數(shù)學(xué)相關(guān)外文翻譯

1、<p><b>　　外文翻譯</b></p><p>　　標(biāo)題：統(tǒng)計(jì)復(fù)變量和信號(hào)-第一部分：變量</p><p><b>　　摘要</b></p><p>　　本文是專門研究高階統(tǒng)計(jì)復(fù)隨機(jī)變量。我們提出一個(gè)一般的框架允許直接操縱復(fù)數(shù)量：分離之間的實(shí)數(shù)和虛數(shù)的部分變量是可以避免的。我們給規(guī)則整合和得出的概率密度函數(shù)

2、和特征函數(shù)，使計(jì)算可以進(jìn)行。在方程的多層面的變量，我們使用的自然結(jié)構(gòu)張量。研究復(fù)變量導(dǎo)致的復(fù)循環(huán)隨機(jī)變量在高斯方程中延伸的概念。</p><p><b>　　1引入</b></p><p>　　高階統(tǒng)計(jì)現(xiàn)在已是一個(gè)密集型領(lǐng)域的信號(hào)和圖像處理的研究。 這一途徑的研究是基于使用的一個(gè)新的特性描述變量和信號(hào)。 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(xié)方差的變量、

3、關(guān)聯(lián)和交叉關(guān)聯(lián)的信號(hào)在時(shí)間域、頻譜功率密度和跨頻譜功率密度的信號(hào)在頻率域。</p><p>　　[30,9,31]后的先鋒性的論文不斷開拓高階統(tǒng)計(jì)的潛力，使其使用密集。 這是很長的完整視圖,使這一領(lǐng)域的新模式正在涌現(xiàn),并支持發(fā)展的大量應(yīng)用程序。 可以找到一個(gè)綜合的[13,14,20,22,25,26]。 此外,有幾個(gè)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]。<

4、/p><p>　　其主要特點(diǎn)是在研究上找到高階統(tǒng)計(jì)的建模和應(yīng)用程序。</p><p>　　在建模、隨機(jī)變量，基本上高階統(tǒng)計(jì)累計(jì)階數(shù)大于2。高階描述信號(hào)是通過多關(guān)聯(lián)在時(shí)間域和多光譜的頻率域中。</p><p>　　應(yīng)用程序正在開發(fā)一個(gè)偉大的數(shù)字—誤碼率的戰(zhàn)場。在高階統(tǒng)計(jì)給所有的經(jīng)典域研究信號(hào)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(dòng)診斷，

5、水聲，雷達(dá)，衛(wèi)星通訊，地震測深，天文學(xué)，等。非線性系統(tǒng)辨識(shí)，是高階統(tǒng)計(jì)一個(gè)基本的工具[ 19]。此外，一個(gè)與之密切相關(guān)的存在著的高階統(tǒng)計(jì)和模仿病的系統(tǒng)[10,17,23]。</p><p>　　這一非常積極和富有成果的研究領(lǐng)域需要有堅(jiān)實(shí)的理論基礎(chǔ)。被那些很久以前研究隨機(jī)變量和信號(hào)的理論的數(shù)學(xué)家和統(tǒng)計(jì)學(xué)家建立起來的。高階統(tǒng)計(jì)特性隨機(jī)變量在許多經(jīng)典教材[12,8,27]中被描述了。在[ 21 ]，我們發(fā)現(xiàn)張量方法的發(fā)

6、展對高階性能多維變量特別適合。該多相關(guān)時(shí)間域和多光譜被描述于[ 9]和[ 31]。</p><p>　　然而，一些作者仍然參與了有關(guān)領(lǐng)域的復(fù)雜隨機(jī)變量和信號(hào)，即使在實(shí)際應(yīng)用中出現(xiàn)這種情況：在頻域傅里葉變換后的處理，特別是在數(shù)組的處理，在單波段系統(tǒng)的通信信號(hào)的分析是常用的，在Wigner - Ville分布頻分析，等等。</p><p>　　高斯復(fù)雜的模型，這是足夠的在經(jīng)典二階的方法，是記錄

