Cache capacity: 16 bytes Associativity: 2 way set associative

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1 EE 26 Qualification Exam Question, January 208 (page ) Problem Caches (5 pts) A memory hierarchy with two levels of inclusive cache (L and L2) is running a test application accesses the following memory addresses (8 bit addresses), shown in hex: 0x2, 0x6, 0xE, 0x3, 0x22, 0x6, 0x3, 0x27, 0xE The L cache has the following parameters: Word size one byte Block size 2 bytes Cache capacity: 8 bytes Associativity: direct mapped Physical address size: 8 bits Write back policy Hit time cycle (clock rate 2 Ghz) The L2 cache has the following parameters: Word size one byte Block size 2 bytes Cache capacity: 6 bytes Associativity: 2 way set associative Physical address size: 8 bits Write back policy Cache Eviction Policy: LRU Hit time: 0 cycle (clock rate 2 Ghz) Miss penalty to memory 00 cycles a.) Calculate the size of the Tag, index and offset fields in bits. Draw a table that translates each address for each cache into binary digits and shows the separate tag, index, and offset fields. (3pts) b) Record each access as a hit or miss in the table above. Draw a diagram showing the final contents of the data and tag portion of both caches. The caches are initially empty. (5pts) c) What is the Miss rate for the L cache? What is the miss rate for the L2 cache? Calculate the average memory access time (AMAT) assuming the access pattern above. (3pt) d) Describe the type of misses observed in each cache (pt) e) A colleague recommends replacing the direct mapped cache in the L with a set associative cache. Will this improve memory performance? What are the downsides and costs of this change? (2pts) f) What problem should we be concerned with if a second processor was added to the system with its own private L and the processor shares the L2 cache? Describe the additional protocol that would need to be implemented. (pt)

2 EE 26 Qualification Exam Question, January 208 (page 2) Problem 2 Pipelining. (0 points) Imagine a new pipeline architecture with two different ALUs available: one ALU is used for integer instructions and takes one cycle (EX), the second ALU is used for a multiply (MULT) instruction and takes two cycles to execute (M, M2), a pipeline diagram is below. Besides the new M, M2 stages, the processor includes the following stages instruction fetch (IF), instruction decode (ID), memory access (MEM), and write back (WB). The system includes forwarding logic from the (EX or M2)/MEM or MEM/WB registers to the input of either ALU. The processor supports simple static branch prediction by always predicting not taken. Branches are resolved in the EX stage, speculative state is flushed. All registers are loaded at the start of the code segment and instructions issue in program order. IF ID ALU MEM WB M M2 LW $t, 0($s0) MULT $t, $t, x08a LW $t2, 4($s0) SUB $t, $t, $t2 MULT $t, $t2, x00 ADD $t, $t2, $t a) How long, in cycles, will it take to complete the code segment on the MIPS processor above? Draw a pipeline diagram and indicate all stalls. (4 pts) b) The forwarding logic will be used pretty extensively. In your pipeline diagram draw an arrow to indicate each instance of forwarding. List each instance of forwarding in a table like the one below, circle the register. Note: the example below is not found in the code. (2pts) Source Instruction and RegisterID Destination Instruction and RegisterID Example SUB, $t0, $t, $t2 ADD, $t4, $t0, $t5 c) As you can see the addition of a second ALU has created an additional structural hazard. Please describe briefly the new hazard check that your pipeline needs. (2pt) d) As you can see your pipeline still has significant stalls due to both data hazards and structural hazards. One way to solve this problem is to expose more instruction level parallelism through the use of loop unrolling. Unroll the loop two times and reschedule the loop to reduce stalls. (2pt)

