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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|傅里叶分析代写Fourier analysis代考|The Discrete Fourier Transform

Transform means change in form. For example, we use the product rule to change the form of the problem of finding the derivative of the product of two functions, so that its derivative can be found easily. The idea of a transform, in signal analysis, is to approximate practical signals, which usually have arbitrary amplitude profiles and difficult to analyze in their original form, adequately in terms of well-defined basis signals, such as the cosine and sine signals. Then, it is easier to interpret, analyze, transmit, and store them. In the representation of a function in the form $x(t)$, variable $t$ is the independent variable in a certain domain, designated as the time domain. Since the time is the independent variable frequently (but not always), it is named as the time domain. In the representation of a function in the form $X(k)$, variable $k$, which represents the frequency index of a frequency component, is the independent variable in the frequency domain. Either representation completely specifies the given function. While the frequency-domain representation of signals and systems looks unnatural, it is convenient and efficient in signal and system analysis. For example, a high-quality recording of a music signal requires frequency components in the range $0-20 \mathrm{kHz}$ and the corresponding recording devices, amplifiers, and speakers should have a good frequency response in that frequency range.

Fourier analysis is an indispensable representation of signals and systems in science and engineering. There are many other representations of various entities. Infinite points in a plane are represented by their $x$-axis and $y$-axis coordinates. A place on earth is represented by its longitude and latitude. Any color can be specified by its red, green, and blue components. With all the mathematics, Fourier analysis looks complex and difficult. But, it is not so. It is similar to finding the amount of a set of coins. Let us say, we have a box of 1 cent, 10 cent, and 50 cent coins. We can take one by one and add its value to a partial sum. We find the amount after the values of all the coins are added. An alternate way is to decompose the coins into the three denominations and count the number of coins in each. Multiplying the number of different coins by their value and adding results in the amount.

数学代写|傅里叶分析代写Fourier analysis代考|The Exponential Function

Fourier analysis is a representation of arbitrary signals in terms of sinusoidal (or its equivalent complex exponential) basis signals. This representation is similar to that of the logarithmic function. An exponential function is of the form
$$x(n)=b^{n}$$
where the base $b \neq 1$ is a positive constant and the exponent $n$ is the independent variable. An important property of the exponential function is that
$$b^{m} b^{n}=b^{m+n}$$
Therefore, the exponential representation reduces a multiplication operation into a relatively simpler addition operation. Similarly, the Fourier representation reduces a convolution operation into a relatively simpler multiplication operation. Considering the importance of the convolution operation in signal and system analysis, this single advantage alone is sufficient enough to make the Fourier analysis an indispensable tool in science and engineering.

While the detailed description of Fourier analysis is the topic of the book, let us continue with the more familiar exponential function. Let us say that we want to multiply 8 by 16. Assume that a table is available to find the exponent of the exponential function with base 2 of any number. Then, $8=2^{3}$ and $16=2^{4}$ and
$$8 \times 16=2^{3} 2^{4}=2^{3+4}=2^{7}=128$$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|The Exponential Function

$$x(n)=b^{n}$$

$$b^{m} b^{n}=b^{m+n}$$

$$8 \times 16=2^{3} 2^{4}=2^{3+4}=2^{7}=128$$

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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