. In the examined case, this criterion value was ~ν = 0.33 < νran=0.42 (n = 4000) so there is no reason to speculate about harmonic oscillations. A visual confirmation of this fact gives a histogram (Fig. 27) of values of oscillations filtered with f0 = 393 Hz. li/4000 0.2

Gauss distribution

0

x

Fig. 27: Histograms of oscillations with δ = 0.05.

The summary of evaluating the impact of broadband noise on decrement estimates is as follows: – presence of combustion noise in the measurement signal increases the value of decrement estimates which needs to be taken into account;

4.4 Contactless measurements of blade oscillation characteristics

75

– deduced formulas are applicable for examining phenomena in the linear elastic system with input (broadband noise) and output (random oscillations) existing in the same physical environment.

4.4 Contactless measurements of blade oscillation characteristics Distortions of GTE inlet air flow may provoke vibrations of compressor blades. If excitation energy dominates dissipation energy in vibrating blades, dangerous oscillations (flutter) may occur. These oscillations lead to unsteady aerodynamic operation of compressors and sometimes, to complete failure of blades. Dynamic blade loads during such vibrations have random nature. Therefore, a proper description of blade vibrations requires utilization of random functions. In case of flutter, the observing oscillations X(t) may be interpreted as a reaction of a hypothetical elastic system with input E(t) being white noise. Such an approach allows to identify flutter by evaluating the value of an oscillation frequency as well as to estimate dynamic blade loads by means of getting decrement estimates. Evaluation of characteristics δ and f0 of blade oscillations is subject to processing experimental data (blade vibrations) which may be received by contacted (intrusive) and contactless measurements. An example of a contactless method is the “Blade Tip Timing” (BTT) measurement. This concept dates back to the end of the 1940s [12]. Typically, the BTT method is realized with two sensors (S0, S1) embedded in nonrotating components of a compressor (Fig. 28).

S1

S0

Fig. 28: Allocation of two sensors.

Sensors mark each passing instance of the blade root and tip. The time interval between two moments is marked by S0 and S1 corresponding to blade tip deflection. During a compressor test procedure, these two sensors measure the tip displacement xij of each ith blade for jth rotor rotation.

76

4 Experimental validation of autoregressive method

Practical applications of BTT measurements began in the 1970s [13]. At present time this technique is widely used for researching blade flutter. Equipment from a company MERA (www.nppmera.ru) is an example of such instruments (Fig. 29).

S1 S0

Ethernet

Shaf t rotation sensor

Fig. 29: Schematic diagram of BTT method implementation.

The main peculiarity of the BTT method is the discretization of blade vibrations Xi(t) is realized by means of natural rotor rotation. In other words, the sampling interval between readouts xij equals Δt = Tr. As a rule, the sampling frequency fr = 1/Tr is less than the vibration frequency f0. It means that sampling of blade oscillations does not correspond to the Nyquist rate. Such specifics of sampling the vibrations Xi(t) create difficulties for implementation of spectral-correlation methods for readouts xij processing due to the spectrum aliasing effect. As a result, the BTT method is mainly used for evaluation of blade vibration magnitudes. Discrete nature of BBT readouts motivates the implementation of autoregressive method for their processing. To do this, an appropriate time domain representation of blade vibrations has to be provided. Such an approach utilizing three sensors S1, S2, and S3 located at a central angle α (Fig. 30) was proposed in [14]:

α

S1

α S2

Fig. 30: Allocation of three sensors.

S3

4.4 Contactless measurements of blade oscillation characteristics

77

Let us take into account only the flexural (bending) oscillations of blades. Their measurements are provided by locating a sensor along the blade chord above the axis of torsional oscillations. Such sensor arrangement leads to availability of threereadout x1ij, x2ij, and x3ij for ith blade during jth rotation (Fig. 31).

Xi (t) x1ij

x2ij Δt

t

x3ij

Fig. 31: Three sensors sampling.

A time interval between these readouts is determined by values of a central angle and a rotor speed rotation: Δt = α · Tr =ð2πÞ. The readouts x1i1, x2i1, x3i1, . . ., x3in collected during n rotations allow to calculate the MLE of oscillation covariances: n X ~i = 1 ~γ0i = D ðx2 + x2 + x2 Þ; 3n j = 1 1ij 2ij 3ij

~γ1i =

n 1 X ðx x2ij + x2ij x3ij Þ; 2n j = 1 1ij

~γ2i =

n 1X ðx x Þ: n j = 1 1ij 3ij

These covariance estimates make it possible to calculate the correlations estimates ~1i and ρ ~1i : ρ ~ i; ~1i = ~γ1i =D ρ ~ i. ~2i = ~γ2i =D ρ

