# 数学代写|傅里叶分析代写Fourier analysis代考|MATH3205

#### Doug I. Jones

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## 数学代写|傅里叶分析代写Fourier analysis代考|2-D Linear Convolution

2-D convolution is a straightforward extension of the 1-D convolution. As the sequences involved are 2-D, folding, shifting, and zero-padding operations have to be carried out along the rows and columns of the 2-D data. If one of the sequences is separable, 2-D convolution can be implemented faster by a set of 1-D convolutions. The convolution of 2-D sequences $x(m, n)$ and $h(m, n)$ is defined as
\begin{aligned} y(m, n) &=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} x(k, l) h(m-k, n-l) \ &=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} h(k, l) x(m-k, n-l)=h(m, n) * x(m, n) \end{aligned}
The same four steps of the 1-D convolution (folding, shifting, multiplying, and summing) are repeatedly carried out in implementing the 2-D convolution in two dimensions.

1. Any one of the two sequences, say $h(k, l)$, is rotated in the $(k, l)$ plane by $180^{\circ}$ about the origin to get $h(-k,-l)$. Of course, the same result is achieved by folding

$h(k, l)$ about the $k$-axis to get $h(k,-l)$ and then folding $h(k,-l)$ about the $l$-axis to get $h(-k,-l)$ or vice versa.

1. The rotated sequence $h(-k,-l)$ is shifted by $(m, n)$ to get $h(m-k, n-l)$ to find the convolution output at coordinates $(m, n)$.
2. The term-by-term products, $x(k, l) h(m-k, n-l)$, of all the overlapping samples are computed.
3. Summing all the products is the convolution output $y(m, n)$ at $(m, n)$.
Let us find the output of convolving the $3 \times 3$ sequence $h(k, l)$ and the $4 \times 4$ sequence $x(k, l)$
$$h(k, l)=\left[\begin{array}{lll} 2 & 1 & 3 \ 1 & 2 & 2 \ 3 & 2 & 1 \end{array}\right] \quad \text { and } \quad x(k, l)=\left[\begin{array}{llll} 3 & 1 & 3 & 2 \ 2 & 1 & 3 & 4 \ 2 & 1 & 2 & 3 \ 1 & 1 & 2 & 2 \end{array}\right]$$
shown in Fig. 5.5. Four examples of computing the convolution output are shown. For example, with a shift of $(0-k, 0-l)$, there is only one overlapping pair (3, 2). The product of these numbers is the output $y(0,0)=6$. The process is repeated to get the complete convolution output $y(m, n)$ shown in the figure.

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

Correlation is a similarity measure between two signals. The correlation output indicates the strength of the relationships between the signals, which may be negative, zero or positive. If the signals are positively correlated, then both increase or decrease together. The more time we walk, the more calories we burn. If the signals are negatively correlated, then one increases and the other decreases. It is an inverse relationship. An increase in the amount of physical effort results in weight loss. Zero correlation implies no discernible relationship between the two variables. If a signal increases with the other remaining constant or increasing half the time and decreasing half the time, it indicates no correlation. In signal processing, object recognition and estimation are typical examples of correlation operation. The most famous and important example, of course, is the determination of the amplitudes of the signal components by correlating the given signal with each of its components in Fourier analysis.
The cross-correlation of two signals $x(n)$ and $y(n)$ is defined as
$$r_{x y}(m)=\sum_{n=-\infty}^{\infty} x(n) y^(n-m)=\sum_{n=-\infty}^{\infty} x(n+m) y^(n), \quad m=0, \pm 1, \pm 2, \ldots$$
Equivalent but alternate definition is also used. The asterisk in the definition indicates complex conjugation operation, which has no effect for real-valued signals. The output is the sum of products of two signals, with one of them shifted. The number of shifts is the independent variable and the sum is the dependent variable.

