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64
Signals and Spectra Chap. 2
TABLE 2–2
SOME FOURIER TRANSFORM PAIRS
Function Time Waveform w(t) Spectrum W( f)
Rectangular ßa t T
b T[Sa( p fT)]
Triangular a t T b T[Sa( fT)] 2 p
Unit step u(t)!e +1, t 7 0
0, t 6 0
Signum sgn(t) ! e +1, t 7 0
-1, t 6 0
1
2 d(f) + 1
j2pf
1
jpf
Constant 1 d( f)
Impulse at t = t 0 d(t - t 0 ) e -j2pft 0
Sin c
Phasor
Sa(2pWt)
e j(v 0t+w)
1
2W ßa f
2W b
e jw d(f - f 0 )
Sinusoid cos (v c t + w)
1
2 ejw d(f - f c ) + 1 2 e-jw d(f + f c )
Gaussian e -p(t/t 0) 2 t 0 e -p(ft 0) 2
Exponential,
one-sided e e-t/ T , t 7 0
0, t 6 0
Exponential,
two-sided
Impulse train
e -|t|/T
k= q
a
k=-q
d(t - kT)
2T
1 + j2pfT
2T
1 + (2pfT) 2
n= q
f 0 a
n=-q
d(f - nf 0 ),
where f 0 = 1/T
in communication systems. In Eq. (2–42), the energy spectral density (ESD) was defined in
terms of the magnitude-squared version of the Fourier transform of the waveform. The PSD
will be defined in a similar way. The PSD is more useful than the ESD, since power-type
models are generally used in solving communication problems.
First, we define the truncated version of the waveform by
w(t), -T/2 6 t 6 T/2
w T (t) = e f = w(t)ß a t 0, t elsewhere
T b
Using Eq. (2–13), we obtain the average normalized power:
(2–64)
P = lim
T: q
1
T L
T/2
-T/2
w 2 (t) dt = lim
T: q
1
T L
q
- q
w 2 T(t) dt