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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
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where 0;: denotés the complex conjugate of Ck. (20) " 7 has 1 3+? 3 H ' I as” _ ’
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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
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‘ (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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 Spring '09

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