A spectral estimation case study in frequency-domain by subspace methods
基于CBD适当选择组件的案例推理系统提议研究(IJITCS-V5-N9-4)
I.J. Information Technology and Computer Science, 2013, 09, 43-55Published Online August 2013 in MECS (/)DOI: 10.5815/ijitcs.2013.09.04A Proposal of Case Based Reasoning System for the Appropriate Selection of Components UsingCBDAbrar Omar Alkhamisi, M. Rizwan Jameel QureshiFaculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi ArabiaE-mil: abrar200828@, anriz@Abstract—Software engineering starts to be less linked to the development, but at the same time it tends to rely on using the component-based software. The community interested in software engineering has proposed what is called software reuse and offered some ways for component-based software development. The basic difficulty encountered when designing component-based systems is the process of searching for and selecting the appropriate set of the current software components. Selecting a component is considered a hard task in the Component Based Software Engineering (referred to as CBSE), particularly with the augmentation of the number of the component development. Hence, it is hard to select component for CBSE. Different ways and approaches were suggested to solve the problem related to software component selection. Validation of the proposed solution in this paper through collecting a sample of people who answer an electronic survey that composed of 15 questions. An electronic survey target distributed to specialists in software engineering through social sites such as twitter and Facebook also by email them. The result of the validation of the proposed solution proved using a new improvement CBR system to do select the suitable component .Index Terms—CBSE: Component Based Software Engineering, CBR: Case Based Reasoning System, OTSO: Off-The-Self SoftwareI.IntroductionSoftware engineering came to be less connected with the development; however, it tended to utilize the component-based software. The community concerned with software engineering has suggested what is called software reuse that is the process of creating software systems from predefined software components and submitted some procedures for component-based software development. Hence, the CBD is interested in developing the software from the pre manufactured components and the possibility of re-utilizing and keeping up those components. Those components are referred to as a software component.Since the software reuse began to have a noticeable effect, CBD has contributed greatly in decreasing the time needed to achieve development, augmenting flexibility, and raised the trustworthiness of the component-based systems. Of course, designing, carrying out, experimenting, fixing and documenting a component take more time than purchasing a component. Also, it is obvious that the self-contained components which present a group of specific functions in a system can be substituted without difficulty or problems. Furthermore, the components utilized in other systems are considered more mature than the new advancements.The basic difficulties faced when designing the component-based systems are represented in choosing and finding out a proper group of accessible software components. This obstacle is referred to as the component choice problem whereas nowadays the task of finding a group of components to perform the needed functions came to be a hard task. After defining the potential candidate components, a subsection of all candidates has to be chosen in such a way that meets the developer's goals. Then the following obstacle is represented in choosing a subgroup where all components are consistent with each other. It is worth noting that it is very hard to carry out the task of choosing and searching for proper components manually, particularly for larger systems.In spite of the fact that there is no commonly approved way for choosing the component, there are no specific rules and procedures for performing the component choice task. Therefore, every project carries out this task on his own way. There are numerous means that can handle the component selection problem. Hence, the improve case-based reasoning system approach is tackled as a solution for this problem.The remainder of this paper is organized as follows: Section 2 describes some brief literature review. Section 3 the authors have described the statement of the problem. Section 4 solution towards that problem is given. Section 5 the solution is validated by the means of a survey. Conclusion and future work are given in the final section.44 A Proposal of Case Based Reasoning System for the Appropriate Selection of Components Using CBDII.Related WorkThe Component-Based Software Engineering is responsible for selecting, designing, and putting such components together. Since this method becomes more common and the number of the commercially accessible software components increases, the selection of a set of components that meet the requirements with low costs becomes very difficult. Hence, this problem makes it important to be concerned with designing effective greedy and genetic algorithms [1]in order to computerize the component selection for software developing organization approaches utilized to handle this problem. On the other hand, other approaches handle this problem through utilizing evolutionary algorithms [2].The component selection is considered a critical problem in CBSE that is interested in gathering the pre-existing software components; the matter that results in creating a software system that is compatible with the client-specific needs and demands. There are two types of the evolutionary approaches utilized [3]. The first one evolutionary depends on components (EAc) while the second one evolutionary depends on requirements (EAr).The difficulty of selecting a component from a group of components is not considered a unique one. Generally, there are various substitute components which might be selected whereas every one of them meets specific needs and requirements. It is significant to select the most efficient substitutes; hence, it is necessary to assess the components. Therefore, these approaches [4, 5]achieve formulate the problem as multiobjective whereas the various metric values have been put into consideration and the principles of evolutionary algorithm have been utilized when handling this problem.Also, there is the difficulty of selecting the most proper components that meet the demands and needs without incurring high costs. The evolutionary algorithms [6]are utilized here to hand this difficulty. The process of re-utilizing the available components when designing a system saves time, but at the same time selecting the proper components that can meet the system requirements is a difficult task for a number of reasons. First of all, the specifics of the current components should embrace all data and information needed by the designers in order to take the proper decisions related to the selection of the most proper components. Also, the system designer faces difficulty when selecting between various components having similar functionality and varying in their quality and performance level. In this research [7], the component selection problem can be overcome through utilizing the simulated annealing method.The component selection is considered a hard task in CBSE. The first matter [8] considered when selecting a component is the cost which calculated basis the component quality and specification. Selecting the component according to this approach is considered a part of OTSO method that was designed primarily for this purpose.Selecting a component from a set of components is a big problem [9]. To face this problem, the system requirements should be reduced in order to diminish the number of the components that will be selected from. Then, a greedy approach should be followed when selecting the component whereas this approach relies on the components characteristics and features.Carrying out [10]the component selection task efficiently contributes greatly to the success of the system. In order to resolve the component selection problem, a comparison is drawn between the case based reasoning system approach and the conventional component selection methods.Table 1: Comparison of Related WorkPaper Title with Reference Number Problems FoundApproximation Algorithms for Software Component Selection Problem [1]. ∙The experiments minimizing the number of components used.∙The experiment's environment uses ActiveX controls as the set of components.∙Each ActiveX control has a set of characteristics. In this study, Each ActiveX control characteristic is assumed to be a requirement which can be satisfied with it.∙Dataset using in experiments a set of 60 components downloaded from the Internet.∙Algorithms use for this problem specific cases in which utility of a component regarding a requirement discrete zero or one values assumed here.∙In the experiment not including non-functional requirements in the set of requirements.Evolutionary Algorithms for the Component Selection Problem [2]. ∙In this experiment discussed not considered the dependencies between the requirements that have to be satisfied by the final system.∙In this experiment better results are obtained with smaller number requirements.Two Evolutionary Multiobjective Approaches for the Component Selection Problem [3]. ∙Limitation number of requirements and available components that used in the experiment. ∙Quality attributes for non-functional requirements do not consider in this experiment.A Metrics-based Evolutionary Approach for the Component Selection Problem [4]. ∙Limitation in this study metrics that use for select component three attributes: cost, reusability and functionality.∙In this experiment limitation number of components and requirements that used in the study.Paper Title with Reference Number Problems FoundPareto dominance - based approach for the Component Selection Problem [5]. ∙Limitation number requirements and components that used.∙Limitation in this experiment does not mention non-functional requirements when select component.An evolutionary multiobjective approach for the Component Selection Problem [6]. ∙In this experiment used dependencies between the requirements of the system. ∙In this experiment limitation number of components and requirements.An Integrated Component Selection Framework for System-Level Design [7]. ∙In this paper limitation proposed applied the problem selecting of components for Network-on-Chip (NOC) architecture.