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數學相關外文翻譯-其他專業.doc

外文翻譯 ① 標題統計復變量和信號-第一部分變量 摘要 本文是專門研究高階統計復隨機變量。我們提出一個一般的框架允許直接操縱復數量分離之間的實數和虛數的部分變量是可以避免的。我們給規則整合和得出的概率密度函數和特征函數,使計算可以進行。在方程的多層面的變量,我們使用的自然結構張量。研究復變量導致的復循環隨機變量在高斯方程中延伸的概念。 1引入 高階統計現在已是一個密集型領域的信號和圖像處理的研究。 這一途徑的研究是基于使用的一個新的特性描述變量和信號。 到目前為止這一定性基本上是基于二階矩陣措施方差和協方差的變量、關聯和交叉關聯的信號在時間域、頻譜功率密度和跨頻譜功率密度的信號在頻率域。 [30,9,31]后的先鋒性的論文不斷開拓高階統計的潛力,使其使用密集。 這是很長的完整視圖,使這一領域的新模式正在涌現,并支持發展的大量應用程序。 可以找到一個綜合的[13,14,20,22,25,26]。 此外,有幾個特殊問題的期刊是專門討論這一議題[ 4-7] 和一系列專門的研究于1989年開始[13]。 其主要特點是在研究上找到高階統計的建模和應用程序。 在建模、隨機變量,基本上高階統計累計階數大于2。高階描述信號是通過多關聯在時間域和多光譜的頻率域中。 應用程序正在開發一個偉大的數字誤碼率的戰場。在高階統計給所有的經典域研究信號與圖像處理介紹了新方法。我們可以列舉盲源分離和盲反卷積問題在各種情況下振動診斷,水聲,雷達,衛星通訊,地震測深,天文學,等。非線性系統辨識,是高階統計一個基本的工具[ 19]。此外,一個與之密切相關的存在著的高階統計和模仿病的系統[10,17,23]。 這一非常積極和富有成果的研究領域需要有堅實的理論基礎。被那些很久以前研究隨機變量和信號的理論的數學家和統計學家建立起來的。高階統計特性隨機變量在許多經典教材[12,8,27]中被描述了。在[ 21 ],我們發現張量方法的發展對高階性能多維變量特別適合。該多相關時間域和多光譜被描述于[ 9]和[ 31]。 然而,一些作者仍然參與了有關領域的復雜隨機變量和信號,即使在實際應用中出現這種情況在頻域傅里葉變換后的處理,特別是在數組的處理,在單波段系統的通信信號的分析是常用的,在Wigner - Ville分布頻分析,等等。 高斯復雜的模型,這是足夠的在經典二階的方法,是記錄在[ 32]和[ 15]。這些作者證明了代數簡化帶來的使用是一個復雜的建模。他們還證明新的特性,如高斯復雜的圓,介紹了這個復雜的建模。 最近,缺少一般復雜的模型在[ 24]中被提出的證據作者指出,“奇怪的是,人們發現在文獻中很少處理復隨機變量和過程”。他們引進了“適當的復隨機過程”的概念,其名稱循環過程。然而,這種方法基本上是有限的二階性質。當在一個有關的雙譜中的復信號的這種特殊的性質,已體現在[ 16]。 隨著越來越多地使用高階統計,現在是需要開發一個通用的建模復雜隨機變量和信號。它的主要的目的是分成兩部分的說明。 在第一部分中,我們關注的是復隨機變量。我們首先定義的概率規律使用復雜的符號。我們的結果,在一般情況三維以及多維隨機變量是否高斯或非高斯,這是已知的高斯情況[ 15,32 ]。然后,我們的張量形式主義發展的實數情況在[ 21 ]中的多維復雜隨機變量。我們表明,對于一個給定的順序,不同種類的累積量可以定義。這一結果是一個延伸的偽協方差在[24]中介紹 。這個模型我們可以給一個一般性的定義,我們表明,在這一特定情況下,許多高階累積量是空的。我們表明之間的直接關系傅里葉變換和圓。復雜的圓形和高斯隨機變量被給出和說明模擬生成算法。說明算法新規則在附錄B中圓高斯例題中給出。 第二部分是用于建模和表示復雜的隨機信號。 為平穩信號,多維隨機變量在使用的結果被給出了,我們定義了復平穩隨機信號多相關性和多譜性。我們表明,完整的表征復雜的信號采用不同的多相關性和多譜在一階的需求中。在正常情況下的實際值信號,這些 多相關性和譜是相同的。不同的情況是信號分析的一些 多相關性和光譜是空空。擴展循環的信號概念,我們表明,這種信號,唯一的非空相關性和光譜在共軛和非共軛條件具有相同數量的。此外,我們表明,限制信號循環達到一定的秩序。我們回到選擇矩和累積量和顯示,除了傳統的利益提出的累積量是由于其加和其表征的高斯知識點,他們可以明確區分的性能在每個秩序和消除奇異的多光譜遙感。這個模型是然后擴展到數字信號和數字信號的時間限制使用離散傅立葉變換。 2起點 本文的目的是介紹復隨機變量的一般模型。有用這種模型的例子加以說明。我們將表明,它導致新的特征在信號的描述并允許發展領域的高階統計量所有的新的理解。 復隨機變量作為輸出大量的加工等 --傅里葉變換 --陣列處理, --希爾伯特變換 當處理復隨機變量可以使用 --考慮一個復隨機變量(CRV)作為二維實隨機變量(RRV), --發展于復隨機變量有關的代數工具 第二種方法具有優勢 --它使所有的推導簡單, --它保留的物理意義相關的復雜性質的數據。 這種方法在[ 15 ]中的在高斯案件中已經發展并引出來復高斯隨機變量的理論(CGRV)。在這種情況下審議的復高斯隨機變量引起了重要的概念---復雜的隨機高斯-西安循環變量。 本文的主要目標是推廣一般情況下高斯和非高斯隨機變量的這些概念。主要的動機是,新算法使用高階統計(HOS)正在開發中,很明顯的是在這個領域中,這是絕對要處理非高斯分布數據。此外,一種理論將制定使用張量構成自然框架的高階統計。因此,本文件的第二項主要的問題是,延長MacCullagh[21]提出的框架的復雜的案件。 在我們的模型建立后定義基本原則,我們將目前的技術作為實現的主要工具。我們將提供一個并說明其效用的這一新形式主義在復循環隨機變量的一般性的定義。 2.1 復隨機變量 復隨機變量定義是著名的。從實隨機變量和,我們定義復雜的隨機變量為 (1) 當。 把概率密度函數(pdf)和復隨機變量聯合在一起是一個轉折點。 在高斯圓案例,這是通過“兩個及其復雜共軛定義為. 有關高斯圓變量的“正規”的概率密度函數在[15]。 因此,它似乎從高斯例子,我們必須考慮與的定義,以便提取所有的統計信息。上述定義說明復高斯變量E[ Z”]等于零。因此,唯一的非空的二階矩陣是E[]。這意味著,這兩個和給出統計也許是不同的信息。因此,一個理論的高階統計量在一般情況下必須考慮的變量及其共軛復數。信息的統計資料不僅是兩個變量,而且在他們的互相統計。我們現在介紹我們的形式主義用更一般的方式來處理復雜的隨機變量。 主要的問題是從前面的討論是一個代數問題,因為這兩個變量、代數計算聯系在一起的。為了克服這一點,我們假設與在其實數區間內,在較大的區間中與不是代數相關。這樣做的一個辦法是考慮與(的實部和虛部)復雜的隨機變量。在這方面,我們將繼續寫和,但盡管符號不同,與不再是復共軛。為了引進一個符號在連續性的古典性和新的之間,使用張量,將不久,我們選擇使用這些模棱兩可的有關和的符號。我們將介紹在下列替代,避免了這一問題。 這個“惡作劇”將使我們把和作為獨立變量。我們將看到,這大大有利于所有的計算。然而,這些復隨機變量的純粹的概念,只有作為容易計算的手段。當我們想要回過頭來談談真實的物理世界,我們一定要限制和屬于所產生的子集的實數和。 2.2規則 為此,我們必須建立一些規則,以便獲得的定義是有意義的。我們將提出兩項法則 1所有的功能必須明確的數學,和所有的算子,如積分,必須收斂、 2當我們考慮到特殊情況的實隨機變量希望能夠恢復經典公式。 這種新的觀點,同時適用于一個二維隨機變量,如多維隨機變量。 我們現在將看到它是如何工作的。 3概率密度函數的功能和特點 我們會考慮先后一維和多維情況下的隨機變量。 3.1一維復隨機變量 我們已經看到,在2.1節的概率密度函數的一個功能 和中有關和的。讓我們嘗試定義第一特征函數。定義和這兩個復雜的變量可以寫 , . 如來自中的和,和可能是實數或虛數。特征函數已被定義[15] 2 和,然而,在指數讀取上有所爭議。 根據第二個規則,特征函數的復雜變量的和是延伸的二維平面復雜的經典特征函數定義為實變量。 因此,相關的整體的數學期望,應是與計算的、子集生成的和的實值。 3 其中2來自于雅可比的轉化,的子集是D中所以關于和中的和的所有值以及 。 關系(3)給出的定義,特征函數是適用于所有的來自與中的和的實數的功能。 在相同的方式,我們可以定義的第二種特征函數 4 其中,在真實的情況下,產生的累積量。 