## 电气工程代写|数字信号过程代写digital signal process代考|ECE4624

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号过程digital signal process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号过程digital signal process代写方面经验极为丰富，各种代写数字信号过程digital signal process相关的作业也就用不着说。

• 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

## 电气工程代写|数字信号过程代写digital signal process代考|Review of Linear Algebra

Linear algebra is a topic in mathematics that deals with calculations involving linear systems of equations. Although linear algebra is not required to understand the fundamentals of digital signal processing and communication engineering, a thorough command of its arithmetic and properties is paramount as we advance our skills beyond basic DSP concepts. The concept of optimization, which includes equalization, approximation, and optimal filter design, as well as the spatial multiplexing MIMO techique are built upon linear algebra.

While in-depth treatments of linear algebra are available in several text books [4], it is the goal of this section to review only those concepts that we will encounter in later chapters. Let’s start by considering a simple linear system of equations describing two lines.
\begin{aligned} &a_{11} \cdot x+a_{12} y=b_1 \ &a_{21} \cdot x+a_{22} y=b_2 \end{aligned}
Although the equation of a line is usually shown in y-intercept form $(y=m x+b)$, the formulation above, called the standard form, will be more convenient to work with. However, to get the first equation back to the more familiar $y$-intercept form, simply subtract $a_{11^{-}} x$ from both sides of the equation and divide by $a_{12}$.
$$y=-\frac{a_{11}}{a_{12}} x+\frac{b_1}{a_{12}}$$
For now, we will stay with the standard form and reformulate the systems of equations into an expressions.using matrices.
$$\left[\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right] \cdot\left[\begin{array}{l} x \ y \end{array}\right]=\left[\begin{array}{l} b_1 \ b_2 \end{array}\right]$$
One of the chief goals of linear algebra is to find the solution to such a system of equations, which, in the case above, means finding the $x$ and $y$ coordinate where the two lines cross (thus satisfying both equations simultaneously). We can rewrite the expression above by replacing the matrices with variables $A, X$, and $B$. Finding $X$ obviously requires that both matrices $A$ and $B$ are known.
$$A \cdot X=B$$
The variable $A$ represents a two row by two column (or $2 \times 2$ ) matrix while the other two variables are column vectors of dimension $2 x 1$. Vectors are matrices that feature either one row or one column.

## 电气工程代写|数字信号过程代写digital signal process代考|Orthogonal Vectors and Matrices

Orthogonal matrices are composed of column vectors that are themselves orthogonal. Geometrically speaking, orthogonal vectors in two and three dimensional space, $R^2$ and $R^3$, feature directions that are perpendicular to one another. The same is true for higher dimensional space, but it is more difficult to visualize. A more generalized terms, two vectors, $v_l$ and $v_2$, of length $N$ are orthogonal if the sum of their entry by entry products is equal to zero.
$$\sum_{n=0}^{N-1} v_1[n] \cdot v_2[n]=0$$
The matrices below feature column vectors that are orthogonal.
The orthogonal column vectors of each matrix – let’s call them $v_1, v_2$, and $v_3-$ are seen in the two and three dimensional coordinate systems below. $A_1$ is an identity matrix, which leaves all input vectors that it transforms unchanged. Matrix $A_2$ would cause an input vector to be rotated by 45 degrees and stretched by a factor equal to the square root of 2 . Similarly, $A_3$ produces a 45 degree rotation around the $z$ axis, stretches the $x$ and $y$ components of an input véctor by the square root of 2 but leaves the $z$ component unchanged. However, regardless of how each matrix affects its input vector, the column vectors in each matrix are perpendicular and therefore orthogonal.Orthogonal matrices, whose column vectors feature unit-length are called orthonormal. Notice that of the three matrices shown above only $A_1$ is orthonormal. We will meet orthonormal matrices later on in this chapter and discover their interesting properties.