78

4 Experimental validation of autoregressive method

~1i and ρ ~1i provide an ability to build the Yule–Walker equation In turn, estimates ρ system: ~1i ; ~1i = a1i + a2i ρ ρ ~2i = a1i ρ ~1i + a2i : ρ ~1i and a ~2i . Knowledge of AR(2) factor The solution to this system provides estimates a values may be used to calculate the decrement and frequency estimates of the ith blade:   ~1i ~f0i = arccos pa ffiffiffiffiffiffiffiffiffiffi ð2παÞ; (42) ~2i 2 −a . ~δi = − lnð− a ~2i Þ ð2~f 0i αÞ. A value of angle α sets a sampling rate of blade oscillations Xi(t) as Δt = α/(2π fр). Hence, this value choice looks challenging due to a possible wide range of the rotational frequency fr. For instance, the rotational speed of a low-pressure compressor in GTE changes by several multiples from “Idle Rate” (IR) to “Maximal Rate” (MR) (100%); in particular, during the IR it may have 20% to 40% of the MR rotation speed. As such, the challenge is related to the choice of angle α which, from on the one hand, makes sure that the sampling frequency satisfies the Nyquist rate and on the other hand, it is in the pseudo-optimal range. As it was mentioned in Section 3.1, the minimal value of κ = 1.5ðΔt = T0 =3Þ. Therefore, the value of the central angle α has to be determined for the IR as follows: αIR = 2πfIR · Δt =

2πfIR f ≈ 2 IR . 3f0 f0

It is an obvious fact that such a choice of the central angle will lead to an increase in the sampling rate during engine regime change from IR to MR. Initially, the accuracy of estimates will increase up κ = 2 (optimal sampling ensured the efficient estimates). After that, the rms error of estimates will increase due to a decrease in the value of determinant |B| of the Yule–Walker system. The longest operating regime of the GTE is “Cruise Rate” (CR) which is characterized by rotation speed in the range of 70–80%. This means that for the CR a value of κ may be in range of 2.6. . .6, that is, accuracy of δ and f0 estimates may drop by 1.3 to 3 times relative to the optimal sampling. Further increasing the rotation speed will lead to a drastic degradation of estimate accuracy. To avoid this, only the readouts of sensors S1 and S3 have to be taken into account after getting the CR. Such an approach will increase the sampling interval by two times Δt = α · Tr =n. As result, sampling with κ = 1.6−3.75 for rates from CR to MR will be available. Accordingly, changing the

4.4 Contactless measurements of blade oscillation characteristics

79

measurement scheme demands an appropriate modification of algorithms of processing the BTT readouts. The two sensors S1 and S3 provide a collection of readouts x1i1, x3i1, . . ., x3in during n rotations. Their availability allows to calculate two covariances for each ith blade: n 1 X ðx2 + x2 Þ; 2n j = 1 1ij 3ij

~γ0i = ~γ1i =

n 1X ðx + x Þ. n j = 1 1ij 3ij

~1i = ~γ1i ~γ0i may be calculated, and the first equation of Consequently, the estimate ρ the Yule–Walker system can be constructed. To compose the second equation the following covariances must be calculated ~γ1i ðTr Þ = ~γ2i ðTr Þ =

n−1 1 X x3ij x1iðj + 1Þ ; n − 1 j=1

n−1 X 1 ðx1ij x1iðj + 1Þ + x3ij x3iðj + 1Þ Þ; 2ðn − 1Þ j = 1

~γ3i ðTr Þ =

n−1 1 X x1ij x3iðj + 1Þ . n − 1 j=1

These estimates relate to values of covariance function Γi (τl) with lags τl = Tr + Δt·(l−2) for l = 1, 2, 3 (Fig. 32).

Γi (τ) 𝛾0i

𝛾2i (Tr) 𝛾1i

𝛾3i (Tr)

𝛾1i (Tr) Tr

τ

Fig. 32: Covariance function of blade oscillations.

Calculated on a basis of ~γ1i , ~γ2i , ~γ13 appropriate correlations allow the composition of the second equation:

