midterm_sp2018_sol.pdf - University of Illinois ECE 310 ‘...

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Unformatted text preview: University of Illinois ECE 310 ‘ Profs. Do and Liang Midterm Exam 7:00—8:30pm, Thursday, March 1, 2018 Name: Section: 9:00 AM 12:00 PM 3:00 PM NetID: Score: Instructions 0 You may not use any books, calculators, or notes other than two handwritten two—sided sheets of 8.5” x 11” paper. 0 Show all your work to receive full credit for your answers. 0 When you are asked to “calculate”, “determine”, or “find”, this means providing closed-form ex- pressions (i.e., without summation or integration signs). 0 Neatness counts. If we are unable to read your work, we cannot grade it. 0 Turn in your entire booklet once you are finished. No extra booklet or papers will be considered. (10 Pts.) . :{I‘TS 84:»ch 1. Answer True or False to each of the following statements: 00 =l {govt-292‘; m (a) Let Z x[n] 6[sin(7rn)] = 2. Then 2 cr[n] = 2. .False (b) Suppose that a:[n] has z—transform X (z) The DTFT of 1:[n] can always be expressed as: Xd(w) = X(z)|z:em. True NO I Okla, 41 ROQX ikaluflfif 'lfie UIAH‘ Gwala (c) The DTFT of the sequence $[n] : cos (gm), —00 < n < 00 is Xd(w) = 7r [5(w — g) + 6(w + 1 , fish for—oo<w<oo. True Wm) 04} 4w 1, éwST (d) The BIBO stability of any system is completely determined by the system’s unit pulse re— sponse. \ K True ‘ fig No , cal? *gow L l _L SVPTQMS’ (e) The output y[n] of a system for an arbitrary input a:[n] is given by y[n] = $[n] * h[n], where h[n] is the unit pulse response of the system. The system must be linear and shift—invariant. yag ) LT I $1: "(‘{b' E COK¢°l LifColx -False (12 Pts.) 2. For each of the systems with input x[n] and output y[n] shown in the table, indicate by “yes” or “no” Whether the properties indicated apply to the system. 4 W. aw)“ Linear Shift- Causal —t V [ JR: Yes No y[n]=‘y[n—1]+ n m[n], 7120,12,... ya: Mo YLS‘ Stable y[n] = $[n — 1] + $[n] + $[n + 1] Mo £019th 310‘} x O v” (8 Pts.) 3. Calculate and plot the results of the following convolution: {i’ 2, 3, 2, 1} * {i’ —1} ‘fCrfl;i|,l,\,41-1,_‘ " (10 Pts.) 7‘ i o ' I 4. Calculate the z—transform (if it exists) and the corresponding region of convergence for each of the following signals. Simplify your expressions. (a) x[n] = 36[n + 1] — 6[n — 100] (b) $[n] = 3"u[n] + 2"u[n + 3] a) XL%):‘ %) Ea, 240°, {20¢ i370, \aér‘o b) ZfluL43+ a“ ULA4’3] : gflvL/l’1k Qflfgrlufin+31 " Znuln] 4”? 2fl*%an+3] 9} M23: + {€23 1'} 2/; (10 Pts.) 5. The z—transform of $[n] is given below: Mai—1:2; (a) Determine all valid ROCs for X (z) fob—3 : 2 = 3L imo ROCs; (4;) H! > (b) Assuming that 3:[n] is a right-sided sequence, determine a:[n] N ROQ ‘5‘“ fli‘ ,QQ ("Ki X(i)= h—“‘ a MD} 39] ”lg (13) MM) $3.50] —é_ £301.40] 4 > ( Likicbv \ i I‘ L: a“ 9‘ Pf; V . (5 Pts.) 6. Determine the sequence x[n] whose discrete—time Fourier transform is: 1 KW) = m (Hint: Consider the relationship between DTFT and z-tmnsform.) iii—ct): /L__— ‘ n l— £24 —7 Bibi]? (E‘) ufn] fig (10Ci5 i'kl 7 VS {Mair/Jag Unii dale) M X In : ._|. —}w w . I) ) rig (S)fle A 2”; EEC—nu)": ’4— : 7e (7 Pts.) 7. Consider a causal LTI system with the following linear constant coefficient difference equation (LC- CDE): y[n] = %y[n — 1] +x[n] — 53:[n— 4], n = 0,1,2,... Determine the transfer function H (z) of the system and sketch its zero-pole plot. We) = yr" w?) + u?) ~52’”x£%) w. t ”if? a we men He): 1%). egg-‘4 {1,5 _ . 2“,§ XL?) l 1,324 2“’~23 13(2—4) \\ p (15 Pts.) 8. Consider the following cascaded system: Assume that the transfer function of overall system is H (z) = $ with ROC Izl > 1/2, and 5'1 is implemented by an LCCDE: 1 ylnl = iyln—1]+:c[n], n=0,1,2,.... (a) Determine the transfer function of 52: H2 (2:) (b) Determine the unit pulse response of $2: h2[n] (c) Express 52 in the form of an LCCDE. 2 \‘Q H PIS-2 0 \79H(—%) 2 H02} H2039: (it-137;" zC‘lj- I43: 2"\ (15 Pts.) 9. The difference equation of a causal LTI system is given by 1 y[n] — Eejgym — 1] = m[n], —00 < n < oo. Determine y[n] for input Mn] = 1 + 2cos(%n), —00 < n < oo. Heb): "—L——'-— [2| > -L .L 3% -\ 7- / t-“le '% Th‘ {yie‘w I‘J Bu» SfiL/e f ‘ ‘ 4 ~}— (uJ: HG? ‘u: “T? 1 ~ “to 6’ Hé PG) Fiche ’ have 5::th -\’L\- {Dykw |‘J h¢+ Po/Q~ VJu—e—A) “K H? Wide 4V0.) ‘9 «(“3 C ‘ 1. 8914154 + Q‘s-gin 34$" ~53an 6W @032 Hm) + le‘éfi + 144%)6 z \ 1 \ e}. (8 Pts.) QB @ 10. The transfer function of a causal LSI system is: H (z) = if}. Find a bounded, real-valued input to the system which will produce an unbounded output 3/ n]. Give an expression for the input in the z—domain, X(z). HQ) gig a 6mm; min “4+ CJrcLe Ar 22-?“ 50/ t3 .‘VQ. an Mn!bM4~€\ow‘iU}K+l 46043 ”Mi flow: a gale oi it: 3‘ Ts makQ m3 HJWM I} 341»le (ALB Ears Oxyxie at :3: +2“ So, (MK st/iék mtg/MAM) M \ lab—4 "flair b 2~l\,._1_\\ ! X6” 2+5 %~‘”2%+4;_ [email protected] 1: ...
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