19th Ave New York, NY 95822, USA

# 统计代写|时间序列分析作业代写time series analysis代考|Linear Time-Invariant Filters

my-assignmentexpert™时间序列分析time series analysis作业代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。my-assignmentexpert™， 最高质量的时间序列分析time series analysis作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此时间序列分析time series analysis作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

my-assignmentexpert™ 为您的留学生涯保驾护航 在统计Statistics作业代写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。我们的专家在时间序列分析time series analysis代写方面经验极为丰富，各种时间序列分析time series analysis相关的作业也就用不着 说。

## 统计代写|时间序列分析作业代写time series analysis代考|Basic Theory of LTI Analog Filters

Let us define a continuous parameter filter $L$ as a mapping, or association, between an input function $x(\cdot)$ and an output function $y(\cdot)$. Symbolically we write
$$L{x(\cdot)}=y(\cdot)$$
Since we regard both $x(\cdot)$ and $y(\cdot)$ as functions of time $t \in \mathbb{R}$, the qualifier “continuous parameter” is appropriate – in the engineering literature a continuous parameter filter is often called an analog filter. In mathematics $L$ is known as a transformation or operator. It is important to realize that a filter is not just an ordinary function: for example, a real-valued function that is defined over $\mathbb{R}$ associates a point in $\mathbb{R}$ with another point in $\mathbb{R}$, whereas a filter associates a function from some – so far unidentified – abstract space of functions with another function in that same space.

For the remainder of this section we need the following special notation. If $\alpha$ is a real or complex-valued scalar and $x(\cdot)$ is a function, the notation $\alpha x(\cdot)$ refers to the function defined by $\alpha x(t)$ for $t \in \mathbb{R}$. If $x_{1}(\cdot)$ and $x_{2}(\cdot)$ are two functions, then $x_{1}(\cdot)+x_{2}(\cdot)$ denotes the function defined by $x_{1}(t)+x_{2}(t)$. Finally, if $\tau$ is a real-valued scalar and $x(\cdot)$ is a function, then $x(\cdot ; \tau)$ denotes the function whose value at time $t$ is given by $x(t+\tau)$; i.e.,
$$x(t ; \tau)=x(t+\tau), \quad t \in \mathbb{R}$$

## 统计代写|时间序列分析作业代写time series analysis代考|Basic Theory of LTI Digital Filters

In the previous section we defined an analog (or continuous parameter) filter as a transformation that maps a function of time to another such function. A parallel theory exists for a transformation that associates a sequence with another sequence – such a transformation is referred to as a discrete parameter filter or digital filter. The theory of linear time-invariant digital filters closely parallels that of LTI analog filters, so we only sketch the key points for sequences in this section.

A digital filter $L$ that transforms an input sequence $\left{x_{t}\right}$ into an output sequence $\left{y_{t}\right}$ is called a linear time-invariant digital filter if it has the following three properties:
[1] Scale preservation:
$$L\left{\left{\alpha x_{t}\right}\right}=\alpha L\left{\left{x_{t}\right}\right}$$
[2] Superposition:
$$L\left{\left{x_{1, t}+x_{2, t}\right}\right}=L\left{\left{x_{1, t}\right}\right}+L\left{\left{x_{2, t}\right}\right}$$
[3] Time invariance:
$$\text { if } L\left{\left{x_{t}\right}\right}=\left{y_{t}\right} \text {, then } L\left{\left{x_{t+\tau}\right}\right}=\left{y_{t+\tau}\right} \text {, }$$

## 统计代写|时间序列分析作业代写time series analysis代考|Convolution as an LTI Filter

We consider in this section some details about an LTI analog filter $L$ of the following form:
$$L{X(t)}=\int_{-\infty}^{\infty} g(u) X(t-u) \mathrm{d} u \stackrel{\text { def }}{=} Y(t)$$
(that this indeed satisfies the properties of an LTI filter is the subject of Exercise [5.2a]). Here the input to the LTI filter is a stationary process ${X(t)}$ that, for simplicity, we take to have zero mean and a purely continuous spectrum with associated SDF $S_{X}(\cdot)$. The output is the stochastic process ${Y(t)}$ that results from convolving ${X(t)}$ with the real-valued deterministic function $g(\cdot)$. The process ${Y(t)}$ is thus formed from an infinite linear combination of members of the process ${X(t)}$. The characteristics of the LTI filter are entirely determined by $g(\cdot)$, which – in the analog (continuous parameter) case – is called the impulse response function for the following reason. Suppose we let the input to the LTI analog filter in Equation (142) be $\delta(\cdot)$, the Dirac delta function with an infinite spike at the origin. By the properties of that function, we have
$$L{\delta(t)}=\int_{-\infty}^{\infty} g(u) \delta(t-u) \mathrm{d} u=g(t)$$

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|BASIC THEORY OF LTI ANALOG FILTERS

X(吨;τ)=X(吨+τ),吨∈R

## 统计代写|时间序列分析作业代写TIME SERIES ANALYSIS代考|BASIC THEORY OF LTI DIGITAL FILTERS

[1] Scale preservation:
$$L\left{\left{\alpha x_{t}\right}\right}=\alpha L\left{\left{x_{t}\right}\right}$$
[2] Superposition:
$$L\left{\left{x_{1, t}+x_{2, t}\right}\right}=L\left{\left{x_{1, t}\right}\right}+L\left{\left{x_{2, t}\right}\right}$$
[3] Time invariance:
$$\text { if } L\left{\left{x_{t}\right}\right}=\left{y_{t}\right} \text {, then } L\left{\left{x_{t+\tau}\right}\right}=\left{y_{t+\tau}\right} \text {, }$$

## 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。