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Course: ENEE 324, Fall 2009
School: Maryland
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Word Count: 239

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324 ASSIGNMENT ENEE 1 Due Tue 09/08 (All geographical information needed can be found on the Web. For part (iv), use the appropriate formula from spherical geometry.) A meteorite strikes Earth at a random point of latitude 1 and longitude 2 , both of which are measured in degrees. The range of 1 is [90, 90], where positive values denote north. The range of 2 is (180, 180], where positive values denote east. (i)...

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324 ASSIGNMENT ENEE 1 Due Tue 09/08 (All geographical information needed can be found on the Web. For part (iv), use the appropriate formula from spherical geometry.) A meteorite strikes Earth at a random point of latitude 1 and longitude 2 , both of which are measured in degrees. The range of 1 is [90, 90], where positive values denote north. The range of 2 is (180, 180], where positive values denote east. (i) (8 pts.) On the (1 , 2 ) rectangle (sample space), sketch each of the events A1 = point of impact is on the western hemisphere A2 = point of impact is on the same meridian as New York City A3 = point of impact is in the Arctic (i.e., north the of Arctic circle) A4 = point of impact is within twenty degrees (north or south) of the equator (ii) (4 pts.) For every pair of events dened in part (i), state whether the two events are disjoint or overlapping, and whether one event is subset of the other. (iii) (4 pts.) Similarly, determine whether the events A5 = point of impact is no farther than 2,000 miles from New York and A6 = point of impact is no farther than 2,000 miles from London are disjoint or overlapping, and whether one event is subset of the other. (iv) (4 pts.) Assuming that Earth is a perfect sphere, determine the fraction of its total surface represented by the event A4 .
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Maryland - ENEE - 324
ENEE 324ASSIGNMENT 2Due Thu 09/10A student enrolled in a degree program takes a total of thirty-two courses, indexed by i = 1, . . . , 32 and also spread evenly over eight semesters. Each course is graded on an A through D scale, and the events Ai , Bi
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 3Due Tue 09/15Problem 3.1 The outcome of a random experiment is a pair of integers (i, j ) in cfw_1, 92 . The probability mass function for the eighty-one outcomes is depicted below, where each symbol represents a dierent probability
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 4Due Thu 09/17A circuit is fabricated by one of three facilities, labeled i = 1, 2, 3. The fraction of circuits on the market produced by Facility i is i . The table below shows the probabilities (likelihoods) that a circuit manufact
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 5Due Tue 09/22Problem 5A In the diagram below, information ows from the transmitter T to receivers R1 and R2 via links controlled by switches 16. Switches operate independently of each other, and the probability that a given switch i
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 6Due Thu 09/24In any given morning, Ill have a mun with probability 0.5, a bagel with probability 0.3 and nothing at all with probability 0.2. My choices vary independently from day to day. (i) (4 pts.) Compute the probability that I
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 7Due Tue 09/29A point on the unit square [0, 1]2 is chosen at random according to a uniform probability assignment, i.e., the probability that it lies in a certain region (within the unit square) equals the area of that region. Let R
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 8Due Tue 10/06The probability density function (pdf) of a continuous random variable X is given by cx , fX (x) =0x1 cx2 , x &gt; 1 0, x&lt;0(i) (2 pts.) Sketch fX (x). What is the range of values of X ? (ii) (2 pts.) What is the value
