Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

PROBLEMS

Ergodicity

8.1 (a) Derive the necessary and sufficient condition in (8.23) for random sequence X[k] to be ergodic in the mean. (b) Verify the sufficient condition in (8.24).

8.2 Let X[k] have zero mean and autocovariance function CXX[m] = α |m|. Determine if both conditions in the first problem are satisfied for this random sequence.

8.3 Suppose that X(t) is ergodic in the mean, and let Y be a random variable that is independent of X(t). Define random process Z(t) = YX(t) and determine if it is also ergodic in the mean. Note that a realization is of the form z(t) = yx(t) such that outcome y remains fixed with time as x(t) varies.

8.4 Random process X(tx) has zero mean and the following autocorrelation function:

(8.332)

Determine if X(t) is ergodic in the mean.

8.5 Random process X(t) has the following periodic autocovariance function:

(8.333)

for some fixed frequency fc. Determine if X(t) is ergodic in the mean.

8.6 For RXX(τ) = exp(−|τ|), use the result in Example 8.3 to determine if the zero-mean Gaussian random process X(t) is ergodic in correlation.

Power Spectral Density

8.7 For the cross-PSD, show that (a) SXY(f) = SYX(−f) and (b) the imaginary part is an odd function.

8.8 Consider the following random process:

(8.334)

where Φ is uniformly distributed on . Find ...