### 数学代写|傅里叶分析代写Fourier analysis代考|MAST20026

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• Statistical Inference 统计推断
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• 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|Unit-Impulse Signal

The unit-impulse and the sinusoidal signals are the most important signals in the study of signals and systems. The continuous unit-impulse $\delta(t)$ is a signal with a shape and amplitude such that its integral at the point $t=0$ is unity. It is defined, in terms of an integral, as
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$
It is assumed that $x(t)$ is continuous at $t=0$ so that the value $x(0)$ is distinct. The product of $x(t)$ and $\delta(t)$ is
$$x(t) \delta(t)=x(0) \delta(t)$$
since the impulse exists only at $t=0$. Therefore,
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$
The value of the function $x(t)$, at $t=0$, is sifted out or sampled by the defining operation. By using shifted impulses, any value of $x(t)$ can be sifted.

It is obvious that the integral of the unit-impulse is the unit-step. Therefore, the derivative of the unit-step signal is the unit-impulse signal. The value of the unit-step is zero for $t<0$ and 1 for $t>0$. Therefore, the unit area of the unit-impulse, as the derivative of the unit-step, must occur at $t=0$. The unit-impulse and the unitstep signals enable us to represent and analyze signals with discontinuities as we do with continuous signals. For example, these signals model the commonly occurring situations such as opening and closing of switches.

The continuous unit-impulse $\delta(t)$ is difficult to visualize and impossible to realize in practice. However, the approximation of it by some functions is effective in practice and can be used to visualize its effect on signals and its properties. While there are other functions that approach an impulse in the limit, the rectangular function is often used to approximate the impulse. The unit-impulse, for all practical purposes, is essentially a narrow rectangular pulse with unit area. Suppose we compress it by a factor of 2 , the area, called its strength, becomes $1 / 2=0.5$. The scaling property of the impulse is given as
$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$
With $a=-1, \delta(-t)=\delta(t)$ implying that the impulse is an even-symmetric signal. For example,
$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$
The discrete unit-impulse signal, shown in Fig. 1.1a, is defined as
$$\delta(n)=\left{\begin{array}{l} 1 \text { for } n=0 \ 0 \text { for } n \neq 0 \end{array}\right.$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal

The discrete unit-step signal, shown in Fig. 1.1b, is defined as
$$u(n)=\left{\begin{array}{l} 1 \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$ For positive values of its argument, the value of the unit-step signal is unity and it is zero otherwise. An arbitrary function can be expressed in terms of appropriately scaled and shifted unit-step or impulse signals. By this way, any signal can be specified, for easier mathematical analysis, by a single expression, valid for all $n$. For example, a pulse signal, shown in Fig. 1.2a, with its only nonzero values defined as $\{x(1)=1, x(2)=1, x(3)=1\}$ can be expressed as the sum of the two delayed unitstep signals shown in Fig. 1.2b, $x(n)=u(n-1)-u(n-4)$. The pulse can also be represented as a sum of delayed impulses. $$x(n)=u(n-1)-u(n-4)=\sum_{k=1}^3 \delta(n-k)=\delta(n-1)+\delta(n-2)+\delta(n-3)$$ The continuous unit-step signal is defined as $$u(t)= \begin{cases}1 & \text { for } t>0 \ 0 & \text { for } t<0 \\ \text { undefined for } t=0\end{cases}$$ The value $u(0)$ is undefined and can be assigned a suitable value from 0 to 1 to suit a specific problem. In Fourier analysis, $u(0)=0.5$. A common application of the unit-step signal is that multiplying a signal with it yields the causal form of the signal. For example, the continuous signal $\sin (t)$ is defined for $-\infty0$.

The discrete unit-ramp signal, shown in Fig. 1.1c, is also often used in the analysis of signals and systems. It is defined as
$$r(n)=\left{\begin{array}{l} n \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$
It linearly increases for positive values of its argument and is zero otherwise.
The three signals, the unit-impulse, the unit-step, and the unit-ramp, are related by the operations of sum and difference. The unit-impulse signal $\delta(n)$ is equal to $u(n)-u(n-1)$, the first difference of the unit-step. The unit-step signal $u(n)$ is equal to $\sum_{k=0}^{\infty} \delta(n-k)$, the running sum of the unit-impulse. The shifted unit-step signal $u(n-1)$ is equal to $r(n)-r(n-1)$. The unit-ramp signal $r(n)$ is equal to
$$r(n)=n u(n)=\sum_{k=0}^{\infty} k \delta(n-k) .$$
Similar relations hold for continuous type of signals.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Impulse Signal

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$

$$x(t) \delta(t)=x(0) \delta(t)$$

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$

$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$

$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$

$\$ \$$Idelta(n)=Veft { 1 for n=00 for n \neq 0 正确的。 ## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal 如图 1.1b 所示，离散单位阶跃信号定义为 \ \$$
$\mathrm{u}(\mathrm{n})=\backslash \mathrm{left}{$
1 for $n \geq 00$ for $n<0$ 、正确的。 Forpositivevaluesofitsargument, thevalueoftheunit – stepsignalisunityanditiszer $x(n)=u(n-1)-u(n-4)=\backslash$ sum_ ${k=1}^{\wedge} 3$ \delta(nk)=ldelta(n-1)+ldelta(n-2)+ \三角洲 $(n-3)$ Thecontinuousunit – stepsignalisdefinedas $u(t)=$ $$\left{\begin{array}{l} 1 \ \text { undefined for } t=0 \end{array} \text { for } t>00 \quad \text { for } t<0\right.$$
$\$ \$$价值 u(0) 是末定义的，可以分配一个从 0 到 1 的合适值以适应特定问题。在傅立叶分析中， u(0)=0.5. 单位阶跃信号的一个常见应用是将一个信号与其相乘产生信号的因果形式。例如，连续信 号 \sin (t) 被定义为 -\infty 0. 离散单位斜坡信号，如图 1.1c 所示，也经常用于信号和系统的分析。它被定义为 \ \$$
$$r(n)=\backslash l \text { eft }{$$
$n$ for $n \geq 00$ for $n<0$

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