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As previously discussed, the design of digital filters with fixed coefficients requires well-defined prescribed specifications. However, there are situations where the specifications are not available, or are time varying. The solution in these cases is to employ a digital filter with adaptive coefficients, known as adaptive filters [10-17].
Since no specifications are available, the adaptive algorithm that determines the updating of the filter coefficients requires extra information that is usually given in the form of a signal. This signal is in general called a desired or reference signal, whose choice is normally a tricky task that depends on the application.

Adaptive filters are considered nonlinear systems, therefore their behavior analysis is more complicated than for fixed filters. On the other hand, because the adaptive filters are self-designing filters, from the practitioner’s point of view their design can be considered less involved than in the case of digital filters with fixed coefficients.

The general setup of an adaptive-filtering environment is illustrated in Fig.1.1, where $k$ is the iteration number, $x(k)$ denotes the input signal, $y(k)$ is the adaptivefilter output signal, and $d(k)$ defines the desired signal. The error signal $e(k)$ is calculated as $d(k)-y(k)$. The error signal is then used to form a performance (or objective) function that is required by the adaptation algorithm in order to determine the appropriate updating of the filter coefficients. The minimization of the objective function implies that the adaptive-filter output signal is matching the desired signal in some sense.

The complete specification of an adaptive system, as shown in Fig. 1.1, consists of three items:

Application: The type of application is defined by the choice of the signals acquired from the environment to be the input and desired-output signals. The number of different applications in which adaptive techniques are being successfully used has increased enormously during the last 3 decades. Some examples are echo cancellation, equalization of dispersive channels, system identification, signal enhancement, adaptive beamforming, noise cancelling, and control [14-20]. The study of different applications is not the main scope of this book. However, some applications are considered in some detail.

Adaptive-filter structure: The adaptive filter can be implemented in a number of different structures or realizations. The choice of the structure can influence the computational complexity (amount of arithmetic operations per iteration) of the process and also the necessary number of iterations to achieve a desired performance level. Basically, there are two major classes of adaptive digital filter realizations, distinguished by the form of the impulse response, namely the finiteduration impulse response (FIR) filter and the infinite-duration impulse response (IIR) filters. FIR filters are usually implemented with nonrecursive structures, whereas IIR filters utilize recursive realizations.

Adaptive FIR filter realizations: The most widely used adaptive FIR filter structure is the transversal filter, also called tapped delay line, that implements an all-zero transfer function with a canonic direct-form realization without feedback. For this realization, the output signal $y(k)$ is a linear combination of the filter coefficients, that yields a quadratic mean-square error (MSE $=$ $\left.E\left[|e(k)|^2\right]\right)$ function with a unique optimal solution. Other alternative adaptive FIR realizations are also used in order to obtain improvements as compared to the transversal filter structure, in terms of computational complexity, speed of convergence, and finite wordlength properties as will be seen later in the book.

Adaptive IIR filter realizations: The most widely used realization of adaptive IIR filters is the canonic direct-form realization [5], due to its simple implementation and analysis. However, there are some inherent problems related to recursive adaptive filters which are structure dependent, such as pole-stability monitoring requirement and slow speed of convergence. To address these problems, different realizations were proposed attempting to overcome the limitations of the direct-form structure. Among these alternative structures, the cascade, the lattice, and the parallel realizations are considered because of their unique features as will be discussed in Chap. 10.

Algorithm: The algorithm is the procedure used to adjust the adaptive filter coefficients in order to minimize a prescribed criterion. The algorithm is determined by defining the search method (or minimization algorithm), the objective function, and the error signal nature. The choice of the algorithm determines several crucial aspects of the overall adaptive process, such as existence of suboptimal solutions, biased optimal solution, and computational complexity.

The basic objective of the adaptive filter is to set its parameters, $\boldsymbol{\theta}(k)$, in such a way that its output tries to minimize a meaningful objective function involving the reference signal. Usually, the objective function $F$ is a function of the input, the reference, and adaptive-filter output signals, i.e., $F=F[x(k), d(k), y(k)]$. A consistent definition of the objective function must satisfy the following properties:

Non-negativity: $F[x(k), d(k), y(k)] \geq 0, \forall y(k), x(k)$, and $d(k)$.

Optimality: $F[x(k), d(k), d(k)]=0$.
One should understand that in an adaptive process, the adaptive algorithm attempts to minimize the function $F$, in such a way that $y(k)$ approximates $d(k)$, and as a consequence, $\boldsymbol{\theta}(k)$ converges to $\boldsymbol{\theta}_o$, where $\boldsymbol{\theta}_o$ is the optimum set of coefficients that leads to the minimization of the objective function.

Another way to interpret the objective function is to consider it a direct function of a generic error signal $e(k)$, which in turn is a function of the signals $x(k), y(k)$, and $d(k)$, i.e., $F=F[e(k)]=F[e(x(k), y(k), d(k))]$. Using this framework,we can consider that an adaptive algorithm is composed of three basic items: definition of the minimization algorithm, definition of the objective function form, and definition of the error signal.

## 自适应算法代写

$$10-17$$

$$14-20$$
·不同应用的研究不是本书的主要范围。但是，某些应用程序在某些细节上得到了考虑。

amountofarithmeticoperationsperiteration 的过程以及实现所需性能水平所需的迭代次数。基本上，有两大类自适应数字漶波器实现，以脉 冲响应的形式区分，即有限持续时间脉冲响应FIR滤波器和无限持续时间的脉冲响应 $I I R$ 过滤器。FIR 滤波器通常使用非递归结构实现，而 IIR滤 波楍吏用递归实现。

5
，由于其简单的实施和分析。然而，存在一些与结构相关的递归自适应滤波器相关的固有问题，例如极稳定性监测要求和收致速度慢。为了解决 这些问题，提出了不同的实现来试图克服直接形式结构的局限性。在这些可供选择的结构中，级联、点阵和并联实现被考虑在内，因为它们具有 独特的特性，这将在第 1 章中讨论。10.

$F=F[e(k)]=F[e(x(k), y(k), d(k))]$. 利用这个框架，我们可以认为自适应算法由三个基本项组成: 最小化算法的定义、目标函数形式的定义 和误差信号的定义。

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。