1. Signals and Systems Lecture 7
Convergence of CTFS
Properties of CTFS

2. Chapter 3 Fourier Series
§3.4 Convergence（收敛） of the Fourier Series
1. Approximation(近似性)
——Error
EN最小 If
, then the series is convergent. ( xN(t)  x(t) )

3. Chapter 3 Fourier Series
Dirichlet Conditions:
Condition 1

4. Chapter 3 Fourier Series
Condition 2.
In any finite interval , is of bounded variation.

5. Chapter 3 Fourier Series
Condition 3.
In any finite interval , there are only a finite number
of discontinuities.

6. The Dirichlet Conditions (cont.)
Chapter Fourier Series
The Dirichlet Conditions (cont.)
Dirichlet conditions are met for most of the signals we will encounter in the real world. Then
The Fourier series = x(t) at points where x(t) is continuous
The Fourier series = “midpoint” at points of discontinuity
Still, convergence has some interesting characteristics:
As N → ∞, exhibits Gibbs’ phenomenon at points of discontinuity.

7. Chapter 3 Fourier Series
Gibbs Phenomenon:
Figure 3.9
Any continuity:
xN(t1)  x(t1)
Vicinity of discontinuity:
ripples
peak amplitude does not seem to decrease
Discontinuity:
overshoot 9%

8. x(t) and y(t) may have the same period T.
Chapter Fourier Series
§3.5 Properties of Continuous-Time Fourier Series
§ Linearity
x(t) and y(t) may have the same period T.
§ Time Shifting

9. Chapter 3 Fourier Series
§ Time Reversal

10. §3.5.6 Conjugation and Conjugate Symmetry
Chapter Fourier Series
§ Conjugation and Conjugate Symmetry
（共轭及共轭对称性）

11. Chapter 3 Fourier Series
Example (more symmetry properties )3.42 (P262)
real even
real even
Purely imaginary odd
real odd
[x(t)real ]
[x(t)real ]

12. The Fourier series representation has changed!
Chapter Fourier Series
§ Time Scaling
The Fourier series representation has changed!
§ Multiplication（相乘）
Convolution
Sum

13. Chapter 3 Fourier Series
§ Parseval’s Relation（帕兹瓦尔关系式）
Average Power of
Average Power
of kth harmonic
§ Differential Property

14. Chapter 3 Fourier Series
Example (Proof Multiplication and Parseral’s Relation )3.46 (P264)

15. Chapter 3 Fourier Series
Example ( Continue )

16. Chapter 3 Fourier Series
Example 3.6 t
Figure of Example 3.5
-T T/2 –T T1 T/ T t
g(t)=x(t-1)-1/2
Based on Property of linear and time-shifting, we may get
dk=bk+ck

17. Chapter 3 Fourier Series
Example 3.7
Figure of Example 3.6 t
Differentiation Property

18. Chapter 3 Fourier Series
Example 3.9
According to Fact 2
According to Fact 3
Synthesis Equation
Symmetry Property
According to Fact 1 So

19. ck=a-k bk=e-jkπ/2ck b0=0, b1=-b-1 Chapter 3 Fourier Series
Example 3.9 Continue
Time-Reversal
Time-Shifting
According to Fact 4
So,
ck=a-k
=>
z(t)=x(-t)
bk=e-jkπ/2ck
=>
y(t)=z(t-1)
=x(-(t-1))
Because bk is odd,
b0=0, b1=-b-1

20. Chapter 3 Fourier Series
Example 3.9 Continue
According to Fact 5
Parseval’s Relation
So, b1=-b-1
=±j/2
so, a0=0, a1=b-1ejπ/2=jb-1
Because bk=e-jkπ/2a-k,
then, x(t)=±cos(πt/2)

21. Chapter 3 Fourier Series

22. Readlist
Signals and Systems:
3.6~3.7
Question:
Calculation of DTFS.

23. Problem Set 3.5 P251 3.8 P252 Reference Example 3.9(P210)
3.40 P261 Reference Table3.1(P206)