EE Qualifying Exam – Summer 2014

Control Theory

Problem 1 (/20):

A rocket is launched in the atmosphere, and follows a theoretical trajectory ( ) ( ) ( ). Its initial conditions are zero altitude and zero vertical speed, i.e. ( ) ( ) . We measure both its vertical speed ( ) and its altitude ( ). The

variable ( ) is an input. 1. Write the state space equation in standard form (/4) 2. What is the order of the system? (/2) 3. Is the system completely controllable? (/3) 4. Is the system completely observable? (/3) 5. Design a controller to stabilize the system at zero altitude and zero vertical velocity, with time constants of 1s. (/3) 6. We assume that two independent sensors (accelerometer and Doppler radar) measure the acceleration ( ) and the vertical speed ( ) of the rocket for all times

t=0,1,…n seconds. Write the problem of estimating the parameter from these n+1 measurements as a linear least squares problem. Find its solution. (/5)

EE Qualifying Exam

Control Theory

Problem 2 (/20):

A linear dynamical system evolves as ( ) ( ) ( ), where [

] and

[ ]

1. Is this system continuous time or discrete time? (/2) 2. What is the condition on a,b,c and d for the system to be stable? Hint: ( )( ) ( ) (/5) 3. Under what condition is the system completely controllable? (/4) 4. We now consider the system with state feedback ( ) ( ). Find the condition on a,b,c,d and the coefficients of k for the closed look system to be stable. (/6) 5. True or false: if a system is unstable, then the norm of its state always increases

exponentially (provided that the initial state is not an equilibrium), i.e. ‖ ( )‖ for some α and p. If true, prove it, if false give a counterexample. (/3)

PhD Qualifier Exam – Summer 2014

Digital Communications

Question No 01: (20 marks)

Part 1: Consider the following signals:

0

2 2cos if 0,

( ) 6

0 otherwise

tt T

x t TT

1

2 2 5cos if 0,

( ) 6

0 otherwise

tt T

x t TT

2

2 2 3cos if 0,

( ) 2

0 otherwise

tt T

x t TT

a) Find a set of orthonormal basis functions for this signal set. Show that they are orthonormal.

(4 marks)

b) Find the data symbols (and plot the constellation points) corresponding to the signals above for the

basis functions you found in (a) and the corresponding average energy. (3 marks)

c) One of the three signals in part (a) is transmitted through an AWGN channel and the received

signal is 0.3 2 0.9 2

( ) cos sin( )t t

y tT TT T

. Which of the three signals 0 ( )x t , 1( )x t and 2 ( )x t is

detected by the minimum probability of error detector. (3 marks)

Part 2: Consider the channel in Fig. 1.

where 1 1s and 2 0.5s . Moreover, 1|p r s and 2|p r s are shown in Fig. 2.

a) Assume the two message are equiprobable. Determine the received signal when

i. 2r ii. 1r iii. 3r iv. 7r (3 marks)

b) Calculate the probability of error assuming that 1( ) 0.75p s and 2( ) 0.25p s . (7 marks)

FIG. 1. Channel For Part 2 FIG. 2. p(r|s1) and p(r|s2) for for Part 2

Question no 02: (20 Marks)

Consider the constellation given in Fig. 3, where the center constellation point has a probability of 1

4

and each corner point occurs with a probability of 3

16.

FIG. 3

a) Comment on the energy efficiency of the constellation. (3 marks)

b) Calculate mind , the average energy, and the average number of nearest neighbors. (4 marks)

c) Draw the decision regions assuming the signals are equi-probable. (4 marks)

d) Find the probability of correct decision given the center point is transmitted (again assume equi-

probable points). (3 marks)

e) Design waveform for the constellation points (assume they are equi-probable) satisfying the

following: (6 marks)

i. Waveforms should use a rectangular pulse shape of unit energy and duration T and should

be transmitted in a channel centered at a center frequency of 1 MHz.

ii. The average energy of the transmitted waveforms should be 5.

iii. Waveforms should be able to transmit 1024 bits per second.

1

PhD Qualifying Exam, June 2014

DSP

1. For this problem, assume that is even.

a- If the output of IDFT, ( ), is real, prove that the input satisfies the relation

( ) ( ) What does this imply for ( ) and (

) (5 marks)

b- Prove that if the even inputs to an IDFT block are zero, i.e, ( ) ( ) ( )

( ) , then the output satisfies the relation ( ) ( ). (5 marks)

c- Consider a system with an IDFT block such that its output is real and its input is zero

for even . Now consider sequence ( ) which is equal to ( ) if ( ) is nonnegative,

and is equal to zero if ( ) is negative. Prove that if we DFT the sequence ( ), the

outputs satisfy the relation ( ) ( ) for odd . (10 marks)

2.

a. How many complex multiplications are required for a 4-point FFT? Do not use the

general complexity expressions or asymptotic results. Just count the actual needed

number of multiplications. (5 marks)

b. A sleep-deprived PhD student mistakenly computed the FFT as:

( )( ) ( )

( ) (

)

( )( )

where ( )

( ) and ( )

( ) are the DFT of the even and odd indexed samples,

respectively. To add insult to injury, the student inadvertently lost ( )

( ) and

( )

( ) How can he or she compute the correct Fourier transform using ( )( )?

(15 marks)

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EE PhD Qualifier Exam – Summer 2014

Probability and Random Processes

Problem 1 (20 Points)

Part 1 (5 Points)

Let X be discrete random variable such that P[X=-1] = p, P[X=0] = q, P[X=1] = r, and P[X=k]=0 for other

values of k. Express the variance of Y in terms of p and r, where Y=Xn for n, a positive even integer.

Part 2 (15 Points)

Suppose (X,Y) has a distribution which is uniform over a unit circle centered at (0,0).

1- (7 Points) Find the joint probability density function (PDF) of (X,Y).

2-(5 Points) Find the marginal PDFs of X and Y.

3- (3 Points) Are X and Y independent ?

Problem 2 (20 Points)

Part 1 (10 Points)

Let X and Y be independent and identically distributed, each with a continuous uniform distribution and

range [0,1]. Let Z= X Y. What is the cumulative distribution function of Z ?

Part 2 (10 Points)

Consider a random process X(t) defined by X(t)= U cos(w t) + V sin(w t), where w is a constant and

U and V are random variables. Find the conditions that U and V have to satisfy in order for X(t) to be a

wide sense stationary random process.