Question

3. Let the following periodic signal : x(t) = m+0 8(t -- 3m) + 8(t-1-3m) + 8(t – 2 – 3m) be the input to a LTI system with a system function: H(s) = es/4 – e-s/4, Let by represent the Fourier series coefficients of the resulting output signal y(t). Determine bk. (5 points)

Answer 1

MO 100 (ME-7-778+ (ME-1-7) 8+ (usa) 3 = (1) n + 514 - 514 (5) It X (L) = & (t) + 6(1-1) + 6(1-1) [in= e) +8 (L-5) +$(6-4) 68 (7-5) +$(t+3) + $(t+2) + S(t +1) +8(4-6) +8 (t-7) +8 (t-8) +S(t+6) +(t+5) +8 (t+4) [+n=1 -- m=2) [m=2 12(11) 2) 1 2 3 4 5 6 7 8 st - -6 -4 -4 -3 - 3 So, 216) is periodic with period T .: Wo = 2 2 TT T = 2T 3 뱅 The focusick Coefficient be for is 8 2 (t) T bx a ? 5 Kyt). Sin Kwot dt = $ (647+ 6(t-1)+8(4-2) sinkwt dt Sale{+ -t) = X(t) fra tieto eta 2/3 17

: bk = Sin (kw. (0))+ sin(kW. (1)) + Sinje wac9)) = FMCo) + Sin (wa)t sin(2xw.) ] 23/35 [ sin kwot sim akwo 3 [sm (Prke ) + Sim (49 k) - ':wo 201 3 2 bx - H(6) = es14 - es14 9W/4 3w/4 H GW) - المان -jW14 .: HBW) =e e X 23 Bj H(JW) =2jsin (ut the y (t) of The Coefficient of imput a (t) is be output coefficient ie-, Coefficient is Toko + (wa) ] ... bru for y(t) br. H (Kwo) Gm 2TIK +Sim bay K. 211 r +Semicork] Hlk. = 3 monk + simu nk). 27 sinf ) 4 [Simak & Simutk . Sim in (177 Coefficient bk of u (t) is boky = 3 6