3 EE Qualifier Problem: Note: if you do not have an equation make an attempt to derive it. Otherwise, indicate how you would solve the problem if you had the information you are missing and describe best you can what the answer should be, form and function. Unless otherwise notes, assume the relative permeability of a material is one. ) It is always crucial to know when an equation applies before using it; otherwise you cannot trust the solutions it provides. For example, using an equation that is only valid in a vacuum to solve for charges flowing through a semiconductor will give you radically incorrect electromagnetic fields. Below is one of many possible versions of Maxwell s Equations, from which virtually every other equation in electromagnetics can be derived. There are at least a dozen assumptions that must be made for these equations to be valid. Please name and describe six of the assumptions implicit in this version of Maxwell s Equations. 2) Many BNC cables are made of the copper wires coaxially jacketed with Kapton followed by an aluminum grounding mesh, but for simplicity treat the mesh as a solid layer. If the copper wire is mm in diameter and the Kapton and aluminum are both mm thick. Using the facts that the dielectric constant of Kapton is 4 and the resistivity of copper, aluminum, and Kapton are ~2E-8 ohmcm, ~2.5E-9 and E7, respectively, answer the following questions: a. What are the lumped element circuit model components (R, L, G, & C) for a unit length of the BNC cable? Solve for their numerical values b. What is the skin depth for copper at GHz? Assume copper is a good conductor. At what frequency would our wire s resistivity double? c. What is the characteristic impedance of this line at GHz? If the cable is m long and terminated with a 50 ohm line, how much power is dissipated in the 50 ohm load? d. What would the impedance of a shorted-stub need to be to impedance match your line to the 50 ohm load?

4 Semiconductor 208 Qualifier Problem: Note: if you do not have an equation make an attempt to derive it. Otherwise, indicate how you would solve the problem if you had the information you are missing and describe best you can what the answer should be, form and function. ) You want to build a silicon diode. You start off using high grade silicon (τ ~0-3 sec). Using the properties listed to the right answer the following questions: a. The data to the right lists mobilities for holes and electrons. Are these valid for all values of doping? In either case, explain the mechanism that limits mobility, both giving the number listed and properties that may reduce it below that. b. For a diode doping of 0 8 and 0 5 on the given sides. i. Which type of doping is it optimal to have larger if you want to collect photons? Why? ii. Is all of the dopant activated? iii. What are the positions of the Fermi levels on each side (p-type and n-type) c. For the diode described in (b), assuming the top portion of the diode is thin (~ diffusion length) and is the higher doped, calculate the following: i. Draw the band diagram for the diode. On the drawing, label the P, N, and depletion regions; as well as, the built in voltage. ii. PROPERTY VALUE UNITS Lattice spacing (a 0 ) at 300K nm Density at 300K g/cm 3 Nearest Neighbour Distance at 300K nm Number of atoms in cm Effective density of states (conduction, N c T=300 K ) Effective density of states (valence, N v T=300 K ) Electron affinity 2.8x0 9.04x cm -3 cm -3 kj / mol Energy Gap E g at 300 K.2 ev (Minimum Indirect Energy Gap at 300 K) Energy Gap E g at ca. 0 K.7 (at 0 K) ev (Minimum Indirect Energy Gap at 0K) Minimum Direct Energy Gap at 300 K 3.4 ev Energy separation (E ΓL ) 4.2 ev Intrinsic Debye length 24 um Intrinsic carrier concentration 0 0 cm -3 Intrinsic resistivity Ω cm Auger recombination coefficient C n cm 6 / s Auger recombination coefficient C p cm 6 / s Mobility electrons 400 cm 2 / (V x s) Mobility holes 450 cm 2 / (V x s) Diffusion coefficient electrons 36 cm 2 /s Diffusion coefficient holes 2 cm 2 /s Electron thermal velocity m/s Electronegativity.8 Pauling` s Hole thermal velocity m/s Optical phonon energy ev Density of surface atoms (00) /cm 2 (0) /cm 2 () /cm 2 Work function (intrinsic) 4.5 ev Donors Sb ev P ev As ev Ionization Energies for Various Dopants Acceptors B Al Ga In 0.6 The size of the space charge region and the respective size on each side of the mechanical junction. Label each part on the drawing The built-in voltage iii. iv. The saturation current v. Sketch the IV curve. Label the knee voltage compared to the built-in voltage and the saturation current ev ev ev ev

