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Problems 483
The power of n 1 (t) is 5 W, the power of n 2 (t) is 10 W, the DC value of n 1 (t) is -2 V, and the DC
value n 2 is +1 V. Find the power of n(t) if
(a) n 1 (t) and n 2 (t) are orthogonal.
(b) n 1 (t) and n 2 (t) are uncorrelated.
(c) The cross-correlation of n 1 (t) and n 2 (t) is 2 for t = 0.
6–10 Assume that x(t) is ergodic, and let x(t) = m x + y(t), where m x = x(t) is the DC value of x(t) and
y(t) is the AC component of x(t). Show that
(a) R x (t) = m 2 x + R y (t).
(b) lim t : qR x (t) = m 2 x .
(c) Can the DC value of x(t) be determined from R x (t)?
★ 6–11 Determine whether the following functions satisfy the properties of autocorrelation functions:
(a) sin v 0 t.
(b) (sin v 0 t)(v 0 t).
(c) cos v 0 t + d(t).
(d) e -a| t | , where a 6 0.
(e) (Note: [R(t)] must also be a nonnegative function.)
6–12 A random process x(t) has an autocorrelation function given by R x (t) = 5 + 8e -3|τ | . Find
(a) The RMS value for x(t).
(b) The PSD for x(t).
★ 6–13 The autocorrelation of a random process is R x (t) = 4e - t2 + 3. Plot the PSD for x(t) and evaluate
the RMS bandwidth for x(t).
6–14 Show that two random processes x(t) and y(t) are uncorrelated (i.e., R xy (t) = m x m y ) if the
processes are independent.
6–15 If x(t) contains periodic components, show that
(a) R x (t) contains periodic components.
(b) x ( f ) contains delta functions.
6–16 Find the PSD for the random process described in Prob. 6–2.
6–17 Determine whether the following functions can be valid PSD functions for a real process:
(a) 2e -2p|f-45| .
(b) 4e -2p|f2 -16|.
(c) 25 + d(f - 16).
(d) 10 + d(f).
★ 6–18 The PSD of an ergodic random process x(t) is
1
x (f) = B (B - |f| ), |f| … B
L
0, f elsewhere
where B 7 0. Find
(a) The RMS value of x(t).
(b) R x (t).
6–19 Referring to the techniques described in Example 6–4, evaluate the PSD for a PCM signal that
uses Manchester NRZ encoding. (See Fig. 3–15.) Assume that the data have values of a n =;1,
which are equally likely, and that the data are independent from bit to bit.