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Random Processes and Spectral Analysis Chap. 6
6–3 Using the random process described in Prob. 6–2,
(a) Evaluate x 2 (t).
(b) Evaluate x 2 (t).
(c) Using the results of parts (a) and (b), determine whether the process is ergodic for these
averages.
6–4 Let x(t) be a sinusoidal random process that has a uniformly distributed phase angle as described
in Case 1 of Example 6–2. Using MATLAB, plot the PDF for the random process where
A = 5 volts.
★ 6–5 A conventional average-reading AC voltmeter (volt-ohm multimeter) has a schematic diagram as
shown in Fig. P6–5. The needle of the meter movement deflects proportionally to the average current
flowing through the meter. The meter scale is marked to give the RMS value of sine-wave
voltages. Suppose that this meter is used to determine the RMS value of a noise voltage. The noise
voltage is known to be an ergodic Gaussian process having a zero mean value. What is the value of
the constant that is multiplied by the meter reading to give the true RMS value of the Gaussian
noise? (Hint: The diode is a short circuit when the input voltage is positive and an open circuit
when the input voltage is negative.)
+
R
AC voltmeter
Test leads
–
+
–
0
50 A
meter
movement
Figure P6–5
6–6 Let x(t) = A 0 sin (v 0 t + u) be a random process, where u is a random variable that is uniformly
distributed between 0 and 2p and A 0 and v 0 are constants.
(a) Find R x (t).
(b) Show that x(t) is wide-sense stationary.
(c) Verify that R x (t) satisfies the appropriate properties.
6–7 Let r(t) = A 0 cos v 0 t + n(t), where A 0 and v 0 are constants. Assume that n(t) is a wide-sense
stationary random noise process with a zero mean value and an autocorrelation of R n (t).
(a) Find r(t) and determine whether r(t) is wide-sense stationary.
(b) Find R r (t 1 , t 2 ).
(c) Evaluate R r (t, t + t), where t 1 = t and t 2 = t + t.
6–8 Let an additive signal-plus-noise process be described by the equation r(t) = s(t) + n(t).
(a) Show that R r (t) = R s (t) + R n (t) + R sn (t) + R ns (t).
(b) Simplify the result for part (a) for the case when s(t) and n(t) are independent and the noise
has a zero mean value.
★ 6–9 Consider the sum of two ergodic noise voltages:
n(t) = n 1 (t) + n 2 (t)