Statistics for complex variables and signals - Part I Variables Abstract This paper is devoted to the study of higher-order statistics for complex random variables. We introduce a general framework allowing the direct manipulation of complex quantities the separation between the real and the imaginary parts of a variable is avoided. We give the rules to integrate and derive probability density functions and characteristic functions, so that calculations may be carried out. In the case of multidimensional variables, we use the natural framework of tensors. The study of complex variables leads to the extension of the notion of complex circular random variables already known in the Gaussian case. 1. Introduction Higher-order statistics HOS are now an intensive field of research in Signal and Image Processing. This avenue of research is based on the use of a new characterization of variables and signals. Up to now this characterization was essentially based on second-order energetic measures variance and covariance for variables, correlation and cross correlation for signals in the time domain, pectral power density and cross-spectral power density for signals in the frequency domain. After the pioneering papers of [30,9,31] the potentialities of HOS are now used intensively. It would be very long to give a complete view of this domain in which new models are emerging that support the development of a large number of applications. A synthesis can be found in [13,14,20,22,25,26]. Furthermore, several special issues of journals have been devoted to this topic [4-7] and a series of specialized workshops began in 1989 [l-3]. The essential features in research on HOS are found in modeling and in applications. In modeling, for random variables, HOS are essentially based on cumulants of order greater than 2. The higher-order description of signals is made through multicorrelations in the time domain, and multispectra in the frequency domain. Applications are being developed in a great number of fields.In nearly all the classical domains of research in Signal and Image processing. HOS are introducing new methodologies. We can cite the blind source separation and blind deconvolution problems in a wide variety of situations vibrations diagnostic, underwater acoustics, radar, satellite communications, seismic sounding, astronomy, etc. In nonlinear systems identification, HOS are a basic tool [19]. Moreover, a close connexion exists between HOS and neuromimetic systems [10,17,23]. This very active and fruitful field of research needs solid theoretical foundations. They were built a long time ago by mathematicians and statisticians who developed the theory of random variables and signals. The higher-order statistical properties of random variables are described in many classical textbooks [12,8,27]. In [21] we found the development of atensorial approach particularly well fitted to the higher-order properties of multidimensional variables. The multicorrelations and multispectra are described in [9] and [31]. However, few authors have been concerned with the domain of complex random variables and signals, even if this situation appears in practical applications in frequency domain processing after Fourier transformation, particularly in array processing, in single band systems of communications where analytic signals are commonly used, in time-frequency analysis by the Wigner-Ville distribution, etc. The Gaussian complex model, which is sufficient in the classical second-order approach, is well documented in [32] and [15]. These authors have shown the algebraic simplifications brought by the use of a complex modeling. They have shown that new properties, like Gaussian complex circularity, are