CHAP. 17]
FOURIER METHOD OF WAVEFORM ANALYSIS
431
Fig. 17-13
Alternatively, Xð f Þ may be shown by its real and imaginary parts, Re ½Xð f Þ and Im ½Xð f Þ , as in Figs. 17-14(a) and (b). a a2 þ 4 2 f 2 2 f Im ½Xð f Þ ¼ 2 a þ 4 2 f 2
Re ½Xð f Þ ¼
ð26aÞ ð26bÞ
Fig. 17-14
EXAMPLE 17.8
Find the Fourier transform of the square pulse 1 for T < t < T xðtÞ ¼ 0 otherwise
From (22a), ðT Xð f Þ ¼
e j2 ft dt ¼
T
Because xðtÞ is even, Xð f Þ is real.
EXAMPLE 17.9
1 h j2 f iT sin 2 fT e ¼ T j2 f f
ð27Þ
The transform pairs are plotted in Figs. 17-15(a) and (b) for T ¼ 12 s.
Find the Fourier transform of xðtÞ ¼ eat uð tÞ; a > 0. ð0 Xð f Þ ¼ 1
eat e j2 ft dt ¼
1 a j2 f
ð28Þ