Signals and Systems Lecture 11 The Continuous-Time Fourier Transformation

Chapter 4 Fourier Transform Fourier Series of CT Periodic Signals Synthesis equation Analysis equation

Chapter 4 Fourier Transform §4.1 Representation of Aperiodic Signals : The Continuous-Time Fourier Transform -T -T/2 –T1 0 T1 T/2 T t

Chapter 4 Fourier Transform Figure 4.2 谱线变密

Chapter 4 Fourier Transform Consider an aperiodic Signals –T1 0 T1 t -T –T1 0 T1 T t

Chapter 4 Fourier Transform Fourier Transform Pair factor Synthesis equation Analysis equation 1. A linear combination of complex exponentials. ——Spectrum（频谱） of 2.

Chapter 4 Fourier Transform Consider a periodic signal Defining The Fourier coefficients of are proportional to samples of the Fourier transform of one period of

Chapter 4 Fourier Transform §4.1.2 Convergence of Fourier Transforms 1. is square integrable 2. Dirichlet Conditions:

Chapter 4 Fourier Transform §4.1.3 Fourier Transforms of Typical Signals Example 4.1

Chapter 4 Fourier Transform Example 4.2 Example 4.3

Chapter 4 Fourier Transform Example 3:(Continue) Do not satisfy the condition of finite energy，so can not get from define directly.

Chapter 4 Fourier Transform Example 4.4, 4.5 –T1 0 T1 t F Useful facts about CTFT’s Example above: Example above: F

Chapter 4 Fourier Transform Excise 4.1(a) 4.2(b) Solution: Solution:

Chapter 4 Fourier Transform Example 1 Example 2

Chapter 4 Fourier Transform Example 3 F

Chapter 4 Fourier Transform §4.2 The Fourier Transforms for Periodic Signals More generally, if x(t)=x(t+T),then Discrete spectra

Chapter 4 Fourier Transform Example 4.6: Periodic square wave

— sampling function Chapter 4 Fourier Transform Example 4.7: Periodic impulses train — sampling function -T 0 T 2T Note in this case, periodic in both time domain (with a period T) and frequency domain (with a period 2π/T) Same function in the frequency-domain! -ω0 0 ω0 2ω0

Chapter 4 Fourier Transform Excise 4.3(a) 4.4(a) Solution: Solution: ∵ ∴

Summary Chapter 4 Fourier Transform Synthesis equation Analysis equation F

Readlist Signals and Systems: Question: 4.3~4.6 Compare properties of CTFT with of CTFS Basic Fourier Transformation Pairs

Problem Set 4.21(a), (c), (g) 4.22(a), (b), (d)