5 EE07 Communications Systems Qualifying Exam 208 (2 pages including Q table) Question. (2.5 points) An M-QAM signal is generated as in the figure below. The pulse shaping filter is a raised cosine with a 50% roll-off factor and the carrier frequency is f c = 700 khz. a) Suppose that the bit rate is 20 Kbps and M = 6. Determine the spectrum of the modulated signal and sketch it. Clearly label all important frequencies. b) Draw the block diagram illustrating the optimum demodulator/detector for the received signal, which is equal to the transmitted signal plus additive white Gaussian noise. Clearly specify the impulse response of the filter used, if any, and the decision rule for detection. Question 2. (2.5 points) a) Four analog sensor signals are sampled, quantized, encoded and multiplexed into digital data. Each signal has a bandwidth of 200 Hz and is sampled at 25% higher than Nyquist frequency. The quantizer has 256 levels. Eight bits of synchronization and coding are added for each multiplexed frame. Sketch a diagram of the system and compute the resulted bit rate. b) The resulting data of part a) is transmitted over an AWGN channel using binary PAM signals with bipolar rectangular pulses of amplitude 0.5V. Assume that input bits and 0 occur with equal probabilities. If the power spectral density of the additive Gaussian noise is N0/2, where N0 = 70 db W/Hz, compute the average error probability for this receiver.

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7 ES4 PHD Qualifying exam Question ) Using a Kmap, minimize the following function. F(w,x,y,z)= (3,4,5,6,7,) + d (0,,8,5) 2) Using a Kmap, minimize the following function in POS format. F(w,x,y,z)= (0,,4,8,9,3,4) 3) Implement the following function using ONLY NOR Gates. The inverters must be shown implemented as NOR gates as well. F(x,y,z)= X+ X YZ + Z

8 Question 6) a) Using a 6x Multiplexor, implement the following function. You must show the mux implementation table and you must use variable C on the input lines. F(A,B,C,D,E) = (3,0,5,20,24,25,26,30,3). Clearly label all inputs and outputs and MSB/LSB on the device. 6 x Mux

9 Probability and Statistics In this problem we model a machine that makes 00 widgets per day. On any given day, the machine is either working properly or is broken. A properly working machine produces faulty widgets with probability /0 while a broken machine produces faulty widgets with probability /2. On any given day, the probability that the machine is broken is. You can assume that the state of the machine on 00 any given day is independent from one day to the next.. Given that the machine is working properly, what is the probability mass function for the number of widgets that will be produced before the next faulty widget? 2. Given that the machine is working properly, and the last 2 widgets are all good, what is the probability that at least three of the next four widgets will also be good? 3. What is the probability mass function for the number of faulty widgets produced on any given day? Let DD be the number of days in a given week when the machine is working properly and NN the number of good widgets produced in a week. 4. What is the moment generating function NN given DD? 5. What is the expected number of good widgets produced in a week?

10 PhD Qualifier Question EE 05 Feedback Control Systems January 208 Consider the following state-space: ẋ = x u, () y = [ 0 0 ] x. (2) Answer the following questions: () Characterize the asymptotic behavior of this system. (2) Is the system controllable? (3) Find the transfer function of this system. (4) What are the poles and zeros of this system? (5) Design a feedback controller to place the closed-loop poles at,, 6.

11 Qualifying Examination EE28 -- Operating Systems ) Given a computer and a device, what are the conditions under which one would need a driver for the device? Under what conditions is a driver unnecessary? Why? 2) A scheduling algorithm is "fair" if all processes at the same priority get roughly the same amount of CPU time. Is round-robin scheduling fair? Why or why not? 3) Describe the difference between system and user time, and give examples of computations that consume one kind of time but not the other.

12 Tufts University Department of Electrical and Computer Engineering EE 23 - Linear Systems ECE PhD Qualifying Exam 208 January 208 PhD Qualifying Exam ID Number : Honor Code: This exam represents only my own work. I did not give or receive help on this exam. Instructions: Please do not turn this page until told to do so. Total time allowed for this test is 20 min. You should concisely indicate your reasoning and show all relevant work for each problem. Your score will be based on an evaluation of your understanding as reflected by what you have written for an answer. All work you want graded must go in the exam booklet provided. Use the extra sheets provided for scratch work only. There are several sheets of formulas and properties of transforms attached with this exam, which you may use to solve the problems.