7、在[ 32]和[ 15]。這些作者證明了代數(shù)簡化帶來的使用是一個(gè)復(fù)雜的建模。他們還證明新的特性，如高斯復(fù)雜的圓，介紹了這個(gè)復(fù)雜的建模。</p><p>　　最近，缺少一般復(fù)雜的模型在[ 24]中被提出的證據(jù)：作者指出，“奇怪的是，人們發(fā)現(xiàn)在文獻(xiàn)中很少處理復(fù)隨機(jī)變量和過程”。他們引進(jìn)了“適當(dāng)?shù)膹?fù)隨機(jī)過程”的概念，其名稱循環(huán)過程。然而，這種方法基本上是有限的二階性質(zhì)。當(dāng)在一個(gè)有關(guān)的雙譜中的復(fù)信號(hào)的這種特殊的性質(zhì)，已體

8、現(xiàn)在[ 16]。</p><p>　　隨著越來越多地使用高階統(tǒng)計(jì)，現(xiàn)在是需要開發(fā)一個(gè)通用的建模復(fù)雜隨機(jī)變量和信號(hào)。它的主要的目的是分成兩部分的說明。</p><p>　　在第一部分中，我們關(guān)注的是復(fù)隨機(jī)變量。我們首先定義的概率規(guī)律使用復(fù)雜的符號(hào)。我們的結(jié)果，在一般情況三維以及多維隨機(jī)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(fā)展的實(shí)數(shù)情況在[ 21

9、 ]中的多維復(fù)雜隨機(jī)變量。我們表明，對于一個(gè)給定的順序，不同種類的累積量可以定義。這一結(jié)果是一個(gè)延伸的偽協(xié)方差在[24]中介紹 。這個(gè)模型我們可以給一個(gè)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(guān)系傅里葉變換和圓。復(fù)雜的圓形和高斯隨機(jī)變量被給出和說明模擬生成算法。說明算法新規(guī)則在附錄B中圓高斯例題中給出。</p><p>　　第二部分是用于建模和表示復(fù)雜的隨機(jī)信號(hào)。<

10、/p><p>　　為平穩(wěn)信號(hào)，多維隨機(jī)變量在使用的結(jié)果被給出了，我們定義了復(fù)平穩(wěn)隨機(jī)信號(hào)多相關(guān)性和多譜性。我們表明，完整的表征復(fù)雜的信號(hào)采用不同的多相關(guān)性和多譜在一階的需求中。在正常情況下的實(shí)際值信號(hào)，這些 多相關(guān)性和譜是相同的。不同的情況是信號(hào)分析的一些 多相關(guān)性和光譜是空空。擴(kuò)展循環(huán)的信號(hào)概念，我們表明，這種信號(hào)，唯一的非空相關(guān)性和光譜在共軛和非共軛條件具有相同數(shù)量的。此外，我們表明，限制信號(hào)循環(huán)達(dá)到一定的秩序。

11、我們回到選擇矩和累積量和顯示，除了傳統(tǒng)的利益提出的累積量是由于其加和其表征的高斯知識(shí)點(diǎn)，他們可以明確區(qū)分的性能在每個(gè)秩序和消除奇異的多光譜遙感。這個(gè)模型是然后擴(kuò)展到數(shù)字信號(hào)和數(shù)字信號(hào)的時(shí)間限制使用離散傅立葉變換。</p><p><b>　　2起點(diǎn)</b></p><p>　　本文的目的是介紹復(fù)隨機(jī)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(dǎo)致新的特征

12、在信號(hào)的描述并允許發(fā)展領(lǐng)域的高階統(tǒng)計(jì)量所有的新的理解。</p><p>　　復(fù)隨機(jī)變量作為輸出大量的加工等：</p><p><b>　　--傅里葉變換</b></p><p><b>　　--陣列處理，</b></p><p><b>　　--希爾伯特變換</b></p

