Signals And Systems

Chapter 3 Fourier Transform 2

3.1 Introduction Signals and systems analysis: Time domain Frequency domain Complex frequency domain Chapter2 Chapter3 Chapter4 t ω s

4  Jean Baptiste Joseph Fourier( ), born in France.  1807,periodic signal could be represented by sinusoidal series.  1829,Dirichlet provided precise conditions.  1960s,Cooley and Tukey discovered fast Fourier transform. Mathematician : Fourier

Applications of FT Fourier Transform Digital signal processing Image processing Cryptology Speech processing Economics Optics

(a) Lena (b) edge image of Lena Applications of FT---Image processing

3.4 Continuous-Time Fourier Transform

9 1. Definition Inverse Fourier transform Fourier transform Fourier transform pair : X(ω) is in fact spectrum-density function But X(ω) is often abbreviated as “spectrum”

10 2.Convergence of Fourier transforms Dirichlet conditions: x(t) is absolutely integrable.

11. Example Examples of Continuous-Time FT Find its FT

12 Example 3.2 Find its FT

13 Sampling Signal ( 抽样信号 )

14 Example 3.3 Find its FT

15 Exercise 1 Use the Fourier transform analysis equation to calculate the Fourier transform of the following signals: (a) (b)