13 Linear Systems PhD Quals Spring 208 Problem [50 pts] Answer the following questions. Note that no points will be awarded unless correct reasoning is provided.. State TRUE or FALSE (with full reasoning): Series interconnection of two Linear and Time Invariant (LTI) system is itself LTI system. 2. State TRUE or FALSE (with full reasoning): The series interconnection of two nonlinear systems is nonlinear. 3. Consider three systems with the following input output relationship. (a) System : y[n] = { x[n/2] if n is even 0 if n is odd () (b) System 2: y[n] = x[n] + /2x[n ] + /4x[n 2] (c) System 3: y[n] = x[2n] Suppose these systems are connected in series like so x[n] System System 2 System 3 y[n] Find the input output relationship for this overall system. Is this system linear? Is this system Time Invariant?

14 2 Linear Systems PhD Quals Spring 208 Problem :

15 Linear Systems PhD Quals Spring Problem 2 [50 pts]. Duality property of the Continuous Time Fourier Transform (CTFT). Suppose the signal x(t) has a CTFT X(ω). Then what is the CTFT of the signal y(t) = X(t)? 2. Evaluate the CTFT of x(t) = e t, where denotes the absolute value. 3. Using the results of the previous two parts, evaluate the CTFT of +t 2.

16 4 Linear Systems PhD Quals Spring 208 Problem 2:

17 Table : Properties of the Continuous-Time Fourier Series a k = T x(t) = T k= a k e jkω 0t = x(t)e jkω 0t dt = T k= T jk(/t )t a k e x(t)e jk(/t )t dt Property Periodic Signal Fourier Series Coefficients x(t) y(t) } Periodic with period T and fundamental frequency ω 0 =/T a k b k Linearity Ax(t)+By(t) Aa k + Bb k Time-Shifting x(t t 0 ) a k e jkω 0t 0 = a k e jk(/t )t 0 Frequency-Shifting e jmω0t = e jm(/t )t x(t) a k M Conjugation x (t) Time Reversal x( t) a k a k Time Scaling x(αt), α > 0(periodicwithperiodT/α) a k Periodic Convolution x(τ)y(t τ)dτ Ta k b k Multiplication Differentiation Integration Conjugate Symmetry for Real Signals Real and Even Signals T x(t)y(t) dx(t) dt t x(t)dt x(t) real x(t) realandeven (finite-valued and periodic only if a 0 =0) l= a l b k l jkω 0 a k = jk T a k ( ) ( a k = jkω 0 a k = a k Re{a k } = Re{a k } Im{a k } = Im{a k } a k = a k <) a k = <) a k a k real and even jk(/t ) Real and Odd Signals x(t) realandodd a k purely imaginary and odd ) a k Even-Odd Decomposition of Real Signals { xe (t) =Ev{x(t)} [x(t) real] x o (t) =Od{x(t)} [x(t) real] Parseval s Relation for Periodic Signals x(t) 2 dt = a k 2 T T k= Re{a k } jim{a k }

18 Table 2: Properties of the Discrete-Time Fourier Series x[n] = a k e jkω0n = jk(/n )n a k e a k = N n=<n> k=<n> x[n]e jkω 0n = N k=<n> n=<n> jk(/n )n x[n]e Property Periodic signal Fourier series coefficients x[n] y[n] } Periodic with period N and fundamental frequency ω 0 =/N a k b k } Periodic with period N Linearity Ax[n]+By[n] Aa k + Bb k Time shift x[n n 0 ] a k e jk(/n )n 0 Frequency Shift e jm(/n )n x[n] a k M Conjugation x [n] Time Reversal x[ n] a k a k Time Scaling x (m) [n] = Periodic Convolution Multiplication { x[n/m] if n is a multiple of m 0 if n is not a multiple of m (periodic with period mn) x[r]y[n r] r= N x[n]y[n] m a k Na k b k l= N ( viewed as periodic with period mn a l b k l First Difference x[n] x[n ] ( e jk(/n ) )a k n ( ) ( ) finite-valued and Running Sum x[k] a periodic only if a 0 =0 ( e jk(/n ) k ) k= a k = a k Re{a k } = Re{a k } Conjugate Symmetry x[n] real Im{a for Real Signals k } = Im{a k } a k = a k <) a k = <) a k Real and Even Signals x[n] realandeven a k real and even Real and Odd Signals x[n] realandodd a k purely imaginary and odd ) Even-Odd Decomposition of Real Signals x e [n] =Ev{x[n]} x o [n] =Od{x[n]} [x[n] real] [x[n] real] Re{a k } jim{a k } Parseval s Relation for Periodic Signals x[n] 2 = a k 2 N n= N k= N

