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1. For k = 1, . . . , n, Yk and Zk are independent, zero-mean Gaussian random variables
with variance σ
2
k
, and ak are real constants. Consider the random process
X(t) := Xn
k=1
Yk cos akt + Zk sin akt.
Obtain the mean µX(t) and autocovariance CXX(t1, t2) of this random process.
(a) Is X(t) a WSS process?
(b) Is X(t) an ergodic process?
(c) Is X(t) a Gaussian process for n = 1?
(d) Is X(t) a Gaussian process for n > 1?
2. N(t) is a Poisson counting process of rate λ. Determine the mean µN(t) and the
autocovariance CNN(t1, t2) of this process. Next consider the process
X(t) = N(t + 1) − N(t)
Determine the mean µX(t) and the autocovariance CXX(t1, t2) of this process.
3. Let Xn be a sequence of zero-mean, uncorrelated random variables with variance σ
2
.
Let wi
, 0 < wi < 1, for i = 0, . . . , k, be a set of constants. The sequence
Yn =
X
k
i=0
wiXn−i
,
is called the weighted moving window average of Xn. Obtain CYY(n1, n2), the autovariance of the sequence Yn.
4. X(t) is a WSS Gaussian process with mean µX(t) = 3 and autocovariance CXX(τ ) =
e
−0.2|τ|
. Find (i) Pr(X(5) ≤ 2) and (ii) Pr(|X(5) − X(8)| ≤ 1).
5. For k = 1, . . . , n, Rk and Θk are independent random variables. The pdf of Rk are
fRk
(r) =
r
a
2
k
e
−r
2/(2a
2
k
)
for r > 0
0 for r ≤ 0
1
for positive constants ak. Θk are uniformly distributed in (0, 2π). bk are arbitrary
positive constants. Define X(t) as
X(t) = Xn
k=1
Rk cos(bkt + Θk).
Determine the mean µX(t) and autocovariance CXX(t1, t2).
6. White noise with unit power spectral density is input to an ideal low pass filter of
bandwidth B. If Y(t) is the output, determine the autocovariance function of the
output and hence determine Pr(Y (1) > 1).
Assignment
1. X(n) is a zero-mean WSS sequence with autocovariance CXX(τ ) for τ = . . . , −2, −1, 0, 1, 2, . . . .
Let Xˆ(n + 1) = aX(n) be the linear estimate for X(n + 1). Find the minimum mean
square error linear estimate for Xn+1, i.e., determine the a that minimizes
E
(X(n + 1) − aX(n))2
.
2. X(t) is a real valued, bandpass random signal with power spectral density for positive
frequencies given by
SXX(ω) = 3 (U(ω − 9000) − U(ω − 11000)) + 400δ(x − 10000)
Here U(ω) is the unit step function. Determine E(X(t)), the mean value of the signal,
and E
X
2
(t)
, the mean power in the random signal.
3. Consider a random sequence that changes as follows.
Qn =
Qn−1 + 1 with probability λ(1 − µ)
max{Qn−1 − 1, 0} with probability (1 − λ)µ
Qn−1 with probability µλ + (1 − λ)(1 − µ)
where 0 < λ, µ ≤ 1. Write a program to simulate this sequence, and obtain the time
average and ensemble average for a given λ and µ. Using the simulation program,
determine which of the following values corresponds to a stationary Qn and to an
ergodic Qn.
(a) λ = 0.7, µ = 0.9.
(b) λ = 0.9, µ = 0.7.
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