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 9Due Thu 10/08The probability density function (pdf) of a continuous random variable X is given by fX (x) = cx2 ex/2 u(x) where u(x) is the unit step at x = 0 and c &gt; 0. (i) (3 pts.) Determine the value of x which maximizes fX (x). S
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 10Due Tue 10/13Problem 10A An appliance uses one of three circuits (i = 1, 2, 3). The lifetimes Xi (in thousands of hours) of the three circuits have Gaussian distributions: X1 N (6.0, (1.5)2 ), X2 N (6.5, 1.5)2 ) and X3 N (7.0, (1.5
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 11Due Thu 10/15Consider the transformation (function) y = g (x) = (i) (2 pts.) Sketch y = g (x). (ii) (6 pts.) In each of the following cases, determine the range SY (of values) of the output random variable Y = g (X ) based on the r
Maryland - ENEE - 324
ENEE 324 Problem 12AASSIGNMENT 12Due Tue 10/20Consider the transformation (function) g (x) = 0, x&lt;0 x, x 0(i) (2 pts.) If X is a continuous random variable with range SX = R, what is the range SY of Y = g (X )? Is the distribution of Y continuous, dis
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 13Due Thu 10/22Let X be an exponential random variable with = 4. (i) (2 pts.) Give a formula for E [X n ], where n is a positive integer. (ii) (4 pts.) Determine E [(X 2)4 ]. (iii) (5 pts.) Determine E [Y ], where Y = maxcfw_e2X , e2
Maryland - ENEE - 324
ENEE 324 Problem 14AASSIGNMENT 14Due Tue 10/27X and Y are random nonnegative integers such that P [X = i, Y = j ] = c 2ij , 0 i j ; 0, otherwise.(i) (4 pts.) Determine, in terms of c, P [X = i] and P [Y = j ] for every i 0 and j 0. (ii) (4 pts.) What
Maryland - ENEE - 324
ENEE 324 Problem 15AASSIGNMENT 15Due Tue 11/03X and Y are independent and identically distributed random variables with pdf fX (t) = fY (t) = 1/(t + 1)2 , t 0; 0, t &lt; 0.(i) (2 pts.) Sketch the region on the (x, y ) plane where the random pair (X, Y )
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 16Due Thu 11/05Consider the curve in the rst quadrant of the (x, y ) plane dened by y= 2 x, 0 x 1; 1/x2 , x &gt; 1.(i) (2 pts.) Sketch the region A in the rst quadrant bounded by the two positive axes (x and y ) and the curve dened abo
Maryland - ENEE - 324
ENEE 324 Problem 17A The density of X is given byASSIGNMENT 17Due Tue 11/10fX (x) =2x, 0 &lt; x &lt; 1; 0, otherwise.The conditional density of Y given X = x is given by fY |X (y | X = x) = cx2 /y 3 , y x; 0, otherwise.(i) (2 pts.) Shade the region on the
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 18Due Thu 11/12Let X1 , X2 , . . . be pairwise uncorrelated random variables with mean and variance 2 , i.e., Cov(Xi , Xj ) = 2, i = j ; 0, i = j .The sequence cfw_Xn , n 1 is the input to a time-varying causal lter which produces a
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 19Due Tue 11/17Let . . . , U0 , U1 , U2 , . . . be pairwise uncorrelated random variables with mean = 3 and variance 2 = 36. For every n, dene Un1 Un2 Xn = Un + 2 3 (i) (4 pts.) Determine Cov(Xn , Xn+k ) for every k Z. (ii) (3 pts.)
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 20Due Tue 11/24Let X1 , X2 , X3 , . . . be independent positive random variables with pdf as shown below, where 0 &lt; b &lt; a.f (x )X?0a- baa+bxThe sequence is recorded using an instrument with dynamic range [0, 5]. Thus the rec
Maryland - ENEE - 324
ENEE 324ASSIGNMENT 21Due Thu 12/03Problem 21A Consider a service model where each day (e.g., between 9 AM and 5 PM), a server processes no more than two jobs. The servers buer can hold up to two jobs. Jobs arrive during the night (e.g., between 12 AM a
Maryland - ENEE - 324
ENEE 324 Problem 22AASSIGNMENT 22Due Thu 12/10Consider a Poisson arrival process cfw_N (t), t &gt; 0 (where t is in seconds) of rate = 2 arrivals/sec. (i) (2 pts.) Determine P [N (4) = 8]. (ii) (3 pts.) Determine P [N (5) = 10 | N (1) = 2]. (iii) (3 pts.)