∙Limitation number of requirements that used.Component Selection for Component based Software Engineering [8]. ∙Limitation in OTSO method domain specific characteristics that would limit the applicability of the method in other domains.∙Limitation in the requirement specification may not be detailed enough for evaluating OTS software alternatives.∙Case study in this experiment relatively small, the evaluation processes and the resulting criteria.Component selection strategies based on system requirements' dependencies on component attributes [9]. ∙Limitation in paper components satisfying the system requirements cannot be found within the existing components and new components need to be developed.∙This approach depended only the given satisfied system requirements.A Study on Software Component Selection Methods [10].∙The drawback of this approach is that firstly they need to have a case base and database of components to use approach that not availability many cases in the beginning.III.Problem StatementSelecting a proper component became a hard task due to the increasing number of the reusable components. Users can easily examine tens of the existing components in order to determine the most suitable ones. However, they cannot examine hundreds or thousands of components in order to accurately select the components they require.This question is taken up in this paper to add the problem reviewers [1-10]: 'How to identify appropriate components to satisfy users' requirements?' is considered one of the main problems connected with the component reuse. The emersion of various component architecture criteria increases the seriousness of this problem. The next section submits the proposed solution to this problem.IV.The Proposed Solution for Component Selection ProblemSome approaches have been provided in order to solve the component selection problem. In this paper propose an approach that can handle this problem through using the improve Case Based Reasoning System in order to contribute in supporting the decision making process. The CBR system [10] sets down a group of functionalities and utilizes similar components to perform those functionalities. Also, it assumes that the user requirements are considered as cases. With the passage of time, the casebase is regarded as a database for components which might be considered as a general knowledge for resolving the problem [10]. Sometimes, cases needed to fully accomplish problem functionalities may be unavailable. In such cases, the database can assist greatly in handling the problem functionalities [10]. In this section, going to give briefly introduce the tasks of improving the CBR system. Case-based reasoning originates from the field of artificial intelligence that is utilized for reasoning and learning [11]. The CBR system utilizes previous experience in order to solve problems [11]. When the CBR is utilized, all the records or elements of the preceding problem will be stored in the database to be easily used and accessed when trying to resolve similar problems [11]. In this paper, the authors add two features to improve CBR system to overcome the drawback as shown below.∙The first one is represented in providing the CBR system with a database that includes all the potential cases that may help in solving the problem. This database is very important because when the CBR system was utilized for the first time, the outcomes have been unsatisfactory due to the lack of sufficient cases that assist in selecting the most suitable component that can satisfy the requirements.∙The second one is represented in adding new task the assurance task. After selecting the proper component from the CBR system stored temporally in the database, this procedure is reiterated again automatically to ensure the selection of the same outcome of the CBR system. If the outcome is different, the improve CBR system will in his turn combine the two components to create a new one meeting the user requirements and demands. Improve CBR system has five tasks: Retrieve, Reuse, Assurance, Revise, Retain. Improve CBR system is shown in figure 1.Fig. 1: The Proposed Case Based Reasoning System∙ As for the retrieve task, it depends on retrieving the component from the set of components in the database [10]. Also, this task consists of a number of subtasks that are shown as follows:1. Determination features: It is important to identify the requirements the problem description base the index of casebase [10].2. Searching: basis of requirements problem description is searched component similar to problem description and returned this component [10]. ∙ As for the reuse task, it depends on one of two ways to copy or to adjust [10]. Copy means to utilize the chosen component solution. On the other hand, adjust means that the selected component solution has changed in such a way that go well with the new problem [10].∙ The assurance task follows directly the reuse task. In other words, after performing the reuse task (i.e. selecting the proper component from the CBR system stored temporally in the database), this procedure is reiterated again to ensure the selection of the same outcome of the CBR system. If the outcome is different, the improve CBR system will in his turn combine the two components to create a new one meeting the user requirements and demands. After that the revise task to evaluation solution.∙ The revise task is a significant one whereas in this stage the user revises the solutions recommended by the system. Also, the revise task can be divided into two stages:1. Assessing the solution: In this stage, the success degree of the component selected to meet the user needs and requirements is assessed [10].2. Correcting mistakes: In this stage, if any fault is discovered in the existing solution, it should be illustrated and corrected [10].∙ The retain task is concerned with saving the problems solved in the system database whereas they are considered a part of a new component [10].V. Validation of the Proposed SolutionValidation of the proposed solution is one of the most important points that need to any research.In this paper the validation of the proposed solution through used an electronic survey. The purpose of using this method it's not too much time consuming and gives the respondent much of time to think and answer questions be credible. Validation of the proposed solution will be through collecting a sample of people who answer an electronic survey that composed of 15 questions. An electronic survey will be target distributed to specialists in software engineering through social sites such as twitter and Facebook also by email them. The Likert scale is the scale will be used in this research to answer questionnaire. Likert scale is given in the following Table 2.Table 2: Likert scale1 Strongly Disagree2 Disagree3 Neither Agreed Nor Disagree4 Agreed 5Strongly AgreedQuestions divided into 3 goals were arranged according to their relevancy to defined goals this goal: Goal 1: Management problem can be faced when selected component manual particularly with the augmentation of the number of the component development.Goal 2: Need prefect and effective automatic to do select components especially when the large number of components available.Goal 3: The prove used to improve the CBR system to do select the suitable component.A statistical analysis is made on the basis of gathering data through the distribution of questionnaires. The analytic form is represented through frequency tables and charts showing the exact degree of analysis. The describe the validation results on the basis of results below. Goal 1:Management problem can be faced when selected component manual particularly with the augmentation of the number of the component development.ProblemSimilar CasesCase BaseSelect ComponentAssu ranceReus eRetriev eRevis eTemporaryBasic DatabaseAdd SolutionRetainGoal: 1Q1: Do you agree the select component is one of the major problems when using reuse component? Results of total respondents 34 of question 1 given in Table 3 showing that 76 % (where 44 % strongly agreed and 32 % agreed) the people were supportive to question 1 that it is agree the select component is one of the major problems when using reuse component whereas 3% of the people were not agreed. The percentage of the people who has a neutral opinion neither agreed nor disagree is 21 %. The conclusion of the survey of this question is that the select component is one of the major problems when using reuse component. Following is the Table 3 showing the results obtained for the question 1.Table 3: Result for Question 1LikertScale FrequencyPercent Cumulative Percent1 0 0% 0%2 1 3% 3%3 7 21% 24%4 11 32% 56%5 15 44%100% Total respondents34100%Fig. 2: Graphical Representation of Question 1Goal: 1Q2: Have the company any problem face when select the wrong component?Results of total respondents 34 of question 2 given in Table 4 showing that 91 % (where 35 % strongly agreed and 56 % agreed) the people were supportive to question 2 that it is agree the company face problem when select the wrong component. The percentage of the people who have a neutral opinion neither agreed nor disagree is 9%. The conclusion of the survey of this question is that the company face problem when select the wrong component. Following is the Table 4 showing the results obtained for the question 2.Table 4: Result for Question 2 LikertScale FrequencyPercent Cumulative Percent1 0 0% 0%2 0 0% 0%3 3 9% 9%4 19 56% 65%5 12 35%100%Total respondents34100%Fig. 3: Graphical Representation of Question 2Goal: 1Q3: How much acceptable for users to look through tens of available components to identify the most appropriate ones?Results of total respondents 34 of question 3 given in Table5 showing that 62 % (where 21 % strongly agreed and 41 % agreed) the people were supportive to question 3 that it is agree the acceptable for users to look through tens of available components to identify the most appropriate ones. The percentage of the people who have a neutral opinion neither agreed nor disagree is 38%. The conclusion of the survey of this question is that acceptable for users to look through tens of available components to