The correlation of $y(n)={2,1,-3}$ and $x(n)={2,1,3,4}$ is shown in Fig. 5.7. The output is $r_{x y}(n)={-6,-1,-\breve{4},-7,10,8}$. The convolution operation without time-reversal is the correlation operation. Convolution of the time-reversed version of the sequence $y(n)$ with $x(n)$ is the same as correlation of $x(n)$ and $y(n)$. Two real signals $x(n)$ and $y(n)$ are said to be orthogonal over the entire time interval if
$$\sum_{n=-\infty}^{\infty} x(n) y(n)=0$$
When two signals to be correlated are the same, the operation is called autocorrelation. The autocorrelation function of real-valued signals is even-symmetric. Unlike convolution, correlation operation, in general, is not commutative. The correlation of a function with an impulse shifts the time-reversed version of the function to the location of the impulse.

# 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|二维线性卷积

2-D卷积是1-D卷积的直接扩展。由于所涉及的序列是二维的，因此必须沿着二维数据的行和列执行折叠、移动和零填充操作。如果其中一个序列是可分离的，二维卷积可以通过一组一维卷积来更快地实现。二维序列$x(m, n)$和$h(m, n)$的卷积定义为
\begin{aligned} y(m, n) &=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} x(k, l) h(m-k, n-l) \ &=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} h(k, l) x(m-k, n-l)=h(m, n) * x(m, n) \end{aligned}

1. 两个序列中的任何一个，比如$h(k, l)$，在$(k, l)$平面上绕原点$180^{\circ}$旋转，得到$h(-k,-l)$。当然，通过折叠

$h(k, l)$关于$k$ -轴得到$h(k,-l)$，然后折叠$h(k,-l)$关于$l$ -轴得到$h(-k,-l)$，反之亦然

1. 将旋转后的序列$h(-k,-l)$移动$(m, n)$，得到$h(m-k, n-l)$，得到坐标$(m, n)$处的卷积输出。
2. 计算所有重叠样本的逐项乘积$x(k, l) h(m-k, n-l)$。
3. 把所有的积加起来就是卷积输出$y(m, n)$ at $(m, n)$。
让我们找到对$3 \times 3$序列$h(k, l)$和$4 \times 4$序列$x(k, l)$
$$h(k, l)=\left[\begin{array}{lll} 2 & 1 & 3 \ 1 & 2 & 2 \ 3 & 2 & 1 \end{array}\right] \quad \text { and } \quad x(k, l)=\left[\begin{array}{llll} 3 & 1 & 3 & 2 \ 2 & 1 & 3 & 4 \ 2 & 1 & 2 & 3 \ 1 & 1 & 2 & 2 \end{array}\right]$$
进行卷积的输出，如图5.5所示。给出了计算卷积输出的四个例子。例如，移位为$(0-k, 0-l)$时，只有一个重叠对(3,2)。这些数字的乘积就是输出$y(0,0)=6$。重复这个过程，得到如图所示的完整卷积输出$y(m, n)$。

## 数学代写|傅里叶分析代写傅里叶分析代考|线性相关

$$r_{x y}(m)=\sum_{n=-\infty}^{\infty} x(n) y^(n-m)=\sum_{n=-\infty}^{\infty} x(n+m) y^(n), \quad m=0, \pm 1, \pm 2, \ldots$$

$y(n)={2,1,-3}$和$x(n)={2,1,3,4}$的相关性如图5.7所示。输出为$r_{x y}(n)={-6,-1,-\breve{4},-7,10,8}$。没有时间反转的卷积运算是相关运算。时间反转版本的序列$y(n)$与$x(n)$的卷积与$x(n)$与$y(n)$的相关相同。如果
$$\sum_{n=-\infty}^{\infty} x(n) y(n)=0$$
，则两个实数信号$x(n)$和$y(n)$在整个时间间隔内是正交的，当两个要相关的信号相同时，这种操作称为自相关。实值信号的自相关函数是均匀对称的。与卷积不同，相关运算通常不是可交换的。一个函数与一个脉冲的相关性将该函数的时间反转版本移到脉冲的位置

## 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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