Discrete-time LTI system : the convolution sum representation
来源:互联网 发布:js elementHTML 编辑:程序博客网 时间:2024/06/07 05:43
1. LTI system
LTI system is the abbreviation of linear ,time-invariant system . Linearity and time invariance are the two distinguishing features of this kind of system .
An example of a LTI system :
if
the response of x1(1) is y1(t)
the response of x2(1) is y2(t)
then
the response of ax(t-t0) + bx2(t-t0) is ay1(t) + by2(t) , where ' a ' and ' b ' are any complex constants
2. Convolution
Convolution is a mathematical operation . The convolutions of two functions in continuous time and discrete time are given by equation (1) and (2)
3. The Representation of Discrete-Time Signals in Terms of Unit Impulse
Any discrete signals can be represented as the sum of scaled and time-shifted unit impulses . This is apparent when we think of a discrete-time signal as the combination of a sequence of individual impulses (figure 1 ). The mathematical representation is given by equation (3) .
4.The Representation of the Discrete-Time Signal Responses in Terms of Unit Impulse Response
Due to the linearity and time-invariance properties of LTI system , we can construct any signal responses by using the response of the unit impulse.
if the response of unit impulse δ[n] is h [n ] and x[n] is defined by eq.(4)
then the response of x[n] (i.e. y[n]) can be constructed as eq.(5)
More simple form is
5. Two important convolution sum representations
From the dissusions above , we finally got two useful representions of the discrete-time LTI system .
- Discrete-time LTI system : the convolution sum representation
- Continuous-Time LTI System : The Convolution Integral Representation
- 第三讲 2.1 Time-Domain Representations for LTI System
- Solving state function of continuous-time LTI system
- The first time to exercise the convolution
- The unit impulse and step functions in Discrete-time
- The UNIX Time-Sharing System
- Neural Discrete Representation Learning-代码分析
- The Evolution of the Unix Time-sharing System
- Explained: The Discrete Fourier Transform
- The Response of LTI systems to complex Exponential Signals
- Time system
- Real-time programming applied to the FreeRTOS operating system
- CSE hands-on#2 The UNIX Time-Sharing System
- get the current system time and print it
- Date and Time Representation in Python
- Date and Time Representation in Python
- 3 The Bayesian Network Representation
- [龙果学院] Spring Boot 教程笔记
- [LeetCode]27. Remove Element
- C++ Primer Plus第五版 第六章 编程练习答案
- C#语言和SQL Server数据库技术_用SQL语句操作数据
- [NOIP模拟] 矩阵
- Discrete-time LTI system : the convolution sum representation
- Canny边缘检测算法原理及其VC实现详解
- 洛谷P1364 医院设置
- 从地址空间看进程和线程
- Hibernate4 和 Hibernate5 创建session的区别
- Qt5 xml转Json
- 生活小记29
- 【日常】sublime text 3配置chrome插件LiveReload
- 习题3.4