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Signals and Systems Lecture 7

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Presentation on theme: "Signals and Systems Lecture 7"— Presentation transcript:

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)


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