Chapter 6 Time and frequency characterization of Signals and Systems
1 Time and Frequency Characterization of Signals and Systems Chapter 6
Chapter 6 Time and Frequency Characterization 56.1 The Magnitude-Phase Representation (幅度-相位) of the fourier transform 产+o Xo olde 2丌 XGo)=rGjoyelzrvo X(io)— Magnitude Energy-Density xX(jo)—Phae 2
2 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
Chapter 6 Time and Frequency Characterization ,(t)=4cos 2+cos 6zt cos 2nt+4 cos 67t
3 x (t) 4cos2t cos6t 1 = + x (t) cos2t 4cos6t 2 = + Chapter 6 Time and Frequency Characterization
Chapter 6 Time and Frequency Characterization X1(o) sin t →x X2(o) 11-10-90910110 带哪带 卜带 sin t 2-cos 1Ot
4 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 =
Chapter 6 Time and Frequency Characterization x(t)=1+cos(2nt+Φ1)+cos(4xt+Φ2)+cos(67t+Φ3) Φ1=①,=①3=0 ①1=4,Φ2=8.3=12 A ①1=6,①2=-27①3=0.93 ①1=12,①2=41,3=-72
5 ( ) 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
Chapter 6 Time and Frequency Characterization 2. A black-and-white picture x(1,)("X(ja4,jn) X(ja 1 )=∫ x\12 pile jonh dt dt Figure 6.2 6
6 Chapter 6 Time and Frequency Characterization 2. A black-and-white picture ( ) ( ) F 1 2 1 2 x t t X j j , , ⎯→ ( ) ( ) 1 1 2 2 1 2 1 2 1 2 , , j t j t X j j x t t e e dt dt − + − − = Figure 6.2
Chapter 6 Time and Frequency Characterization I X(O,,j@2) (a) ∠X(m,jo2) 7
7 1 2 | ( , ) | X j j 1 2 X j j ( , ) Chapter 6 Time and Frequency Characterization
Chapter 6 Time and Frequency Characterization (d) Magnitude: X(j@,,ja2) Magnitude: X(ja, jO,=1 Phase:∠X( yO1,)a2)=0 Phase:∠X(jm,jo2) 8
8 :| ( , ) | : ( , ) 1 2 1 2 0 Magnitude X j j Phase X j j = :| ( , ) | : ( , ) 1 2 1 2 Magnitude X j j 1 Phase X j j = Chapter 6 Time and Frequency Characterization
Chapter 6 Time and Frequency Characterization Magnitude Phase: LX(O,,jo,)
9 Magnitude Chapter 6 Time and Frequency Characterization : ( , ) 1 2 + Phase X j j
Chapter 6 Time and Frequency Characterization 56.2 The Magnitude-Phase Representation of the Frequency response of LTI Systems Y(ja)=X(joh(jo Y(jo)=X(jo)H (jo) ∠Y(j)=∠X()+∠H() H(ja)--Gain ∠H(jio) Phase Shift
10 Chapter 6 Time and Frequency Characterization §6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems Y j X j H j ( ) = ( ) ( ) Y j X j H j ( ) = ( ) ( ) = + Y j X j H j ( ) ( ) ( ) H j ( ) ——Gain H j ( ) ——Phase Shift