identify the most appropriate ones. Following is the Table 5 showing the results obtained for the question 3.Table 5: Result for Question 3Likert Scale FrequencyPercent Cumulative Percent1 0 0% 0%2 0 0% 0%3 13 38% 38%4 14 41% 79%5 7 21%100%Total respondents34100%Fig. 4: Graphical Representation of Question 3Goal: 1Q4: How much acceptable for users to look through hundreds, or thousands of candidate components to select what they really need?Results of total respondents 34 of question 4 given in Table 6 showing that 97 % (where 88 % strongly disagree and 9 % disagree) the people were not supportive question 4 that it is not agree the acceptable for users to look through hundreds, or thousands of candidate components to select what they really need. The percentage of the people who has a neutral opinion neither agreed nor disagree is 3%. The conclusion of the survey of this question is that not acceptable for users to look through hundreds, or thousands of candidate components to select what they really need. Following is the Table 6 showing the results obtained for the question 4.Table 6: Result for Question 4LikertScale FrequencyPercent Cumulative Percent1 30 88% 88%23 9% 97% 3 1 3% 100%4 0 0% 100%5 0 0%100% Total respondents34100%Fig. 5: Graphical Representation of Question 4Goal: 1Q5: How much difficult and time consuming to find a perfect component by present manual when not used automate component selection software?Results of total respondents 34 of question 5 given in Table 7 showing that 94 % (where 47 % strongly agreed and 47 % agreed) the people were supportive to question 5 that it is agree that difficult and time consuming to find a perfect component by present manual when not used automate component selection software. The percentage of the people who have a neutral opinion neither agreed nor disagree is 6%. The conclusions of the survey of this question are that difficult and time consuming to find a perfect component by present manual when not used automate component selection software. Following is the Table 7 showing the results obtained for the question 5.Table 7: Result for Question 5LikertScale FrequencyPercent Cumulative Percent1 0 0% 0%2 0 0% 0%3 2 6% 6%4 16 47% 53%5 16 47%100% Total respondents34100%Fig. 6: Graphical Representation of Question 55.1 Cumulative Survey of Goal 1Questions divided into 3 goals the first goal covers that management problem can be faced when selected component manual particularly with the augmentation of the number of the component development. That’s show 17.64% are strongly disagreed and 2.35% are disagree 15. 29% are neither agreed nor disagree 35.29% are agreed and 29.41% are strongly agreed.Table 8: Frequency Table of Cumulative Goal 1Q.NoStrongly DisagreeDisagreeNeither Agree nor DisagreeAgree Strongly Agree1 1 7 11 1523 19 12 3 13 14 7 4 30 3 15 2 16 16 Total 30 4 26 60 50 Avg.17.642.3515.2935.2929.41Fig. 7: Graphical representation of Goal 1Goal: 2Need prefect and effective automatic to do select components especially when the large number of components available. Goal: 2Q6: Do you believe there is a level of need for automatic to select component especially when the large number of components available?Results of total respondents 34 of question 6 given in Table 9 showing that 91% (where 56 % strongly agreed and 35 % agreed) the people were supportive to question 6 that it is agree that believe there is a high level of need for automatic to select component especially when the large number of components available. The percentage of the people who have a neutral opinion neither agreed nor disagree is 9%. The conclusion of the survey of this question is that believe there is a high level of need for automatic to select component especially when the large number of components available. Following is the Table 9 showing the results obtained for the question 6.Table 9: Result for Question 6Likert Scale FrequencyPercent Cumulative Percent1 0 0% 0%2 0 0% 0%3 3 9% 9%4 12 35% 44%5 19 56%100% Total respondents34100%Fig. 8: Graphical Representation of Question 6Goal: 2Q7: Do you believe the must used automatic method more effectively to do select components?Results of total respondents 34 of question 7 given in Table 10 showing that 85% (where 44% strongly agreed and 41 % agreed) the people were supportive to question 7 that it is agree that believe the must used automatic method to become more effective to do select components whereas 3% of the people were not agreed. The percentage of the people who has a neutral opinion neither agreed nor disagree is 12%. The conclusion of the survey of this question is that believe the must used automatic method to become more effective to do select components. Following is the Table 10 showing the results obtained for the question 7.Table 10: Result for Question 7Likert Scale FrequencyPercent Cumulative Percent1 0 0% 0%2 1 3% 3%34 12% 15% 4 14 41% 56%5 15 44%100%Total respondents34100%Fig. 9: Graphical Representation of Question 7Goal: 2Q8: How do you rate the possibility of problems when using chooses not perfect automated software for component selection?Results of total respondents 34 of question 8 given in Table 11 showing that 94% (where 59% strongly agreed and 35 % agreed) the people were supportive to question 8 that it is agree that height rate problems when using chooses not perfect automated software for component selection whereas 3% of the people were not agreed. The percentage of the people who has a neutral opinion neither agreed nor disagree is 3%. The conclusion of the survey of this question is that height rate problem when using chooses not perfect automated software for component selection. Following is the Table 11 showing the results obtained for the question 8. Table 11: Result for Question 8LikertScale FrequencyPercent Cumulative Percent1 0 0% 0%2 1 3% 3%3 1 3% 6%4 12 35% 41% 520 59%100% Total respondents34100%Fig. 10: Graphical Representation of Question 85.2 Cumulative Survey of Goal 2In the second goal define need prefect and effective automatic to do select components especially when the large number of components available. That’s show 1.96% are disagree 7.84% are neither agreed nor disagree 37.25% are agreed and 52.94% are strongly agreed.Table 12: Frequency Table of Cumulative Goal 2Q.NoStrongly DisagreeDisagreeNeither Agree nor DisagreeAgree Strongly Agree6 3 12 197 1 4 14 158 1 1 12 20 Total 2 8 38 54 Avg.1.967.8437.2552.94Fig. 11: Graphical representation of Goal 2Goal: 3The prove used improve CBR system to do select the suitable component.Goal: 3Q9: How much using improve Case Based Reasoning system for the software component selection problem is acceptable to you as compared with other methods? Results of total respondents 34 of question 9 given in Table 13 showing that 88% (where 47% strongly agreed and 41 % agreed) the people were supportive to question 9 that it is agree improve Case Based Reasoning system for the software component selection problem is acceptable to user as compared with other methods whereas 3% of the people were not agreed .The percentage of the people who have a neutral opinion neither agreed nor disagree is 9%. The conclusion of the survey of this question is that improve Case Based Reasoning system for the software component selection problem is acceptable to user as。
基于广义柯西分布的最大后验准则频谱估计方法
基于广义柯西分布的最大后验准则频谱估计方法作者：宋俊才张曙来源：《现代电子技术》2010年第07期摘要:经典的频谱估计方法和现代的频谱估计方法在低信噪比及小数据量的情况下,谱估计的分辨率和方差性能不能满足实际应用需要。
因此,提出一种高分辨率、高精度DFT变换的新方法,此方法特别适用于线性频谱的估计。
该方法基于最大后验概率准则,建立广义柯西-高斯分布模型,克服了短数据情况下的DFT变换分辨率低的缺点,具有收敛快、频率分辨率高、频率精度高的优点。
计算机仿真结果证实了新方法的有效性。
关键词:最大后验概率; 离散傅里叶变换; 频谱估计; 广义柯西分布中图分类号:TN911.6 文献标识码:A文章编号:1004-373X(2010)07-0017-04New Method for Spectrum Estimation Based on Generalized Cauchy Distribution and MAPSONG Jun-cai, ZHANG Shu(College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China)Abstract: In the low SNR and small amount of data, the resolution and variance performance of spectral estimation can not meet the actual requirement by using classical or modern spectrum estimation methods. Therefore, a new high-resolution and high-precision method of DFT transform is proposed. It is suitable for estimation of linear spectra. Based on maximum a posteriori probability criterion, a generalized Cauchy-Gaussian distribution model to overcome the low resolution of DFT in the case of short data is established. The proposed method has advantages of fast convergence, high efficiency and high accuracy.The results of computer simulation show that the novel method is effective.Key words: maximum a posterior probability; discrete Fourier transform; spectrum estimation; generalized Cauchy distribution0 引言信号的频谱分析是研究信号特征的重要手段之一,该技术在雷达、通信、震动、地震信号处理及电子监测领域有着广泛的应用。
The_Spectral_Analysis_of_Random_Signals
7The Spectral Analysis of Random Signals Summary.When one calculates the DFT of a sequence of measurements of a random signal,onefinds that the values of the elements of the DFT do not tend to“settle down”no matter how long a sequence one measures.In this chapter, we present a brief overview of the difficulties inherent in analyzing the spectra of random signals,and we give a quick survey of a solution to the problem—the method of averaged periodograms.Keywords.random signals,method of averaged periodograms,power spectral den-sity,spectral estimation.7.1The ProblemSuppose that one has N samples of a random signal1,X k,k=0,...,N−1,and suppose that the samples are independent and identically distributed(IID). Additionally,assume that the random signal is zero-mean—that E(X k)=0. The expected value of an element of the DFT of the sequence,a m,isE(a m)=EN−1k=0e−2πjkm/N X k=0.Because the signal is zero-mean,so are all of its Fourier coefficients.(All this really means is that the phases of the a m are random,and the statistical average of such a m is zero.)On the other hand,the power at a given frequency is(up to a constant of proportionality)|a m|2.The expected value of the power at a given frequency 1In this chapter,capital letters represent random variables,and lowercase letters represent elements of the DFT of a random variable.As usual,the index k is used for samples and the index m for the elements of the DFT.In order to minimize confusion,we do not use the same letter for the elements of the sequence and for the elements of its DFT.587The Spectral Analysis of Random Signalsis E(|a m|2)and is non-negative.If one measures the value of|a m|2for some set of measurements,one is measuring the value of a random variable whose expected value is equal to the item of interest.One would expect that the larger N was,the more certainly one would be able to say that the measured value of|a m|2is near the theoretical expected value.One would be mistaken.To see why,consider a0.We know thata0=X0+···+X N−1.Assuming that the X k are real,wefind that|a0|2=N−1n=0N−1k=0X n X k=N−1n=0X2k+N−1n=0N−1,k=nk=0X n X k.Because the X k are independent,zero-mean random variables,we know that if n=k,then E(X n X k)=0.Thus,we see that the expected value of|a0|2isE(|a0|2)=NE(X2k).