# 数学代写|傅里叶分析代写Fourier analysis代考|MTH418

#### Doug I. Jones

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## 数学代写|傅里叶分析代写Fourier analysis代考|Convolution and Correlation

Practical systems are analyzed using mathematical models. Convolution and transfer function are two of the frequently used models in the study of LTI systems. In this chapter, we study the convolution operation and we study the transfer function in later chapters. Both the models are based on decomposing an arbitrary input signal in terms of well-defined basis signals, impulse in the case of convolution and complex exponential signals in the case of transfer function. After the decomposition of the input signal, the system output can be found using the response of the system to the basis signals and the linearity and time-invariance properties of the LTI systems. Convolution operation relates the input and output of a system through its impulse response. The impulse response of a system is its response to the unit-impulse signal, assuming that the system is initially relaxed (zero initial conditions). Convolution expresses the output of a system in terms of its input only.

The important concepts, such as convolution and Fourier analysis, are easier to understand and remember through their physical interpretations. Convolution operation is the same thing as finding the current balance for our deposits in a bank. Assuming compound interest, the interest is computed on the principal at regular intervals and it is added to the principal. Let the interval be one year and the interest rate per year be $10 \%$. Let $n=0$ be the starting time and the deposit made at that time is designated as $x(0)$. Then, $x(n)$ is the deposit made after the $n$th year. Assuming $N$ number of years. the deposits are
$${x(0), x(1), x(2), \ldots, x(N)}$$
The compound interest rates $h(n)$ for $n$ years of deposit are
$${h(0), h(1), h(2), \ldots, h(N)}=\left{1,1.1,1.21, \ldots,(1.1)^N\right}$$
Therefore, the balance in the deposit $y(N)$ at the $N$ th year is \begin{aligned} &y(N)=1 x(N)+1.1 x(N-1)+1.21 x(N-2)+, \cdots,+(1.1)^N x(0) \ &{h(0) x(N)+h(1) x(N-1)+h(2) x(N-2)+, \cdots,+h(N) x(0)} \ &=y(N)=\sum_{k=0}^N h(k) x(N-k) \end{aligned}
Alternately,
\begin{aligned} &{x(0) h(N)+x(1) h(N-1)+x(2) h(N-2)+, \cdots,+x(N) h(0)} \ &\quad=y(N)=\sum_{k=0}^N x(k) h(N-k) \end{aligned}
After time reversing and shifting one of the two sequences, the output is the sum of the product of the corresponding terms. The time-reversal is required, as each other’s index is running in opposite directions. The generalization of this problem is the convolution operation. In system analysis, the interest rate is called the impulse response of the system. The deposits are called the input. The balance at intervals is the output.

## 数学代写|傅里叶分析代写Fourier analysis代考|Linear Convolution

In the convolution operation, the input $x(n)$ is decomposed into scaled and delayed impulses. At each point, the contribution of all the impulses is summed to find the output $y(n)$. The input, in terms of impulses, is
\begin{aligned} x(n)=& \cdots+x(-2) \delta(n+2)+x(-1) \delta(n+1) \ &+x(0) \delta(n)+x(1) \delta(n-1)+x(2) \delta(n-2)+\cdots \ =& \sum_{k=-\infty}^{\infty} x(k) \delta(n-k) \end{aligned}
Let the impulse response of the system be $h(n)$. Then, due to the time-invariance of the LTI system, the response to a delayed impulse $\delta(n-k)$ is $h(n-k)$. Due to the linearity of the LTI system, the response to $x(k) \delta(n-k)$ is $x(k) h(n-k)$ and the total response of the system is the sum of the contributions to all the scaled and shifted impulses. The 1-D linear convolution of two aperiodic sequences $x(n)$ and $h(n)$, again due to linearity, is defined as
$$y(n)=\sum_{k=-\infty}^{\infty} x(k) h(n-k)=\sum_{k=-\infty}^{\infty} h(k) x(n-k)=x(n) * h(n)=h(n) * x(n)$$
The convolution operation relates the input $x(n)$, the output $y(n)$, and the impulse response $h(n)$ of a system.

Figure 5.1 shows the convolution of the signal $x(n)={2,1,3,4}$ and $h(n)=$ ${1,-2,3}$.
The output $y(0)$, from the definition, is
$$y(0)=x(k) h(0-k)=(2)(1)=2,$$
where $h(0-k)$ is the time-reversal of $h(k)$. Shifting $h(0-k)$ to the right, we get the remaining outputs as

# 傅里叶分析代写

## 数学代写|傅里叶分析代写傅里叶分析代考|卷积与相关

$${x(0), x(1), x(2), \ldots, x(N)}$$

$${h(0), h(1), h(2), \ldots, h(N)}=\left{1,1.1,1.21, \ldots,(1.1)^N\right}$$

\begin{aligned} &{x(0) h(N)+x(1) h(N-1)+x(2) h(N-2)+, \cdots,+x(N) h(0)} \ &\quad=y(N)=\sum_{k=0}^N x(k) h(N-k) \end{aligned}

## 数学代写|傅里叶分析代写傅里叶分析代考|线性卷积

\begin{aligned} x(n)=& \cdots+x(-2) \delta(n+2)+x(-1) \delta(n+1) \ &+x(0) \delta(n)+x(1) \delta(n-1)+x(2) \delta(n-2)+\cdots \ =& \sum_{k=-\infty}^{\infty} x(k) \delta(n-k) \end{aligned}

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

## 有限元方法代写

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## 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。

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