80

4 Experimental validation of autoregressive method

~i3 ðTr Þ = ai1 ρ ~i2 ðTr Þ + ai2 ρ ~1i ðTr Þ. ρ Estimates of factors a1i and a2i will be yielded by a solution of the following equation system: ~i1 ; ~i1 = ai1 + ai2 ρ ρ ~i3 ðTr Þ = ai1 ρ ~i2 ðTr Þ + ai2 ρ ~1i ðTr Þ. ρ Utilizing estimates of factors a1 and a2 in expressions (42) gives values of δi and f0i estimates for each ith blade. There is a particular case when two sensors may provide measurements for entire rate of operation of the GTE – when the speed of rotation during IR fr ≥ 40% of the MR speed rotation. Actually, an association of κ = 1.5 with the IR will provide the oscillations discretization featuring κ = 1.5–3.75 for all rates from the IR to the MR. The proposed scheme of sensor allocation and a rule of central angle value determination provide δ and f0 estimates with a priori known expressions for their confidence intervals. Examining the value of the determinant |H| (Fig. 14) demonstrates that |B| ≥ 0.75 when κ = 1.5−3. In other words, the estimate accuracy degrades in the range 0–25% when 3–6 samples for an oscillation period are provided. Hence, confidence intervals (P = 0.95) determined by the expressions (39) have to be corrected by the factor 1.25. The Yule series model is applicable for estimating blade oscillation characteristics when a research phenomenon may be simulated by a linear elastic system and the first oscillation mode is under consideration. Besides that, the decrement and natural frequency of blade oscillations may be used for the purposes of blades “health” monitoring. Together with the vibration magnitude they compose a quantified vibration pattern of a blade. It other words, these three characteristics are diagnostic factors whose analysis allow to estimate the technical conditions of the blades. The vibration decrement, natural frequency, and magnitude must be measured in different GTE regimes during the production tests. Measured values will create the initial points of blade performance lifespan trend. It is a well-known fact there is a strong correlation of oscillation characteristics with structural variation of the blades (cracks, material property degradation, etc.). Therefore, monitoring of vibration decrement, natural frequency, and magnitude estimates allows to diagnose the current technical conditions of blades.

4.5 Application of tracking filter to vibration signals Tracking analysis of GTE vibrations is an effective means of engine diagnostics. Checking vibration levels on frequencies correlated with turbine shaft rotations

4.5 Application of tracking filter to vibration signals

81

allows to detect an appearance of structural defects and to prevent engine failures. The tracking analysis is usually implemented by a bandpass filter with a central frequency f0 that is correlated with the shaft rotation frequency. Such an examination of vibrations signals is conducted during steady as well as transient regimes of GTE operation. As it was discussed in Section 4.1, digital processing starts with the sampling of a continuous vibration signal x(t): xi = xðti Þ = xðΔt · iÞ. As the central frequency of the filter must be varied synchronously to engine rotor frequency fr, the sampling interval Δt is as follows: Δt = 1=ðkfr Þ, where coefficient k > 2. Therefore, the relative value of the filter central frequency: fr = fr · Δt = 1 = const. k This means that the digital filtering will be done with a constant value of the filter’s central frequency. The value of k depends on a range of possible values of rotor frequency fmin < fr < fmax, and its choice has to eliminate the spectrum aliasing effect during the discretization of vibration signal. The AR(2) model, being a discrete model of the elastic system, may be successfully applied for a digital realization of the tracking filter. A digital bandpass filtering is realized by AR(2) model in the form of IIR filter: yi = a1 yi − 1 + a2 yi − 2 + xi , where xi is a readout of the vibration signal; yi, yi−1, yi−2 and a1, a2 are readouts of the filter output and filter factors, correspondingly. A filter algorithm utilizing its previous outputs is called recursive. Their realization requires a much smaller size of memory as well as a number of mathematical operations compared to FIR filters. For instance, the bandpass FIR filter (q = 128) presented in Section 3.4 requires a memory for its 128 coefficients and 128 input readouts plus 128 multiplication and 127 addition operations. Whereas the implementation of abovementioned recursive filter requires a memory for only two coefficients and two previous output readouts as well as just two multiplication and two addition operations. At the same time, the recursive filter has specifics related to its stability. As far as AR(2) is concerned, this problem was solved in Section 1.2, and expression (19) established relationships between parameters of AR(2) model and characteristics of the elastic system

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4 Experimental validation of autoregressive method

1 f0 = f0 · Δt = 1 arccos paffiffiffiffiffiffiffiffiffi ; 2 − a2 2π

δ≈

(43)

1 + a2 . 2f0

It is necessary to remark that a change of Δt value (synchronously to fr changes) will lead to fr ¼ const: Accordingly, values of a1 and a2 will be constant and the transient processes in the filter will be excluded. A value of δ is correlated with bandwidth Δf (–3dB level) of the filter via earlier presented expression: Δf δ = . f0 π Thus, conformity of recursive filter factors and its amplitude-frequency performance may be established. The choice of factors a1 and a2 will be done from the standpoint of a filter realization in binary arithmetic and its good selectivity. The first condition allows to increase the calculation speed providing a real-time operation of the filter and to utilize simple microprocessors. A requirement of a high selectivity is required due to multicomponent nature of GTE vibration signals. A spectrum of a vibration signal observed in a steady (“Idle”) regime of a real engine is presented in Fig. 33.

