6 Time and frequency characterization of s&s 6. Time and Frequency Characterization of Signals and Systems 6.1 The Magnitude-phase Representation of the Fourier transform For signal x(t):x(o<X(jo) X(o=Y(o)le/zoo LXGOI Magnitude spectrum Phase spectrum
6 Time and frequency characterization of S&S 6.1 The Magnitude-phase Representation of the Fourier Transform For signal x(t) : 6. Time and Frequency Characterization of Signals and Systems x(t) X( j) ⎯F → ( ) ( ) | ( )| j X j X j X j e = e Phase Spectrum X j Magnitude Spectrum j X j − − − − − − ( ) | ( )|
6 Time and frequency characterization of s&s x(t)=l+cos(2t+i+cos(2nt +2)+cos(2t+3) ①1=Φ,=Φ2=0 ①1=4,Φ2=8①3=12 AAA△AA 1=6,①2=-2.3=093 AM △=122=41=-72
6 Time and frequency characterization of S&S cos(2 ) 3 2 cos(2 ) cos(2 ) 2 1 ( ) =1+ + 1 + + 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 = = = −
6 Time and frequency characterization of s&s P(On,jo2) (a) ∠P(jo1,j02)
6 Time and frequency characterization of S&S ( , ) 1 2 P j j | ( , )| 1 2 P j j
6 Time and frequency characterization of s&s Magnitude: P(O,,jO,) Phase: 0 (d) Magnitude: I Phase:∠P(jo12j2)
6 Time and frequency characterization of S&S : 0 :| ( , )| 1 2 Phase Magnitude P j j : ( , ) :1 1 2 Phase P j j Magnitude
6 Time and frequency characterization of s&S Magnitude Phase:∠P(O1,jo2)
6 Time and frequency characterization of S&S : ( , ) : 1 2 Phase P j j Magnitude
6 Time and frequency characterization of s&s 6.2 The Magnitude-phase Representation of the frequency response of lti System System characterization Impulse response: h(t)F>H(o Frequency response H(@)= r(jo) X(o H(0)=H(o)e i∠H(O) IHG@- Magnitude response eJ2H(o Phase response
6 Time and frequency characterization of S&S 6.2 The Magnitude-phase Representation of the Frequency Response of LTI System System characterization: h(t) H( j) ⎯F → ( ) ( ) | ( )| j H j H j H j e = e Phase Response H j Magnitude Response j H j − − − − − − ( ) | ( )| Impulse response: ( ) ( ) ( ) X j Y j Frequency response: H j =
6 Time and frequency characterization of s&s 6.2.1 Linear and nonlinear phase Linear phase ∠H(jo)=ko Nonlinear phase: ZH(o)=Nonlinear function EXample: y(t=x(t-to) HGo=e oo HGo=-@to (Linear phase) Effect: Linear phase means non-distortion of signal transmission
6 Time and frequency characterization of S&S 6.2.1 Linear and Nonlinear Phase ( ) ( ) ( ) ( ) ( ) 0 0 0 H j t Linear phase H j e y t x t t j t = − = = − − Linear phase: H( j) = k Nonlinear phase: H( j) = Nonlinear function Example: Effect: Linear phase means non-distortion of signal transmission
6 Time and frequency characterization of s&s (Linear phase) Original signal Nonlinear phase
6 Time and frequency characterization of S&S ( Linear phase ) ( Nonlinear phase ) ( Original signal )
6 Time and frequency characterization of s&S 6.2.2 Group delay Definition ( o ∠H(jO) d EXample: y(t)=x(t-to) Ho=e ∠H()=-ot0 r(o=to (signal delay) Distortionless system T(o)is flat
6 Time and frequency characterization of S&S 6.2.2 Group Delay ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 t signal delay H j t H j e y t x t t j t = = − = = − − Definition: ( ) ( ) H j d d = − Example: Distortionless system: is flat. ()
6 Time and frequency characterization of s&s 7,000 6.000 5.000 Medium 4.000 Medium 2000 1.000 60012001,8002.4003.0003600 Frequency in(Hz)
6 Time and frequency characterization of S&S
