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# 物理代写|声学代写Acoustics代考|MUS1008 Geometrical Interpretation on the Argand Plane

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## 物理代写|声学代写Acoustics代考|Geometrical Interpretation on the Argand Plane

To develop and exploit this geometric interpretation of exponential functions, which contain complex numbers within their arguments (hereafter referred to as complex exponentials), we can represent a complex number on a two-dimensional plane known as the “complex plane” or the Argand plane. In that representation, we define the $x$ axis as the “real axis” and the $y$ axis as the “imaginary axis.” This is shown in Fig. 1.7. In this geometric interpretation, multiplication by $j$ would correspond to making a “left turn” [12], that is, making a $90^{\circ}$ rotation in the counterclockwise direction. Since $j * j=j^2=-1$ would correspond to two left turns, a vector pointing along the real axis would be headed backward, which is the equivalent of multiplication by $-1$.

In this textbook, complex numbers will be expressed using bold font. A complex number, $z=x+j y$, where $x$ and $y$ are real numbers, would be represented by a vector of length, $|\vec{r}|=\sqrt{x^2+y^2}$, from the origin to the point, $z$, on the Argand plane, making an angle with the positive real axis of $\theta=\tan ^{-1}(y / x)$. The complex number could also be represented in polar coordinates on the Argand plane as $z=A e^{j \theta}$, where $A=|\vec{r}|$. The geometric and algebraic representations can be summarized by the following equation:
$$\mathbf{z}=x+j y=|\mathbf{z}|(\cos \theta+j \sin \theta)=|\mathbf{z}| e^{j \theta}$$

## 物理代写|声学代写Acoustics代考|Phasor Notation

In this textbook, much of our analysis will be focused on response of a system to a single-frequency stimulus. We will use complex exponentials to represent time-harmonic behavior by letting the angle $\theta$ increase linearly with time, $\theta=\omega_o t+\phi$, where $\omega_{\mathrm{o}}$ is the frequency (angular velocity) which relates the angle to time and $\phi$ is a constant that will accommodate the incorporation of initial conditions (see Sect. 2.1.1) or the phase between the driving stimulus and the system’s response (see Sect. 2.5). As the angle, $\theta$, increases with time, the projection of the uniformly rotating vector, $\vec{x}=|\vec{x}| e^{j \omega_o t+j \phi t} \equiv \widehat{\mathbf{x}} e^{j \omega_{\omega_o} t}$, traces out a sinusoidal time dependence on either axis. This choice is also known as phasor notation. In this case, the phasor is designated $\widehat{\mathbf{x}}$, where the “hat” reminds us that it is a phasor and its representation in bold font reminds us that the phasor is a complex number.
$$\widehat{\mathbf{x}}=|\widehat{\mathbf{x}}| e^{j \theta}$$
Although the projection on either the real or imaginary axis generates the time-harmonic behavior, the traditional choice is to let the real component (i.e., the projection on the real axis) represents the physical behavior of the system. For example, $x(t) \equiv \Re e\left[\widehat{\mathbf{x}} e^{j \omega_o t}\right]$.

## 物理代写|声学代写声学代考|阿根平面上的几何解释

$$\mathbf{z}=x+j y=|\mathbf{z}|(\cos \theta+j \sin \theta)=|\mathbf{z}| e^{j \theta}$$

## 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。