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52
Signals and Spectra Chap. 2
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
Wa f |a| a b
W * (-f)
Conjugation
w *
(t)
Duality W(t) w(-f)
Real signal
w(t) cos(w c t + u)
1
2 Ceju W(f - f c ) + e -ju W(f + f c )D
frequency
translation
[w(t) is real]
Complex signal
w(t) e jv ct
W(f - f c )
frequency
translation
Bandpass signal ReEg(t) e jv ct
F
1
Differentiation
Integration
d n w(t)
dt n
t
w(l)dl
L
-q
2 CG(f - f c) + G * (-f - f c )D
(j2pf) n W(f)
(j2pf) -1 W(f) + 1 2 W(0) d(f)
Convolution
Multiplication b
q
w 1 (t)*w 2 (t) = w 1 (l)
L
# w 2 (t - l) dl
w 1 (t)w 2 (t)
-q
W 1 (f)W 2 (f)
q
W 1 (f)*W 2 (f) = W 1 (l) W 2 (f - l) dl
L
-q
Multiplication
t n w(t) (-j2p) -n dn W(f)
by t n df n
a
w c = 2pf c .
b * Denotes convolution as described in detail by Eq. (2–62).
Dirac Delta Function and Unit Step Function
DEFINITION.
The Dirac delta function d(x) is defined by
L
q
-q
w(x) d(x) dx = w(0)
(2–45)
where w(x) is any function that is continuous at x = 0.