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56
Signals and Spectra Chap. 2
DEFINITION. Let ß( # ) denote a single rectangular pulse. Then
ßa t T b !
1, |t| … T 2
e
0, |t| 7 T 2
(2–52)
)
DEFINITION. Sa( # ) denotes the function †
DEFINITION. Let ( denote the triangular function. Then
Sa (x) = sin x
x
#
(2–53)
1 - |t|
T , |t| … T
a t L
(2–54)
T b !
0, |t| 7 T
These waveshapes are shown in Fig. 2–5. A tabulation of Sa(x) is given in Sec. A–9
(Appendix A).
Example 2–6 SPECTRUM OF A RECTANGULAR PULSE
The spectrum is obtained by taking the Fourier transform of w(t) =ß(t/T).
T/2
W(f) = 1 e -jvt dt = e-jvT/2 - e jvT/2
L
-T/2
-jv
sin (vT/2)
= T
vT/2
= T Sa (pTf)
Thus,
ßa t b 4 T Sa (pTf)
T
(2–55)
For magnitude and phase plots, see Example2_06.m. A numerical evaluation of this FT integral
is given by file Example2_06 FT.(m). The preceding Fourier transform pair is shown in Fig. 2–6a.
Note the inverse relationship between the pulse width T and the spectral zero crossing 1/T. Also,
by use of the duality theorem (listed in Table 2–1), the spectrum of a (sin x)/x pulse is a rectangle.
That is, realizing that ß(x) is an even function, and applying the duality theorem to
Eq. (2–55), we get
† This is related to the sin c function by Sa(x) = sin c(x/p) because sin c(l) !(sin pl)>pl. The notation
Sa(x) and sin c(x) represent the same concept, but can be confused because of scaling. In this book, (sin x)/x will
often be used because it avoids confusion and does not take much more text space.