563489578934

TABLE 2–1
SOME FOURIER TRANSFORM THEOREMS a
Operation Function Fourier Transform
Linearity a 1 w 1 (t) + a 2 w 2 (t) a 1 W 1 (f ) + a 2 W 2 (f)
Time delay w(t - T d ) W(f) e -jvT d
Scale change
w(at)
1
|a| Wa f a b
Conjugation w * (t) W * (-f )
Duality W(t) w(-f )
Real signal
w(t) cos (v c t + u) 1 2 [eju W(f - f c ) + e -ju W(f + f c )]
frequency
translation
[w(t) is real]
Complex signal
w(t) e jv ct
W(f - f c )
frequency
translation
Bandpass signal Re{g(t) e jvct }
1
2 [G(f - f c) + G * (-f - f c )]
Differentiation
d n w(t)
dt n
(j2pf) n W(f)
Integration
L
t
-q
w(l) dl
(j2pf) -1 W(f) + 1 2 W(0) d (f)
Convolution w 1 (l) * w 2 (t) = W 1 (f )W 2 (f)
L
q
-q
w 1 (l) # w 2 (t - l) dl
q
Multiplication b w 1 (t)w 2 (t) W 1 (f) * W 2 (f) = W 1 (l) W 2 (f - l) dl
L
Multiplication
t n w(t) (-j2p) -n dn W(f)
df n
by t n
a v c = 2pf c .
b * denotes convolution as described in detail by Eq. (2–62).
-q