(7.1) We would like to examine the variance of|a0|2.First,consider E(|a0|4). Wefind thatE(|a0|4)=NE(X4i)+3N(N−1)E2(X2i).(See Exercise5for a proof of this result.)Thus,the variance of the measure-ment isE(|a0|4)−E2(|a0|2)=NE(X4i)+2N2E2(X2i)−3NE2(X2i)=Nσ2X2+2(N2−N)E2(X2i).Clearly,the variance of|a0|2is O(N2),and the standard deviation of|a0|2is O(N).That is,the standard deviation is of the same order as the measure-ment.This shows that taking larger values of N—taking more measurements—does not do much to reduce the uncertainty in our measurement of|a0|2.In fact,this problem exists for all the a m,and it is also a problem when the measured values,X k,are not IID random variables.7.2The SolutionWe have seen that the standard deviation of our measurement is of the same order as the expected value of the measurement.Suppose that rather than taking one long measurement,one takes many smaller measurements.If the measurements are independent and one then averages the measurements,then the variance of the average will decrease with the number of measurements while the expected value will remain the same.Given a sequence of samples of a random signal,{X0,...,X N−1},define the periodograms,P m,associated with the sequence by7.3Warm-up Experiment59P m≡1NN−1k=0e−2πjkm/N X k2,m=0,...,N−1.The value of the periodogram is the square of the absolute value of the m th element of the DFT of the sequence divided by the number of elements in the sequence under consideration.The division by N removes the dependence that the size of the elements of the DFT would otherwise have on N—a dependence that is seen clearly in(7.1).The solution to the problem of the non-decreasing variance of the estimates is to average many estimates of the same variable.In our case,it is convenient to average measurements of P m,and this technique is known as the method of averaged periodograms.Consider the MATLAB r program of Figure7.1.In the program,MAT-LAB takes a set of212uncorrelated random numbers that are uniformly dis-tributed over(−1/2,1/2),and estimates the power spectral density of the “signal”by making use of the method of averaged periodograms.The output of the calculations is given in Figure7.2.Note that the more sets the data were split into,the less“noisy”the spectrum looks.Note too that the number of elements in the spectrum decreases as we break up our data into smaller sets.This happens because the number of points in the DFT decreases as the number of points in the individual datasets decreases.It is easy to see what value the measurements ought to be approaching.As the samples are uncorrelated,their spectrum ought to be uniform.From the fact that the MATLAB-generated measurements are uniformly distributed over(−1/2,1/2),it easy to see thatE(X2k)=1/2−1/2α2dα=α331/2−1/2=112=0.083.Considering(7.1)and the definition of the periodogram,it is clear that the value of the averages of the0th periodograms,P0,ought to be tending to1/12. Considering Figure7.2,we see that this is indeed what is happening—and the more sets the data are split into,the more clearly the value is visible.As the power should be uniformly distributed among the frequencies,all the averages should be tending to this value—and this too is seen in thefigure.7.3Warm-up ExperimentMATLAB has a command that calculates the average of many measurements of the square of the coefficients of the DFT.The command is called psd(for p ower s pectral d ensity).(See[7]for more information about the power spectral density.)The format of the psd command is psd(X,NFFT,Fs,WINDOW)(but note that in MATLAB7.4this command is considered obsolete).Here,X is the data whose PSD one would like tofind,NFFT is the number of points in each607The Spectral Analysis of Random Signals%A simple program for examining the PSD of a set of%uncorrelated numbers.N=2^12;%The next command generates N samples of an uncorrelated random %variable that is uniformly distributed on(0,1).x=rand([1N]);%The next command makes the‘‘random variable’’zero-mean.x=x-mean(x);%The next commands estimate the PSD by simply using the FFT.y0=fft(x);z0=abs(y0).^2/N;%The next commands break the data into two sets and averages the %periodograms.y11=fft(x(1:N/2));y12=fft(x(N/2+1:N));z1=((abs(y11).^2/(N/2))+(abs(y12).^2/(N/2)))/2;%The next commands break the data into four sets and averages the %periodograms.y21=fft(x(1:N/4));y22=fft(x(N/4+1:N/2));y23=fft(x(N/2+1:3*N/4));y24=fft(x(3*N/4+1:N));z2=(abs(y21).^2/(N/4))+(abs(y22).^2/(N/4));z2=z2+(abs(y23).^2/(N/4))+(abs(y24).^2/(N/4));z2=z2/4;%The next commands break the data into eight sets and averages the %periodograms.y31=fft(x(1:N/8));y32=fft(x(N/8+1:N/4));y33=fft(x(N/4+1:3*N/8));y34=fft(x(3*N/8+1:N/2));y35=fft(x(N/2+1:5*N/8));y36=fft(x(5*N/8+1:3*N/4));y37=fft(x(3*N/4+1:7*N/8));y38=fft(x(7*N/8+1:N));z3=(abs(y31).^2/(N/8))+(abs(y32).^2/(N/8));z3=z3+(abs(y33).^2/(N/8))+(abs(y34).^2/(N/8));z3=z3+(abs(y35).^2/(N/8))+(abs(y36).^2/(N/8));z3=z3+(abs(y37).^2/(N/8))+(abs(y38).^2/(N/8));z3=z3/8;Fig.7.1.The MATLAB program7.4The Experiment61%The next commands generate the program’s output.subplot(4,1,1)plot(z0)title(’One Set’)subplot(4,1,2)plot(z1)title(’Two Sets’)subplot(4,1,3)plot(z2)title(’Four Sets’)subplot(4,1,4)plot(z3)title(’Eight Sets’)print-deps avg_per.epsFig.7.1.The MATLAB program(continued)FFT,Fs is the sampling frequency(and is used to normalize the frequency axis of the plot that is drawn),and WINDOW is the type of window to use.If WINDOW is a number,then a Hanning window of that length is e the MATLAB help command for more details about the psd command.Use the MATLAB rand command to generate216random numbers.In order to remove the large DC component from the random numbers,subtract the average value of the numbers generated from each of the numbers gener-ated.Calculate the PSD of the sequence using various values of NFFT.What differences do you notice?What similarities are there?7.4The ExperimentNote that as two ADuC841boards are used in this experiment,it may be necessary to work in larger groups than usual.Write a program to upload samples from the ADuC841and calculate their PSD.You may make use of the MATLAB psd command and the program you wrote for the experiment in Chapter4.This takes care of half of the system.For the other half of the system,make use of the noise generator imple-mented in Chapter6.This generator will be your source of random noise and is most of the second half of the system.Connect the output of the signal generator to the input of the system that uploads values to MATLAB.Look at the PSD produced by MATLAB.Why does it have such a large DC component?Avoid the DC component by not plotting thefirst few frequencies of the PSD.Now what sort of graph do you get?Does this agree with what you expect to see from white noise?Finally,connect a simple RC low-passfilter from the DAC of the signal generator to ground,and connect thefilter’s output to the A/D of the board627The Spectral Analysis of Random SignalsFig.7.2.The output of the MATLAB program when examining several different estimates of the spectrumthat uploads data to MATLAB.Observe the PSD of the output of thefilter. Does it agree with what one expects?Please explain carefully.Note that you may need to upload more than512samples to MATLAB so as to be able to average more measurements and have less variability in the measured PSD.Estimate the PSD using32,64,and128elements per window. (That is,change the NFFT parameter of the pdf command.)What effect do these changes have on the PSD’s plot?7.5Exercises63 7.5Exercises1.What kind of noise does the MATLAB rand command produce?Howmight one go about producing true normally distributed noise?2.(This problem reviews material related to the PSD.)Suppose that onepasses white noise,N(t),whose PSD is S NN(f)=σ2N through afilter whose transfer function isH(f)=12πjfτ+1.Let the output of thefilter be denoted by Y(t).What is the PSD of the output,S Y Y(f)?What is the autocorrelation of the output,R Y Y(τ)? 3.(This problem reviews material related to the PSD.)Let H(f)be thefrequency response of a simple R-Lfilter in which the voltage input to thefilter,V in(t)=N(t),enters thefilter at one end of the resistor,the other end of the resistor is connected to an inductor,and the second side of the inductor is grounded.The output of thefilter,Y(t),is taken to be the voltage at the point at which the resistor and the inductor are joined.(See Figure7.3.)a)What is the frequency response of thefilter in terms of the resistor’sresistance,R,and the inductor’s inductance,L?b)What kind offilter is being implemented?c)What is the PSD of the output of thefilter,S Y Y(f),as a function ofthe PSD of the input to thefilter,S NN(f)?Fig.7.3.A simple R-Lfilter647The Spectral Analysis of Random Signalsing Simulink r ,simulate a system whose transfer function isH (s )=s s +s +10,000.Let the input to the system be band-limited white noise whose bandwidth is substantially larger than that of the fie a “To Workspace”block to send the output of the filter to e the PSD function to calcu-late the PSD of the output.Plot the PSD of the output against frequency.Show that the measured bandwidth of the output is in reasonable accord with what the theory predicts.(Remember that the PSD is proportional to the power at the given frequency,and not to the voltage.)5.Let the random variables X 0,...,X N −1be independent and zero-mean.Consider the product(X 0+···+X N −1)(X 0+···+X N −1)(X 0+···+X N −1)(X 0+···+X N −1).a)Show that the only terms in this product that are not zero-mean areof the form X 4k or X 2k X 2n ,n =k .b)Note that in expanding the product,each term of the form X 4k appears only once.c)Using combinatorial arguments,show that each term of the formX 2k X 2n appears 42times.d)Combine the above results to conclude that (as long as the samplesare real)E (|a 0|4)=NE (X 4k )+6N (N −1)2E 2(X 2k ).。
