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# 电子代写|数字信号处理代写Digital Signal Processing代考|ECE2026 Stationary Random Processes

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## 电子代写|数字信号处理代写Digital Signal Processing代考|Stationary Random Processes

Let us once again consider our symbol generator and the accompanying random process $X$, which is composed of a certain number of random variables. Each one of these random variables has a mean, variance and potentially some effect on the other random variables in the process $X$. If these statistics don’t change over time – meaning they are the same now, in five minutes and next month – then the process is said to be stationary. Furthermore, if the linear time-invariant or LTI system we showed earlier also remains unchanged, then the random process $Y$ is also considered to be stationary.

Assuming that a process is in some way stationary is a very reasonable hypothesis in many situations, and it leads to a simpler formulation of the different functions we have listed earlier. If, however, the characteristics of the LTI system were to change with time, then the process $Y$ would no longer be stationary; dealing with such situations requires adaptive signal processing concepts, which we discuss in chapter four.

The autocorrelation and autocovariance functions for a stationary random process will no longer depend on the exact time index $a$ and $b$ inside its vector of random variables but on the time difference between them. The variable $x_0$ influences $x_1$ in the same way as variable $x_{100}$ influences $x_{101}$. Thus, for any time offset of $c$ the following expression holds true.
$$r_x(a, b)=r_x(a+c, b+c)$$
Better yet, we express the functions in terms of a difference in time index which we will call $\tau$, where $\tau=a-b$. The location or index, $a$, of the random variable inside our vector is arbitrary.
$r_x(\tau)=r_x(a, a+\tau) \quad \leftarrow$ Autocorrelation Function (stationary $R P$ )
$c_x(\tau)=c_x(a, a+\tau) \quad \leftarrow$ Autocovariance Function (stationary $\left.R P\right)$

## 电子代写|数字信号处理代写Digital Signal Processing代考|Correlation and Covariance expressed as Matrices

To find the projections of the measurement along principle component vectors, we recall the geometric interpretation of the dot product of two vectors $A$ and $B$, which states the following.
$$\operatorname{Dot}(A, B)=a_1 v_1^+a_2 b_2^+a_3 b_3^+\ldots+a_N b_N^=|A| \cdot|V| \cdot \cos (\theta)$$
Observe the figure below that illustrates the geometric interpretation of the dot product using the vectors A (one of our measurements), B (one of the two principle component vectors), and ProjA_B $\cdot B_{\text {unit }}$ (A projected onto B). Simple algebra proves that the length ProjA_B is equal to the ||$A|| \cdot \cos (\theta)$, which leads us to the equation for the projection of $A$ onto $B$. Note that the unit vector of $B=B / \mid B |$, where $|B|$ represents the length of norm of $B$.

The projection of the measurement, $\mathrm{A}$, onto the principle component vector, $\mathrm{B}$, has the following form, which is valid for both real and complex vectors.
\begin{aligned} \text { ProjA_B }_{\text {Unit }} &=|A| \cdot \cos (\theta) \cdot \frac{B}{|B|} \ &=|A| \cdot \cos (\theta) \cdot \frac{|B|}{|B|} \cdot \frac{B}{|B|} \ &=\frac{|A| \cdot|B| \cdot \cos (\theta)}{|B| \cdot|B|} \cdot B \ &=\frac{\operatorname{Dot}(A, B)}{\operatorname{Dot}(B, B)} \cdot B \end{aligned}

## 电子代写数字信号处理代写Digital Signal Processing代考|Stationary Random Processes

$$r_x(a, b)=r_x(a+c, b+c)$$

$r_x(\tau)=r_x(a, a+\tau) \quad \leftarrow$ 自相关函数 $($ 平稳 $R P)$
$c_x(\tau)=c_x(a, a+\tau) \quad \leftarrow$ 自协方差函数 $($ 平稳 $R P)$

## 电子代写数字信号处理代写Digital Signal Processing代考|Correlation and Covariance expressed as Matrices

$$\operatorname{Dot}(A, B)=a_1 v_1^{+} a_2 b_2^{+} a_3 b_3^{+} \ldots+a_N b_N|A| \cdot|V| \cdot \cos (\theta)$$

## 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。