【数字信号处理】线性时不变系统 LTI “ 输入 “ 与 “ 输出 “ 之间的关系 ( 线性卷积计算方法列举 | 线性卷积计算案例一 | 根据 线性卷积 定义直接计算 卷积 )

1.线性卷积计算方法

线性卷积计算方法:

  • 直接法:根据线性卷积的定义直接计算;
  • 图解法:
  • 无进位乘法:
  • 编程计算:

2.线性卷积计算示例1(直接法)

给定以下两个序列:

x%28n%29%20%3D%20%5C%7B%201%20%2C%20-1%2C%202%20%5C%7D_%7B%5B0%2C2%5D%7D

h%28n%29%20%3D%20%5C%7B%203%2C%200%2C%20-1%5C%7D_%7B%5B0%2C2%5D%7D

y%28n%29%20%3D%20x%28n%29%20%2A%20h%28n%29

x%28n%29可表示为以下序列:

x%28n%29%20%3D%20%5Cdelta%28n%29%20-%20%5Cdelta%28n%20-%201%29%20%2B%202%5Cdelta%28n%20-%202%29

当输入为%5Cdelta%28n%29时,输出为h%28n%29%20%3D%20%5C%7B%203%2C%200%2C%20-1%5C%7D

%5Cdelta%28n%29%20%5Crightarrow%20h%28n%29%20%3D%20%5C%7B%203%2C%200%2C%20-1%5C%7D

当输入为-%20%5Cdelta%28n%20-%201%29,输出为-h%28n%20-%201%29,先将h%28n%29右移一位变成h%28n%20-%201%29%20%3D%20%5C%7B0%2C%203%2C%200%2C%20-1%5C%7D,再取负-h%28n%20-%201%29%20%3D%20%5C%7B0%2C%20-3%2C%200%2C%201%5C%7D

%5Cdelta%28n%29%20%5Crightarrow%20-h%28n%20-%201%29%20%3D%20%5C%7B0%2C%20-3%2C%200%2C%201%5C%7D

当输入为2%20%5Cdelta%28n%20-%202%29时 , 输出为2h%28n%20-%202%29, 先将h%28n%29右移 2 位变为h%28n%20-%202%29%20%3D%20%5C%7B0%2C%200%2C%203%2C%200%2C%20-1%5C%7D, 然后再将其乘以 2 得到2%20h%28n%20-%202%29%20%3D%20%5C%7B0%2C%200%20%2C%206%2C%200%2C%20-2%5C%7D;

2%20%5Cdelta%28n%20-%202%29%20%5Crightarrow%202%20h%28n%20-%202%29%20%3D%20%5C%7B0%2C%200%20%2C%206%2C%200%2C%20-2%5C%7D

x%28n%29%20%3D%20%5Cdelta%28n%29%20-%20%5Cdelta%28n%20-%201%29%20%2B%202%5Cdelta%28n%20-%202%29

对应的输出序列:

y%28n%29%20%3D%20h%28n%29%20-%20h%28n%20-%201%29%20%2B%202h%28n%20-%202%29

%5C%7B%203%2C%200%2C%20-1%5C%7D
%5C%7B0%2C%20-3%2C%200%2C%201%5C%7D
%5C%7B0%2C%200%20%2C%206%2C%200%2C%20-2%5C%7D

三个序列相加的结果是%5C%7B3%2C%20-3%2C%205%20%2C%201%2C%20-2%5C%7Dn的取值范围是【数字信号处理】线性时不变系统 LTI “ 输入 “ 与 “ 输出 “ 之间的关系 ( 线性卷积计算方法列举 | 线性卷积计算案例一 | 根据 线性卷积 定义直接计算 卷积 )0~4

在线性时不变系统中,先变换后移位和先移位后变换得到的输出序列是相同的;

最终结果是:

y%28n%29%20%3D%20h%28n%29%20-%20h%28n%20-%201%29%20%2B%202h%28n%20-%202%29%20%3D%20%5C%7B3%2C%20-3%2C%205%20%2C%201%2C%20-2%5C%7D_%7B%5B0%2C%204%5D%7D

以上是根据“线性卷积”的定义直接计算的;

“输出序列”等于“输入序列”与“系统单位脉冲响应”的卷积;

输入序列为:x%28n%29%20%3D%20%5Cdelta%28n%29%20-%20%5Cdelta%28n%20-%201%29%20%2B%202%5Cdelta%28n%20-%202%29

系统脉冲响应为:h%28n%29%20%3D%20%5C%7B%203%2C%200%2C%20-1%5C%7D_%7B%5B0%2C2%5D%7D

输出序列:是x%28n%29%20%2A%20y%28n%29的卷积;

在这里找到“输出序列”,得到x%28n%29%20%2A%20y%28n%29的卷积结果;

版权声明:本文为博主韩曙亮原创文章,版权归属原作者,如果侵权,请联系我们删除!

原文链接:https://blog.csdn.net/shulianghan/article/details/123064686

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