Exam2SolnSP2015

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Nov 24, 2024

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SolLU TioeN EE301 Signals and Systems Spring 2015 Exam 2 Tuesday, Mar. 31, 2015 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 3,4 with emphasis on Chap. 4 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators NOT allowed. All work should be done on the sheets provided. You can NOT do work on the back of a page unless permission is granted. No work on the back of a page will be graded unless permission is granted. | You must show all work for each problem to receive full credit.

Problem 1. The sum of (infinite-duration) sinewaves below z(t) = 1+ 7 cos (nt) + 7 cos (27t) is input to an LTI system with impulse response given by h(t) = cos(mt)rect(t) Determine and write a closed-form expression for the output y(t) = z(t)*xh(t). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. N . ‘PL m + J x (k)= 2, @ € 6 =—1 T _ Y QGA_, = a\$ ‘-"-l:‘., 0\’1 = QZ' ry g € SHymo | - + (1 + Co&(ﬂ{>> "“e(’{fi — (D . S = .z rect (JC> = 2 ———-—@r—\]—-_g—-— B —_ ou—-n) rect (t> C°3<ﬂ{’>c/“+‘5 E\‘\ ( -

Additional Space for Problem 1 answer and work. S I (’ww-: 5"Y‘(—T—T—5‘n> N gs%)/? o ™o §\r\<‘fi"‘w =0 N— A S (_\[4-9;?].)’(,‘\) O/ZU\ /Q o ket we Jravmned 'n— class 0\\>ou¢ S:T o+ \C\'m'-\fi\{se v\é+tv Sﬂh@wc‘\\«eg A4 T - ! 2T - 0 g\ ( - i feny= ST : B 21T —© 1+ 4+ T 1 1 sfv(?-:‘-‘ =L - o v 2Z 7 2 - ) _ = - H(—’Zﬂ) el s e— - 'gTY' 2 1T 2 e 2

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+2 Problem 2 (a). The Gaussian pulse z(t) = fie—;‘f, with o7 = 62, is multiplied by a complex-valued sinewave to form y(t) = €’*'z(¢). Find the numerical value of F = 1 00 o / Y (w)|*dw. Show all work, state which Fourier Transform pairs and/or properties T J—0 you are using, and clearly indicate your final answer. C‘ovxst\ovf "{:0:0 A nd SqQUavre T cle)f et«evg2~' - _ v 2 (%) * ST )= T =0 §f€wo\vcv\5 rec\uces S’i‘é, dev, loé. \FZ’ T hug &~ 'k]_ Z o ‘ ’5&—1—) le= Jem & =47 6 1 — 02 | £z \ = « and -~ L f zot Q)'E/ 207 6° =) \C 1 — co - —— 2 s o _t T hus * — \ J \ Zat ~ — Yy S Rwevey of o2 Q 2 o o0 B = | dt = Now ZTEOJ \(Cuo>) Qw J 82(t5§ — NS ra T A P ) AN s wey = \ = T —

Problem 2 (b). The Gaussian pulse z(t) = ;7=¢ 291 | with 02 = 62, is time-shifted to 1 00 form y(t) = (¢t — 3). Find the numerical value of £ = . / Y (w)|*dw. Show all work, ™ J—00 state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. A ‘tiw\e-s\r\if“} does n o change ¥ he Q\\'S*V(Lu%’éc\ A, A o Fal erheved ")us)- how The RS-V 'S F(Ar\c“f\‘on ol Fime. (o éd . fo<2<£> e = a2 (e-dt =k _ oo — 0o

+2 Problem 2 (c¢). The Gaussian pulse z(t) = 21 with o2 = 62, is differentiated 1 6v2r o to form y(t) = 2£z(t). Find the numerical value of A = / y(t)dt. Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. T . < X (-E) HJWX(W>:Y(W>

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+2 Problem 2 (d). The Gaussian pulse z(t) = fie_fi, Woich 0? = 62, is multiplied by ¢ to form y(¢t) = tz(t). Find the numerical value of A = / y(t)dt. Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. G | 2177 , = w 0™ — ——z (+) .CLszwe b = ("2 , w o ) T Tz ot w(l - J — —_

_ 2 Problem 2 (e). The Gaussian pulse (t) = ;7= 221 with o2 = 62, is compressed in time by a factor of 2 to form y(t) = z(2t). Find the numerical value of the energy £ = / % (t)dt. Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. l (2 t>2_ , | . ?({\-_—. x(2t)= e 6 ( -~ 29 Q\IZTV\ Jc:Z w heve — T O"‘:‘S _t e 7 =2 aofar ; .Y ?V\P‘(‘é%\fi 20_(5_ = 2'34’%— Vel 4 wer: N = - \ 4 2 ) ¢ {m 24