一种考虑多次谐波的行波自然频率测距方法
一种考虑多次谐波的行波自然频率测距方法李金泽;李宝才;翟学明【摘要】在基于行波自然频率的输电线路单端故障定位方法中，主自然频率值的准确度是进行故障点精确定位的关键。
目前的主自然频率的提取大多采用小波变换、MUSIC 方法，小波分析受所选小波基影响较大，MUSIC 的参数选择对频谱估计影响较大，它们都未能很好地解决这一问题。
提出一种基于故障线路自然频率的单端测距新方法。
该方法在提取主自然频率过程中首先对行波信号进行EEMD 分解，并用 ICA 方法进行正交化处理，从而抑制 WVD 本身存在交叉项的问题，然后对各个分量进行 WVD 转换并叠加，获得正交的自然频率谱；进而综合考虑基波和多次谐波求取全局主自然频率。
EMTDC 仿真实验验证了该算法在不同故障类型、故障距离、过渡电阻和噪声情况下的可行性及其精度。
%In the single terminal fault locating method of transmission line based on traveling wave natural frequency, the accuracy of extracting primary natural frequency is the keyto caring out to pinpoint trouble spots in. Currently, wavelet transform and MUSIC method are commonly used for extracting primary natural frequency. Wavelet analysis is influenced by the selected wavelets and the parameters’ selection greatly impacts spectral estimation in MUSIC, which can’t solve this problem well. A new single ended fault location method of extracting faulted line natural frequencies is described. The traveling wave signal is decomposed by EEMD and orthogonal process is made with ICA method to suppress the WVD's problem of cross-term, and then each component of WVD is converted and superimposed to obtain the natural frequency spectrum orthogonal. Then the global primarynatural frequency is obtained considering the fundamental and harmonics. Simulation experiment by EMTDC confirms the feasibility and accuracy of the proposed algorithm under different fault types, fault distance, transition resistance and noise situation.【期刊名称】《电力系统保护与控制》【年(卷),期】2016(044)011【总页数】7页(P9-15)【关键词】主自然频率的提取;全局主自然频率;集合经验模态 WVD;输电【作者】李金泽;李宝才;翟学明【作者单位】华北电力大学控制与计算机工程学院，河北保定 071003;保定学院信息技术系，河北保定 071000;华北电力大学控制与计算机工程学院，河北保定071003【正文语种】中文高压输电线路是大容量、远距离送电的主要方式，因此，在发生故障时准确定位，快速排除故障对电力系统的安全运行具有重大意义。
高三英语学术研究方法创新思路探讨单选题30题
高三英语学术研究方法创新思路探讨单选题30题1. In academic research, a method that involves collecting and analyzing data from a large number of people is called:A. Qualitative researchB. Quantitative researchC. Descriptive researchD. Exploratory research答案：B。
本题主要考查学术研究方法的定义。
选项A“Qualitative research”定性研究，侧重于对事物的性质、特征和意义进行描述和解释。
选项C“Descriptive research”描述性研究，主要是对现象进行详细的描述。
选项D“Exploratory research”探索性研究，旨在探索新的领域或问题。
而选项B“Quantitative research”定量研究，就是通过收集和分析大量的数据来得出结论。
2. Which of the following is NOT a type of academic research method?A. Historical researchB. Experimental researchC. Hypothetical researchD. Comparative research答案：C。
在学术研究中，选项A“Historical research”历史研究，通过研究过去的事件和情况来获取知识。
选项B“Experimental research”实验研究，通过控制变量来验证假设。
选项D“Comparative research”比较研究，对不同的事物进行对比分析。
而“Hypotheticalresearch”不是常见的学术研究方法类型。
3. The research method that focuses on understanding the meaning and experience of individuals is:A. Mixed-method researchB. Grounded theory researchC. Action researchD. Phenomenological research答案：D。
unit 4 Electronic information science and technology
4.1 Introduction to Information Science and Technology
Data retrieval
4.1 Introduction to Information Science and Technology
Research vectors and applications
The following areas are some of those that information science investigates and develops. ◆ Information access ◆ Information architecture ◆ Information management ◆ Information retrieval ◆ Information seeking ◆ Knowledge representation and reasoning
New words and phrases
ASIC（application-specific integrated circuit ） 特定用途集成芯片 FPGA（field-programmable gate arrays ) 现场可编辑门阵列 stream processors 流处理器
4.1 Introduction to Information Science and Technology
New words and phrases
Electronic Information Science and Technology 电子信息科学与技术
interdisciplinary [,ɪntɚ‘dɪsəplɪnɛri] adj. 各学科间的 ,跨领域的 inter- pref. 表示“在一起， 交互”之义；表示“在… 中间、在…之间”之义 disciplinary [‗disiplinəri] adj. 规律的；训练的 discipline [‗disiplin] n. 学科；纪律；训练 vt. 训练，训导
作者姓名：阿布都瓦斯提·吾拉木
作者姓名：阿布都瓦斯提·吾拉木论文题目：基于n维光谱特征空间的农田干旱遥感监测作者简介：阿布都瓦斯提·吾拉木，男，1975年2月出生，于2006年7月获北京大学理学博士学位。
2006年12月至今任美国圣路易斯大学环境科学中心Geospatial Analyst/Research Professor。
中文摘要农田生态系统是一个水分、土壤、植被、大气等诸多因素耦合的复杂系统（SPAC，Soil-Plant-Atmosphere Continuum）。
在农田生态系统水循环中，水分亏缺的积累使农田供水量在一定的时间段内不能满足作物需水量，导致农田干旱的发生。
农田干旱直接和间接地影响人类生存、社会稳定、农业生产、资源与环境可持续发展。
正确评价或预防农田干旱，对促进农业生产和区域可持续发展具有重要的现实意义。
遥感具有客观反映农田水分时空变化的监测能力。
国内外农田遥感干旱监测研究表明：在复杂地表环境下，单纯采用可见光、近红外、热红外或微波波段都无法全面、准确反映农田水分信息，其方法在农田水分监测中暴露出诸多问题，如水分监测的滞后效应、模型复杂、参数的不确定性和过度依赖于田间和气象观测资料等，不能适应全面、动态的农田干旱监测与农田水分信息提取的迫切需求。
利用定量遥感方法，实现准确的农田干旱信息提取一直是遥感应用领域亟待解决的重要科学问题之一。
基于多维光谱特征空间的农田干旱信息提取，可以综合多源遥感的优势，为干旱监测提供更丰富、更高分辨率的农田水分信息，有望去除以往的遥感干旱模型带来的监测效果滞后、模型复杂、参数的不确定性等问题，形成农田干旱遥感监测新方法。
本论文以可见光近红外2维光谱空间干旱建模为切入点，通过加入短波红外，进一步拓宽遥感干旱监测的波段和地表生态物理参数，构建了反演土壤水分、叶片/冠层含水量（EWT）和叶片/冠层相对含水量（FMC）等参数的遥感模型，针对农田干旱最关键的两个指标土壤水分和叶片/冠层含水量，建立了多个干旱监测模型，形成了以n维光谱特征空间为基础的农田遥感干旱监测的新方法。
信号平稳性检验新方法
TESTING STATIONARITY WITH SURROGATES—A ONE-CLASS SVM APPROACHJun Xiao,Pierre Borgnat,Patrick Flandrin ´Ecole Normale Sup´e rieure de Lyon46all´e e d’Italie69364Lyon Cedex07FranceC´e dric RichardUniversit´e de Technologie de Troyes 12rue Marie Curie10010Troyes Cedex FranceABSTRACTAn operational framework is developed for testing stationar-ity relatively to an observation scale,in both stochastic and deterministic contexts.The proposed method is based on a comparison between global and local time-frequency features. The originality is to make use of a family of stationary surro-gates for defining the null hypothesis and to base on them a statistical test implemented as a one-class Support Vector Ma-chine.The time-frequency features extracted from the sur-rogates are considered as a learning set and used to detect departure from stationnarity.The principle of the method is presented,and some results are shown on typical models of signals that can be thought of as stationary or nonstationary, depending on the observation scale used.Index Terms—Stationarity Test,Time-Frequency Anal-ysis,Support Vector Machines,One-Class Classification1.REVISITING STATIONARITY Considering stationarity is central in many signal processing applications,either because its assumption is a pre-requisite for applying most of standard algorithms devoted to steady-state regimes,or because its breakdown conveys specific in-formation in evolutive contexts.Testing for stationarity is therefore an important issue,but addressing it raises some dif-ficulties.The main reason is that the concept itself of“station-arity”,while uniquely defined in theory,is often interpreted in different ways.Indeed,whereas the standard definition of sta-tionarity refers only to stochastic processes and concerns the invariance of statistical properties over time,stationarity is also usually invoked for deterministic signals whose spectral properties are time-invariant.Moreover,while the underlying invariances(be they stochastic or deterministic)are supposed to hold in theory for all times,common practice allows them to be restricted to somefinite time interval[1,2,3],possibly with abrupt changes in between[4,5].As an example,we can think of speech that is routinely“segmented into station-ary frames”,the“stationarity”of voiced segments relying in fact on periodicity structures within restricted time intervals. Those remarks call for a better framework aimed at dealing with“stationarity”in an operational sense,with a definition that would both encompass stochastic and deterministic vari-ants,and include the possibility of its test relatively to a given observation scale.This is the purpose of the present study.2.FRAMEWORK2.1.A time-frequency approachAs far as only second order evolutions are to be tested,time-frequency(TF)distributions and spectra are natural tools[6]. Well-established theories exist for justifying the choice of a given TF representation.In the case of stationary processes, the Wigner-Ville Spectrum(WVS)is not only constant as a function of time but also equal to the Power Spectrum Density (PSD)at each instant.From a practical point of view,the WVS is a quantity that has to be estimated.In this study,we choose to make use of multitaper spectrograms[7]defined as S x,K(t,f)=1Ksidered over a given duration,a process will be referred to as stationary relatively to this observation scale if its time-varying spectrum undergoes no evolution or,in other words, if the local spectra at all different time instants are statistically similar to the global spectrum obtained by marginalization.3.TEST3.1.SurrogatesRevisiting stationarity within the TF perspective has already been pushed forward[2],but the novelty is to address the sig-nificance of the difference“local vs.global”by elaborating from the data itself a stationarized reference serving as the null hypothesis for the test.Indeed,distinguishing between stationarity and nonstationarity would be made easier if,be-sides the analyzed signal itself,we had at our disposal some reference having the same marginal spectrum while being sta-tionary.Since such a reference is generally not available,one possibility is to create it from the data:this is the rationale be-hind the idea of“surrogate data”,a technique which has been introduced and widely used in the physics literature,mostly for testing linearity[8,9](up to some proposal reported in [10],it seems to have never been used for testing stationar-ity).For an identical marginal spectrum over the same obser-vation interval,nonstationary processes are expected to differ from stationary ones by some structured organization in time, hence in their time-frequency representation.A set of J“sur-rogates”is thus computed from a given observed signal x(t), so that each of them has the same PSD as the original signal while being“stationarized”.In practice,this is achieved by destroying the organized phase structure controlling the non-stationarity of x(t),if any.To this end,x(t)isfirst Fourier transformed to X(f),and the modulus of X(f)is then kept unchanged while its phase is replaced by a random one,uni-formly distributed over[−π,π].This modified spectrum is then inverse Fourier transformed,leading to as many station-Fig.2.Surrogates.Thisfigure compares the TF structure of the nonstationary FM signal of Fig.1(1st column),of one of its surrogates(2nd column)and of the mean over J=40 surrogates(3rd column).The spectrogram is represented in each case on the1st line,with the corresponding marginal in time on the2nd line.The marginal in frequency,which is the for the three spectrograms,is displayed on the far right of the 1st line.ary surrogate signals as phase randomizations are operated. Fig.1shows a nonstationary