b09exam4sol

# b09exam4sol - ECE2311 EXAM 4 B09 Name Mia:2” ECE Box...

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Unformatted text preview: ECE2311 EXAM # 4 B09 Name : Mia-:2” ECE Box Number: ______________ \ Clearly write the ﬁnal answer on the line provided on the exam sheets. When you ﬁnish your exam7 be sure that your name and ECE box number is on this sheet and turn in this entire exam with the answers placed where requested. Problem 1 — Fourier Transform Integral 1a) Obtain the Fourier transform of the function x(t) = 56‘3tu(t — 1) by evaluating the intergral for the Fourier Transform (Le. do not use the table method or Fourier properties). Show all work. 0:) 7‘” 00 -‘w‘fi — -Y‘f: ' f ’ Ylw) ’* )ixt’ﬂe ’ d“. ‘ J58 e“’on— = Cfe‘czrﬂyedi. (OK) I l , - ~ - ﬂu) - 5* e"(’wm 00. 5‘ o'e‘oml : 5'8 3'8 I ‘ ' ,‘wq' ” for/“*3 jw+3 ‘e ,qu) —(9,°"+3) 59' 2 re Xe) : #__.L03_________ 1b) Obtain the Fourier transform of the function y(t) 2 u(t+ l) —u(t — 1) by evaluating the intergral for the Fourier Transform (i.e. do not use the table method). Show all work. W l l , r , , r t NH” 5”“- e’” 6 ﬂour 3 v)” = C...— 1 Xau) IKE)? all e 6“? [7w '2’“) ’90 v! l _' 2; 3% (V) 25.3 (V) N 1w ‘0 (jaw—e ’ 1:04 (a) v . 1' ’77—; Y(w) = 9‘“) _ Problem 2 A Fourier Transform Tables and Properties 2a) Obtain the Fourier transform of the function x(t) 2 6t2e_2tu(t) +46 (t —— 1) by using the provided tables of Fourier transforms and Fourier transform properties . Show all work. 1%) : ()ﬁle'ltudfl + We-» / \ -n é.21 :12 a ("naval3 (Two’le I .1!) IL " +46 X(w) = _. ..(.. L_*3“’_)_E 2b) Obtain the Fourier transform of the function y(t) 2: 6‘2‘u(t) * e2‘u(—t) by using the provided tables of Fourier transforms and Fourier transform properties . Show all work. , Li“ "drum ’64 e uf'i‘l EH)”; e Problem 2 — Fourier Transform Tables and Properties 2a) Obtain the Fourier transform of the function :56) = 6tze“2‘u(t)+46(t— 1) by using the provided tables of Fourier transﬁmrns and Fourier transform properties . Show all work. 7/“ : 6tle'ltuﬂ') + #SR-t) / '\ w the 2b) Obtain the Fourier transform of the function y(t) : e”2iu(t) * e2‘u(—t) by using the provided tables of Fourier transforms and Fourier transform properties . Show all work. ,‘L+ g Lf (wt) f Z Z. 1‘; M) 1.4!») ’ x a r i, \/(‘V\ Iv (’21—;&J>’(/2.*JQB * \$14.11 ‘i i ——-j:"‘“"1“1‘ « ; YOU): aha/Vlvii (a +V ’- Problem 3 — More Fourier Transform Properties Use the basic properties of the Fourier Transform to ﬁnd the following transforms. 3a) Let x(t) 2 8‘21”. Then its Fourier transform is given by .7-"[a:(t)] : X(s) = “ﬂip If y(t) : 3:1:( then its Fourier transform is given by: 9 Mr W 2:) 3 ’4 Lav em = Exec] a 3 {3| MW) ; axe/(m) 9”)” were a eme‘w : 7 wme'w , -7;ch , «2nd ‘, . Ym = or." ’8 ~ .363 (ECLAL‘H-f (aw-L +L’ -2414) ?é €___, 0, U? 1 LI 3b) Let x(t) : 6"”. If y(t) :2 (21722536) then its Fourier transform is given by: _tz/ZG—2 _—— A14 6”. From» Jaw ﬁIMzw/«Ww: C HOW/3713 :1 a 7/47) a 0%.We‘zi—i :JFQ “if 'L f I ' 9R” ,— oqtﬂr—a (W) XEA *3 “(we wteomfw MC1 -52" Va»: ~ wLVTTT‘f-f " , 'L -v‘\/7€'¥ 30) Let = 64"”. If y(t) ‘2 a:(t)ej30‘ then its Fourier transform is given by: 0‘ M ; e—E’H .4?) >61») =47? ' \$— 9M a 7cm e’“ a X/w‘uai - OI ..«,. “ELM ' (wadxxuyq cal— 60:.) +5107 4 a “1—! 3b) Let \$(t) : 642. If y(t) = \$226) then its Fourier transform is given by: -14} Frau—y 4J1; ‘H #5111 We WW )(l‘u)? V?— e 7. dc ~ L‘JZ‘ 7:8,” W'L 3c) Let; z(t) : 6—3;“- If y(t) = \$(t)ej3°‘ then its Fourier transform is given by: ‘ ‘X 7C/q :/ Q'KHA érﬁ ‘vzjyﬁ "10+ 914:) ’1 761+)e) X/U’UOB _ "I __ q (w,3d\t+q 5.1% bag-M07 ,Lm. r.in Hm («Jami-+4 W’sou‘r‘im yt = __ "ﬁn—- 3d) Let x(t) : L What is the Fourier transform of :c(t) given by? 3+jt' 413% La “kiwi—a Ylu) VA», YIiJé—% 7n7¢/_w) -31‘: g M( é—é " J 3+Ju ? {are :2—9 ZTT e M "V J Problem 4 e Inverse Fourier Transforms Use the basic properties of the Fourier Transform and the Fourier tables to ﬁnd the following inverse transforms. 4a) X(w) : 1+ 61"” ~ 6‘23” 4b) X(w) = 46”?“ 64—3jw g2» L/ Q I i 8—21'W’ I WT... : 3 2~3U 6 ng‘u W 7 885(41th “Ki/3514M: “r 2“ ’73 (2* “A ’3' 6 awe—03 : ﬁe a (we) f'1[X(w)] = ____ Problem 4 — Inverse Fourier Transforms Use the basic properties: of the Fourier Transform and the Fourier tables to ﬁnd the following inverse transforms. ‘ 5H) +5695) ~5/t'l) W E E, I l y la) Lt e 1 ’ §f_€“lj‘~” ’1‘) ' 2—) 6 PZJV W 7 sum/«Q iﬁégbljmc "£925 4* 2‘ (1+ “0 rim»): E 6— L‘ (’(‘e"‘)\) : Zie “(1%) ...
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b09exam4sol - ECE2311 EXAM 4 B09 Name Mia:2” ECE Box...

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