QRD-RLS自适应滤波技术：现代通信中的应用与展望

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"QRD-RLS自适应过滤技术由José Antonio Apolinário Jr.编著，本书深入探讨了该领域的重要算法和应用。LMS算法在过去的许多年里一直是自适应过滤领域的首选，但随着电子技术的快速发展，对稳定且高效的RLS（递归最小二乘）算法的需求正在迅速增长，特别是在3G移动通信、认知无线电和高速成像等现代商业领域中。QRD-RLS（QR分解递归最小二乘）算法作为RLS的一种变体，因其高效性和稳定性而受到关注。本书由一众权威作者共同撰写，涵盖了该主题下的广泛话题和概念。" QRD-RLS自适应过滤是一种高级的信号处理技术，主要用于在噪声环境中识别和估计未知的线性系统参数。QR分解是这一算法的核心，它将权重矩阵分解为一个正交矩阵和一个上三角矩阵，从而简化了RLS算法的计算过程，提高了算法的效率和稳定性。 QRD-RLS算法相比于传统的RLS，主要优势在于它可以降低计算复杂度，这对于处理大量数据和实时系统尤其关键。在3G移动通信中，QRD-RLS能够帮助优化无线信道的性能，提高数据传输的准确性和速度。在认知无线电中，该算法有助于快速适应不断变化的频谱环境，实现更有效的频谱利用率。而在高速成像领域，QRD-RLS可以用于图像恢复和噪声抑制，提升图像质量。本书的作者阵容强大，他们深入探讨了QRD-RLS算法的理论基础、实现细节以及在不同领域的应用实例。读者将从中了解到如何设计和实现QRD-RLS滤波器，以及如何将其应用于实际问题中。此外，书中还可能涵盖了与其他自适应算法的比较，如LMS算法，以及针对不同应用场景的优化策略。 "QRD-RLS自适应过滤"是一个重要的信号处理工具，随着硬件技术的发展，其在各种高科技领域的应用潜力正在被不断挖掘。这本书对于科研人员、工程师和研究生来说是一份宝贵的资源，它不仅提供了理论知识，还提供了实践经验，有助于读者理解和掌握这一关键技术。

List of Contributors

Jos

e Antonio Apolin

ario Jr. (Editor)

Department of Electrical Engineering (SE/3)

Military Institute of Engineering (IME)

Prac¸a General Tib

urcio 80, Rio de Janeiro, RJ, 22290-270 – Brazil

e-mail: apolin@ieee.org

Richard C. Le Borne

Department of Mathematics

Tennessee Technological University

Box 5054, Cookeville, TN 38505 – USA

e-mail: rleborne@tntech.edu

Marcello L. R. de Campos

Electrical Engineering Program, COPPE

Federal University of Rio de Janeiro (UFRJ)

P. O. Box 68504, Rio de Janeiro, RJ, 21941-972 – Brazil

e-mail: campos@lps.ufrj.br

Shiunn-Jang Chern

Department of Electrical Engineering

National Sun-Yat Sen University

70 Lienhai Road, Kaohsiung, Taiwan 80424 – R.O.C.

e-mail: chern@mail.ee.nsysu.edu.tw

PauloS.R.Diniz

Electrical Engineering Program, COPPE

Federal University of Rio de Janeiro (UFRJ)

P. O. Box 68504, Rio de Janeiro, RJ, 21941-972 – Brazil

e-mail: diniz@lps.ufrj.br

xvii

2 Marcello L. R. de Campos and Gilbert Strang

often arising as consequence of least-squares minimization [3], and the solution of

the eigenvalue–eigenvector problem [4, 5]

Ax =

x. (1.2)

These problems may also be seen as particular cases of the problem of Fredholm [6]

in the matrix format (see, e.g., [7]),

x−

Ax = b, (1.3)

which simpliﬁes to the eigenproblem for b = 0 and to the solution of the linear

system of equations for

→ ∞. The methods to be outlined here are systematic

procedures to solve the problem, offering a more economical and sometimes a more

stable alternative to matrix inversion or to the Liouville-Neumann expansion

x =(I−

−1

b =(I+

A+

+ ···)b. (1.4)

For an account of the early developments in numerical linear algebra related to

the different methods for matrix inversion and eigenproblem solution see,

e.g., [8–11].

In the following sections, the QR decomposition and the QR algorithm, which

have become a standard subject in all linear algebra books (see, e.g., [3, 12–16]),

will be placed in the context of the solutions proposed for the Equation (1.2). We

will describe the solutions trying to give credit to key researchers that contributed to

the development of the methods, as well as their reﬁnement and dissemination.

1.2 Eigenvalues and Eigenvectors

The eigenproblem, as the eigenvalue–eigenvector problem is usually referred to,

can be traced back more than 150 years ago to the pioneering work of Jacobi [17]

(see also [18–21]). He has not described the problem in matrix notation, which was

invented shortly after [22–24]. Jacobi indeed proposed an ingenious solution to the

Equation (1.2) for the particular case of symmetric matrices.

A non-zero vector x ∈ C

is an eigenvector of A ∈ C

N×N

and

∈ C is the asso-

ciated eigenvalue if the Equation (1.2) is true. Since (A−

I)x = 0, the eigenvalues

are the roots of the characteristic polynomial, given by

℘

(z)=det(zI−A), (1.5)

where I is the identity matrix [25]. A scalar

is an eigenvalue of A if and only if

℘

(

)=0 [19, 26]. But the roots of an Nth-degree polynomial have no closed-form

formula for N > 4 [27–29], and a direct approach yields an ill-conditioned problem.

Therefore the solution of the eigenproblem must resort to iterative methods that

reveal the eigenvalues of A as they factor or transform the matrix.

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