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NEW MEXICO STATE UNIVERSITY THE KLIPSCH SCHOOL OF ELECTRICAL AND COMPUTER

ENGINEERING Ph.D. QUALIFYING EXAMINATION

January 10, 2011 9:00 AM - 1:00 PM CLOSED BOOK

Exam Instructions:

a. Write the last four digits of your Banner ID number on the top of every page.

b. Work six (6) problems from the three (3) areas of specialization

selected at the time of registration. Do not work more than two (2) problems in any one area.

A Circuits and Electronics B Communications C Computers D Control Systems E Digital Signal Processing F Electric Energy Systems G Electromagnetics H Photonics

c. Check the boxes below indicating which six (6) problems you

want graded. (You must work two problems from each of the three areas you specified at the time of registration.)

a b c d e f

A Circuits and Electronics.

B Communications…………..

C Computers…………………

D Control Systems.…...........

E Digital Signal Processing

F Electric Energy Systems.

G Electromagnetics.............

H Photonics………………......

Design a fully differential folded cascade operational amplifier for a minimum gain-bandwidth product GB=200MHz and a load capacitance CL=5pF. Assume following 0.13μm CMOS technology technology parameters: knn=340μA/V2, Vthn0.40V , knp=50μA/V2, Vthp=0.40V, λn=λp=0.25V-1, Cox=10fF/μm2 and a capacitive load CL=10pF (kn=μCox/2). Determine all transistor sizes and cascode biasing voltages including those in the biasing branch. Assume a single supply voltage VDD=1.2V The design requires to determine bias currents and W/L sizes for all transistors. Estimate for your design: a) location of the dominant pole and second (high frequency) pole taking into account only gate-source (Cgs) parasitic capacitances (do not take into account Cgd , Cbd and Csb parasitic capacitances) the open loop gain, the slew rate, the common mode input range and the output swing of the op-amp

A (a) Circuits and Electronics

Design a single ended two stage operational amplifier for a minimum gain-bandwidth product GB=30MHz and a load capacitance CL=20pF. Assume following 0.18μm CMOS technology technology parameters: knn=170μA/V2, Vthn0.45V , knp=25uA/V2, Vthp=0.45V, λn=λp=0.1V-1, Cox=4.2fF/μm2 (kn=μCox/2). Determine all transistor sizes including those in the biasing branch. Assume a single supply voltage VDD=1.8V The design requires to determine bias currents, W/L size all transistors, and to determine the value of the compensation capacitance. Estimate for your design: a) location of the dominant pole and second (high frequency) pole taking into account only gate-source (Cgs) parasitic capacitances (do not take into account Cgd , Cbd and Csb parasitic capacitances) the open loop gain and the slew rate of the op-amp. Determine the Bandwidth, the common mode input range and the output impedance in unity gain voltage follower configuration.

A (b) Circuits and Electronics

Explain briefly 1) List at least four of the main design constraints/challenges faced by analog designers in modern deep submicrometer (fine line) CMOS technology 2) Explain the three types of random mismatch errors for analog circuit can be classified, b) what layout techniques are used to minimize the effect of each type of mismatch error?. Justify your answers

A (c) Circuits and Electronics

(i) The random variable pair (X, Y ) has a joint pdf

fX,Y (x, y) =

2e−xe−2y x > 0, y > 0

0 otherwise

Find the probability P [X − Y ≤ 10].

(ii) Find the correlation of X and Y . [Useful relation:∫

xeaxdx = 1

axeax − 1

a2 eax]

B (a) Communications

(i) Let X(t) = A cosωt + B sin ωt, where A and B are iid Gaussian random variables with

zero mean and variance σ2. Find the mean and the autocovariance of X(t). Explain if the

process X(t) is wide sense stationary. What is the average power of the process X(t)?

(ii) Let Un be a sequence of iid zero-mean, unit variance Gaussian random variables. A

“low-pass filter” takes the sequence Un and produces the sequence

Xn =1

2(Un + Un−1)

Explain if the sequence {Xn} converges in the mean-square sense and in distribution.

B (b) Communications

(i) Let X = [X1 X2]T be a jointly Gaussian random vector with mean [1, 1]T , and covariance

matrix

KX =

5 2

2 5

Find the correlation E[X1X2]. Next, suppose we define a new random variable Y as Y =

X1 −X2. Write down the expression for the probability density function (pdf) of Y . [Please

note that the notation [·]T denotes the transpose of a matrix or vector.]

(ii) Suppose the input into a filter is zero-mean white noise with noise power spectral density

of N0/2. The filter has the transfer function,

H(f) =1

1 + j2πf

Find the average power of the output. [Useful result: The Fourier transform of e−a|t|, for

a > 0, is 2aa2+4π2f2 ]

B (c) Communications

Election exit polls show candidate A winning 52% of the vote, and candidate B 48%, with a sample size of 2000. We know that the normal quartile at 90% is 1.282, at 97% is 1.881. Can one predict A wins the election with 90% confidence level? How about at 97% level?

