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
)()(?jXtx F
)(|)(|)( jXjejXjX
S p e c t r u mP h a s ee
S p e c t r u mM a g n it u d ejX
jXj

)(
|)(|
6 Time and frequency characterization of S&S
)2c o s (32)2c o s ()2c o s (211)( 321 ttttx
0321
12,8,4 321
93.0,7.2,6 321
2.7,1.4,2.1 321
6 Time and frequency characterization of S&S
),( 21 jjP?|),(| 21 jjP
6 Time and frequency characterization of S&S
0:
|),(|,21
P h a s e
jjPM a g n it u d e
),(:
1:
21 jjPP h a s e
M a g n it u d e
6 Time and frequency characterization of S&S
),(:
:
21 jjPP h a s e
M a g n it u d e
6 Time and frequency characterization of S&S
6.2 The Magnitude-phase Representation of
the Frequency Response of LTI System
System characterization:
)()(?jHth F
)(|)(|)( jHjejHjH
R e s p o n s eP h a s ee
R e s p o n s eM a g n i t u d ejH
jHj

)(
|)(|
Impulse response:
)(
)()(

jX
jYjH?Frequency response:
6 Time and frequency characterization of S&S
6.2.1 Linear and Nonlinear Phase
)()(
)(
)()(
0
0
0
p h a s eLi n e a rtjH
ejH
ttxty
tj




Linear phase, kjH )(
Nonlinear phase,f u n c t io nNo n lin e a rjH )(?
Example:
Effect,Linear phase means non-distortion
of signal transmission.
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
)()(
)(
)(
)()(
0
0
0
0
de lays ig na lt
tjH
ejH
ttxty
tj




Definition:
)()( jHdd
Example:
Distortionless system,is flat.)(
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6.2.3 Log-Magnitude and Bod Plots
Magnitude spectrum:
)(lo g~|)(|lo g20
~|)(|
1010 p lo t sB o djH
jH


Phase spectrum:
)(lo g~)(
~)(
10 p lo t sB o djH
jH


6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6.3 Time-Domain Properties of Ideal Frequency-
selective Filters
Lowpass filter:
(1) Continous time:


c
cjH


||,0
||,1)(
t
tth cF
s in)(
(2) Discrete time:





||,0
||,1)(
c
cjH
n
nnh cF
s in][
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6.4 Time-Domain and Frequency-domain Aspects
of Non-ideal Filters
Basic parameter of lowpass filter:
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
6 Time and frequency characterization of S&S
Problems,6.5 6.23 6.27