中英文互译

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 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 Furthermore, several special issues of journals have been devoted to this topic and a series of specialized workshops began in 1989.

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. Moreover, a close connexion exists between HOS and neuromimetic systems.

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 . We found the development of atensorial approach particularly well fitted to the higher-order properties of multidimensional variables. The multicorrelations and multispectra are described.

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. 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: the authors noted that “paradoxically, one finds in the literature very few treatments of com plex 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.

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 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 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. 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 order 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 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 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) x and y , we define the complex random variable z by z x jy =+.(1)where 21j =.

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 z and its complex conjugate z * defined as z x jy *=-.

The …formal? pdf of the Gaussian circular variable is then

()2,1,zz z z P z z e σπσ*

*-*=

Thus, it appears from the Gaussian example that we must consider both z and z*in the definitions in order to extract all the statistical information. The preceding definition for the complex Gaussian variable shows that E[''z] equals zero. Hence, the only nonnull second-order moment is E[z z*]. This means that both z and z*give 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 variables z and z*are algebraically linked. In order to overcome this, we propose to include the real world of z and z*in a larger space in which z and z*are not algebraically dependent.One way to do this is to consider x and y(real and imaginary parts of z)as complex random variables.In this context we will continue to

*=-,but despite the notations,z and z*are no longer write z x jy

=+and z x jy

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 z and z*.We will introduce in the following an alternative presentation that avoids this problem.

This …trick? will allow us to treat z and z*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 z and z*to belong to the subset generated by the real numbers x and y.

2.2The rules

For this, we must establish some rules in order to obtain definitions which make sense. We will propose two laws:

1. All the functions used must be well-defined mathematically, and all the operators, like integrals, must converge.

2. We want to be able to recover the classical formulae when we consider the particular case of RRV.

This new point of view applies to both one-dimensional random variables as multidimensional random variables.

We will now see how it works.

3. PDF and characteristic functions

We will consider successively the one- and the multi-dimensional cases of random complex variables.

3. 1One-dimensional complex random variables

We have seen in Section 2.1 that the pdf is a function (),,z z P z z **of z and z *.

Let us try to define the first characteristic function. Let ωand ω*错误！未找到引用源。 be two complex variables that can be written:

u jv ω=+,

u jv ω*=-.

Like x and y for z ,u and v can be real valued or complex valued. The characteristic

function has already been defined by ()(*)2,,z

z j z z E e ωωωω***+*??Φ=??????

with z x jy =+and z x jy *=-.Then,the argument in the exponential reads 2

z z

ux vy ωω**+=+. According to the second rule,the characteristic function of the (),,z z ωω**Φ=complex variables ωand ω* is the extension,to the

two-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 z and z * generated by x and y real.

()()()()2/2,,(3),,2,j z z j uv vy z z z z D R e P z z dzdz e f x y dxdy

ωωωω?? ???****++*

**==Φ??

where 2 comes from the Jacobian of the transformation, 2R is the subset of D of all the values of z and z * forwhich x and y are real valued and

()()(),,,,,z z f x y P z x y z x y **??=??.

Relation (3) gives a definition of the characteristic function which is valid for all real values of u and v if (),,z z P z z ** is a well-behaved function.

In the same way we can define the second characteristic function by

()(),,,log ,z z z z ωωωω****??ψ=Φ??

(4) which, as in the real case, generates the cumulants.

外文翻译

统计复变量和信号-第一部分：变量

摘要

本文是专门研究高阶统计复随机变量。我们介绍了一个通用的框架，允许复杂的数量直接操纵：避免部分变量的实数和虚数之间的分离。我们把规则将得到的概率密度函数和特征函数的计算，从而使计算可以进行。在多维变量的情况下，我们利用张量的自然框架。研究复变量导致的复循环随机变量在高斯方程中延伸的概念。

1引入

高阶统计现在已是一个密集型领域的信号和图像处理的研究。这一途径的研究是基于使用的一个新的特性描述变量和信号。到目前为止这一定性基本上是基于二阶矩阵措施：方差和协方差的变量、关联和交叉关联的信号在时间域、频谱功率密度和跨频谱功率密度的信号在频率域。

之后的先锋性的论文不断开拓高阶统计的潜力，使其使用密集。这是很长的完整视图,使这一领域的新模式正在涌现,并支持发展的大量应用程序。可以找到一个综合的领域。此外,有几个特殊问题的期刊是专门讨论这一议题和一系列专门的研究于1989年开始。