## 电气工程代写|数字信号过程代写digital signal process代考|Review of Linear Algebra

$$a_{11} \cdot x+a_{12} y=b_1 \quad a_{21} \cdot x+a_{22} y=b_2$$

$$y=-\frac{a_{11}}{a_{12}} x+\frac{b_1}{a_{12}}$$

$$\left[\begin{array}{llll} a_{11} & a_{12} & a_{21} & a_{22} \end{array}\right] \cdot\left[\begin{array}{ll} x & y \end{array}\right]=\left[\begin{array}{ll} b_1 & b_2 \end{array}\right]$$

$$A \cdot X=B$$

## 电气工程代写|数字信号过程代写digital signal process代考|Orthogonal Vectors and Matrices

$$\sum_{n=0}^{N-1} v_1[n] \cdot v_2[n]=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 电气工程代写|数字信号过程代写digital signal process代考|ENEE425

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号过程digital signal process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号过程digital signal process代写方面经验极为丰富，各种代写数字信号过程digital signal process相关的作业也就用不着说。

• 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

## 电气工程代写|数字信号过程代写digital signal process代考|Roots of Polynomials

The two expressions below show a quadratic (degree 2 ) and a cubic (degree 3 ) polynomial function. The roots or solutions to these expressions may be found by setting them to zero and solving for the dependent variable $x$, or by plotting them and observing where $f(x)$ crosses zero.
$$\begin{array}{llrl} f(x) & =a \cdot x^2+b \cdot x+c & & \leftarrow \text { quadratic } \ f(x) & =a \cdot x^3+b \cdot x^2+c \cdot x+d & & \leftarrow \text { cubic } \end{array}$$
In the examples below we factor two polynomials thus revealing the roots algebraically and then verify those results graphically.

The roots, which are clearly revealed when we factor the polynomials, naturally fall at the zero crossings of the curves. While the above quadratic is easy to factor using mental math, more sophisticated expressions require the use of the well-known quadratic formula.
$$a z^2+b z+c=0 \quad \rightarrow \quad \text { Roots }=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$$
All was well in the world of mathematics until someone tried to apply the equation above to an innocuous expression like $f(x)=s^2-4 s+8$. The initial curiosity was to fathom why the curve never crosses zero; moreover, applying the coefficients $a=1, b=-4$, and $c=8$ to our quadratic equation forces us to take the square root of a negative number.
$$\text { Roots }=\frac{+4 \pm \sqrt{-16}}{2}=\frac{+4 \pm 4 \sqrt{-1}}{2}=2 \pm j 2$$
Given what we have learned so far, accepting the fact that roots can be complex is no longer an obstacle, but the fact that the curve does not cross zero remains confusing.

## 电气工程代写|数字信号过程代写digital signal process代考|Complex Exponentials and Euler’s Formulas

In the study of communication systems and digital signal processing, few equations are as mysterious and useful as Euler’s formula, which establishes a relationship between trigonometric and complex exponential functions. His formula states the following.
\begin{aligned} e^{j \theta} &=\cos (\theta)+j \sin (\theta) \ & \text { and therefore } \ \operatorname{Mag} \cdot e^{j \theta}=& \text { Mag } \cdot(\cos (\theta)+j \sin (\theta)) \end{aligned}
This formula allows us to express a complex number in polar format, $\operatorname{Mag} \angle \theta$, as a value, Mag $\cdot e^{\prime \theta}$, that can be easily manipulated in equations. It simplifies complex multiplication, since multiplying exponential functions of the form $e^{j a \cdot} e^{j b}=e^{j(a+b)}$ involves the mere addition of exponents.

How an exponentially increasing function can be linked to trigonometric expressions (sine and cosine) that are oscillatory in nature is mysterious indeed. Although we won’t recreate his approach used to arrive at the formula, we will show that it is true.

Leonhard Euler not only introduced the formula, for which he is now famous but also established the number $e$, which he defined as a series [3].
$$e=1+\frac{1}{1}+\frac{1}{1 \cdot 2}+\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{1 \cdot 2 \cdot 3 \cdot 4} \cdots=\sum_{n=0}^{\infty} \frac{1}{n !}=2.7182818$$
The Taylor series expansions for the more general case of $e^\theta$, as well as for $\sin (\theta)$ and $\cos (\theta)$ are shown next and can be looked up in any calculus text book [3].