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 1Due Fri 01/08Consider the complex numbers z1 = 1 + j 3 and z2 = 1 j(i) (3 pts.) Plot both numbers on the complex plane. (ii) (3 pts.) Evaluate |zi | and zi for both values of i (i = 1, 2).2 2 3 (iii) (6 pts.) Evaluate
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 2Due Mon 02/04Consider the complex numbers u = ej (/7) , v = ej (6/7) , and w = ej (2/5) (i) (2 pts.) Plot the three numbers on the complex plane. (ii) (3 pts.) Plot the straight line described by the equation |z u| = |z
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 3Due Wed 02/06Use your calculator for algebraic and trigonometric functions only, as indicated. Trial-and-error or graphical solutions to parts (i)(iii) are not acceptable. Consider the sinusoid x(t) = A cos(t + ), where
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 4Due Fri 02/08Consider the discrete-time sinusoids x[n] = cos 3n 2 + 4 3 and y [n] = cos n 2 + 5 3(i) (4 pts.) What is the period N of x[]? Use MATLAB to generate a discrete plot of the rst four periods of x[], (i.e., f
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 54 3 2 1 0 1 2 3 4Due Mon 02/1100.10.20.30.40.50.60.7 The rst period of the sinusoid x(t) = A cos(t + ) is plotted above, where x(0) = 2 3. (i) (5 pts.) Determine A, and . In what follows: A and are as found in
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 6Due Wed 02/13(i) (5 pts.) Compute the sum of the two discrete-time sinusoids: x1 [n] = cos(2n/5 + /3) and x2 [n] = cos(8n/5 + /4)(Hint : Start with reducing the angular frequency of x2 [n] to a value between 0 and , th
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 7Due Fri 02/15(i) (2 + 2 pts.) Suppose a 3 3 matrix A satises 1 4 0 3 A 0 = 2 , A 1 = 2 0 1 0 2 Find A and compute 1 A 0 1 010 B= 0 0 1 100 and 0 0 A 0 = 1 . 1 2(ii) (10 pts.) Ifand x = [ 1 2 3 ]T , determine the fol
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 8Due Mon 02/18(i) (10 pts.) It is known that 1 4 A 1 = 5 , 0 6 Determine the matrix A. 2 1 A 1 = 4 0 0and 2 7 A 1 = 6 2 2(ii) (10 pts.) Without using your calculator, determine so that the product of the ve matrices
Maryland - ENEE - 241
ENEE 241 02* Consider the matrixHOMEWORK ASSIGNMENT 9 abcd A= e f g h wxyz Due Wed 02/20In each of the following cases, nd matrices P and Q such that PAQ equals the matrix shown. (Note that if either P or Q is superuous, it trivially equals I.) wxyz e
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 10Due Fri 02/22Solve by hand without using calculator matrix functions. Show all intermediate steps. Let 1 0 U= 0 0 2 1 0 1 2 1 0 1 2 0 0 1(i) (4 pts.) Solve Ux = b for arbitrary x and b. Display U1 . (ii) (6 pts.) Show
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 11Due Wed 02/27Solve by hand without using calculator matrix functions. Show all intermediate steps. Consider the 4 4 system Ax = b, 2 1 4 0 A= 1 3 2 6 9 (i) (14 pts.) Determine x. (ii) (6 pts.) Express A in the form LU,
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 12Due Fri 02/29Solve by hand without using calculator matrix functions. Show all intermediate steps. Let 1 2 A= 1 1 4 2 1 3 4 0 3 1 4 2 3 2(i) (8 pts.) Obtain the LU factorization A = LU. (ii) (12 pts.) Determine A1 by s
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 13Due Mon 03/03Solve by hand without using calculator matrix functions. Show all intermediate steps. Consider the following 1 1 1 1 A= 1 2 1 3 three matrices: 1 1 1 1 B= 4 8 6 91 2 3 5 234 4 7 8 5 9 12 7 12 201 2 1 2
