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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 continuoustime
signals over the range —00 < t < 00. (b) Write the convolution formula y = hm: where h , a: are discretetime 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(n2) (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 zdomain to frequency domain in general. 21. [4 pts.] In freespace, 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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