Fourier Series Representations Chapter 3 Fourier Series Representations of Periodic Signals

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.

§3.2 The Response of LTI Systems to Complex Exponentials Chapter 3 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

§3.2 The Response of LTI Systems to Complex Exponentials Chapter 3 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

§3.2 The Response of LTI Systems to Complex Exponentials Chapter 3 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.

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

Chapter 3 Fourier Series Example 3.1 Consider an LTI system :

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

3.3 Fourier Series Representation of Continuous-time Periodic Signals Chapter 3 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 )

: Fundamental components Chapter 3 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）

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

(2) Representation for Real Signal Chapter 3 Fourier Series (2) Representation for Real Signal Real periodic signal: x(t)=x*(t) So a*k=a-k Let (A)

Chapter 3 Fourier Series Let (A) (B)

3.3.2 Determination of the Fourier Series Chapter 3 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 )

Fourier Series Representation: Chapter 3 Fourier Series Fourier Series Representation:

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

Chapter 3 Fourier Series Example 3.5 Periodic square wave defined over one period as -T -T/2 –T1 0 T1 T/2 T t Defining

Example 1: Periodic Square Wave (P135) Defining

Chapter 3 Fourier Series

Chapter 3 Fourier Series Figure 4.2 谱线变密

Example Periodic Impulse Trains (周期冲激串) Chapter 3 Fourier Series Example Periodic Impulse Trains (周期冲激串) -ω0 0 ω0 2ω0

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

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