563489578934

Problems 123
where A 1 , A 2 , v1, v2, u1, and u2 are constants,
Find the autocorrelation for w(t) as a function of the constants.
2–47 Referring to Prob. 2–46,
(a) Find the PSD function for w(t).
(b) Sketch the PSD for the case of v 1 Z v 2 .
(c) Sketch the PSD for the case of v 1 = v 2 and u 1 = u 2 + 90°.
(d) Sketch the PSD for the case of v 1 = v 2 and u 1 = u 2 .
★ 2–48 Given the periodic voltage waveform shown in Fig. P2–48,
(a) Find the DC value for this waveform.
(b) Find the RMS value for this waveform.
(c) Find the complex exponential Fourier series.
(d) Find the voltage spectrum for this waveform.
s(t)
A
Sections of a sinusoid s(t) > 0
– A
1 2 3 4
5 6
t
Figure P2–48
2–49 Determine if s 1 (t) and s 2 (t) are orthogonal over the interval (- 5 where
2 T 2 6 t 6 5 2 T 2 2 ,
s 1 (t) = A 1 cos (v 1 t + w 1 ), s 2 (t) = A 2 cos (v 2 t + w 2 ), and v 2 = 2p/T 2 for the following cases.
(a) v 1 = v 2 and w 1 = w 2 .
(b) v 1 = v 2 and w 1 = w 2 + p/2.
(c) v 1 = v 2 and w 1 = w 2 + p.
(d) v 1 = 2v 2 and w 1 = w 2 .
(e) v 1 = 4 and w 1 = w 2 .
5 v 2
(f) v 1 = pv 2 and w 1 = w 2 .
★ 2–50 Let s(t) = A 1 cos(v 1 t + w 1 ) + A 2 cos(v 2 t + w 2 ). Determine the RMS value of s(t) in terms of
A 1 and A 2 for the following cases.
(a) v 1 = v 2 and w 1 = w 2 .
(b) v 1 = v 2 and w 1 = w 2 + p/2.
(c) v 1 = v 2 and w 1 = w 2 + p.
(d) v 1 = 2v 2 and w 1 = w 2 = 0.
(e) v 1 = 2v 2 and w 1 = w 2 + p.
2–51 Show that
q
a
k=-q
d(t - kT 0 ) 4 f 0
where f [Hint: Expand a q 0 = 1/T 0 .
k=-q d(t - kT 0) into a Fourier series and then take the
Fourier transform.]
q
a
k=-q
d(f - nf 0 )