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EXAM # 1, EE5352, Spring 2008

1. Let the power spectral density of the stationary random process x(n) be Pxx(w).Then cov[IN(w1),IN(w2)] is approximately

The Bartlett estimate of Pxx(w) is

where N = KM.

(a) Find the approximate frequencies where var(Bx(w)) has local minima.(b) Let Dx(m) denote the autocorrelation estimate corresponding to Bx(w). Give an

expression for Dx(m) in terms of K and Cxx(i)

(m), and define Cxx(i)

(m).

(c) Find an approximation for var(Dx(m)) in terms of the symbolvar(Cxx

(i)(m)).

(d) Let x(n) = h(n)*e(n) for white, zero-mean Gaussian noise e(n) with unit

variance. v(n) denotes the finite energy autocorrelation of h(n). Find an

approximation for var(Dx(m)) in terms of K, M, and v(n).

2. Let x(n) = n + e(n) where e(n) is zero-mean and stationary, with the

autocorrelation ree(m).(a) Find the autocorrelation of x(n).

(b) Find the autocovariance of x(n).

(c) Let e(n) = w(n) - .5e(n-1) for all n where w(n) is zero-mean white noise with

variance 2. Find a power spectral density of x(n).

3. x(n) is a zero-mean, independent, stationary random process, where each

individual x(n) is uniformly distributed between - 1 and 1.(a) Find E[x2(n)].

(b) Find E[x4(n)].

(c) Using the results from (a) and (b), find an expression for E[x(n)x(m)x(i)x(j)] in

terms of n,m,i,j, .

sin sin

sin sin

2 21 2 1 2xx xx1 2

1 2 1 2

(( + )N/2) (( - )N/2)w w w w( ) ( )[( + ( ]) )w wP P

N (( + )/2) N (( - )/2)w w w w

(w)IK

1=(w)B

(i)M

K

=1i

x

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4. It is known that the DTFT of Cxx(m) is IN(w).

(a) State Parseval's equation for Cxx(m) and IN(w).

(b) Find E[Cxx(m)] and E[IN(w)], in terms of Pxx(w) and rxx(m), and state how theyare related.

(c) Using part (a), give an equation which relates var(Cxx(m)) to var(IN(w)). You do

not need to use your expressions from part (b).

5. Cross-correlations are often used to estimate the time delay between two signals,

in radar, sonar, and various oil industry applications. However, sometimes there are

more than two signals that need to be correlated. In a sonic well-logging tool, four

equally spaced receivers pick up signals xi(n) where 1 i 4. Here x

i(n) denotes the

ith signal, rather than x(n) to the ith power. Each receiver detects a delayed version

of the same sonic wave, so xi(n) is modeled as s(n (i-1)nd). where nd represents thetime delay between two adjacent receivers, due to the speed of sound in a given typeof rock. The four-fold correlation between the received signals is defined as

1 2 3 4( ) 2 3c m = E[x (n)x (n m)x (n m)x (n m)]+ + +

(a) Assuming that s(n) is a zero-mean, stationary random process, evaluate c(m) interms of the autocorrelation rss(m).

(b) Assuming that the highest frequency in s(t) is c radians/sec, what is the highest

frequency in radians seen in s(n) and rss(m) ? Give the answer in terms of c andthe sampling period T.

(c ) Recall that the highest value allowed in part (b) is radians. Now, what is the

largest sampling period used to construct c(m), in terms of T ?(d) Given your answer in part (c ), what is the largest sampling period allowed if

c(m) is not aliased ? Give your answer in terms ofc .

6. Let x(n) = s(n)e(n) where the stationary random processes s(n) and e(n) haveautocorrelations rss(m) and ree(m) respectively. s(n) and e(n) are statistically

independent of each other.(a) Find the autocorrelation of x(n).

(b) Give an expression for the PSD of x(n) in terms of Pss(w) and Pee(w), assuming

that s(n) and e(n) are zero-mean.(c) Here, s(n) and e(n) are not zero-mean. They can be represented as s(n) = t(n)+ms

and e(n) = v(n)+me where t(n) and v(n) are zero-mean stationary randomprocesses. Find the PSD of x(n) in terms of Ptt(w), Pvv(w), me, and ms .