334
Practical MATLAB
® Applications for Engineers
where a is a real nonzero constant. If a < 1, then f(t) is expanded in time, whereas
F(w) is compressed in frequency by a similar factor.
Note that if a > 1, then the function f(at) is compressed in time, whereas F(w) is
expanded in frequency by a similar factor.
c. Duality or symmetry
F t
f w
( )
( )
↔ 2 Ϫ
obtained by the following subtitution t → w and w → t.
Observe that this property can be used to double the table of transform pairs,
by simply interchanging the time and frequency variables.
d. Time shifting
f t t
F w e
jt w
(
)
( )
Ϫ
Ϫ
0
0
↔
where a time shift translates as a linear phase shift in frequency.
e. Frequency shifting
f t e
F w w
jw t
( )
(
)
0
0
↔
Ϫ
This property is also referred to as the modulation property. It is a fundamental property in communication theory, and is used to prove the modulation
theorem that states
f t
w t
F w w
F w w
( )
{ (
)
(
)}
и
ϩ
ϩ
Ϫ
cos 0
0
0
↔
1
2
Observe that multiplying an arbitrary time function f(t) by cos(w 0 t) shifts the
spectrum of f(t), so that half the original spectrum is centered at w 0 , and the other
half is centered at −w 0 . The signal f(t) is referred to as the modulating signal,
whereas cos(w 0 t) is referred as the carrier signal (in practical applications the
frequency w 0 must be much higher than the highest frequency of f(t)).
f. Time differentiation
df t
dt
jwF w
( )
( )
↔
or the more general relation given by
d f t
dt
jw F w
n
n
n
( )
( ) ( )
↔
g. Frequency differentiation
( ) ( )
( )
Ϫjt f t
dF w
dw
↔
CRC_47760_CH004.indd 334
CRC_47760_CH004.indd 334
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7/28/2008 12:25:54 PM