13、><p>　　當(dāng)處理復(fù)隨機(jī)變量可以使用:</p><p>　　--考慮一個(gè)復(fù)隨機(jī)變量（CRV）作為二維實(shí)隨機(jī)變量（RRV）, </p><p>　　--發(fā)展于復(fù)隨機(jī)變量有關(guān)的代數(shù)工具</p><p>　　第二種方法具有優(yōu)勢：</p><p>　　--它使所有的推導(dǎo)簡單，</p><p>　　--它保

14、留的物理意義相關(guān)的復(fù)雜性質(zhì)的數(shù)據(jù)。</p><p>　　這種方法在[ 15 ]中的在高斯案件中已經(jīng)發(fā)展并引出來復(fù)高斯隨機(jī)變量的理論（CGRV）。在這種情況下審議的復(fù)高斯隨機(jī)變量引起了重要的概念---復(fù)雜的隨機(jī)高斯-西安循環(huán)變量。</p><p>　　本文的主要目標(biāo)是推廣一般情況下高斯和非高斯隨機(jī)變量的這些概念。主要的動(dòng)機(jī)是，新算法使用高階統(tǒng)計(jì)（HOS）正在開發(fā)中,很明顯的是在這個(gè)領(lǐng)域中,這

15、是絕對要處理非高斯分布數(shù)據(jù)。此外，一種理論將制定使用張量構(gòu)成自然框架的高階統(tǒng)計(jì)。因此,本文件的第二項(xiàng)主要的問題是,延長MacCullagh[21]提出的框架的復(fù)雜的案件。</p><p>　　在我們的模型建立后定義基本原則，我們將目前的技術(shù)作為實(shí)現(xiàn)的主要工具。我們將提供一個(gè)并說明其效用的這一新形式主義在復(fù)循環(huán)隨機(jī)變量的一般性的定義。</p><p><b>　　2.1 復(fù)隨機(jī)變量

16、</b></p><p>　　復(fù)隨機(jī)變量定義是著名的。從實(shí)隨機(jī)變量和，我們定義復(fù)雜的隨機(jī)變量為：</p><p><b>　?。?）</b></p><p><b>　　當(dāng)。</b></p><p>　　把概率密度函數(shù)（pdf）和復(fù)隨機(jī)變量聯(lián)合在一起是一個(gè)轉(zhuǎn)折點(diǎn)。</p>

17、<p>　　在高斯圓案例,這是通過“兩個(gè)及其復(fù)雜共軛定義為. 有關(guān)高斯圓變量的“正規(guī)”的概率密度函數(shù)在[15]。</p><p>　　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(tǒng)計(jì)信息。上述定義說明復(fù)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(gè)和給出統(tǒng)計(jì)(也許是不同)的信息。因此，一個(gè)理論的高階統(tǒng)計(jì)量在一般情況下必須考慮的變量及其共軛復(fù)數(shù)。信息的統(tǒng)計(jì)

18、資料不僅是兩個(gè)變量，而且在他們的互相統(tǒng)計(jì)。我們現(xiàn)在介紹我們的形式主義用更一般的方式來處理復(fù)雜的隨機(jī)變量。</p><p>　　主要的問題是從前面的討論是一個(gè)代數(shù)問題,因?yàn)檫@兩個(gè)變量、代數(shù)計(jì)算聯(lián)系在一起的。為了克服這一點(diǎn)，我們假設(shè)與在其實(shí)數(shù)區(qū)間內(nèi)，在較大的區(qū)間中與不是代數(shù)相關(guān)。這樣做的一個(gè)辦法是考慮與（的實(shí)部和虛部）復(fù)雜的隨機(jī)變量。在這方面，我們將繼續(xù)寫和，但盡管符號(hào)不同，與不再是復(fù)共軛。為了引進(jìn)一個(gè)符號(hào)在連續(xù)性的

19、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(guān)和的符號(hào)。我們將介紹在下列替代，避免了這一問題。</p><p>　　這個(gè)“惡作劇”將使我們把和作為獨(dú)立變量。我們將看到，這大大有利于所有的計(jì)算。然而，這些復(fù)隨機(jī)變量的純粹的概念，只有作為容易計(jì)算的手段。當(dāng)我們想要回過頭來談?wù)務(wù)鎸?shí)的物理世界,我們一定要限制和屬于所產(chǎn)生的子集的實(shí)數(shù)和。</p><p><b>　　2