signal and one surrogate result-ing from this operation.The effect of the surrogate procedure is further illustrated in Fig.2,displaying both signal and sur-rogate spectrograms,together with their marginals in time and frequency.It clearly appears from thisfigure that,while the original signal undergoes a structured evolution in time,the recourse to phase randomization in the Fourier domain ends up with stationarized(i.e.,time unstructured)surrogate data with identical spectrum.3.2.One-class SVMOnce a collection of stationarized surrogate data has been synthesized,different possibilities are offered.Thefirst one is to extract from them some features such as distances between local and global spectra,and to characterize the null hypothe-sis of stationarity by the statistical distribution of their varia-tion in time.This approach is the subject of current investiga-tions that will be reported elsewhere[11].We will here rather focus on an alternative viewpoint rooted in statistical learn-ing theory:the collection of surrogates will be considered as a learning set and used to detect departure from station-arity.In this context,the classification task is fundamentally a one-class classification problem and differs from conven-tional two-class pattern recognition problems in the way howFig.3.One-class SVM with kernelκ(z i,z j)depending only on z i−z j.the classifier is trained.The latter uses only target data to estimate a boundary which encloses most of them.The ma-chinery of one-class Support Vector Machines(1-class SVM), which was introduced for outlier detection[12],can be used. This technique has been successfully applied to a number of problems,including audio and biomedical signal segmenta-tion[4,13].Let Z={z1,...,z J}be a set of J surrogate signals(or a collection of features derived from it).Letκ:Z×Z→R be a kernel function that satisfies Mercer conditions.The lat-ter can be used to map the z j’s into a feature space denoted by H viaϕ:Z→H defined asϕ(z)=κ(z,·).The space H is shown to be a reproducing kernel Hilbert space of functions with dot product ·,· H.The reproducing kernel property states that κ(z i,·),κ(z j,·) H=κ(z i,z j),which means thatκ(z i,z j)can be interpreted as the dot product be-tween z i and z j mapped to H byϕ(·).A classic example of Mercer kernel is the Gaussian kernel defined asκ(z i,z j)= exp(− z i−z j 2/2σ20),whereσ20is a bandwidth parameter. Note that it maps any data point onto a hypersphere of radius 1sinceκ(z j,z j)=1for all z j.The learning strategy adopted by1-class SVM is to map the data into the feature space corresponding to the kernel function,and determine the hyperplane w,ϕ(z) H−ρ=0 which separates them from the origin with maximum mar-gin.The decision function d(z)=sgn( w,ϕ(z) H−ρ)then gives on which side of the hyperplane any new point z falls in feature space,and determine if it may be considered as an outlier.For kernelsκ(z i,z j)depending only on z i−z j such as the Gaussian kernel,which map data onto a hypersphere, this strategy is equivalent tofinding the minimum volume hy-persphere enclosing the data[14];See Fig.3.Now,let us fo-cus on the optimization problem solved to get the hyperplane parameters w andρ.On the one hand,the distanceρ/ wthat separates the hyperplane from the origin must be maxi-mized.But on the other hand,the number of target samples wrongly classified as outliers must be minimized.Such sam-ples z j satisfy inequalities of the form w,ϕ(z j) H≥ρ−ξj withξj>0.Based on these results,the decision function is found by minimizing the weighted sum of a regularizationterm w 2,and an empirical error term depending on the mar-gin variableρand individual errorsξjmin w,ρ,ξ12 wνJ∞S(t,f)d f{F2−(F)2}}Fig.4.Two examples.Signals,spectrograms and space (P,F )of the TF features in AM (left)and FM (right)situations.From top to bottom,T 0=T/20,T and 20T ,with T =1600.In each case,the red circle corresponds to the (P,F )pair of one test signal used to draw the surrogates.Those surrogates (J =40in the experiments reported here)are plotted as green dots which,with 1-class SVM,de fine the domain of stationarity represented here as the gray shaded region,the blue circles corresponding to the support vectors.Magenta dots are independent realizations of the same test model.Other parameters are as follows —number of tapers:K =5,length of tapers:T h =387,modulation indices:α=0.5(AM)and 0.02(FM),signal-to-noise ratio:SNR =10dB —SVM kernel κis Gaussian,σ=0.07,ν=0.05.The first one (P )is a measure of the fluctuations in time of the local power of the signal,whereas the second one (F )op-erates the same way with respect to the local mean frequency.These characteristics are used as features,z =(P,F ),for the 1-class SVM whose output is displayed in Fig.4.The SVM toolbox proposed in [15]was used for this illustration.The results are shown for T 0=T/20,T and 20T ,allow-ing to consider stationarity relatively to the ratio between the observation time T and the modulation period T 0.They can be summarized as follows;Macroscale —For a small modulation period (or a large ob-servation time,i.e.,when T 0 T ),the situation can be considered as stationary,due to the observation of many similar oscillations over the observed time scale.This is re flected by a test signal (P,F )feature (red cir-cle,see caption)which lies inside the region de fined by the 1-class SVM for the stationary surrogates.Mesoscale —For a medium observation time (T ≈T 0),thelocal evolution due to the modulation is prominant and the red circle for the modulated signal is well outside the stationary region,in accordance with a situation that can be referred to as nonstationary.Microscale —Finally,if T 0 T ,the result turns back tostationarity because no signi ficative change in the am-plitude or the frequency is observed over the considered time scale.Two remarks can be made with respect to these results.First,although both AM and FM signals are seen as nonsta-tionary in the mesoscale regime,in the AM case the nonsta-tionarity manifests through a deviation of the local power P ,whereas in the FM case,it is the local frequency F that is mostly different from the stationary class.Second,it turns out that,in the microscale regime,the deterministic stationar-ity (FM case)naturally ends up with a much larger dispersion in P than in F since,by construction,the spectrum is narrow-band.Moreover,the randomization which underlies the con-struction of surrogates necessarily ends up with more power fluctuations in the stationarized data than in the original test signal,and hence with a (P,F )pair which,at best,lies on the border of the support of the stationary class.This sug-gests that,in some sense,the position of the (P,F )pair with respect to the stationary region does not only give an infor-mation about a possible nonstationarity but also an indication about its type.5.CONCLUSIONTesting for stationarity in signal processing and data analy-sis has already received some attention,but maybe not as much as might be expected from its ubiquitous nature.In this paper,we proposed an operational framework to mea-sure and test departures from stationarity.Its originality is that it makes use of a family of stationarized realizations of the analyzed signal,called surrogates,for de fining the nullhypothesis.Time-frequency features are then extracted from surrogates,and used as a learning set to train a1-class SVM which encompasses what may be considered stationary.A number of extensions to the present work are possi-ble.Here,we use in Section4specific bidimensional fea-tures,but one can think of directly using general TF represen-tations emerging from the use of SVM machinery for Time-Frequency,such as in[16].Another possibility is to general-ize the present approach to other forms of stationarity.This requires to define new specific stationarizing tools and signal representations,in the spirit of[17,18].6.REFERENCES[1]S.Mallat,G.Papanicolaou,and Z.Zhang,“Adaptivecovariance estimation of locally stationary processes,”Ann.of Stat.,vol.24,no.1,pp.1–47,1998.[2]W.Martin and P.Flandrin,“Detection of changes of sig-nal structure by using the Wigner-Ville spectrum,”Sig-nal Proc.,vol.8,pp.215–233,1985.[3]R.A.Silverman,“Locally stationary random processes,”IRE Trans.on Info.Theory,vol.3,pp.182–187,1957.[4]M.Davy and S.Godsill,“Detection of abrupt signalchanges using Support Vector Machines:An application to audio signal segmentation,”in Proc.IEEE ICASSP-02,Orlando(FL),2002.[5]urent and C.Doncarli,“Stationarity index forabrupt changes detection in the time-frequency plane,”IEEE Signal Proc.Lett.,vol.5,no.2,pp.43–45,1998.[6]P.Flandrin,Time-Frequency/Time-Scale Analysis,Academic Press,1999.[7]M.Bayram and R.G.Baraniuk,“Multiple window time-varying spectrum estimation,”in Nonlinear and Non-stationary Signal Processing,W.J.Fitzgerald et al.,Ed.2000,Cambridge Univ.Press.[8]J.Theiler et al.,“Testing for nonlinearity in time series:the method of surrogate data,”Physica D,vol.58,no.1–4,pp.77–94,1992.[9]T.Schreiber and A.Schmitz,“Improved surrogate datafor nonlinearity tests,”Phys.Rev.Lett.,vol.77,no.4, pp.635–638,1996.[10]C.J.Keylock,“Constrained surrogate time series withpreservation of the mean and variance structure,”Phys.Rev.E,vol.73,pp.030767.1–030767.4,2006.[11]J.Xiao,P.Borgnat,and P.Flandrin,“Testing stationaritywith time-frequency surrogates,”in Proc.EUSIPCO-07, Poznan(PL),Sept.2007,to appear.[12]B.Sch¨o lkopf,J.C.Platt,J.S.Shawe-Taylor,A.J.Smola,and R.C.Williamson,“Estimating the support of a high-dimensional distribution,”Neural Computation,vol.13, no.7,pp.1443–1471,2001.[13]A.B.Gardner, A.M.Krieger,G.Vachtsevanos,andB.Litt,“One-class novelty detection for seizure analy-sis from intracranial eeg,”Journal of Machine Learning Research,vol.7,pp.1025–1044,2006.[14]D.M.J.Tax and R.P.W.Duin,“Support vector data de-scription,”Machine Learning,vol.54,pp.45–66,2004.[15]S.Canu,Y.Grandvalet,V.Guigue,and A.Rakotoma-monjy,“SVM and Kernel Methods Matlab Toolbox,”Perception Syst`e mes et Information,INSA de Rouen, http://asi.insa-rouen.fr/˜arakotom/toolbox/index.html,2005.[16]P.Honeine, C.Richard,and P.Flandrin,“Time-frequency learning machines,”IEEE Trans.on Signal Proc.,vol.55,no.7(Part2),pp.3930–3936,2007. [17]P.Flandrin,P.Borgnat,and P.-O.Amblard,“Fromstationarity to self-similarity,and back:Variations on the Lamperti transformation,”in Processes with Long-Range Correlations:Theory and Applications,G.Ra-ganjaran and M.Ding,Eds.June2003,vol.621of Lec-tures Notes in Physics,pp.88–117,Springer-Verlag. [18]P.Borgnat,P.-O.Amblard,and P.Flandrin,“Stochasticinvariances and Lamperti transformations for stochastic processes,”J.Phys.A:Math.Gen.,vol.38,no.10,pp.2081–2101,Feb.2005.。
