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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

## 电子工程代写|数字系统设计作业代写Digital System Design代考|DISCRETE TIME CONVOLUTION

Convolution is a mathematical operation involving two functions to produce a third function that is typically considered to be a modification of one of the original functions. It can be used to compute the zero state response of a linear, shift invariant, discrete time system. The zero state refers to the state of a system when all of the initial conditions are equal to zero. Thus, a discrete time convolution can be used to compute the output sequence $y(n)$, for a linear, shift invariant, discrete time system with an impulse response $h(n)$, an input sequence $x(n)$, and with all initial conditions equal to zero.
The convolution summation, or superposition summation, is defined as [4]
$$y(n)=\sum_{k=-\infty}^{\infty} h(k) x(n-k)$$
where $h(k)$ is the discrete time system impulse response and $x(n-k)$ is the input sequence shifted by $k$ time intervals. Note that either $h(n)$ or $x(n)$ can be shifted to form the convolution summation (2.107). Thus, the convolution of $h(n)$ and $x(n)$ can also be represented as
$$y(n)=\sum_{k=-\infty}^{\infty} h(n-k) x(k) .$$
Eigenfunctions have the property that if an eigenfunction is an input to the system, then the output will be the same eigenfunction multiplied by a real or complex constant that does not depend on the independent variable. The input $x(n)=C_{1} a^{n}$, where $C_{1}$ is either a real or complex constant, is an eigenfunction for a linear, shift invariant, discrete time system as an example. If an input, $C_{1} a^{n}$, is applied to a linear, shift invariant, discrete time system at $n=0$, then this input can be represented as the eigenfunction
$$x(n)=C_{1} a^{n} u(n) .$$
Assume that the impulse response of the system in question can be represented by
$$h(n)=C_{2} b^{n} u(n), \quad 0 \leq n \leq \infty .$$
Note that $a$ and $b$ can be either real or complex numbers. The discrete time system impulse response, $h(n)$, shifted as shown in Eq. (2.107), is given by
$$h(n-k)=C_{2} b^{n-k} u(n-k) .$$

## 电子工程代写|数字系统设计作业代写Digital System Design代考|OUTPUT RESPONSE OF A DISCRETE TIME SYSTEM

The discrete time impulse response for a linear, shift invariant discrete time system can be used with the convolution summation to compute the output of the system to an arbitrary discrete time input. If the discrete time impulse $\delta(n)$ if applied to a linear, shift invariant, discrete time system, then its output will be the impulse response $h(n)$. The computation of the output response of a digital system for a given input sequence $x(n)$ in terms of the convolution summation is given by
$$y(n)=\sum_{k=-\infty}^{\infty} h(k) x(n-k)=\sum_{k=-\infty}^{\infty} h(n-k) x(n)$$
where $h(k)$ is the impulse response for a discrete time impulse input applied at $k=0$. The convolution summation uses the linear shift property for discrete time systems. If an input $x(n)$ is applied at time $k$, then the corresponding output will be $h(k) x(n-k)$. The convolution summation formula uses this result to compute the output for a given input sequence as a sum of shifted discrete time impulses $\delta(n-k)$ with magnitude $x(n)$.

The effects of initial conditions on the system are often ignored. When all of the initial conditions of a system are equal to zero, then the system is said to be relaxed. However, in many practical applications, initial conditions as well as the input sequence must also be considered to determine the output of a discrete time system. If the system is linear and shift invariant, the principle of superposition can be used to compute the output as a linear combination of the output due to the initial state (initial conditions) and the output due to the input sequence.

Discrete time systems are often characterized in terms of the nature of their impulse responses. If the impulse response is of finite duration, then the system is called a finite impulse response (FIR) system. If the system has an infinite impulse response, then the system is called an infinite impulse response (IIR) system. The following is an impulse response for a FIR system:
$$h(n)={0.125,0.25,0.5,0.25,0.125}$$
where the symbol ( $\uparrow$ ) identifies the sample at $n=0$.
Infinite impulse response systems are usually recursive. The definition of a recursive system is given by [4]

Definition: A system whose output $y(n)$ at time $n$ depends on any number of past output values $y(n-1), y(n-2), \ldots$, is called a recursive system.
The following is an impulse response for an IIR system:
$$h(n)=\left[(0.76)^{n}+(-0.59)^{n}\right] \quad \forall 0 \leq n \leq \infty .$$

# 数字系统设计代考

## 电子工程代写|数字系统设计作业代写Digital System Design代考|DISCRETE TIME CONVOLUTION

$$y(n)=\sum_{k=-\infty}^{\infty} h(n-k) x(k) .$$

$$x(n)=C_{1} a^{n} u(n) .$$

$$h(n)=C_{2} b^{n} u(n), \quad 0 \leq n \leq \infty .$$

$$h(n-k)=C_{2} b^{n-k} u(n-k) .$$

## 电子工程代写|数字系统设计作业代写Digital System Design代考|OUTPUT RESPONSE OF A DISCRETE TIME SYSTEM

$$y(n)=\sum_{k=-\infty}^{\infty} h(k) x(n-k)=\sum_{k=-\infty}^{\infty} h(n-k) x(n)$$

$$h(n)=0.125,0.25,0.5,0.25,0.125$$

$$h(n)=\left[(0.76)^{n}+(-0.59)^{n}\right] \quad \forall 0 \leq n \leq \infty .$$

## 有限元方法代写

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## 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。

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