C (a) Computers

The sign table of a 2^2 factorial design is provided below. Build the regression model and compute the variations explained by each factor.

I A B AB y 1 -1 -1 1 2 1 1 -1 -1 2 1 -1 1 -1 1 1 1 1 1 3

C (b) Computers

Consider an M/M/∞ system with customer arriving rate of 2 per second and service time of 0.25 second. Using a birth-death process model, draw a state transition diagram for the system. What are the average number of customers and mean response time of the system? What is the probability that there are five customers in the system?

C (c) Computers

A processor that implements speculation in hardware typically has better performance than a processor without hardware speculation. Explain why and in your explanation, include an assembly language instruction sequence that definitively shows the advantages of speculation. State any assumptions you make with respect to issue width, number of execution units, and instruction execution latencies.

C (d) Computers

What two changes in the design of a computer architecture would allow the CPI to be decreased? Note that the ISA stays the same. Indicate why these changes allow the CPI to decrease.

a. Design change 1 to decrease CPI:

b. Why this can result in decreased CPI:

c. Design change 2 to decrease CPI:

d. Why this can result in decreased CPI:

 

C (e) Computers

Consider a cache of size 1MB with cache blocks of size 128 bytes. a. Suppose that this cache is a direct-mapped cache. Given that the least-significant bit of the 32-bit address is bit 0, indicate the bits that represent

i. the byte offset in the word ii. the index, used to address the cache iii. the tag, which differentiates between the memory blocks that can be resident in the set

b. Now consider a cache of size 1MB with cache blocks of size 128 bytes. But suppose that this cache is a two-way set-associative cache. Given that the least- significant bit of the 32-bit address is bit 0, indicate the bits that represent

i. the byte offset in the word ii. the index, used to address the cache iii. the tag, which differentiates between the memory blocks that can be resident in the set

C (f) Computers

Find the largest value of a > 0 such that G s( ) = a

s2 + 12 s + 2

has G ∞ ≤ 1 .

D (a) Control Systems

Given the nonlinear system

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

sin 2 x2 − x1 + π4( )( )

cos 2 x2 − x1 + π4( )( ) −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a) Determine any equilibrium point for the system. b) Using Lyapunov’s Indirect method, determine if the nonlinear system is stable.

D (b) Control Systems

Given the LTV discrete-time system

x t +1( ) =2 −3 01 − 3

2 08 − 2t 2t+1 −16 8 − 2t

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥x t( ) +

74

1+ −12( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

u t( )

choose the reachable set on [0,4) from among the following.

a) span74

1+ −12( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

,21

8 −12( )t − −1( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

b) span210

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,112

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

c) span210

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,7478

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

d)

span210

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,7478

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,

72

−74

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪∈3

e) span7478

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,7412

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

f)

span74

1+ −12( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

,21

8 −12( )t − −1( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

,112

−2( )t −16 −1( )t + 64 −12( )t

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

∈3

D (c) Control Systems

Problem A

Consider the digital filtering system shown below with sampling rate fs = 1000 Hz.

x(t) y(t)x(n) y(n)Ideal

Sampling

Ideal

ReconstructionH(z)

Let

x(t) = cos(250πt), (1)

and

H(z) =1

1− 12z−1

, |z| > 1/2. (2)

Determine the output y(t).

E (a) Digital Signal Processing

Problem B

Listed below are six combinations of causality/stability system properties

(i) BIBO stable and causal (ii) BIBO stable and anti-causal(iii) BIBO stable and mixed (iv) not BIBO stable and causal(v) not BIBO stable and anti-causal (vi) not BIBO stable and mixed

where BIBO stands for “Bounded Input, Bounded Output” and mixed implies the systemhas both causal and anti-causal parts.

For each system function and ROC, determine which combination(s) apply.

(a) H1(z) = 11− 1

2z−1 , |z| < 1

2.

(b) H1(z) = 11− 1

2z−1 , |z| > 1

2.

(c) H2(z) = 11−2z−1 , |z| < 2.

(d) H2(z) = 11−2z−1 , |z| > 2.

(e) H3(z) = 1(1− 1

2z−1)(1−2z−1)

, |z| < 12

(f) H3(z) = 1(1− 1

2z−1)(1−2z−1)

, |z| > 2

E (b) Digital Signal Processing

Problem C

Let the impulse response of a discrete-time, linear, shift-invariant system be given by

h [n] = βn

∞∑k=0

δ [n− kP ]

where 0 < β < 1 and real-valued and P is a positive integer.

(a) Sketch the impulse response (be sure to carefully label the important values).

(b) Determine a rational expression for the system function, H(z). Include a Region ofConvergence (ROC). The following relation may be useful:

∞∑k=0

αk =1

1− α, |α| < 1.

(c) Determine the difference equation of the system.

(d) Sketch a Direct Form I implementation of the system.

