Signals and Systems - Phase

Signals and SystemsLecture 18·The Magnitude-Phase Representation of FT.The Magnitude-Phase Representation of theFrequency Response of LTI Systems.Time-Domain Properties of Ideal Filter.Time-Domain Properties and Frequency-Domainof Nonideal Filter

Signals and Systems Lecture 18 •The Magnitude-Phase Representation of FT •The Magnitude-Phase Representation of the Frequency Response of LTI Systems •Time-Domain Properties of Ideal Filter •Time-Domain Properties and Frequency-Domain of Nonideal Filter

Chapter6Time and Frequency Characterization6.1 The Magnitude-Phase Representation（幅度-相位）of the Fourier Transformx(0)-2元 xG0)endoX(jo)-[X(jo)eizX(o)X(jo) -Magnitude[X(jo)~-Energy-Density<X(jo) -Phase

Chapter 6 Time and Frequency Characterization §6.1 The Magnitude-Phase Representation （ 幅度-相位） of the Fourier Transform ( ) ( ) ( )   j X j X j X j e  = ( ) ( )    x t X j e d j t 2 1  + − = X(j) ——Magnitude ( ) 2 X j ——Energy-Density X j ( ) ——Phase

Chapter6Time and Freguency CharacterizationThe magnitude X(jo)l (or X(ejo)) describes the relative magnitude ofthe individual frequency components that make up the signalx, (t) = 4cos 2元t + cos6元tx, (t)= cos 2元t + 4cos6元t

x (t) 4cos2t cos6t 1 = + x (t) cos2t 4cos6t 2 = + Chapter 6 Time and Frequency Characterization The magnitude |X(jω)| (or |X(ejω)|) describes the relative magnitude of the individual frequency components that make up the signal

Chapter6Time and Frequency Characterization1 X(io)sin tx(0) =元t-1010X,(jo)0-11 -10 -99100110wsin tx2(0)=2cos10t元t

Chapter 6 Time and Frequency Characterization -1 0 1 ω 1 X (j) 1 ( ) t t x t  sin 1 = -11 -10 –9 0 9 10 11 ω 1 X (j) 2 ( ) t t t x t cos10 sin 2 2  =

Timeand Frequency CharacterizationChapter 6The phase angle Z X(jo) provide the relative phase information of thefrequency components2x(t) =1+cos(2元t + Φ,) + cos(4元t +Φ,)+cos(6元t +Φ,)3Φ=Φ, =Φ, =0aΦ, = 4,Φ, =8,Φ, =12bΦ =6,Φ, =-2 .7,Φ, = 0.93Φ, =1.2,Φ, = 4.1, Φ, =-7.2(d)

( ) cos( ) cos( ) cos( ) 1 2 3 1 2 1 2 4 6 2 3 x t t t t = + + + + + +       0 1 = 2 = 3 = 4, 8, 12 1 = 2 = 3 = 6, 2 .7, 0.93 1 2 3  =  = −  = 1.2, 4.1, 7.2 1 2 3  =  =  = − Chapter 6 Time and Frequency Characterization The phase angle ∠X(jω) provide the relative phase information of the frequency components