introduced by this complex modeling. More recently the lack of a general complex modeling was put in evidence in [24] the authors noted that “paradoxically, one finds in the literature very few treatments of complex random variables and processes”. They introduce the notion of “proper complex random processes” which is their denomination for circular processes. However, this approach is essentially limited to the second-order properties. This particular character of complex signals, when one is concerned with the bispectrum,has been exemplified in [16]. With the increasing use of higher-order statistics, it is now necessary to develop a general modeling for complex random variables and signals. It is the aim of this exposition, which is divided into two parts. In the first part, we are concerned with complex random variables. We begin by the definition of the probability laws using complex notations. We extend the results, which are already known for the Gaussian case [15, 32], to the general situations of monodimensional and multidimensional complex random variables, whether Gaussian or non-GausSian. Then, we extend the tensorial formalism developed in the real case in [21] to the multidimensional complex random variables. We show that, for a given order, different kinds of cumulants can be defined. This result is an extension of the pseudo-covariance introduced in [24]. With this modeling we can give a general definition of circularity, and we show that, in this specific case, many higher-order cumulants are null. We show the direct relation between the Fourier transform and circularity. Algorithms for the generation of complex circular non-Gaussian random variables are given and illustrated on simulations. An illustration of the new rules of calculation is given in Appendix B in the circula Gaussian case. Part II is devoted to the modeling and representation of complex random signals. For stationary signals, using the results given for the multidimensional random variables, we define the multicorrelations and multispectra for complex random stationary signals. We show that the complete characterization of complex signals at an order pdemands the introduction of different multicorrelations and multispectra. In the usual case of real valued signals these multicorrelations and spectra are identical. The situation is different for analytic signals for which some multicorrelations and spectra are null. Extending the concept of circularity to the signals, we can show that, for this kind of signal, the only nonnull multicorrelations and spectra possess the same number of conjugated and nonconjugated terms. Furthermore, we show that band limited signals are circular up to a certain order. We come back to the choice between moments and cumulants and show that, except for the classical interest presented by cumulants due to their additivity and to their characterization of the Gaussian property, they make it possible to distinguish clearly between the properties at each orde and to eliminate singularities in the multispectrum. This modeling is then extended to digital signals and to digital time-limited signals used in the Discrete Fourier Transform. 