(完整版)数字信号处理英文文献及翻译
数字信号处理一、导论数字信号处理（DSP）是由一系列的数字或符号来表示这些信号的处理的过程的。
数字信号处理与模拟信号处理属于信号处理领域。
DSP包括子域的音频和语音信号处理，雷达和声纳信号处理，传感器阵列处理，谱估计，统计信号处理，数字图像处理，通信信号处理，生物医学信号处理，地震数据处理等。
由于DSP的目标通常是对连续的真实世界的模拟信号进行测量或滤波，第一步通常是通过使用一个模拟到数字的转换器将信号从模拟信号转化到数字信号。
通常，所需的输出信号却是一个模拟输出信号，因此这就需要一个数字到模拟的转换器。
即使这个过程比模拟处理更复杂的和而且具有离散值，由于数字信号处理的错误检测和校正不易受噪声影响，它的稳定性使得它优于许多模拟信号处理的应用（虽然不是全部）。
DSP算法一直是运行在标准的计算机，被称为数字信号处理器（DSP）的专用处理器或在专用硬件如特殊应用集成电路（ASIC）。
目前有用于数字信号处理的附加技术包括更强大的通用微处理器，现场可编程门阵列（FPGA），数字信号控制器（大多为工业应用，如电机控制）和流处理器和其他相关技术。
在数字信号处理过程中，工程师通常研究数字信号的以下领域：时间域（一维信号），空间域（多维信号），频率域，域和小波域的自相关。
他们选择在哪个领域过程中的一个信号，做一个明智的猜测（或通过尝试不同的可能性）作为该域的最佳代表的信号的本质特征。
从测量装置对样品序列产生一个时间或空间域表示，而离散傅立叶变换产生的频谱的频率域信息。
自相关的定义是互相关的信号本身在不同时间间隔的时间或空间的相关情况。
二、信号采样随着计算机的应用越来越多地使用，数字信号处理的需要也增加了。
为了在计算机上使用一个模拟信号的计算机，它上面必须使用模拟到数字的转换器（ADC）使其数字化。
采样通常分两阶段进行，离散化和量化。
在离散化阶段，信号的空间被划分成等价类和量化是通过一组有限的具有代表性的信号值来代替信号近似值。
GMM3
Bent E.SørensenJanuary23,20071Teaching notes on GMM III.1.1Variance estimation.Most of the material in this note builds on Anderson(1971),chapters8and9.[This book is now available in the Wiley Classics series].First recall thatΩ=limJ→∞Jj=−JE[f t f t−j].Notice,that for any L dimensional vector a we havea Ωa=Jj=−Ja f t(a f t−j) ,so,since the quadratic formΩis characterized by the bilinear mapping a→a Ωa(and similar for estimatesˆΩ,you see that the behavior of the estimators are characterized by the actions of the estimator on the univariate processes a f t.In the following I will therefore look at the theory for spectral estimation for univariate processes,and in this section we will ignore that f t is a function of an estimated parameter.Under the regularity conditions that is normally used,this is of no consequence asymptotically.Defining the k’th autocorrelationγ(k)=Ef t f t−k,our goal is to estimate ∞k=−∞γ(k).Define the estimate(based on T observations)of the k’th autocorrelation byc(k)= Tt=k[f t f t−k]T;k=0,1,2,....Notice that we do not use the unbiased covariance estimate of the autocovariances(this is obtained by dividing by T−k rather than T).We will use estimators of the formˆΩ=Jj=−Jw j c(k),1where the w j are a set of weights.(The reason for these and how to choose them is the subject of most of the following).The dependence of f t on the estimated parameter will be suppressed in the following,but it is always evaluated at our estimate.The spectral density isf(λ)=12π∞k=−∞γ(k)cos(λk).NOTE:f now denotes the spectral density as is common in the literature,it is NOT the moment condition!!!Now it is easy to show thatπ−πcos(λh)f(λ)dλ=12γ(h),since π−πcos(λh)cos(λj)dλ=πδhj(whereδhj is Kronecker’s delta[1for h=j,0other-wise]).You can easily see that the spectral density isflat(i.e.constant)if there is no autocorrelation at all,and that f(λ)becomes very steep near0,if all the autocovariances are large and positive(the latter is called the”typical spectral shape”for economic time series by Granger and Newbold).In any event,since we want to estimate only f(0),this is the all the intuition you need about this.The Sample Spectral DensityDefineI(λ)=12πTk=−Tc(k)cos(λk).I(λ)is that sample equivalent of the spectral density and is denoted the sample spectral density.It is fairly simple to show(you should do this!)thatI(λ)=12πT|Tt=1f t e iλt|2.The importance of this is that it shows that the sample spectral density is positive.We do not want spectral estimators that can be negative(or not positively semi-definite in the multivariate case).Anderson(1971),p.454shows thatEI(0)= π−πk T(ν)f(ν)dν,2wherek T(ν)=sin212νT2πT sin212νis called Fejer’s kernel.Notice that the expected value is a weighted average of the values of f(λ)in a neighborhood of0.If the true spectral density isflat then the sample spectrum is unbiased but otherwise not in general.Anderson also shows(page457)that if the process is normal thenV ar(I(0))=2[E{I(0)}]2(for non-normal processes there will be a further contribution involving the4th order cu-mulants).If|γ(k)|<∞then on can show thatlimT→∞EI(λ)=f(λ),and for normal processes on can show thatlimT→∞V arI(0)=2f(0)2,(and again there is a further contribution from4th order cumulants for non-normal pro-cesses).One can also show that(for normal processes)limT→∞Cov{I(λ)I(ν)}=0,forλ=ν,so that the estimates for even neighboringλs are independent.This independence together with the asymptotic unbiasedness is the reason that one can obtain consistent estimates of the spectral density by“smoothing”the sample spectrum.For a general(and extremely readable)introduction to smoothing and other aspects of density estimation(these methods are not specific for spectral densities),see B.Silverman:“Density Estimation for Statistics and Data Analysis”,Chapman and Hall,1986.3Consistent estimation of the spectral densityOne can obtain consistent estimates of the spectral density function by using weights,i.e.for a sequence of weights w jˆf (γ)=1πT −1 r =−T +1cos(γr )w r c (r ).If you definew ∗(λ|ν)=1πT −1r =−T +1cos(λr )cos(νr )w r ,it is easy to see thatˆf (ν)= π−πw ∗(λ|ν)I (λ)dλ.We will only use these formula’s for ν=0,but the important thing to see is that our estimate of the spectral density is a smoothed estimate of the sample spectral density.Also note that the usual way to show that a set of weights result in a positive density estimate is to check that the implied w ∗(.|0)function is positive.Anderson (page 521)shows thatlim E ˆf (0)= π−πw ∗(λ|0)f (λ)dλ.This means that the kernel smoothed estimate is not in general consistent for a fixed set of weights.Of course if the true spectral density is constant the smoothed estimate will be consistent (since the weights will integrate to 1in all weighting schemes you would actually use),but the more “steep”the actual spectral density is,the more bias you would get.We will show how one can obtain an asymptotically unbiased estimate of the spectral density by letting the weights be a function of T,but the above kind of bias is still what you would expect to find in finite samples,which is why it is worth keeping in mind.In most cases the weights take the formw j =k (j K T),where k is a continuous function,k (0)=1,k (x )=k (−x ),normalized such that the implied w ∗satisfies π−πw ∗(λ|ν)dλ=1for all ν.We will always assume that K T tends to infinity with T .4For the asymptotic theory the smoothness of the function k near 0is important,define k q aslim x →01−k (x )|x |=k q ,where q is the largest exponent for which k q is finite.Various ways of choosing the function k to generate the weights result in different values of q and k q .Under regularity conditions (most importantly ∞r =−∞|r |q γ(k )<∞)you find that for K T →∞such that the q -thpower grows slower than T ,K q T /T →0,thenlim K q T [E ˆf (ν)−f (ν)]=−k q 2π∞ r =−∞|r |q cos(νr )γ(k ).Note that this implies that the smoothed estimate is consistent,and the most important isthe rate of convergence,which is faster the larger K q T (subject to being less than T).It is easy to verify that q =1for the Bartlett kernel,and q =2for most other kernel schemes used.For the variance one can show thatlim T →∞T K T var {ˆf T (0)}=2f 2(0) 1−1k 2(x )dx(for the estimate at points not equal to zero or πthe factor 2disappears -this is due to the fact that the spectral density is symmetric around 0,so at 0a symmetric kernel will in essence smooth over only half as many observations of the sample spectral density).So wenotice that the variance does not go to zero at the usual parametric rate 1T ,but only at theslower rate K T /T .So in order to get low variance you would like K T to grow very slowly,but in order to obtain low bias you would like K T to grow very fast.You can also see that asymptotically the kernel with higher values of q will totally dominate the ones with lower values of q since you for the same order of magnitude of the variance get a lower order of magnitude of the bias.In practice this may no be so relevant,however,since the parameter q only depends on the kernel near 0,which only really comes into play in extremely large samples.Andrews (1991)shows the consistency of various kernel smoothed spectral density esti-mates (at 0frequency),when the covariances are estimated via estimated orthogonality conditions (or as you would usually say,when you use the error terms rather than the un-observed innovations).In this case some more regularity conditions,securing that the error term varies smoothly with the estimated parameters,are clearly necessary.The only kernels that allow for a q larger than 2are kernels that do not necessarily give positive density estimates,which people tend to avoid (although Lars Hansen have used5the truncated kernel,which belongs to those).Among the kernels that have q =2Andrews show that the optimal kernel is the one which minimizes k 2q ( 1−1k 2(x )dx )4.