Signals and Systems Lecture 7 Convergence of CTFS Properties of CTFS

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) )

Chapter 3 Fourier Series Dirichlet Conditions: Condition 1

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

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

The Dirichlet Conditions (cont.) Chapter 3 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.

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%

x(t) and y(t) may have the same period T. Chapter 3 Fourier Series §3.5 Properties of Continuous-Time Fourier Series §3.5.1 Linearity x(t) and y(t) may have the same period T. §3.5.2 Time Shifting

Chapter 3 Fourier Series §3.5.3 Time Reversal

§3.5.6 Conjugation and Conjugate Symmetry Chapter 3 Fourier Series §3.5.6 Conjugation and Conjugate Symmetry （共轭及共轭对称性）

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 ]

The Fourier series representation has changed! Chapter 3 Fourier Series §3.5.4 Time Scaling The Fourier series representation has changed! §3.5.5 Multiplication（相乘） Convolution Sum

Chapter 3 Fourier Series §3.5.7 Parseval’s Relation（帕兹瓦尔关系式） Average Power of Average Power of kth harmonic §3.5.8 Differential Property

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

Chapter 3 Fourier Series Example ( Continue )

Chapter 3 Fourier Series Example 3.6 -2 -1 0 1 2 t Figure of Example 3.5 -T -T/2 –T1 0 T1 T/2 T t g(t)=x(t-1)-1/2 Based on Property of linear and time-shifting, we may get dk=bk+ck

Chapter 3 Fourier Series Example 3.7 Figure of Example 3.6 -2 -1 0 1 2 t Differentiation Property

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

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

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)

Chapter 3 Fourier Series

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

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