19 Table 3: Properties of the Continuous-Time Fourier Transform x(t) = X(jω) = X(jω)e jωt dω x(t)e jωt dt Property Aperiodic Signal Fourier transform x(t) y(t) X(jω) Y (jω) Linearity ax(t)+by(t) ax(jω)+by (jω) Time-shifting x(t t 0 ) e jωt 0 X(jω) Frequency-shifting e jω0t x(t) X(j(ω ω 0 )) Conjugation x (t) X ( jω) Time-Reversal x( t) X( jω) ( ) jω Time- and Frequency-Scaling x(at) a X a Convolution x(t) y(t) X(jω)Y (jω) Multiplication x(t)y(t) X(jω) Y (jω) d Differentiation in Time dt x(t) jωx(jω) t Integration x(t)dt jω X(jω)+πX(0)δ(ω) Differentiation in Frequency tx(t) j d dω X(jω) X(jω) =X ( jω) Re{X(jω)} = Re{X( jω)} Conjugate Symmetry for Real x(t) real Im{X(jω)} = Im{X( jω)} Signals X(jω) = X( jω) <) X(jω) = <) X( jω) Symmetry for Real and Even x(t) realandeven X(jω) realandeven Signals Symmetry for Real and Odd x(t) realandodd X(jω) purelyimaginaryandodd Signals Even-Odd Decomposition for x e (t) =Ev{x(t)} [x(t) real] Re{X(jω)} Real Signals x o (t) =Od{x(t)} [x(t) real] jim{x(jω)} Parseval s Relation for Aperiodic Signals + x(t) 2 dt = + X(jω) 2 dω

20 Table 4: Basic Continuous-Time Fourier Transform Pairs Fourier series coefficients Signal Fourier transform (if periodic) k= e jω 0t a k e jkω 0t k= δ(ω ω 0 ) a k δ(ω kω 0 ) cos ω 0 t π[δ(ω ω 0 )+δ(ω + ω 0 )] sin ω 0 t x(t) = Periodic { square wave, t <T x(t) = T 0, T < t 2 and x(t + T )=x(t) π j [δ(ω ω 0) δ(ω + ω 0 )] δ(ω) k= T 2sinkω 0 T δ(ω kω 0 ) k ( δ ω k ) T a k a = a k =0, otherwise a = a = 2 a k =0, otherwise a = a = 2j a k =0, otherwise a 0 =, a k =0,k 0 ( ) this is the Fourier series representation for any choice of T>0 ω 0 T π δ(t nt ) a k = T n= { k=, t <T 2sinωT x(t) 0, t >T ω { sin Wt, ω <W X(jω) = πt 0, ω >W δ(t) u(t) jω + πδ(ω) δ(t t 0 ) e jωt 0 e at u(t), Re{a} > 0 te at u(t), Re{a} > 0 t n (n )! e at u(t), Re{a} > 0 a + jω (a + jω) 2 (a + jω) n ( ) kω0 T sinc π for all k = sin kω 0T kπ

