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

2022年9月26日

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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

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