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# 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|ECE457A Applications

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In this section, we discuss some possible choices for the input and desired signals and how these choices are related to the applications. Some of the classical applications of adaptive filtering are system identification, channel equalization, signal enhancement, and prediction.

In the system identification application, the desired signal is the output of the unknown system when excited by a broadband signal, in most cases a white-noise signal. The broadband signal is also used as input for the adaptive filter as illustrated in Fig. 1.2. When the output MSE is minimized, the adaptive filter represents a model for the unknown system.

The channel equalization scheme consists of applying the originally transmitted signal distorted by the channel plus environment noise as the input signal to an adaptive filter, whereas the desired signal is a delayed version of the original signal as depicted in Fig. 1.3. This delayed version of the input signal is in general available at the receiver in a form of standard training signal. In a noiseless case, the minimization of the MSE indicates that the adaptive filter represents an inverse model (equalizer) of the channel.

In the signal enhancement case, a signal $x(k)$ is corrupted by noise $n_1(k)$, and a signal $n_2(k)$ correlated with the noise is available (measurable). If $n_2(k)$ is used as an input to the adaptive filter with the signal corrupted by noise playing the role of the desired signal, after convergence the output error will be an enhanced version of the signal. Figure $1.4$ illustrates a typical signal enhancement setup.

Finally, in the prediction case the desired signal is a forward (or eventually a backward) version of the adaptive-filter input signal as shown in Fig. 1.5. After convergence, the adaptive filter represents a model for the input signal and can be used as a predictor model for the input signal.

Further details regarding the applications discussed here will be given in the following chapters.

## 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|Signal Representation

A deterministic discrete-time signal is characterized by a defined mathematical function of the time index $k$, ${ }^1$ with $k=0, \pm 1, \pm 2, \pm 3, \ldots$. An example of a deterministic signal (or sequence) is
$$x(k)=\mathrm{e}^{-\alpha k} \cos (\omega k)+u(k)$$
where $u(k)$ is the unit step sequence.
The response of a linear time-invariant filter to an input $x(k)$ is given by the convolution summation, as follows [7]:
\begin{aligned} y(k) & =x(k) * h(k)=\sum_{n=-\infty}^{\infty} x(n) h(k-n) \ & =\sum_{n=-\infty}^{\infty} h(n) x(k-n)=h(k) * x(k) \end{aligned}
where $h(k)$ is the impulse response of the filter. ${ }^2$
The $\mathcal{Z}$-transform of a given sequence $x(k)$ is defined as
$$\mathcal{Z}{x(k)}=X(z)=\sum_{k=-\infty}^{\infty} x(k) z^{-k}$$

## 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代 考|Signal Representation

$$x(k)=\mathrm{e}^{-\alpha k} \cos (\omega k)+u(k)$$

$$y(k)=x(k) * h(k)=\sum_{n=-\infty}^{\infty} x(n) h(k-n) \quad=\sum_{n=-\infty}^{\infty} h(n) x(k-n)=h(k) * x(k)$$

$$\mathcal{Z} x(k)=X(z)=\sum_{k=-\infty}^{\infty} x(k) z^{-k}$$

## 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。