ECE111_2010SPRING_EXAM1__[0]

ECE111_2010SPRING_EXAM1__[0] - The Cooper Union Department...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Exam I March 26, 2010 Time: 2 hours. Closed book, closed notes. No calculators. 1. [8 pts.] Make sure your formulas are PRECISE here! (a) Write the convolution formula y = h * a: where h (t) ,x (t) are continuous-time signals over the range —00 < t < 00. (b) Write the convolution formula y = hm: where h , a: are discrete-time signals over the range ~00 < n < 00. (c) Write the DTFT formula. ((1) Write the IDTFT formula. (e) Let a: have DTFT X Starting from the DTFT formula (do not invoke properties you may have memorized), find the DTFT of ejwmm’“ 2. [2 pts.] What do FIR and HR stand for? (Do not define them— just write what the letters stand for) 3. [2 pts.] Let h be the impulse response of a discrete—time LTI system. Write an explicit condition on h that holds iff the system is BIBO stable (write it out— don’t try to describe or define this condition in words) 4. [5 pts.] True or false, no justification: (a) An FIR filter is necessarily causal. (b) An FIR filter is necessarily BIBO stable. (0) An FIR filter can only be realized nonrecursively. (d) An IIR filter, if it can be realized, can only be realized recursively. (e) The system 3/ = :1: (2n) + :3 (2n — 1) is causal. 5. [4 pts.] Refer to the block diagram in Figure P5. Find an explicit formula for the overall transfer function (do not try to simplify it). - 6. [3 pts.] Given the following transfer function: 3(z — 2)2 (z + 1/3) Hm = (z — 1/2)3(z+ 1/5) (a) List all the poles and zeros along with their multiplicities. (b) Identify all the possible regions of convergence (ROG). 7. [6 pts.] Given the following difference equation: y(n) = 2w(n) —3:v(n—I) +4zc(n—2) ——0.3y(n—1)+0.6y(n-2) (a) Write the transfer function H (z) as a ratio of polynomials in z (in particular, clear out any negative powers of z). (b) Sketch direct form II and direct form II transposed realizations. Be sure to indi- cate which is which, and indicate direction of signal flow in your sketches. 8. [6 pts.] The DTFT X (w) of a discrete—time signal is shown in Figure P8. Let Yo(w) = X (w/Z) and Y1(w) = X(w/2 + 71'), and let: Y(w)=Yb(w)+Y1(w) =X(w/2)+X(w/2+7r) (3) Find Yo (w) and Y1 (w) at w = 0,.+_7r/2, :l:7r, :l:31r/2, i21r [just by plugging in val- ues; do not attempt a sketch, yet] (b) Using the results of part (a) as a guide, sketch Yo (La) and Y1 (w) over the range ~27r S w S 27L (c) Sketch Y (M) over the range ——27r S w s 271'. ((1) Based on your sketches (do not attempt a formal proof), which, if any, among YE) (w) ,Yl (w) , Y (to) are periodic with period 27r? 9. [6 pts.] Given the following signal: :c(t) = A€_t/T cos (27r (f0 — Af)t+ (#0) Take f0 as the reference carrier frequency. (a) Write the baseband equivalent signal [EBB (b) Starting from your expression for 0033 (do not go back up to m) find the in—phase and quadrature components ,q (25) respectively. 10. [5 pts.] Given the following signal, which is a special case of an FM signal: A cos (27rfot + E sin (27rfmt)> m (3) Compute the instantaneous frequency finst (b) Specify conditions on f0, [3 and/ or fm so that finst (t) > 0 always. (c) Assuming the conditions in part (b) hold, specify the minimum and maximum values attained by finst 11. [2 pts.] Evaluate the following integral: [00 (26 (t + 1) + 36’ (t — 4)) etht 12. [2 pts.] Let a: (t) = e‘tu (t). On separate axes, sketch x (t) and x (2 —— t) + u(t — 2). 13. [4 pts.] Let x(t) = e‘tu (t), h(t) = 2 for 0 S t g 1 (and 0 otherwise), and y = h * ac. Do not compute 3; except as noted below!! (a) Draw a sketch indicating the flip—and—slide procedure for determining y (2) only, in which h (not 3:) is the signal that you fiip—and—slide. (b) Using your sketch as a guide, compute y 14. [6 pts.] The finite length discrete—time signals h, a: are specified as follows (where the underscore denotes n = 0): :1: {—2,3,4, 1,2} h = {2, —_—_1_, 5, 1, 3} (a) Specify the support and lengths of h and x. (b) Let y = h * so. Specify its support and length. (c) Write an expression for y (2) where indices are shown explicitly (i.e., a sum of terms of the form hm); only terms with nonzero value should be listed! ((1) Use the result of part (c) to actually compute y 15. [4 \pts.] Given the following: H (z) = 3 — 52‘1 + 724 + 62—3 (a) Sketch a transversal filter realization. 