New Basis Functions. Section 8. Complex Fourier Series


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1 Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting btwn th two typs of rprsntation. Exampls ar givn of computing th complx Fourir sris and convrting btwn complx and ral sriss. Rcall that th Fourir sris builds a rprsntation composd of a wightd sum of th following basis functions. (i.. a constant trm) cos(t) cos(2t) cos(3t) cos(4t)... sin(t) sin(2t) sin(3t) sin(4t)... Computing th wights a n, b n and c oftn involvs som nasty intgration. W now prsnt an altrnativ rprsntation basd on a diffrnt st of basis functions: (i.. a constant trm) it 2it 3it 4it... it 2it 3it 4it... Ths can all b rprsntd by th trm int with n taking intgr valus from to +. Not that th constant trm is providd by th cas whn n = 0. 2
2 Sris of Complx Exponntials A rprsntation basd on this family of functions is calld th complx Fourir sris. c n int Th cofficints, c n, ar normally complx numbrs. It is oftn asir to calculat than th sin/cos Fourir sris bcaus intgrals with xponntials in ar usually asy to valuat. W will now driv th complx Fourir sris quations, as shown abov, from th sin/cos Fourir sris using th xprssions for sin() and cos() in trms of complx xponntials. d + = d + = d + = whr c n = Complx Fourir Sris a n cos(nt) + b n sin(nt)] a n ( int + int 2 (a n ib n ) int + 2 c n int ) ( int int )] + b n 2i (a n + ib n ) int 2 d, n = 0 (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... Not that a n and b n ar only dfind whn n is ngativ. 3 4
3 a n = cos(nt)f(t) dt b n = sin(nt)f(t) dt d = 2 f(t) dt thus for n positiv c n = 2 (a n ib n ) = cos(nt) isin(nt)] f(t) dt 2 = 2 int f(t) dt for n ngativ c n = 2 (a n + ib n ) = cos( nt) + isin( nt)] f(t) dt 2 = 2 int f(t) dt Complx Fourir Sris Summary c n = 2 int f(t) dt c n int and for n = 0 c 0 = d = 2 0 f(t) dt 5 6
4 Complx Sris Exampl Find th complx Fourir sris to modl sin(t). c n = 2 int f(t) dt = 2 int sin(t) dt = in in ] 2 n 2 Which is zro whn n dos not qual or. For ths two spcial cass w hav to st n = + ǫ and calculat th limit of c n as ǫ tnds to zro. This givs us c = 2i c = 2i Which mans th complx Fourir sris for sin(t) is = it it 2i c n int 7 Finding th limit as n tnds to c n = 2 in in ] n 2 St n = + ǫ and lt ǫ tnd to zro. c = i(+ǫ) i(+ǫ) 2 ( + ǫ) 2 = iǫ + iǫ ] 2 ( + ǫ) 2 ] iǫ + iǫ 2 + 2ǫ ] 2iǫ 2 2ǫ i 2 2i 8
5 Complx Sris Exampl 2 Find th complx Fourir sris to modl f(x) that has a priod of 2 and is whn 0 < x < T and zro whn T < x < 2. f(x) c n = 2 int f(t) dt = i 2n T int ], whn n 0 = 2 ara = T 2, whn n = 0 So th Fourir sris is c n int = 2 T + + i n i n int ] int int ] int 2 9 x Convrting c to a, b and d From our xampl on th prvious pag. i 2n int ], whn n 0 c n = 2 ara = 2 T, whn n = 0 W wish to calculat th cofficints for th quivalnt Fourir sris in trms of sin() and cos(). Clarly d = c 0 = T 2. For n > 0 c n = (a n ib n )/2 a n = 2 R{c n } and b n = 2 Im{c n } convrting our xprssion for c n into sin() and cos(): so 2c n = i cos(nt) isin(nt) ] n = sin(nt) + i(cos(nt) )] n a n = sin(nt) n and b n = cos(nt). n 20
6 Complx Fourir Sris 2 T + Ral Fourir Sris T i n int ] int i ] int int n sin(nt) cos(nt) n cos(nt) + sin(nt) n Both sriss convrg as /n. 2 Convrting from Ral to Complx Convrt th ral Fourir sris of th squar wav f(t) to a complx sris. 2 f(t) For th ral sris, w know that d = a n = 0 and b n = 4 sin(nt)f(t) dt = n, n odd giving 4 sin(t) + sin(3t) 3 + sin(5t) ] To convrt to a complx sris, us d, n = 0 c n = (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... so w hav c 0 = 0 c n = 2i/(n), n positiv and odd c n = 2i/( n), n ngativ and n odd 2i it 5 + 3it 3 + it t 2 + it + 3it 3 + 5it ]
7 Gnral Complx Sris For priod of 2 c n = 2 int f(t) dt 2 0 Similarly, for priod L c n = L f(x) = L c n int 0 inx2 L f(x) dx c n inx2 L Th fraction 2 L is oftn writtn as ω 0 and calld th fundamntal angular frquncy. Exampl A vn function f(t) is priodic with priod L = 2, and cosh(t ) for 0 t. Find a complx Fourir sris rprsntation for f(t). f(t) c n = L = 2 L 0 int2 L f(t) dt 2 0 int cosh(t ) dt = sinh() + n 2 2 t 23 24
8 Hnc th complx Fourir sris is = c n int2 L sinh() int + n 2 2 W can chck this answr by computing th quivalnt ral Fourir sris which w calculatd at th start of sction 7. a n = 2 R{c n }, n =,2,3,... b n = 2 Im{c n }, n =,2,3,... d = c 0 In this cas, as c n is ntirly ral, a n = 2c n = 2sinh() + n 2 2, n =,2,3,... b n = 0 d = sinh() Exampl 2 Find th complx Fourir sris of th th squar wav f(x). L f(x) Not that th man of th function is zro, so c 0 = 0. c n = L L 0 inx2 L f(x) dx = L/2 ] L inx2 L dx L 0 L/2 inx2 L dx = 2in + 2 in ] 2in in ] f(x) = inx2 L in n = n 0 f(x) = ix2 L i 5 + ix2 L + 3ix2 L 3 + 3ix 2 L 3 + 5ix2 L 5 + ix2 L L +... x 25 26
9 Convrting to a Ral Sris W wish to convrt th complx gnral rang squar wav sris into a sris with ral cofficints. { 2/(in), n odd c n = 0, n vn Clarly d = c 0 = 0. For a and b us: c n = (a n ib n )/2 a n = 2 R{c n } = 0 and b n = 2 Im{c n } = 4 n, n odd Which givs us th ral sris: 4 sin ( x 2 L ) + sin ( 3x 2 ) L 3 + sin ( 5x 2 L 5 ) +... For priod L Sction 8: Summary c n = L f(x) = L 0 inx2 L f(x) dx c n inx2 L Rlationship with th cos/sin Fourir sris. d, n = 0 c n = (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... a n = 2 R{c n }, n =,2,3,... b n = 2 Im{c n }, n =,2,3,... d = c
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