## 电气工程代写|数字信号过程代写digital signal process代考|Roots of Polynomials

$f(x)=a \cdot x^2+b \cdot x+c \quad \leftarrow$ quadratic $f(x)=a \cdot x^3+b \cdot x^2+c \cdot x+d \leftarrow$ cubic

$$a z^2+b z+c=0 \rightarrow \text { Roots }=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$$

$$\text { Roots }=\frac{+4 \pm \sqrt{-16}}{2}=\frac{+4 \pm 4 \sqrt{-1}}{2}=2 \pm j 2$$

## 电气工程代写|数字信号过程代写digital signal process代考|Complex Exponentials and Euler’s Formulas

$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \quad \text { and therefore } \operatorname{Mag} \cdot e^{j \theta}=\mathrm{Mag} \cdot(\cos (\theta)+j \sin (\theta))$$

$$e=1+\frac{1}{1}+\frac{1}{1 \cdot 2}+\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{1 \cdot 2 \cdot 3 \cdot 4} \cdots=\sum_{n=0}^{\infty} \frac{1}{n !}=2.7182818$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 电气工程代写|数字信号过程代写digital signal process代考|ECE714

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号过程digital signal process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号过程digital signal process代写方面经验极为丰富，各种代写数字信号过程digital signal process相关的作业也就用不着说。

• 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

## 电气工程代写|数字信号过程代写digital signal process代考|The Development of Numbers

Complex numbers have confounded the greatest minds in the history of mathematics for the very same reason they confound us today. It is difficult to picture a complex amount of stuff. In our mind we can imagine 4 potatoes and $21 / 2$ oranges, but $2+j$ lemons just doesn’t make much sense. Even the idea of zero lemons becomes awkward. After all, numbers were invented to express the amount of something that we can see, touch, or otherwise appreciate.

Mathematicians brought complex numbers to life in their quest to find solutions to algebraic equations. To be more specific, they tried to find solutions to polynomial equations of the following form.
$$a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1 x^1+a_0=0$$
In the expression above, $x$ represents the variable that we wish to find, while the quantities $a_0$ through $a_n$ are constants. The degree, or order, of the polynomial is defined as the highest exponent, $n$, in the expression. For lower orders, polynomials reduce to simple algebraic expressions used for everyday calculations. Here are examples of first and second order polynomial equations.
$$\begin{gathered} x+4=0 \ x^2+2 x+1=0 \end{gathered}$$
To get a better appreciation of numbers in general and complex numbers in particular, let’s examine how their use evolved over time and how they helped find the solution for polynomial equations with which mathematicians seemed to be so enamored.

## 电气工程代写|数字信号过程代写digital signal process代考|Expanding Our Concept of Numbers as Mere Quantities

The fact that complex numbers have given closure to the task of finding all roots of polynomials may be reassuring to a whole host of long-dead mathematicians, but it still doesn’t help us appreciate the meaning of negative two sacks of grain or $2+j$ dollars. Luckily, the work of English mathematician John Wallis (1616-1703) paved the way for the use of the number line, which provides numbers with geometric meaning that had not previously been available to the layperson.

The number lines $a$ through $c$ above take us from the original counting or whole numbers through integers and finally to real numbers. While we still can’t put a concrete meaning to negative numbers, as the integers and real number lines seem to allow, we have nevertheless seen and dealt with these lines often enough to feel comfortable handling the numbers that lie on them.

To better understand positive and negative numbers, we think of them as vectors, which feature two separate parameters: a magnitude and a direction. The two vectors lying on number line $d$ represent numbers $1.5$ and $-3$. Their magnitudes, $1.5$ and 3 , are quantities that have a direct and palpable meaning that we as human beings can understand. Their direction conveys a quality about the object that is abstract in nature. The direction in the case-of minus 2 sacks of grain could indicate their state of ownership. The positive sign means that you own it, while the negative sign indicates that you owe it. Just take a look at the negative sign of your bank account balance after you’ve spent more money than you should have. Clearly, in the realm of integers and real numbers there are only two directions, featuring opposite bearings. Since a direction in 2D space is equivalent to an angle, we assign 0 degrees to the positive sign and $180 /-180$ degrees to the negative sign. Let’s reformulate the two numbers $1.5$ and $-3$ in terms of magnitude and direction (angle).
$$+1.5=1.5 \angle 0^{\circ} \quad-3.0=3.0 \angle 180^{\circ}$$

## 电气工程代写|数字信号过程代写digital signal process代考|The Development of Numbers

$$a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1 x^1+a_0=0$$

$$x+4=0 x^2+2 x+1=0$$

## 电气工程代写|数字信号过程代写digital signal process代考|Expanding Our Concept of Numbers as Mere Quantities

+1.5=1.5∠0∘−3.0=3.0∠180∘

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。