1. Signals and Systems Lecture 13
Convolution Property
Basic Fourier Transform Pairs

2. Chapter 3 Fourier Series
Fourier Series and LTI Systems

3. ( ) w j H X Y = Chapter 4 Fourier Transform
§4.4 The Convolution Property ( ) w j H X Y =

4. Chapter 4 Fourier Transform
LPF filter
Series Systems
The Frequency response cannot be defined for every LTI system.

5. Chapter 4 Fourier Transform
Example 4.15
Linear Phase
Example 4.16

6. Chapter 4 Fourier Transform
Example 4.17
This is an unstable system.

7. Chapter 4 Fourier Transform
Example 4.18
1. Ideal LPF filter
Not a causal system
Not easy to implement
Oscillatory behavior may be unaccepted

8. Chapter 4 Fourier Transform
2. Causal LPF filter

9. Chapter 4 Fourier Transform
The usefulness of the Convolution Property
Example 4.19
Solution

10. Chapter 4 Fourier Transform
Example 4.20

11. Chapter 4 Fourier Transform
The Fourier Analysis of periodic signals

12. - Chapter 4 Fourier Transform 例 图1所示的LTI系统中，子系统的频率响应分别如图2和图3 所示。
⑴ 试求整个系统的频率响应；
⑶ 若输入信号 ，试求系统的输出。 + - 图1 图2 图3

13. §4.7 Frequency-domain analysis of LTI systems
Chapter Fourier Transform
§4.7 Frequency-domain analysis of LTI systems
LTI系统的频域分析
1. Stable System
2. Linear Constant-Coefficients Differential Equations

14. Frequency Response of LTI Systems
Chapter Fourier Transform
Frequency Response of LTI Systems

15. ——Strictly proper rational function
Chapter Fourier Transform
3. Partial-Fraction Expansion
——Strictly proper rational function
真分式 ②
真分式

16. Determine the unit impulse response
Chapter Fourier Transform
真分式的部分分式展开(P909)
① 分母具有n个不同的单根
Example 4.25
Consider a stable LTI system
Determine the unit impulse response

17. Chapter 4 Fourier Transform
可用数学归纳法证明：

18. Chapter 4 Fourier Transform
Example 4.25
Consider the LTI system
If the input ,determine the output

19. Chapter 4 Fourier Transform
Example
Determine the unit impulse response of the LTI system

20. Excise 4.33(P345) Chapter 4 Fourier Transform (a) Because then
(b) Because
and , so

21. h(t):Partial-Fraction Expansion
Chapter Fourier Transform
Summary
Convolution Property and Filtering
Ideal LPF and Filter Design
Calculation of H(jω) and h(t)
h(t):Partial-Fraction Expansion

22. Readlist Signals and Systems: Question: 4.5,Chapter8
Implement of Communication Systems

23. Problem Set
4.34
4.35