### 数学代写|傅里叶分析代写Fourier analysis代考|MAT3105

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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|Sinusoids and Complex Exponentials

The impulse and the sinusoid are the two most important signals in signal and system analysis. The impulse is the basis for convolution and the sinusoid is the basis for transfer function. The cosine and sine functions are two of the most important functions in trigonometry. As these functions are the basis functions in Fourier analysis, we have study them in detail.

The unit circle, defined by $x^2+y^2=1$ and shown in Fig. 1.3, is a circle with its center located at the origin and radius 1 . For each point on the circle defined by the coordinates $(x, y)$, starting at $(1,0)$ and moving in the counterclockwise direction, with $\theta \geq 0$ (the angle subtended by the $x$-axis and the line joining the point and the origin), the sine (sin) and cosine (cos) functions are defined in terms of its coordinates $(x, y)$ as
$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$
If the point lies on a circle of radius $r$, then
$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$
Clearly, the sinusoids are of periodic nature. Any function defined on a circle will be a periodic function of an angular variable $\theta$. Therefore, the trigonometric functions are also called circular functions. The argument $\theta$ is measured in radians or degrees. The radian is defined as the angle subtended between the $x$-axis and the line between the point and the origin on the unit circle. One radian is defined as the angle subtended by unit arc length. Since the circumference of the unit circle is $2 \pi$, one complete revolution is $2 \pi \mathrm{rad}$. In degree measure, $2 \pi=360^{\circ}$ and $\pi=180^{\circ}$. One radian is approximately $180 / \pi=57.3^{\circ}$.

A linear combination of sine and cosine functions is a sinusoid, in rectangular form, given by
$$a \cos (\theta)+b \sin (\theta)$$
where $a$ and $b$ are real numbers with $a \neq 0$ or $b \neq 0$. With $c=\sqrt{a^2+b^2}$, and $\cos (d)=a / c$ and $\sin (d)=b / c$,
$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$
is called the polar form of the sinusoid.

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

By using sine and cosine functions, signals can be represented. But it involves two basic functions and the two associated constants. It is found that an equivalent representation of signals is obtained using the complex exponential function, in which only one basic function and one associated constant is involved. The compact representation and the ease of manipulating the exponential functions make its use mandatory in the analysis of signals and systems. However, practical devices generate sine and cosine functions. Euler’s formula is the bridge between the theory and the practice. With $b$ any positive real number except 1 ,
$$x(t)=b^t$$
is called the exponential function with base $b$. Our primary interest, in this book, is the complex exponential function of the form
$$x(\theta)=A e^{j \theta}$$
The base is $e$, which is approximately $2.71828$. The exponent is a complex number with its real part zero (pure imaginary number). The coefficient of the exponential $A$ is a complex number.

The exponential $e^{j \theta}$, shown in Fig. 1.5, is a unit rotating vector, rotating in the counterclockwise direction. The exponential carries the same information about a sinusoid in an equivalent form, which is advantageous in the analysis of signals and systems. In combination with the exponential $e^{-j \theta}$, which rotates in the clockwise direction, a real sinusoidal waveform can be obtained. Since
$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$
solving for $\cos (\theta)$ and $\sin (\theta)$ results in
$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Sinusoids and Complex Exponentials

$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$

$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$

$$a \cos (\theta)+b \sin (\theta)$$

$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$

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

$$x(t)=b^t$$

$$x(\theta)=A e^{j \theta}$$

$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$

$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

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## MATLAB代写

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