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126
Signals and Spectra Chap. 2
2–63 For the waveform shown in Fig. P2–62, find the quadrature Fourier series.
w(t)
2
•
••
•
••
–2
2 4 6
t
Figure P2–62
q
★ 2–64 Given a periodic waveform s(t) = a n=-q p(t - nT 0), where
p t (t) = e At, 0 < t < T
0, t elsewhere
and T … T 0 ,
(a) Find the c n Fourier series coefficients.
(b) Find the {x n , y n } Fourier series coefficients.
(c) Find the {D n , w n } Fourier series coefficients.
2–65 Prove that the polar form of the Fourier series, Eq. (2–103), can be obtained by rearranging the
terms in the complex Fourier series, Eq. (2–88).
2–66 Prove that Eq. (2–93) is correct.
2–67 Let two complex numbers c 1 and c 2 be represented by c 1 = x 1 + jy 1 and c 2 = x 2 + jy 2 , where
x 1 , x 2 , y 1 , and y 2 are real numbers. Show that Re{·} is a linear operator by demonstrating that
Re{c 1 + c 2 } = Re{c 1 } + Re{c 2 }
2–68 Assume that y(t) = s 1 (t) + 2s 2 (t), where s 1 (t) is given by Fig. P2–48 and s 2 (t) is given by
Fig. P2–61. Let T = 3, b = 1.5, and t 0 = 0. Find the complex Fourier coefficients {c n } for y(t).
2–69 Evaluate the PSD for the waveform shown in Fig. P2–2.
v(t)
1
–2 –1 0 1 2 3
–1
t (sec)
Figure P2–70