20、.2規(guī)則</b></p><p>　　為此，我們必須建立一些規(guī)則，以便獲得的定義是有意義的。我們將提出兩項(xiàng)法則：</p><p>　　1所有的功能必須明確的數(shù)學(xué)，和所有的算子，如積分，必須收斂、</p><p>　　2當(dāng)我們考慮到特殊情況的實(shí)隨機(jī)變量希望能夠恢復(fù)經(jīng)典公式。</p><p>　　這種新的觀點(diǎn),同時(shí)適用于一個(gè)二維隨機(jī)變

21、量,如多維隨機(jī)變量。</p><p>　　我們現(xiàn)在將看到它是如何工作的。</p><p>　　3概率密度函數(shù)的功能和特點(diǎn)</p><p>　　我們會(huì)考慮先后一維和多維情況下的隨機(jī)變量。</p><p>　　3.1一維復(fù)隨機(jī)變量</p><p>　　我們已經(jīng)看到，在2.1節(jié)的概率密度函數(shù)的一個(gè)功能 和中有關(guān)和的。讓我們嘗

22、試定義第一特征函數(shù)。定義和這兩個(gè)復(fù)雜的變量可以寫：</p><p><b>　　, </b></p><p><b>　　.</b></p><p>　　如來自中的和,和可能是實(shí)數(shù)或虛數(shù)。特征函數(shù)已被定義[15]</p><p><b>　　(2)</b></p>

23、<p>　　和,然而，在指數(shù)讀取上有所爭議。</p><p>　　根據(jù)第二個(gè)規(guī)則,特征函數(shù)的復(fù)雜變量的和是延伸的二維平面復(fù)雜的經(jīng)典特征函數(shù)定義為實(shí)變量。 因此,相關(guān)的整體的數(shù)學(xué)期望,應(yīng)是與計(jì)算的、子集生成的和的實(shí)值。</p><p><b>　　(3)</b></p><p>　　其中2來自于雅可比的轉(zhuǎn)化，的子集是D中所以關(guān)于和中的

24、和的所有值以及 。</p><p>　　關(guān)系（3）給出的定義，特征函數(shù)是適用于所有的來自與中的和的實(shí)數(shù)的功能。</p><p>　　在相同的方式，我們可以定義的第二種特征函數(shù)</p><p><b>　　(4)</b></p><p>　　其中，在真實(shí)的情況下，產(chǎn)生的累積量。</p><p>　　

25、Statistics for complex variables and signals - Part I: Variables</p><p><b>　　Abstract </b></p><p>　　This paper is devoted to the study of higher-order statistics for complex random v

26、ariables. 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 deri

27、ve 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 </p><p>　

28、　1. Introduction</p><p>　　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 variabl

29、es 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

30、 power density and cross-spectral power density for signals in the frequency domain. </p><p>　　After the pioneering papers of [30,9,31] the potentialities of HOS are now used intensively. It would be very lo

31、ng 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 i

32、ssues of journals have been devoted to this topic [4-7] and a series of specialized workshops began in 1989 [l-3]. </p><p>　　The essential features in research on HOS are found in modeling and in application

33、s. </p><p>　　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, a

34、nd multispectra in the frequency domain. </p><p>　　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 introdu

35、cing 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 soun

36、ding, astronomy, etc. In nonlinear systems identification, HOS are a basic tool [19]. Moreover, a close connexion exists bet</p><p>　　This very active and fruitful field of research needs solid theoretical f

37、oundations. 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

38、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 multisp</p>&

39、lt;p>　　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 transform

40、ation, 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.</p><p>　　The Gaus

41、sian 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 h

42、ave shown that new properties, like Gaussian complex circularity, are introduced by this complex modeling. </p><p>　　More recently the lack of a general complex modeling was put in evidence in [24]: the auth

43、ors 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 ci

44、rcular 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 b</p><p>　　With th