21 Table 5: Properties of the Discrete-Time Fourier Transform x[n] = X(e jω )e jωn dω X(e jω )= n= x[n]e jωn Property Aperiodic Signal Fourier transform x[n] } X(e jω ) Periodic with y[n] Y (e jω ) period Linearity ax[n]+by[n] ax(e jω )+by (e jω ) Time-Shifting x[n n 0 ] e jωn 0 X(e jω ) Frequency-Shifting e jω0n x[n] X(e j(ω ω0) ) Conjugation x [n] X (e jω ) Time Reversal x[ n] { X(e jω ) x[n/k], if n =multipleofk Time Expansions x (k) [n] = X(e jkω ) 0, if n multipleofk Convolution x[n] y[n] X(e jω )Y (e jω ) Multiplication x[n]y[n] X(e jθ )Y (e j(ω θ) )dθ Differencing in Time x[n] x[n ] ( e jω )X(e jω ) n Accumulation x[k] e jω X(ejω ) k= +πx(e j0 ) k= δ(ω k) Differentiation in Frequency nx[n] j dx(ejω ) dω X(e jω )=X (e jω ) Re{X(e jω )} = Re{X(e jω )} Conjugate Symmetry for x[n] real Im{X(e jω )} = Im{X(e jω )} Real Signals X(e jω ) = X(e jω ) <) X(e jω )= <) X(e jω ) Symmetry for Real, Signals Even Symmetry for Real, Odd Signals Even-odd Decomposition of Real Signals x[n] realandeven x[n] realandodd x e [n] =Ev{x[n]} x o [n] =Od{x[n]} [x[n] real] [x[n] real] X(e jω )realandeven X(e jω )purely imaginary and odd Re{X(e jω )} jim{x(e jω )} Parseval s Relation for Aperiodic Signals x[n] 2 = X(e jω ) 2 dω n=

22 Table 6: Basic Discrete-Time Fourier Transform Pairs Fourier series coefficients Signal Fourier transform (if periodic) k= N e jω 0n cos ω 0 n sin ω 0 n x[n] = jk(/n )n a k e Periodic { square wave, n N x[n] = 0, N < n N/2 and x[n + N] =x[n] k= δ[n kn] a n u[n], a < x[n] π π j k= l= l= l= l= k= {, n N sin[ω(n + 2 )] 0, n >N sin(ω/2) ) sin Wn πn = W π sinc ( Wn π 0 <W <π ( a k δ ω k ) N a k (a) ω 0 = { m N, k = m, m ± N,m ± 2N,... δ(ω ω 0 l) a k = 0, otherwise ω (b) 0 irrational The signal is aperiodic (a) ω 0 = { m N {δ(ω ω 0 l)+δ(ω + ω 0 l)} a k = 2, k = ± m, ± m ± N,± m ± 2N,... 0, otherwise ω (b) 0 irrational The signal is aperiodic (a) ω 0 = r N 2j, k = r, r ± N,r ± 2N,... {δ(ω ω 0 l) δ(ω + ω 0 l)} a k = 2j, k = r, r ± N, r ± 2N,... 0, otherwise ω (b) 0 irrational The signal is aperiodic {, k =0, ± N,± 2N,... δ(ω l) a k = 0, otherwise ( a k δ ω k ) N ( δ ω k ) N a k = N N k= ae jω {, 0 ω W X(ω) = 0, W < ω π X(ω)periodic with period δ[n] u[n] e jω + πδ(ω k) k= δ[n n 0 ] e jωn 0 (n +)a n u[n], a < ( ae jω ) 2 (n + r )! a n u[n], a < n!(r )! ( ae jω ) r a k = sin[(k/n)(n+ 2 )] N sin[k/2n], k 0, ± N,± 2N,... a k = 2N+ N, k =0, ± N,± 2N,... for all k

23 Table 7: Properties of the Laplace Transform Property Signal Transform ROC x(t) X(s) R x (t) X (s) R x 2 (t) X 2 (s) R 2 Linearity ax (t)+bx 2 (t) ax (s)+bx 2 (s) At least R R 2 Time shifting x(t t 0 ) e st 0 X(s) R Shifting in the s-domain e s0t x(t) X(s s 0 ) Shifted version of R [i.e., s is in the ROC if (s s 0 )isin R] Time scaling x(at) ( s ) a X a Conjugation x (t) X (s ) R Scaled ROC (i.e., s is in the ROC if (s/a) is in R) Convolution x (t) x 2 (t) X (s)x 2 (s) At least R R 2 Differentiation in the Time Domain Differentiation in the s-domain Integration in the Time Domain t d x(t) dt sx(s) At least R tx(t) d ds X(s) R x(τ)d(τ) X(s) s At least R {Re{s} > 0} Initial- and Final Value Theorems If x(t) =0fort<0andx(t) containsnoimpulsesorhigher-ordersingularitiesatt =0,then x(0 + )=lim s sx(s) If x(t) =0fort<0andx(t) hasafinitelimitast,then lim t x(t) =lim s 0 sx(s)

