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Signals and Spectra Chap. 2
★ 2–52 Three functions are shown in Fig. P2–52.
(a) Show that these functions are orthogonal over the interval (-4, 4).
(b) Find the corresponding orthonormal set of functions.
★ 2–53 Expand the waveform
w 1 (t) = e 1, 0 … t … 4
0, t elsewhere
into an orthonormal series by using the orthonormal set found in Prob. 2–52 (b).
★ 2–54 Evaluate the mean square error for the orthogonal series obtained in Prob. 2–53 by evaluating
4
e = Jw(t) - a a j w j (t) K
2dt
L -4 j=1
★ 2–55 Expand the waveform
cos( 1 pt), -4 … t … 4
4
w 1 (t) = e
0, t elsewhere
into an orthonormal set of functions found in Prob. 2–52 (b).
★ 2–56 Evaluate the mean square error for the orthogonal series obtained in Prob. 2–55 by using the
integral shown in Prob. 2–54.
3
1 (t)
1
–4
–2
2 4
t
2 (t)
1
4
–4
–2
2
t
–1
w 3 (t)
–4
1
4
–3
–1
–1
1 3
t
Figure P2–52