S( f )

0

fr

f, Hz 300

Fig. 33: Frequency spectrum of GTE vibrations.

Analysis of the first expression (43) demonstrates dependence of value f0 mainly on а1 because the value of а2 is close to −1, providing a small decrement value (big selectivity). In turn, the value of the factor а2 will determine the filter bandwidth Δf, as it follows from the second expression in (43). When the value k = 7ðf0 = 0, 143Þ is chosen, the corresponding factor а1 = 1.25, and the multiplication operation may be transformed to division and addition a1 yi − 1 = 1, 25yi − 1 = yi − 1 +

yi − 1 : 22

4.5 Application of tracking filter to vibration signals

83

Since operations of dividing by 22 are realized by shift of binary words by 2 bits, the algorithm has a logical (binary) implementation. As result, multiplying yi−1 by а1 requires just three binary operations – two shifts and an addition. The value of а2 may be chosen in the same way a2 = − 1 +

1 : 2l

Particularly, l = 7 yields a value of а2 = −0.9922 (δ = 0.027). It means that the relative bandwidth of the filter Δf/f0 = δ/π ≈ 0.9%, and its frequency response (FR) presented in Fig. 34.

0

0,5

fΔt

–25 –50 A, dB Fig. 34: FR of a recursive filter.

A critical issue of utilizing binary arithmetic is data scaling. The discussed case was related to a design of a processing device using 16-bit words. This means that a dynamic range of data was limited by ±90.31dB (−32768 . . . 32767). The above-presented bandpass filter increases the level of a vibration component with a frequency fr by 42.14 dB (in ≈128 times). Hence, the range of readouts xi must be (90.31 – 42.14) = 48.17 dB (−256 . . . 256), that is, the digitized value of xi must be presented by an 8-bit word. As it was mentioned at the beginning of this section, a diagnostic factor of structural defects is the level of vibrations. The evaluation of the latter demands rectifying the filtered signal which may be accomplished by two consistent actions – detecting the envelope of the signal with frequency fr and following low-pass filtering of this envelope. In digital processing, the detection operation is realized by evaluating an absolute value of numbers: mi = jyi j: In binary arithmetic, an absolute value of positive numbers is evaluated by ignoring their sign bit, for negative numbers it is by subtracting from 0. Low-pass filtering may be provided by the IIR filter of a first order [15]: Ari = bArði − 1Þ + ð1 − bÞmi ;

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4 Experimental validation of autoregressive method

where Ari, Ar(i−1) are outputs of the low-pass filter (vibration levels) at moments ti, ti−1; b = expð − Δt=τf Þ, and τf is the time constant of the filter. The value of the factor b must be chosen with the approach proposed earlier b=1−

1 . 2l

Particularly, if l = 8, dividing by 28 may be accomplished by just one operation – shifting the most-significant byte of the 16-bit word to a position of the least-significant one. The value of l = 8 provides the time constant of the filter τf ≈ 36Tr. Summarizing the above, the filtering procedure may be described by the following system of equations:   i ; xi = x 7fP yi = yi − 1 + mi =

yi − 1 yi − 2 − yi − 2 + 7 + xi ; 22 2 ( yi , yi ≥ 0; ð0 − yi Þ, yi < 0;

Ari = Arði − 1Þ −

Arði − 1Þ − mi . 28

An experimental approbation of the designed algorithm was done with a real vibration signal whose spectrum was demonstrated in Fig. 33. As the main level of vibrations is correlated with the high-pressure turbine shaft, we shall look through this fr component of the vibration signal. The main challenge of the tracking filter approach is a selection of the rotor’s components during transient regimes of the engines. Initially, the designed algorithm was examined during a smooth increase of shaft rotations (run up) from IR to MR (Fig. 35).

fr, Hz 240 220 200 180 0

60

120 t, s

Fig. 35: Curve of fr changing in time (run up).

The value of fr required for synchronization of the x(t) discretization was determined from the signal Z(t) of the high-pressure turbine rotation sensor. Its first

4.5 Application of tracking filter to vibration signals

85

harmonic was extracted by a low-pass filter with a cutoff frequency equal to 250 Hz. The filtered signal was sampled with Δt = 1/850 s. Therefore, signal discretization was done with values κ = 1.7 − 2.3, that is, in the neighborhood of the optimal value κ = 2. A current value of rotation sensor signal zi may be described by the AR(2) model with the factor а2 = −1 (no damping): zi = a1 zi − 1 − zi − 2 + εi . The presence of readouts εi is dependent on rotor speed fluctuation due to limited accuracy of the engine control system (