Punjab Technical University 2005 M.Tech Electronics and Communication Engineering DETECTION AND ESTIMATION MODULATION THEORY ECE 507/519 - Question Paper

DETECTION AND ESTIMATION MODULATION THEORY ECE 507/519 2^nd Sem

 

Max marks 100

 

Note: Attempt any Five questions. All questions carry equal marks.

 

1. Discuss in detail the orthogonal representation of signals. Describe with suitable example the construction of orthogonal basis functions.

 

2. (a) Ding the Karhunen-Loeve expansion of the white Gaussian noise process.

 

(b) The process x(t) = e^at is a family of exponentials depending on the random variable a. Express the mean, autocorrelation and the first order density f(x,t) of x(t) in terms of density f_a(a) of a.

 

3. (a) Consider the random process:

 

X(t) = cos (w_ot + Q)

 

Y(t) = sin (w_ot + Q)

 

where Q is a random variable that is uniformly distributed over [-p, +p]. find the mean and cross covariance of X(t) and Y(t).

 

(b) Find R(t) if

 

4. (a) Show that if y(t) =x(t+a)-x(t-a), then R_y(t)= 2R_x(t) - R_x(t+2a) )- R_x(t-2a)

and S_y(w) = 4S_x(w) sin² (aw)

 

(b) Write a short note on Markov Process.

 

5. (a) Derive the matched filter for estimating deterministic signal f(t) in the presence of coloured noise n(t) with zero mean.

 

(b) Given the channel model Y_t = X_t + Z_t where R_x(t) = e^-|t| cos (t )

R_z(t) = e^-|t|

 

R_z(t) = e^-|t|

 

R_xz(t) = 0

 

Find the transfer function H(w) for causal Wiener filter.

 

6. (a) Define wide sense stationary, jointly wide sense stationary and cyclo-stationary random processes.

 

(b) Consider a random amplitude sinusoid with period T:

 

X(t) = A cos (2 p fT)

 

Is X(t) cyclostationary, wide sense cyclostationary?

 

7. (a) Consider the random process:

 

X(n) = sin (2p f_on + Q)

 

Where Q is uniformly distributed over [0, 2p]. Find the S_x(f).

 

(b) Prove that a process is mean ergodic iff:

 

 

8. Consider the following binary hypothesis testing problem:

 

where s and n are independent random variables.

 

 

(a) Prove that likelihood test reduces to

 

 

(b) Find d for the optimum Bayes test as a function of costs and a priori probabilities.