Problem 2 (f). The Gaussian pulse z(t) = 6—\—71——%6_—2712—, with 0;% = 6%, is input to an LTI ot system with a Gaussian shaped impulse response, h(t) = g—l\/-é—-;e Eg, with o2 = 82, Determine a simple, closed-form expression for the output y(t) = x(t) x h(t). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. !’L - \ T a2 .8° ey e C —= * el T t gs'nce VA vy el e dd \ 6 2.40°" . - — 2 T (o J2rm (o= 8 T& 2. ‘ , . 2 "S pvaeé i~ clase UsyAg 5 et > ;%; — = r70~{‘/ : \ €~_—20-L4/:\___>€ o d 2w ):T FKOFQV"&‘) - Q (€)= "K€) e (8) g———}\[/Cw)‘:X(w) “Hw)

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_ 12 -6-—1-\/5—;6 201 with 0? = 62, is input to an LTI t2 system with a Gaussian shaped impulse response, h(t) = fie_%g, with o5 = 8%. Determine Problem 2 (g). The Gaussian pulse z(t) = the numerical value of the area, A = / b y(t)dt, under the output y(t) = z(t) x h(t). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. _ W 0 /_\.\j%(t)Qk - \(0)) " e - = 4 ‘I A=A 10

t2 Problem 2 (h). The Gaussian pulse z(t) = 6\/12— , with o7 = 62 is input to an LTI +2 system with a Gaussian shaped impulse response, h(t) 2 \/—- _"g with o2 = 8%, Determine the numerical value of the energy E = / t)dt of the output y(t) = x(t) * h(t). Show all work, state which Fourier Transform pairs and/ or properties you are using, and clearly indicate your final answer. Frewm 2la) ) Linerao Of s 525 e _ J7w o 2 7 dm 11

_(t—2)? 1 20 : 2 2 : Nk 1, with o7 = 67, is input to an _ (t—3)? LTT system with a Gaussian shaped impulse response, h(t) = -é—\/l—%e 2, with o2 = 8. Determine the numerical value of the energy £ = / b y?(t)dt of the output y(t) = z(t)*h(t). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. Problem 2 (i). The Gaussian pulse z(t) = (e - T rvav aince C(\\C}a\;‘@,g ‘i"kc}‘ Ohouwev IS Afi(‘&"‘fi) £, 4+ Ty (/L\\Qv( ;0\' (Ji3 [ < ahguwev RS 2(f> 12

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Problem 2 (j). The signal z(t) = z(t)y(t) is the PRODUCT of the Gaussian pulse z(t) = 6—\/1-5—-7;6_5:’? +2 with 02 = 62, and the Gaussian pulse y(t) = fie_ 272, with o5 = 8%, Determine a simple expression for the Fourier Transform, Z(w), of 2(t) = z(¢)y(¢). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. | 2 (4 =% (O ?(k) £ | 26+ 649 (= i = - N (A/l\QW" Ta ~= S 2¢ + ¢4 v Oo° , A 3¢9 68 A 23 B | &0 22 | (23.69) 2 | 13

Problem 3. The signal 1 1 1Y . z(t) = FoRT cos(20t) — {m * ;r—t} sin(20t) is input to an L'TT system with impulse response h(t) — {_71 sin(5¢) sin(15t)} 5 ot 7t Determine the output y(t) = x(t) * h(t). Show all work, state which Fourier Transform pairs and/or properties you are using, and clearly indicate your final answer. AT JW({V\E eXam Lrom lable . O\\H\/}; 24 4o, a4 e wrat ) I } C <> = A 0\\\.'\" . A a T - (‘ub) 7_2 - 27 € + + 4 | | K L 5 7 on boTh >0 des, Vv \ N ) VA T 2T ] \w\ Y == T 6 ~ - o 5(((») | —r ~ > 14

Additional Space for Problem 3 answer and work. Shihn <§fi> S) v\()g“') j Used on ~at) o 9)‘4,.5‘ L\PC): S T ar T A ’"fi”k _20 N - o 2 T hus, . \((w)—:. ><(Uu> k’%(uD = 0O overlap M (E)=0 Z Note: Lot (€)= 'X () = Xo (O *~_.\_ L+t TEC X <pc@g<w6{>gz;> A ><(w\w0> Pl K (weew) KX (£) sin(w, JC>< >—-—- (w w>\ .__\X (ufi-u? cubabituter X (@.. - :5v\<w>><®cw\ Vs Tn (@) <2S L1 sgn(@-w) Y (w-w) Thast | x sanCoriy) X0ty (O Cos (W, ) c\ (\ +98"\ w - w}\%(u) cu} _ q/ZO (£ g{»\(%aﬂ -+ <\~§gr\<w+%>><<w+w> | J - (*kScw\(w-—w}) { wvw — ‘—;8h<w+w\9 { w:’wo o, W< W, |

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