7 (b) Express the impulse response h as a superposition of impulses. (c) Write an explicit formula for the frequency response of the filter; do not try to simplify it. 16. [6 pts.] Given the following: 2cos (27rf0t — 7r/4) + —4— cos (27rf0t + 7r/2) \/§ (3.) Write the phasor respresentation of each term, and use that to compute the phasor representation of the sum. You are NOT permitted to evaluate any sine / cosine values as you perform the necessary complex arithmetic. Express the final phasor BOTH in rectangular form and polar form (with phase in radians, not degrees). (b) Rom the phasor representation of the sum, express the sum back in the time- domain in the form acos (27rf0t) + bsin (27rf0t). No credit if you use another method (e.g., trig identities to expand cos (A‘+ B) etc). 17. [8 pts.] The signals 3 (t) ,¢1 (t) , ¢2 (t) are shown in Figure P17. In this problem, define (M) = If; f (age) dt. (a) Verify that (151 (t) , ¢2 (t) are orthonormal. You must show work, in particular you must be sure that ALL the necessary conditions are verified. Hint: There is more than one condition to cheek“ (b) Let s (t) E span {qfihqfiz} be the best approximation to 3(t) in the sense that [[3 — is minimized. Find s (t) and sketch it. (c) Find (d) Without performing any more integration, find [[3 — 18. [7 pts.] The impedance of a circuit at 100kH z is Z = 2 + 339. (a) Find R, X, Y, G', B, with their correct units. (b) NAME what R, X, Y, G, B stand for. (c) Model the circuit as either a resistor in series with an inductor, or a resistor in series in capacitor: which is correct? (d) Compute the component values for the series circuit (either R, L or R, 0); write explicit formulas ready to plug into a calculator, and include units. (e) Repeat for the case of a parallel circuit. 19. [5 pts.] A real continuous—time signal with period T has Fourier series coefficients cm (complex exponential form); the coefficients am, bm correspond to the trigonometric form of the Fourier series, using the standard conventions. Given that co = 4, cl=2+j andcz=3—4j. (a) Compute any other cm coefficients that can be determined from the given infor— mation. (b) Without actually computing them, identify the ambm coefficients that can be determined from the given information (again, don’t actually compute them). (c) Compute the total power in the second harmonic. 20. [6 pts.] Given h with z—transform H (z), the paraconjugate 1:1 is defined as the z—transform of h* (a) Starting from the formula for the z—transform, find an expression for F1 (b) You should know what the DTF T of h* (—n) is in terms of H From your expression for H (2:), convert from z—domain to frequency domain (directly, i.e. Without going through the time domain) and confirm the result matches. Note: Even if you couldn’t get part (a), you will get partial credit here if you tell me how to go from z-domain to frequency domain in general. 21. [4 pts.] In free-space, the power of an electromagnetic wave is ~1/r2, where r is distance from the transmitter. However, this result often does not apply to "real" channels which are much more complicated, for example multipath fading channels in which multiple reflections cause interference patterns that lead to potentially severe attenuation. In such cases, power versus distance is often modeled empirically with a path loss exponent a via the relation: PU") = P0 ' (To/7")“ where P0 is a reference power at distance To, and P (r) is the power at distance r > To (here power in Watts). Suppose we measure power at r = 10% that is 24dB below the power level at 7‘0. Compute a. Hint: If you did it correctly, you would not need a calculator to determine the precise value of a! ...
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This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.

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ECE111_2010SPRING_EXAM1__[0] - The Cooper Union Department...

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