1. Fourier Series Representations
Chapter 3
Fourier Series Representations
of Periodic Signals

2. Chapter 3 Fourier Series
Jean Baptiste Joseph Fourier, born in 1768, in France.
1807,periodic signal could be represented by sinusoidal series.
1829,Dirichlet provided precise conditions.
1960s,Cooley and Tukey discovered fast Fourier transform.

3. §3.2 The Response of LTI Systems to Complex Exponentials
Chapter Fourier Series
§3.2 The Response of LTI Systems to Complex Exponentials
LTI 系统对复指数信号的响应
1. Continuous-time system

4. §3.2 The Response of LTI Systems to Complex Exponentials
Chapter Fourier Series
§3.2 The Response of LTI Systems to Complex Exponentials
LTI 系统对复指数信号的响应
2. Discrete-time system

5. §3.2 The Response of LTI Systems to Complex Exponentials
Chapter Fourier Series
§3.2 The Response of LTI Systems to Complex Exponentials
1. Continuous-time system
Eigenfunction
特征函数
——Eigenvalue (特征值）
2. Discrete-time system
Eigenfunction
特征函数
——Eigenvalue (特征值）
If set z=ejΩ, we can get the same conclusion:
ejΩn is eigenfunction of DT LTI systems.

6. (3) Input as a combination of Complex Exponentials
Chapter Fourier Series
(3) Input as a combination of Complex Exponentials
Continuous time LTI system:
Discrete time LTI system:

7. Chapter 3 Fourier Series
Example 3.1
Consider an LTI system :

8. Chapter 3 Fourier Series
Example :
Consider an LTI system for which the input
and the impulse response determine the output

9. 3.3 Fourier Series Representation of Continuous-time Periodic Signals
Chapter Fourier Series
3.3 Fourier Series Representation of Continuous-time
Periodic Signals
3.3.1 Linear Combinations of Harmonically Related
Complex Exponentials
(1) General Form
The set of harmonically related complex
exponentials:
Fundamental period: T ( common period )

10. : Fundamental components
Chapter Fourier Series
: Fundamental components
: Second harmonic components
: Nth harmonic components
So, arbitrary periodic signal can be represented as
( Fourier series )
——Fourier Series Coefficients
Spectral Coefficients （频谱系数）

11. Chapter 3 Fourier Series
Example 3.2
Consider a real periodic signal
real periodic

12. (2) Representation for Real Signal
Chapter Fourier Series
(2) Representation for Real Signal
Real periodic signal: x(t)=x*(t)
So a*k=a-k
Let (A)

13. Chapter 3 Fourier Series
Let (A)
(B)

14. 3.3.2 Determination of the Fourier Series
Chapter Fourier Series
3.3.2 Determination of the Fourier Series
Representation of a Continuous-time Periodic
Signal
( Orthogonal function set )
Determining the coefficient by orthogonality:
( Multiply two sides by )

15. Fourier Series Representation:
Chapter Fourier Series
Fourier Series Representation:

16. Chapter 3 Fourier Series
§3.3.2 Determination of Fourier Series Representation
Synthesis equation
综合公式
Analysis equation
分析公式
——Fourier Series Coefficients
Spectral Coefficients

17. Chapter 3 Fourier Series
Example Periodic square wave defined over one period as
-T T/2 –T T1 T/ T t
Defining

18. Example 1: Periodic Square Wave (P135)
Defining

19. Chapter 3 Fourier Series

20. Chapter 3 Fourier Series
Figure 4.2
谱线变密

21. Example Periodic Impulse Trains (周期冲激串)
Chapter Fourier Series
Example Periodic Impulse Trains (周期冲激串)
-ω ω0 2ω0

22. Readlist Signals and Systems: Question: 3.4~3.5
Proof properties of Fourier series.

23. Problem Set
3.1 P250
3.3 P251
3.22(a.a) P255