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fall10-mid-term2-solutions - BEL 3123c TEST DURATXQK magnum...

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Unformatted text preview: BEL 3123c:- TEST? - DURATXQK: magnum; ‘ { ‘ 1114/2013” “ _ 1. You are given a square wave with time perimi TmlO having an ampfitude that swings 6 units. Place the waxisyaggd gjvaxisappropflatgiy gathat you, cm; expioi’t computatiaml simpiifigations/fmm qumter’wave symmetry. Then, compate' a.” a)“ and big. {50) 2 Loss at the ssprsssisn for Ck (complex form of Fourier series} in your chsat sheet Show that (I; z 01;? where 0;: denotés the complex conjugate of Ck. (20) " 7 has 1 3+? 3 H ' I as” _ ’ 0 3T to ' , Ts,“ - W A xA+jB V‘LlizyzAij’ 2% 32%? f{t)coslcwot«ij:°+T f(t) )sinkwgt as 5:” mm mam so“ f(t)sin(—e as) :3 c; /o 3 Show that tbs family 0f functions ,1 msin (mat) is orthsficnai; {an the istervai t = G to 1: ...~ Zs/w) _ $20 the famzly ei‘ fimstions f2 :2 sin. (mat) for any integers m and n as iong as m # n Hint: use trigonomsfiric identities as needed to simplify the integratisn (30) ’ 231/5: _ 27r/w 7 , / sin (mat) sin (mast) dt x g— ] [cos (n - m} wt —‘- cas {71 + m) wt} dt 0 a W ‘ (m) , 1 (w) 3 I 2 MM 27",“ W 2‘7th 2: 2 (n 4. m) as sin {n~ m) Latin , ~21“ + 721%.: sin (n + m) wtifl O (103 This IS the integral of two sinusoids over an integral multiple of a cycis and therefore amounts to zero. Thus, these {missions are orthogonai ...
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