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 14Due Wed 03/05Solve by hand without using calculator matrix functions. Show all intermediate steps. Consider the three-dimensional vectors v(1) = [ 1 2 2 4 ]T and v(2) = [ 4 2 2 1 ]T . (i) (4 pts.) Compute v(1) and v(2)
Maryland - ENEE - 241
ENEE 241 02HOMEWORK PROBLEM 15Due Fri 03/07Solve by hand without using calculator matrix functions. Show all intermediate steps. Let v(1) = [ 5 3 1 1 ]T , v(2) = [ 5 1 7 3 ]T and v(3) = [ 1 1 10 0 ]T . Also, let s = [ 3 1 4 4 ]T . (i) (5 pts.) Determin
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 16Due Mon 03/10This problem investigates the least-squares approximation of the function s(t) = 1 (t 1)2 1 + 4(t 1)2over a discrete interval of values t using a sum of three sinusoidal functions: f1 (t) = sin(t/2) , f2 (
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 17Due Wed 03/12This problem refers to the range estimation example in the textbook (p. 113). Suppose f1 = 3 MHz, f2 = 2.5 MHz and = 1.45. The vector s in the le dataB17.txt contains 101 samples of the noisy waveform s(t)
Maryland - ENEE - 241
ENEE 241 02 Consider the 4 3 matrix V =HOMEWORK PROBLEM 18 v(1) v(2) v(3) given byDue Fri 03/14 1.3702 4.6799 0.6379 1.8877 2.5466 4.7986 V= 2.0596 5.5447 2.1429 2.5134 3.7242 0.9872 (i) (3 pts.) Compute (e.g., in MATLAB) the inner product matrix VT V,
Maryland - ENEE - 241
ENEE 241 02HOMEWORK PROBLEM 19 v(1) v(2) v(3) given byDue Mon 03/24Consider the complex-valued matrix V = 3 a + jb 6j 3 a + jb V = 6j a + jb 6j 3 (i) (4 pts.) Determine the only pair of real values a and b such that the columns of V are orthogonal. F
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 20Due Wed 03/26Exam 2 (Monday 03/31) will cover Assignments 10-20. Let V= v(0) v(1) v(2) v(3) v(4) v(5) be the matrix of Fourier sinusoids of length N = 6. (This is the same matrix as in the last example of Reading Assig
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 21Due Fri 03/28Solve without using MATLAB, except to verify your results. All vectors have length N = 10. (i) (4 pts.) The time-domain vector x(1) has DFT X(1) = 0 1 0 0 0 0 0 0 0 1TWhich two Fourier frequencies does x
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 22Due Wed 04/02The real-valued signal vector s has DFT S= 6 z1 3 + 2j 7 z2 5 + 3jT(i) (2 pts.) What are the values of z1 and z2 ? (ii) (3 pts.) Without using complex algebra (or MATLAB), determine the sum of the entries
Maryland - ENEE - 241
ENEE 241 02 The signal vector s is given byHOMEWORK PROBLEM 24Due Mon 04/07s=abcdefghT(i) (8 pts.) Display the following vectors: s(1) = P2 Rs s(2) = R2 Ps s(3) = P4 s s(4) = F2 s (ii) (12 pts.) Express the following vectors in terms of P, R, F and
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 26Due Fri 04/11Single-step the MATLAB script below, keeping a gure window open. Answer questions (i)-(viii). Print and submit the last two graphs, i.e., bar(n, real(X4) and bar(n, real(X5), only. (Make sure that your MAT
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 27 y ofTDue Mon 04/14(i) (7 pts.) Determine the circular convolution s = x x= and y= (ii) (7 pts.) If 3 1 1 5 20 1 2TUsing the FFT function in MATLAB, verify that the DFTs X, Y and S satisfy X Y = S. 1 3 2 1 4 x0 x1
Maryland - ENEE - 241
ENEE 241 02 The signalHOMEWORK ASSIGNMENT 28TDue Wed 04/16x= has DFT X given by X=000abc000X0 X1 X2 X3 X4 X5 X6 X7 X8TExpress the following DFTs in terms of the entries of X: (i) (5 pts.) The DFT X(1) of x(1) = (ii) (5 pts.) The DFT X(2) of x(2) =