其主要特点是在研究上找到高阶统计的建模和应用程序。

在建模、随机变量，基本上高阶统计累计阶数大于2。高阶描述信号是通过多关联在时间域和多光谱的频率域中。

应用程序正在开发一个伟大的数字—误码率的战场。在高阶统计给所有的经典域研究信号与图像处理介绍了新方法。我们可以列举盲源分离和盲反卷积问题在各种情况下：振动诊断，水声，雷达，卫星通讯，地震测深，天文学等。非线性系统辨识，是高阶统计一个基本的工具。此外，还有一个与之密切相关的存在着的高阶统计和模仿病的系统。

这一非常积极和富有成果的研究领域需要有坚实的理论基础。被那些很久以前研究随机变量和信号的理论的数学家和统计学家建立起来的。高阶统计特性随机变量在许多经典教材中被描述了。我们发现张量方法的发展对高阶性能多维变量特别适合。

然而，一些作者仍然参与了有关领域的复杂随机变量和信号，即使在实际应用中出现这种情况：在频域傅里叶变换后的处理，特别是在数组的处理，在单波段系统的通信信号的分析是常用的，在Wigner - Ville分布频分析，等等。

高斯复杂的模型，这是足够的在经典二阶的方法。这些作者证明了代数简化带来的使用是一个复杂的建模。他们还证明新的特性，如高斯复杂的圆，介绍了这个复杂的建模。

最近，缺少一般复杂的模型的证据被提出：作者指出，“奇怪的是，人们发现在文献中很少处理复随机变量和过程”。他们引进了“适当的复随机过程”的概念，其名称循环过程。然而，这种方法基本上是有限的二阶性质。当在一个有关的双谱中的复信号的这种特殊的性质。

随着越来越多地使用高阶统计，现在是需要开发一个通用的建模复杂随机变量和信号。它的主要的目的是分成两部分的说明。

在第一部分中，我们关注的是复随机变量。我们首先定义的概率规律使用复杂的符号。我们的结果，在一般情况三维以及多维随机变量是否高斯或非高斯，这是已知的高斯情况。然后，我们的张量形式主义发展的实数情况的多维复杂随机变量。我们表明，对于一个给定的顺序，不同种类的累积量可以定义。这一结果是一个延伸的伪协方差。这个模型我们可以给一个一般性的定义，我们表明，在这一特定情况下，许多高阶累积量是空的。我们表明之间的直接关系傅里叶变换和圆。复杂的圆形和高斯随机变量被给出和说明模拟生成算法。

第二部分是用于建模和表示复杂的随机信号。

为平稳信号，多维随机变量在使用的结果被给出了，我们定义了复平稳随机信号多相关性和多谱性。我们表明，完整的表征复杂的信号采用不同的多相关性和多谱在一阶的需求中。在正常情况下的实际值信号，这些多相关性和谱是相同的。不同的情况是信号分析的一些多相关性和光谱是空空。扩展循环的信号概念，我们表明，这种信号，唯一的非空相关性和光谱在共轭和非共轭条件具有相同数量的。此外，我们表明，限制信号循环达到一定的秩序。我们回到选择矩和累积量和显示，除了传统的利益提出的累积量是由于其加和其表征的高斯知识点，他们可以明确区分的性能在每个秩序和消除奇异的多光谱遥感。这个模型是然后扩展到数字信号和数字信号的时间限制使用离散傅立叶变换。

2起点

本文的目的是介绍复随机变量的一般模型。有用这种模型的例子加以说明。我们将表明，它导致新的特征在信号的描述并允许发展领域的高阶统计量所有的新的理解。

复随机变量作为输出大量的加工等：

--傅里叶变换

--阵列处理，

--希尔伯特变换

当处理复随机变量可以使用:

--考虑一个复随机变量（CRV ）作为二维实随机变量（RRV ）,

--发展于复随机变量有关的代数工具

第二种方法具有优势：

--它使所有的推导简单，

--它保留的物理意义相关的复杂性质的数据。

这种方法在高斯案件中已经发展并引出来复高斯随机变量的理论（CGRV ）。在这种情况下审议的复高斯随机变量引起了重要的概念---复杂的随机高斯-西安循环变量。

本文的主要目标是推广一般情况下高斯和非高斯随机变量的这些概念。主要的动机是，新算法使用高阶统计（HOS ）正在开发中,很明显的是在这个领域中,这是绝对要处理非高斯分布数据。此外，一种理论将制定使用张量构成自然框架的高阶统计。因此,本文件的第二项主要的问题是,延长MacCullagh[21]提出的框架的复杂的案件。