(See Andrews (1991),Theorem 2,p.829).This turns out to minimized by the Quadratic Spectral (QS)kernel which have the formk QS (x )=2512πx sin(6πx/5)6πx/5−cos(6πx/5) .The usual way the bias and the variance is traded offis by minimizing the asymptotic Mean Square Error.For simplicity definef (q )=12π∞ r =−∞|r |q γ(r ).It is simple to show that the MSE isK T T f 2(0) 1−1k 2(x )dx + 1K q T 2k 2q [f (q )]2Now in order to minimize the MSE,differentiate with respect to K T ,set the resulting expression equal to 0,solve for K T and obtainK T =2qk 2q [f (q )]2f (0)2 k 2 12q +1T 12q +1For example for the Bartlett kernel you can find k (0)=1andk 2=2/3.Andrews define α(q )=2[f (q )]2f (0)2and the optimal bandwidthK ∗T =qk 2qk 2(x )dx 12q +1(α(q )T )12q +1so you findK ∗T =1.1447[α(1)T ]1for the Bartlett kernel,andK ∗T =1.3221[α(2)T ]156for the QS kernel.Theαparameter depends on the(unknown)spectral density function at frequency0,but Andrews suggest that one assume a simple form of the model,e.g.an AR(1)or an ARMA(1,1),or maybe a VAR(1)in the vector case,and use this to obtain an initial estimate of f(0)which one then uses for an estimate of theαparameter.Notice that the important thing here is to get the order of magnitude right,so it is not necessary that the approximating AR(1)(say)model is the“correct”model.In case you knew the correct parametric model for the long run variance you would obtain more efficiency using this model directly rather than relying on non-parametric density estimators.In any event you can show for example for an AR(1)model with autoregressive parameterρthatα(1)=4ρ2(1−ρ)6(1+ρ)2/1(1−ρ)4.You should derive the one given here in order to get a feel for it(we will do it in class if there is time).More formulas are giving in Andrews(1991),you will need for exampleα(2) to use the QS kernel.Andrews also gives formulas for bothα(1)andα(2),for the case where the approximating model is chosen to be an ARMA(1,1),an MA of arbitrary order or a VAR(1)model.Typically the simple AR(1)model is used.In a typical GMM application you would run an initial estimation,maybe using the identity weighting matrix,then you would obtain an estimate of the orthogonality conditions(in other word,you would get some error terms)and on those you would estimate an AR(1) model,obtaining an estimateˆρ,and you would thenfindˆα(1)=4ˆρ2(1−ˆρ)6(1+ˆρ)2/1(1−ˆρ)4.which you would plug into your formula for the optimal bandwidth[this would be for the Bartlett kernel,for the QS kernel you would obviously have tofindα(2)].Usually you will have multivariate models and you would have to estimate either a multi-variate model for the noise(e.g.a VAR(1)),although I personally estimate an AR(1)for each component series and then use the average(i.e.setting the weights w a in Andrews’article to1)-this is the way the GMM program that I gave you is set up.In my experience,the choice between(standard)k-functions matters little,while the choice of band-width(K T)is important.I am not quite sure how much help the Andrews’formulae are in practice,but at least they have the big advantage that if you use a formula then the reader know you didn’t data mine K T.7Pre-whiteningSince the usual weighting scheme gives the autocorrelations less than full weight it is easy to see,in the situation where they are all positive,that the spectral density estimate is always biased downwards.Alternatively,remember that the spectral density estimate is a weighted average of the sample spectral density for neighboring frequencies,so if the sam-ple spectral density is not“flat”,the smoothed estimate is biased.Therefore Andrews and Monahan(1992)suggest the used of so-called“pre-whitened”spectral density estimators. The idea is simple(and not new-see the references in Andrews and Monahan)-if one can perform an invertible transformation that makes the sample spectrumflatter,then one should do that,then use the usual spectral density estimator,andfinally undo the initial transformation.This may sound a little abstract but the way it is usually implemented is quite simple:Assume you have a series of“error”terms f t and you suspect(say)strong positive autocorrelation.Then you may want tofit an VAR(1)model(the generalization to higher order VAR models is trivial)to the f t terms and obtain residuals,which we will denote f∗t,i.e.f t=ˆAf t−1+f∗t.More specifically the process offinding the f∗t s from the f t is denoted pre-whitening.It is easy to see that in large samples this implies(approximately)(I−ˆA)1TT1f t=1TT1f∗t,so we see thatV ar{1TT1f t}=(I−ˆA)−1V ar{1TT1f∗t}(I−ˆA )−1,and tofind your estimate of V ar{1T T1f t}youfind an estimate of V ar{1TT1f∗t}and usethis equality.This is denoted“re-coloring”.The reason that this may result in less biased estimates is that f∗t has less autocorrelation and therefore aflatter spectrum around0.On the other hand the pre-whitening operation may add more noise and one would usually only use pre-whitening in the situation where strong positive auto-correlation is expected.Also be aware that in this situation the VAR estimation is not always well behaved and you may risk that I−ˆA will be singular.Therefore Andrews suggests that one use a singular value decomposition ofˆA and truncate all eigenvalues larger than.97to.97(and less than-.97 to-.97)-see Andrews and Monahan(1992)for the details.Andrews and Monahan supply Monte Carlo evidence that shows that for the models they8consider,pre-whitening results in a significant reduction in the bias,at the cost of an in-crease(sometimes a rather large increase)in the variance.In many applications you may worry more about bias than variance of your t-statistics,and pre-whitening may be pre-ferred.An alternative endogenous lag selection schemeIn a recent paper Newey and West(1994)suggest another method of choosing the lag length endogenously.Remember that the optimal lag-length depends onα(q)=2f(q)f(0)2.Newey and West suggest estimating f(q)byˆf(q)=12πnr=−n|r|q c(r)which you get by taking the definition and plugging in the estimated autocorrelations and truncating at n.Similarly they suggestˆf(0)=12πnr=−nc(r).Note that this is actually the truncated estimator of the spectral density that we want to estimate but they suggest only to use this estimate in order to getˆα(q)=2ˆf(q)ˆf(0)2,and then proceed tofind the actual spectral density estimator using a kernel which guar-antees positive semi-definiteness.Newey and West show that one has to choose n of order less than T2/9for the Bartlett kernel and order less than T2/25for the QS kernel.Note that there still is an arbitrary constant(namely n)to be chosen,but one may expect that the Newey-West lag selection scheme will be superior to the Andrews scheme in very large sam-ples,(if you let n grow with the sample)since it does not rely on an arbitrary approximating parametric model.In Newey and West(1994)they perform some Monte Carlo simulations, that show that their own lag selection procedure is superior to Andrews’but only marginally so.In the paper Andersen and Sørensen(1996)we do,however,find a stronger preference for the Newey-West lag selection scheme in a model with high autocorrelation and high kurtosis.9。
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In this section, we review the subspace algorithms developed in [10] which consistently estimate the cross-PSD and the auto-PSD matrices from the samples in Eqs. (9) and (10). Algorithm 5.1 (Cross-PSD Estimation Algorithm): 1) Expand the Welch cross-PSD samples in Eq. (9) W −jω W (ejω ) to according to the relation Sxy (e ) = Sxy obtain samples of lengths M . 2) Fix the parameters p and r as p > nx +ny , r ≥ nx +ny , p + r ≤ M and for k = 1, · · · , p; l = 1, · · · , r, compute the Hankel matrix Y defined blockwise by Ykl = s ˆxy [k + l − 1] + s ˆxy [M + k + l − p − r − 1] (11) where for k = 0, · · · , M − 1, with zk = ej s ˆxy [k ] = 3) Calculate the SVD 1 M
Abstract— In this paper, the properties of two recently proposed frequency-domain subspace-based algorithms to estimate discrete-time cross-power spectral density (cross-PSD) and auto-power spectral density (auto-PSD) matrices of vector auto-regressive moving-average and moving-average (ARMAMA) models from sampled values of the Welch cross-PSD and auto-PSD estimators on uniform grids of frequencies, are illustrated by numerical and real-life application examples. The latter is concerned with the modeling of acoustic spectra for detecting faults in induction motors. Index Terms— power spectrum, subspace identification, Welch estimator, dynamic factor model, acoustic spectra.
II. T HE V ECTOR ARMAMA M ODEL Let x[k ] ∈ Rmx and y [k ] ∈ Rmy be observable output sequences of two jointly stationary processes. The discrete-time vector ARMAMA model is defined as x[ k ] y [k ] = = G x ( z ) e [ k ] + Hx ( z ) e x [ k ] , Gy ( z ) e [ k ] + Hy ( z ) e y [ k ] (1) (2)
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is divided into L disjoint segments of M samples each so that N = LM , i.e., the segments x(i) [k ] = x[k + iM ], 0 ≤ k < M, 0 ≤ i < L (3)
ห้องสมุดไป่ตู้
V. E STIMATION OF THE CROSS -PSD AND THE AUTO -PSD
2013 European Control Conference (ECC) July 17-19, 2013, Zürich, Switzerland.
A spectral estimation case study in frequency-domain by subspace methods
H¨ useyin Akc ¸ ay and Semiha T¨ urkay Department of Electrical and Electronics Engineering, Anadolu University Eskisehir 26470, Turkey
I. I NTRODUCTION The cross-spectral analysis is a fundamental and powerful technique to investigate an unknown relationship between two time series in frequency-domain. It is widely used in many engineering problems; e.g., time delay estimation of spatial sensors [1], [2], blind equalization in communications [3], speech enhancement [4], analysis of feedback systems [5], system identification of mechanical vibration systems [6], atmospheric problems [7], and vibration testing of structures against earthquake and wind loads [8]. The cross-PSD function has mostly been computed in a nonparametric way by using the fast Fourier transform (FFT). A parametric approach was recently proposed in [9] for the cross-spectral analysis of the so–called ARMAMA models. See Section 2, for the definition of vector ARMAMA models. In [10], the cross-PSD and the auto-PSD matrices were estimated by subspace methods using the averaged periodograms (the Welch auto and cross-PSD estimates) computed on uniform grids of frequencies from observed time series. The proposed algorithms are realization based and the AR parameters are computed by Hankel matrix factorization followed by extraction of the so-called observability range space, and the MA parameters are estimated from the data by a least-squares procedure. An additional step, which is implemented as solution to a convex semidefinite programming problem, comes into action when the lack of positivity of the estimated auto-PSD has been detected. The purpose of this paper is to illustrate in a case study the properties of the identification algorithms developed in [10] by means of numerical and real-life application examples. The latter is concerned with the modeling of acoustic spectra for detecting faults in induction motors.
MATRICES BY SUBSPACE METHODS
are formed and the modified periodograms 1 T X ( ω ; M ) X ( i) ( − ω ; M ) M ( i) are computed for i = 0, · · · , L − 1 where
- Gx ( z ) e[k ] - Gy ( z )
- x[ k ]
+ - h +6
- y [k ]
ey [ k ]
- Hy ( z )
Fig. 1.
Block diagram of the ARMAMA model.
The model defined by Eqs. (1) and (2) is actually a very special case of the dynamic factor model [11], [12] or errors-in-variables model [13]. III. T HE W ELCH METHOD Welch introduced a method [14] that is particularly well suited to direct computation of a power spectrum estimator using the FFT. In his method, a data record x[k ], 0 ≤ k < N