E (c) Digital Signal Processing

  

Figure 1 shows a single-line diagram of a three-phase, 60-Hz synchronous generator, connected through a transformer and parallel transmission lines to an infinite bus. All reactances are given in per-unit on a common system base.

a. If the infinite bus receives 1 per unit real power at 0.9 p.f. leading, determine the equation for the electrical power delivered by the generator versus its power angle δ.

b. The generator is initially operating in the stead-state condition when a permanent three-phase-to-ground bolted short circuit fault occurs at point F. The fault is then cleared by opening circuit breakers B13 and B22. These circuit breakers then remain open. Calculate the critical clearing angle.

Figure 1: Single-line diagram for transient stability analysis

F (a) Electric Energy Systems

 

The figure below shows a representative power system with conventional notations. Base MVA is 100 MVA, 230 kV in the transmission line. Transformers are rated 15 kV/ 230 kV.

1) Assume a line-to-ground fault occurs on bus 2. Answer the following two questions:

a) Find the total fault current in Amperes. b) Find the current through phase-A of generator G1 in Amperes.

2) What is the symmetrical short-circuit MVA at bus 2?

F (b) Electric Energy Systems

 

In the system shown, the voltage source on the left is an ideal, wye‐grounded, 60 Hz, three‐phase voltage source with line to ground voltages Vag=2400/0o, Vbg= 2440/120o and Vcg= 2400/‐120o V. 

The transformer is a wye‐grounded – delta, 500 KVA,60 Hz,  three‐phase, 4160 YG ‐480V Delta, with positive, negative and zero sequence impedances of  j0.05 pu each.  

The load on the right is a balanced, wye‐ungrounded, resistive load consisting of 27.7 ohm resistors per phase. 

 

 

A. Calculate the line currents on the high‐ and the low‐voltage voltage sides of the transformer 

B. Knowing that the source on left is likely to be unbalanced, would you recommend the wye‐grounded –delta transformer connection for serving the load?  Explain why or why not. 

F (c) Electric Energy Systems

G (a) Electromagnetics

G (b) Electromagnetics

G (c) Electromagnetics

Consider a planar, Lambertian, circular optical source of radius R and radiance L. A detector is a distance d from the source. The detector lies on a normal vector from the center of the source. Suppose a flux of φ is collected by the detector at the distance d from the source. Now suppose the detector moves along the center normal vector, either toward or away from the source. At what distance from the source will the collected flux to drop to φ /2 ?

d Detector R

L

H (a) Photonics

Consider the following specifications for the design of an imaging system with a single lens followed by a detector:

• Radiance of the source is L = 1 mW/cm2-sr • The source is a long distance from your system (relative to the lens focal length) • The detector diameter is 1 mm • The field-of-view (full angle) of the system should be at least 1 degree • The irradiance on the detector from the source must be greater than 3x10-5 W/cm2

(a) What is the constraint for the lens focal length? (b) What is the constraint for the system’s f/# (image space)? (c) What is the constraint for the lens diameter?

H (b) Photonics

Using known transform pairs and theorems, find the Fourier transforms of the following:

(a) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

wy

wx

2rect

2rect

(b) ⎟⎠⎞

⎜⎝⎛⎟

⎠

⎞⎜⎝

⎛ −wy

wxx

2rect

2rect 0

(c) ⎟⎟⎠

⎞⎜⎜⎝

⎛ +− 2

22

expw

yx

Perform the following convolutions by applying the convolution theorem:

(d) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛⊗⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

wy

wx

wy

wx

2rect

2rect

2rect

2rect

(e) ( ) ( )yxyx sinc4

sincsinc2

sinc ⎟⎠⎞

⎜⎝⎛⊗⎟

⎠⎞

⎜⎝⎛

Find the autocorrelation of the following:

(f) ⎟⎟⎠

⎞⎜⎜⎝

⎛ +− 2

22

expw

yxπ

H (c) Photonics

A circular aperture of radius w (shown below) is illuminated by a monochromatic plane wave of wavelength λ. (a) Find an expression for the Fraunhofer pattern (electric field) for this arrangement. (b) Find an expression for the Fraunhofer intensity pattern. (c) Suppose the aperture is actually the pupil function of a lens of focal length f. Write an expression for the intensity pattern of the light at the focus (focal plane) of the lens. (d) Derive an expression for the diameter of the central lobe for the pattern in part (c). The central lobe is defined by the first “zero” ring in the intensity pattern. Note that J1(1.22π) = 0.

w

H (d) Photonics

(a) Derive the ABCD matrix (ray transfer matrix) for the interface between the two materials shown below.

(b) Find the ray transfer matrix for the lens

H (e) Photonics

A TEM0,0 beam has a farfield divergence angle of 1 miliradian at λ0=633nm. (a) Find the minimum spot size (w0) (b) What is the maximum distance that this beam can travel before the spot size is

1cm? (c) What is the amplitude of the electric field at r=0 and z=0?

H (f) Photonics