在我们的模型建立后定义基本原则，我们将目前的技术作为实现的主要工具。我们将提供一个并说明其效用的这一新形式主义在复循环随机变量的一般性的定义。

2.1 复随机变量

复随机变量定义是著名的。从实随机变量x 和y ，我们定义复杂的随机变量z 为：

z x j y =+ （1） 当21j =。

把概率密度函数（pdf ）和复随机变量联合在一起是一个转折点。

在高斯圆案例,这是通过“两个z 及其复杂共轭z *定义为z x jy *=-. 有关高斯圆变量的“正规”的概率密度函数。

()2,1,zz z z P z z e σπσ**-*= 因此，它似乎从高斯例子，我们必须考虑z 与z *的定义，以便提取所有的统计信息。上述定义说明复高斯变量E[ Z ”]等于零。因此，唯一的非空的二阶矩阵是E[z z *]。这意味着，这两个z 和z *给出统计(也许是不同)的信息。因此，一个理论的高阶统计量在一般情况下必须考虑的变量及其共轭复数。信息的统计资料

不仅是两个变量，而且在他们的互相统计。我们现在介绍我们的形式主义用更一般的方式来处理复杂的随机变量。

主要的问题是从前面的讨论是一个代数问题,因为这两个变量z 、z *代数计算联系在一起的。为了克服这一点，我们假设z 与z *在其实数区间内，在较大的区间中z 与z *不是代数相关。这样做的一个办法是考虑x 与y （z 的实部和虚部）复杂的随机变量。在这方面，我们将继续写z x jy =+和z x jy *=-，但尽管符号不同，z 与z *不再是复共轭。为了引进一个符号在连续性的古典性和新的之间，使用张量，将不久，我们选择使用这些模棱两可的有关z 和z *的符号。我们将介绍在下列替代，避免了这一问题。

这个“恶作剧”将使我们把z 和z *作为独立变量。我们将看到，这大大有利于所有的计算。然而，这些复随机变量的纯粹的概念，只有作为容易计算的手段。当我们想要回过头来谈谈真实的物理世界,我们一定要限制z 和z *属于所产生的子集的实数x 和y 。

2.2规则

为此，我们必须建立一些规则，以便获得的定义是有意义的。我们将提出两项法则：

1所有的功能必须明确的数学，和所有的算子，如积分，必须收敛、 2当我们考虑到特殊情况的实随机变量希望能够恢复经典公式。

这种新的观点,同时适用于一个二维随机变量,如多维随机变量。

我们现在将看到它是如何工作的。

3概率密度函数的功能和特点

我们会考虑先后一维和多维情况下的随机变量。

3.1一维复随机变量

我们已经看到，在2.1节的概率密度函数的一个功能 和(),,z z P z z **中有关z

和z *的。让我们尝试定义第一特征函数。定义ω和ω*这两个复杂的变量可以写： u jv ω=+

u jv ω*=-.

如来自z 中的x 和y ,u 和v 可能是实数或虚数。特征函数已被定义[15] ()(*)2,,z z j z z E e ωωωω***+*??Φ=??????

(2) z x jy =+和z x jy *=-,然而，在指数读取上有所争议。

2z z

ux vy ωω**+=+

根据第二个规则,特征函数(),,z z ωω**Φ=的复杂变量的ω和ω*是延伸的二维平

面复杂的经典特征函数定义为实变量。 因此,相关的整体的数学期望,应是与计算的z 、z *子集生成的x 和y 的实值。

()()()()2/2,,(3),,2,j z z j uv vy z z z z D R e P z z dzdz e f x y dxdy

ωωωω?? ???****++*

**==Φ??

其中(2)来自于雅可比的转化，2R 的子集是D 中所以关于z 和z *中的x 和y 的所

有值以及 ()()(),,,,,z z f x y P z x y z x y **??=??。

关系（3）给出的定义，特征函数是适用于所有的来自与(),,z z P z z **中的u 和v 的

实数的功能。

在相同的方式，我们可以定义的第二种特征函数

()(),,,log ,z z z z ωωωω****??ψ=Φ??

(4) 其中，在真实的情况下，产生的累积量。