数学代写|傅里叶分析代写Fourier analysis代考|MECH4424

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

The circular convolution is also known as cyclic or periodic convolution. While the linear convolution is used most of the times in the analysis of LTI systems, the circular convolution is also important for the reason that signals are considered as periodic in DFT computation and the DFT is the tool to implement the linear convolution faster. The linear convolution is the periodic convolution in the limit, when the periods of the signals to be convolved become infinite. Both the circular and linear convolution are based on the same four operations (folding, shifting, multiplication, and summing). The difference is that the folding and shifting operations are carried out along a line in the linear convolution, whereas it is carried out around a circle in the circular convolution. Due to this difference, some number of output values of linear and circular convolutions, for the same inputs, are different at the borders.

The circular convolution of the two sequences $x(n)$ and $h(n)$, both of period $N$, is defined as

$$y(n)=\sum_{m=0}^{N-1} x(m) h(n-m)=\sum_{m=0}^{N-1} h(m) x(n-m), n=0,1, \ldots, N-1$$
resulting in the periodic output sequence $y(n)$ with the same period. Consider the circular convolution output $y(n)$ of the sequences
$$\begin{gathered} h(n)={\check{1}, 2,-1,3} \text { and } x(n)={2,1,-3,4} \ y(n)={\check{1} 6,-8,9,3} \end{gathered}$$
shown in Fig. 5.3. This is the same as the linear convolution with the sequences periodic. Consequently, the first three output values are different from that of the linear convolution and the fourth value is the same. The linear convolution output $y(n)$ of the same sequences is
$$y(n)={2,5,-3,3,14,-13,12}$$
The last three values get added to the first three values to form the circular convolution output. The sum of the shifted, by 4 samples, copies of linear convolution output is the circular convolution output. In circular convolution, the periods of the two sequences to be convolved are assumed to be the same. Let $x(n)$ is a sequence of length $N$ and its length also $N$. Then, the circular convolution of $x(n)$ and $h(n)$ yields $N$ output values. The first $(M-1)$ output values are not the corresponding linear convolution output values, while the rest of the $(N-M+1)$ values are the same. In the last example, with $N=M=4$, the last value $y(3)=3$ only is the same in both the outputs.

数学代写|傅里叶分析代写Fourier analysis代考|Overlap–Save Method

In practical applications, the input sequence is often very long and the impulse response is relatively short. Even if the required memory is available, the output will be delayed too long. In these cases, due to the limited availability of the memory in digital systems and the desirability of fast output, the input signal is segmented into blocks to suit the memory availability and the speed of response. Each block is convolved with the impulse response and the convolution outputs of the successive blocks are assembled to form the total convolution output. There are two equivalent methods to carry out convolution in this way. One of it, called the overlap-save method, is described.

The overlap-save method of convolution of long sequences is shown in Fig. 5.4. Let the length of the input sequence $x(n)$ be $N$. Let the length of the impulse response $h(n)$ be $Q$ and the block length be $B$. Then, for efficient implementation of the method, the following condition should be met.
$$N>>B>>Q$$
For illustrative purposes, short sequences are used in the example. Let $x(n)$ and $h(n)$ be
$$x(n)={2,1,-3,4} \quad \text { and } h(n)={1,2,-1,3}$$
The output of linear convolution of $x(n)$ and $h(n)$ is
$$y(n)={2 ้, 5,-3,3,14,-13,12}$$
Therefore, there are $N+Q-1=7$ output values have to be computed. Let the block length $B$ be 8 and $N=Q=4$. As first $Q-1=3$ output values are corrupted, the input data has to be prepended by 3 zeros. Since the block length $B$ is 8 , the data has to be appended by one zero. The first block of extended $x(n)$ is ${0,0,0,2,1,-3,4,0}$. The DFT of this block, with a precision of 2 digits, is
$${4,-0.29+j 0.46,-3+j 5,-1.71-j 7.54,6,-1.71+j 7.54,-3-j 5,-0.29-j 0.46}$$
The extended $h(n)$ is ${1,2,-1,3,0,0,0,0}$. The DFT of this data, which is compuled only once and stored, is
$${5,0.29-j 2.54 .2+j 1.1 .71-j 4.54,-5,1.71+j 4.54 .2-j 1,0.29+j 2.54}$$

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|循环卷积

.

$$y(n)=\sum_{m=0}^{N-1} x(m) h(n-m)=\sum_{m=0}^{N-1} h(m) x(n-m), n=0,1, \ldots, N-1$$

$$\begin{gathered} h(n)={\check{1}, 2,-1,3} \text { and } x(n)={2,1,-3,4} \ y(n)={\check{1} 6,-8,9,3} \end{gathered}$$

$$y(n)={2,5,-3,3,14,-13,12}$$

数学代写|傅里叶分析代写傅里叶分析代考|重叠保存方法

. txt 在实际应用中，输入序列往往很长，而脉冲响应相对较短。即使所需的内存可用，输出也会延迟太长时间。在这些情况下，由于数字系统中内存的可用性有限和快速输出的需求，输入信号被分割成块，以适应内存可用性和响应速度。每个分块与脉冲响应进行卷积，并将相邻分块的卷积输出集合起来，形成卷积输出的总值。用这种方法进行卷积有两种等价的方法。其中一种方法称为重叠保存方法(overlap-save) 长序列卷积的重叠保存方法如图5.4所示。设输入序列$x(n)$的长度为$N$。设脉冲响应的长度$h(n)$为$Q$，块长度为$B$。那么，为了有效地实现该方法，需要满足以下条件。
$$N>>B>>Q$$

$$x(n)={2,1,-3,4} \quad \text { and } h(n)={1,2,-1,3}$$
$x(n)$和$h(n)$的线性卷积输出
$$y(n)={2 ้, 5,-3,3,14,-13,12}$$

$${4,-0.29+j 0.46,-3+j 5,-1.71-j 7.54,6,-1.71+j 7.54,-3-j 5,-0.29-j 0.46}$$

$${5,0.29-j 2.54 .2+j 1.1 .71-j 4.54,-5,1.71+j 4.54 .2-j 1,0.29+j 2.54}$$

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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