quiz03_2_sol

# quiz03_2_sol - ECE 108 Signals and Systems Spring 2007...

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Unformatted text preview: ECE 108 Signals and Systems Spring 2007, Instructor: Tiffany Li QUIZ. 3 (50 minutes) Name: 1. (24 point) Deﬁnitions of Fourier series and Fourier transform: Trigonometric Fourier Series (8 points): LO \ ﬁt): 620+ % (an 506(I’Jl/002i1) 'f" 5915!“ (‘4 W91“) I [102“???[n 3”er— .2, an: 7 73 fatmosrnwnwacs who “rt/await ﬁt) — gm DM 6 77?; D”=’~%; in frrMﬁw/CM 2. (30 points) Given f (t) = cos(w0t), graph the spectrum of ﬁt) cosmet)" Assume we > 7.05“ fa?) CD imp-k!) (\$29 ‘3 C to we) +F( La eryﬂ I {Jr/DENISUM) W I ﬁx) (3(a) (I {3* C0 ‘6‘) ‘ 7? 9 mm ) 5( cf)\$\$3[§{wwm~wc) +5m+wnvcoij +5lLO“ wéi+w6) ("3 'f (“)0 H’Jcﬂ :2? 1h \‘ga any dg/mz‘e,’ 4 5/)663'741 bile; ZE- lowﬂof ‘ ‘ l } W 4i [/00 Hue! pug—mug , qwﬂpwo/ i ... Wwa AMP WW Sfacfmw TT W‘Taﬂ‘wc '“U’c, "HA 0 PM SR S’pédmw .. 3. (30 points) Signal f (t) has a Fourier transform F(w) m 005(2w). (3 points) Is ﬁt) real?( T F ) f a I A Vea( (3 points) 15 m) periodico @) fut) teal \$9 an (an (=9 Frau) MM? ) ? ( T (3 points) Is ﬁt) even? ((1) F (t (spoimsusf )odd?( T F ) JCHT) reﬁt 05m \$1? FM) puffy (harm?! 51; MM fut) fﬁm'd’b (:9 RIWComstf; “ff 5”th SfE-C'ﬁ’id (18 points)Plot the amplitude and phase spectra of f , Fa" F012)":— C°S(?,o=3): ’02; (2,03), £4 ( ) M1125 (gm-«nay /JNV\{7((,M.Q : (®S(lw}) (9/1662 ._ 0 idwa gunﬁgo, 7T {MW-1 tins awaits? /"* I PM” // Mm «ﬁmwﬁy / 4. (16 points) Prove the foliowing Fourier transform pair 27rf1(t) f2(t) 42¢ Flue) * F2(w) Rm) av ﬁrm) sf: gm ﬂea) dv - “I _ ____ f “ft/30w _ gilﬁt JL [(—[Cwahv] momma a q an,» M ' «OD _. i w» “W” _. . '43th I “Rico .ogoF'W) fl(wwv)dv‘ e 06‘“ \ , (ﬂea-"Wi— WO 15m JVﬂl _, div) ...._L‘ , «u dv "zﬁi,uo ENROQ 'i’LU’B )Q T K dlw«V) [gt W’W’V W {.69 turn“ \Vi—W grwik WEE HWEJ {glove cit/LL10); ~ ""5 Am WW m o‘vﬁ m e. ,\ "foo." govt ] wt .. , d’ by] rzn‘ [EL “00 at? 3275900 jeeﬁt) Table 1: Fourier Transform Operations Addition f1(t) + fga) F1 (71)) + F2(‘UJ) Seaiar multiplication kf (t) 19F (to) Symmetry F(t) 27rf(~w~w) Scaling (a real) f (at) Time shift ﬁt — to) F(w)e‘jw‘° Frequency shift f (t)eju""t F (w w 100) Time convolution f1(tW2(t) F101;) e F2('LU) Time differentiation 9% (jw)"F(w) Time integration Kw f(\$)d.’l’: + 7rF(0)6(w) ...
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