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Sec. 2–2 Fourier Transform and Spectra 55
u(f) = e -p/2, f 7 0
-90°, f 7 0
f radians = e
+p/2, f 6 0 90°, f 6 0 f
Plots of these spectra are shown in Fig. 2–4. As shown in Fig. 2–4a, the weights of the delta functions
are plotted, since it is impossible to plot the delta functions themselves because they have infinite
values. It is seen that the magnitude spectrum is even and the phase spectrum is odd, as expected
from Eqs. (2–38) and (2–39).
Now let us generalize the sinusoidal waveform to one with an arbitrary phase angle u0. Then
w(t) = A sin (v 0 t + u 0 ) = A sin [v 0 (t + u 0 /v 0 )]
and, by using the time delay theorem, the spectrum becomes
W(f) = j A 2 eju 0(f/f 0 )
[d(f + f 0 ) - d(f - f 0 )]
The resulting magnitude spectrum is the same as that obtaind before for the u 0 = 0 case. The new
phase spectrum is the sum of the old phase spectrum plus the linear function (u 0 /f 0 )f. However,
since the overall spectrum is zero except at f =;f 0 , the value for the phase spectrum can be arbitrarily
assigned at all frequencies except f =;f 0 . At f = f 0 the phase is (u 0 - p/2) radians, and
at f =-f 0 the phase is -(u 0 - p/2) radians. See Example2_5.m for MATLAB plots.
From a mathematical viewpoint, Fig. 2–4 demonstrates that two frequencies are present
in the sine wave, one at f =+f 0 and another at f =-f 0 . This can also be seen from an expansion
of the time waveform; that is,
which implies that the sine wave consists of two rotating phasors, one rotating with a frequency
f =+f 0 and another rotating with f =-f 0 . From the engineering point of view it is said that
one frequency is present, namely, f = f 0 , because for any physical (i.e., real) waveform,
Eq. (2–35) shows that for any positive frequency present, there is also a mathematical negative
frequency present. The phasor associated with v(t) is c = 0 - jA = Al -90°. Another interesting
observation is that the magnitude spectrum consists of lines (i.e., Dirac delta functions).
As shown by Eq. (2–109), the lines are a consequence of v(t) being a periodic function. If the
sinusoid is switched on and off, then the resulting waveform is not periodic and its spectrum is
continuous as demonstrated by Example 2–10. A damped sinusoid also has a continuous
spectrum, as demonstrated by Example 2–4.
Rectangular and Triangular Pulses
v(t) = A sin v 0 t = A j2 ejv 0t - A j2 e-jv 0t
The following waveshapes frequently occur in communications problems, so special symbols
will be defined to shorten the notation.