2. Starting point The purpose of this paper is to introduce a general model of complex random variables. The usefulness of this modeling will be illustrated with examples. We will show that it leads to new characteristics in the description of signals and allows a new insight into the developing field of higher-order statistics. Complex random variables CRV appear as the output of a great number of processingr such as - Fourier transforms, - Array processing, -- Hilbert transforms. When dealing with CRV two approaches can be used -to consider a CRV as a two-dimensional real random variable RRV, -to develop algebraic tools directly with CRV. The second approach has two advantages -it makes all the derivations simpler, -it preserves the physical sense related to the complex nature of the data. This approach has been developed [15] in the Gaussian case leading to the theory of complex Gaussian random variables CGRV. In this situation the consideration of the CGRV has given rise to the important notion of complex random Gaussian circular variables. The principal aim of this paper is to generalize these notions to the general case of Gaussian and non-Gaussian random variables. The primary motivation is that new algorithms using higher-order statistics HOS are being developed, and it is clear that in this field, it is absolutely necessary to deal with non-Gaussian data. Furthermore, a theory will be developed using tensors which constitute he natural framework of higher-order statistics. Hence, the second main issue of this paper is the extension of the framework introduced by MacCullagh [21] to the complex case. After a definition of the basic principles on which our modeling is built, we will present the technical realization of the principal tools. We will give a general definition of CRV and illustrate the usefulness of this new formalism in the context of complex circular random variables. 2.1. Complex random variables The definition of CRV is well-known. From two real random variables RRV and, we define the complex random variable by .1where . The turning point is to associate a probability density function pdf with this CRV. In the Gaussian circular case, this is done by onsidering both and its complex conjugate defined as . The ‘formal’ pdf of the Gaussian circular variable is then [15] Thus, it appears from the Gaussian example that we must consider both and in the definitions in order to extract all the statistical information. The preceding definition for the complex Gaussian variable shows that E[] equals zero. Hence, the only nonnull second-order moment is E[]. This means that bothandgive statistical and perhaps different information. Therefore, a theory of higher-order statistics in a general case must consider the variable and its complex conjugate. The information is in the statistics of the two variables, but also in their cross-statistics. We now introduce our formalism to handle the complex random variables in a more general way. The main problem which arises from the preceding discussion is an algebraic one, since the variablesand are algebraically linked. In order to overcome this, we propose to include the real world of and in a larger space in which and are not algebraically dependent. One way to do this is to consider and real and imaginary parts of as complex random variables. In this context we will continue to write and , but despite the notations, and are no longer complex conjugates. In order to introduce a continuity between the classical notations and the new ones, using tensors, that will be presented shortly, we have chosen to use these ambiguous notations of and. We will introduce in the following an alternative presentation that avoids this problem. This ‘trick’ will allow us to treat and as algebraically independent variables. We will see that this greatly facilitates all the calculations. However, these purely conceptual CRV are only used as means for easier calculations. When we want to come back to the real physical world, we have to restrict and to belong to the subset generated by the real numb

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