24 Table 8: Laplace Transforms of Elementary Functions Signal Transform ROC. δ(t) All s 2. u(t) s Re{s} > 0 3. u( t) s Re{s} < 0 4. t n (n )! u(t) s n Re{s} > 0 5. tn (n )! u( t) s n Re{s} < 0 6. e αt u(t) 7. e αt u( t) 8. t n (n )! e αt u(t) 9. tn (n )! e αt u( t) s + α Re{s} > Re{α} s + α Re{s} < Re{α} (s + α) n Re{s} > Re{α} (s + α) n Re{s} < Re{α} 0. δ(t T ) e st All s. [cos ω 0 t]u(t) 2. [sin ω 0 t]u(t) 3. [e αt cos ω 0 t]u(t) s s 2 + ω 2 0 ω 0 s 2 + ω0 2 s + α (s + α) 2 + ω 2 0 ω 0 Re{s} > 0 Re{s} > 0 Re{s} > Re{α} 4. [e αt sin ω 0 t]u(t) (s + α) 2 + ω0 2 Re{s} > Re{α} 5. u n (t) = dn δ(t) dt n s n All s 6. u n (t) =u(t) u(t) }{{} s n Re{s} > 0 n times

25 Table 9: Properties of the z-transform Property Sequence Transform ROC x[n] X(z) R x [n] X (z) R x 2 [n] X 2 (z) R 2 Linearity ax [n]+bx 2 [n] ax (z)+bx 2 (z) At least the intersection of R and R 2 Time shifting x[n n 0 ] z n 0 X(z) R except for the possible addition or deletion of the origin Scaling in the e jω0n x[n] X(e jω 0 ( z) R z-domain z0x[n] n z X z 0 R z 0 ) a n x[n] X(a z) Scaled version of R (i.e., a R =the set of points { a z} for z in R) Time reversal x[ n] X(z ) Inverted R (i.e., R =thesetofpoints { z where z is in R) x[r], n = rk Time expansion x (k) [n] = X(z 0, n rk ) R /k for some integer r (i.e., the set of points z /k Conjugation x [n] X (z ) R where z is in R) Convolution x [n] x 2 [n] X (z)x 2 (z) At least the intersection of R and R 2 First difference x[n] x[n ] ( z )X(z) At least the intersection of R and z > 0 Accumulation n k= x[k] z X(z) At least the intersection of R and z > Differentiation nx[n] z dx(z) dz in the z-domain R Initial Value Theorem If x[n] =0forn<0, then x[0] = lim z X(z)

26 Table 0: Some Common z-transform Pairs Signal Transform ROC. δ[n] All z 2. u[n] 3. u[ n ] z z > z z < 4. δ[n m] z m All z except 0(ifm>0) or (if m<0) 5. α n u[n] αz 6. α n u[ n ] αz 7. nα n u[n] αz ( αz ) 2 8. nα n u[ n ] αz ( αz ) 2 z > α z < α z > α z < α 9. [cos ω 0 n]u[n] 0. [sin ω 0 n]u[n]. [r n cos ω 0 n]u[n] 2. [r n sin ω 0 n]u[n] [cos ω 0 ]z [2 cos ω 0 ]z +z 2 z > [sin ω 0 ]z [2 cos ω 0 ]z +z 2 z > [r cos ω 0 ]z [2r cos ω 0 ]z +r 2 z 2 z >r [r sin ω 0 ]z [2r cos ω 0 ]z +r 2 z 2 z >r

27 ES3/EE2 Circuit Theory Problem : Op-Amp Circuits Consider the Op-Amp circuit shown in Figure 2 with two ideal op-amps. The resistor values are given as R = 50Ω, R2 = 2.5kΩ, R3 = 0kΩ, R4 = 300Ω, R5 = 00Ω Figure 2. Op-Amp Circuit (a) Discuss briefly the type of feedback for this circuit. Specifically describe the relative increase or decrease in the voltage at the non-inverting terminal of the first op-amp when the input voltage, Vin, is increased. (b) Write an expression for the output voltage of the first op-amp, Vx, and the second op-amp, Vout, in terms in the input voltage, Vin. Vx = Vout =