Maryland - ENEE - 241
ENEE 241 02HOMEWORK PROBLEM 29Due Fri 04/18Generate the 20-point signal vector x using the MATLAB script below. n = (0:19). ; f0 = 0.15 + 0.25*rand(1) ; A = 10*rand(1) ; phi = pi*rand(1) ; x = A*cos(2*pi*f0*n + phi) + 0.3*A*randn(size(n) ; Following th
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 30Due Mon 04/28Each of the innite sequences x = x[ ], y = y [ ] and u = u[ ] (shown below) contains seven nonzero terms.4 3 2 1 1 x 2 1 4 3 y 2 4 3 un=0n=0n=0(i) (3 pts.) Express u[ ] as a linear combination of seve
Maryland - ENEE - 241
ENEE 241HOMEWORK ASSIGNMENT 32Due Wed 04/30Consider the FIR lter given by the input-output relationship y [n] = x[n] x[n 2] + x[n 4] x[n 6] , (Note the missing, i.e., zero-valued, coecients.) (i) (4 pts.) Show that the input sequences dened for all n b
Maryland - ENEE - 241
ENEE 241 02*HOMEWORK ASSIGNMENT 33Due Fri 05/02Consider the FIR lter whose input-output relationship is given by 1 3 3 y [n] = x[n] x[n 1] + x[n 3] x[n 4] , 2 4 8 nZ(i) Determine the response y (i) [ ] of the lter to each of the input signals given by
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 34 4 1 0 1 4TDue Mon 05/05 .Consider the FIR lter with coecient vector b =The following MATLAB script computes a segment of the lter output sequence y (1) [ ] for an input sequence x(1) [ ] which has period L = 5. Speci
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 35Due Wed 05/07Use the MATLAB script below to generate the coecient vector b of a bandpass FIR lter of order M = 44. f1 = 0.19 + .03*rand(1) ; f2 = 0.34 + .03*rand(1) ; F = [0 f1 f1+0.03 f2 f2+0.03 0.5]*2 ; G = [1 1 0 0 1
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 36Due Fri 05/09(Part A) (5 pts.) Consider the FIR lter with impulse response given by h1 [n] = [n] [n 1]. Compute the outputs y1 [n] and y2 [n] of this system when the inputs are (A.1) (2 pts.) x1 [n] = cos(n + ), and (A.
Maryland - ENEE - 241
ENEE 241 02HOMEWORK ASSIGNMENT 37Due Mon 05/12A FIR lter acts on the input sequence given by x[0 : 5] = 1 3 5 2 6 2T;x[n] = 0 for n &lt; 0 and n &gt; 5to produce the output sequence given by y [0 : 11] = 2 3 0 10 23 27 61 51 54 14 2Tand y [n] = 0 for n
Maryland - ENEE - 241
HW 1 SOLUTION_(i) Plot (-1,sqrt(3) and (1,-1) on the (x,y) plane.(ii) |z1| = sqrt(-1)^2+(sqrt(3)^2) = sqrt(1+3) = 2angle(z1) = arctan(-sqrt(3) + pi = 2*pi/3 = 2.0944|z2| = sqrt(1+1) = sqrt(2) = 2.828angle(z2) = arctan(-1/1) = -pi/4(iii) z1^3 =
Maryland - ENEE - 241
HW 2 SOLUTION_(i) The polar coordinates are |u| = 1, angle(u) = pi/7 |v| = 1, angle(v) = 6*pi/7 |w| = 1, angle(u) = 2*pi/5(ii) |z-u| = |z-v| is tha perpendicular bisector of the segmentjoining u and v. This straight line is the y-axis (x = 0) (ii
Maryland - ENEE - 241
HW 3 Solution_(i) We know that the value 2.5 occurs at t=0, 0.160 and 0.180 sec andat no other intermediate times. Since a sinusoid crosses any givenlevel (except its maximum and minimum) twice in each period, it followsthat the period T equals exact
Maryland - ENEE - 241
HW 4 Solution_(i) For x[.], we have w=3*pi/4 = (3/8)*(2*pi). Thusthe fundamental period of x[.] equals N = 8.N = 8;n = 0 : 4*N-1;x = cos(3*pi*n/4 + 2*pi/3);stem(n,x)(ii) The values of w such that cos(w*n) is periodic with fundamental period N
Maryland - ENEE - 241
HW 5 Solution_(i) Clearly, A = 4. The period T equals 0.7 sec, therefore f = 10/7 Hz andW = 20*pi/7cos(phi) = 2*sqrt(3)/A = sqrt(3)/2, therefore phi = pi/6 or -pi/6. By inspection, t=0 corresponds to a point in the last quarter-cycle of the cosine wa
Maryland - ENEE - 241