45、e 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. </p><

46、p>　　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 [

47、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 mu

48、ltidimensional complex random variables. We show that, for a given </p><p>　　Part II is devoted to the modeling and representation of complex random signals. </p><p>　　For stationary signals, us

49、ing 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

50、 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 signal

51、s for which some multicorrelations</p><p>　　2. Starting point</p><p>　　The purpose of this paper is to introduce a general model of complex random variables. The usefulness of this modeling will

52、 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. </p><p>　　Complex

53、 random variables (CRV) appear as the output of a great number of processingr such as:</p><p>　　- Fourier transforms,</p><p>　　- Array processing, </p><p>　　-- Hilbert transforms.

54、 </p><p>　　When dealing with CRV two approaches can be used: </p><p>　　-to consider a CRV as a two-dimensional real random variable (RRV),</p><p>　　-to develop algebraic tools direc

55、tly with CRV. </p><p>　　The second approach has two advantages: </p><p>　　-it makes all the derivations simpler, </p><p>　　-it preserves the physical sense related to the complex na

56、ture of the data. </p><p>　　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 g

57、iven rise to the important notion of complex random Gaussian circular variables. </p><p>　　The principal aim of this paper is to generalize these notions to the general case of Gaussian and non-Gaussian rand

58、om 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 da

59、ta. Furthermore, a theory will be developed using tensors which constitute he natural framework of higher-order statistics. Hence, the second main issue of this</p><p>　　After a definition of the basic p

60、rinciples 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 con

61、text of complex circular random variables. </p><p>　　2.1. Complex random variables </p><p>　　The definition of CRV is well-known. From two real random variables (RRV) and, we define the com

62、plex random variable by .(1)where . </p><p>　　The turning point is to associate a probability density function (pdf) with this CRV. </p><p>　　In the Gaussian circular case, this is done b

63、y onsidering both and its complex conjugate defined as . </p><p>　　The ‘formal’ pdf of the Gaussian circular variable is then [15] </p><p>　　Thus, it appears from the Gaussian example that

64、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-orde

65、r 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 inf

66、ormation is </p><p>　　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 re

67、al 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 w

68、rite and , but despite the notations, and are no longer complex conjugates. In order to intr</p><p>　　This ‘trick’ will allow us to treat and as algebraically independent variables. We will see that thi

69、s 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 t

70、o the subset generated by the real numbers and.</p><p>　　2.2The rules </p><p>　　For this, we must establish some rules in order to obtain definitions which make sense. We will propose two laws:

71、</p><p>　　1. All the functions used must be well-defined mathematically, and all the operators, like integrals, must converge. </p><p>　　2. We want to be able to recover the classical formulae w

72、hen we consider the particular case of RRV. </p><p>　　This new point of view applies to both one-dimensional random variables as multidimensional random variables. </p><p>　　We will now see how

73、it works. </p><p>　　3. Pdf and characteristic functions </p><p>　　We will consider successively the one- and the multi-dimensional cases of random complex variables. </p><p>　　3. 1O

74、ne-dimensional complex random variables </p><p>　　We have seen in Section 2.1 that the pdf is a function of and. Let us try to define the first characteristic function. Letand be two complex variables that

75、 can be written: </p><p><b>　　,</b></p><p><b>　　.</b></p><p>　　Like and for, and can be real valued or complex valued. The characteristic function has a

76、lready been defined [15] by with and .Then,the argument in the exponential reads.</p><p>　　According to the second rule,the characteristic function of the complex variables and is the extension,to the two

77、-dimensional complex plane,of the classical characteristic function defined for real variables. Thus, the integral associated with the mathematical expectation should be calculated on the subset of values of and genera

78、ted by and real.</p><p><b>　?。?）</b></p><p>　　where 2 comes from the Jacobian of the transformation, is the subset of D of all the values of and forwhich and y are real valued a

79、nd .</p><p>　　Relation (3) gives a definition of the characteristic function which is valid for all real values of and if is a well-behaved function. </p><p>　　In the same way we can define