28 Problem 2: Active Filter Circuits Your goal is to design a first-order active filter using an ideal operational amplifier, resistors and a capacitor to attenuate low-frequency bass signals from an audio recording. The filter should have the following specifications:. Passband gain of Attenuate a 0Hz signal by a factor of 00 relative to the high frequency components of the audio signal. (a) Sketch your circuit and label the input signal, Vaudio(t), and the filtered output signal, Vout(t). Label all component values and include units. (b) Write an expression for the transfer function for your circuit, H(ω). (c) Sketch the Bode diagram for the magnitude of the transfer function H(ω). Label two points on the bode diagram.

29 PhD Exam Analog Electronics January 208 MOSFET equations for this problem: II DD = kk[(vv GGGG VV TT )VV DDDD VV 2 DDDD 0 2 kk(vv GGGG VV TT ) 2 (+λλvv DDDD ) 2 ] For the MOSFETs in the current mirror circuit below, use k=0. ma/v 2 and V T = volt. Channel length modulation may be ignored during DC analysis. For AC analysis you should use λ = 0.0 v -. Part. Find the DC current through element X if the circuit is operating as a proper current mirror. State your assumptions clearly. Part 2. Find the small signal model and model parameters for the MOSFETs using the information provided. Part 3. Draw the Norton equivalent circuit of this current mirror as determined at the two terminals of element X. Label the numerical values for the Norton current and Norton resistance. Part 4. Assume that the element X is a resistor. What are the upper and lower bounds of the resistance of X that allow the current mirror to function as intended? VDD = 5 volts, R = 00kΩ

30 ECE PhD Qualifying Exam Programming Question January 2, 208 The simplest way to implement a matrix is as two-dimensional array of floating point numbers. In C/C++, it might look something like this: float matrix [20][30]; There are two problems with this approach. First, two dimensional arrays only work when the dimensions are known at compile time. Second, for some applications the matrices are very sparse that is, many of the entries are zero. Using the array representation above wastes a lot of space storing zero values. In this series of problems we will ask you to explore alternative representations of matrices that solve these problems. () Using the following struct definition, write a function to create an N by M matrix, where N and M are parameters to the function. For this problem, assume all matrices are dense (i.e., make space for every possible element). struct Matrix { int N; int M; float * data ; }; Matrix * creatematrix ( int N, int M) (2) Write functions to get and set values in the matrix. Make sure you check the bounds on the indices. You can decide how you want to handle out-of-bounds indices; just don t allow memory corruption to happen. float getelement ( Matrix * m, int i, int j); void setelement ( Matrix * m, int i, int j, float value ); (3) Write a function to multiply two matrices together (C = A * B) void multiply ( Matrix * A, Matrix * B, Matrix * C); What is the running time of this algorithm? (4) A sparse matrix representation only stores the non-zero values, saving space when the matrix has many zero values. Design a new struct SparseMatrix to represent sparse matrices. There are a few different strategies you could consider. One is to make a linked list or vector of the non-zero

31 values and their locations in the matrix. Another is to make an array of pointers to linked lists that represents the matrix rows each entry of the array is a pointer to a list of the non-zero elements of that row. Note: the goal is to save space by not storing zero values. You will mostly likely need to define at least one other struct to represent the nodes in the list(s). Feel free to sketch out a picture of your representation to help guide your design. (5) Define a getelement function for sparse matrices. It should have the same interface as the one above, but take a SparseMatrix instead of a Matrix. Remember that any element not explicitly stored in the matrix is a zero (i.e., if getelement does not find the requested element it should return 0). What is the worst-cast running time of this function? (6) Can you use the algorithm defined in Problem (3) to multiply two sparse matrices together? If so, what is the running time for sparse matrix multiply? (7) Assuming that pointers, integers, and floating point numbers all occupy 8 bytes, how much space does a 0 by 0 matrix use is the dense representation? Assuming all values are non-zero (a worst-case scenario) how much space would a 0 by 0 sparse matrix use? (7) How many non-zero values does a 0 by 0 matrix need to have before using the dense representation uses less space than the sparse representation? (What is the threshold?) 2