Question

Find the trigonometric Fourier series (FS) and the exponential FS of the following: 2 *(1) 1.5 1 0.5 O -0.5 1 -1 2 -1 0 2 b) 2 3 4 6 exponential FS Cnejnwot f(t) = En=-00 Where Cn 7Se+ f(t)e-inwot dt trigonometric f(t)= a, +Ža, cos(n6,t)+b, sin(n0,1 ao 1 T. 2 to an S f(t)dt sº f(t)cos(n0,1)dt f(t)sin (no,t)dt To 2 pt b,

Answer 1

hwo 9 Sin.. Solution (0) 0 (3,4) (40) -) This signal is every signal then we caculate the ao and qui 9-y= -4 (1-3) bo o for even signal + 16 To y-ep=-un To foodt f (t) dt = ) t 1/2 =) 9.2 ( -2+3) 2 + 1 ( 2-1)dt Jln 2 39= 2** **) 212-4 "/

7 9. [- +² +38]} + ()ا a (4= --- - (31-(4) Iti 4 2 (0 792 = Ć - 3 4 2 9t3 2 for calculating the as >> Oy focos not dt an 2 To=1 To 0 - au Tola f(t) cosh Notat t iTo f(t) Eushwat dit T Toln 12 an= 2 (24+3) Coswobot du + (27- 1) Lush Wolat VN

an=2 (Jéz+4360 (-2++3) Cashouldt & (2+4) Coshoot at Vz 1/2 au 24+3) / Sinot Coshoot .) - (-) 6- hwol (ww.)? 201) Sinnwort 2 - Sognwol (hw.)? hwo On = 2 -1+3) / 1+3) Sinnwo 2 [toshoo hwol (hw.)? + 2 (hw.)? + Sinnwo +2 Coshoot (hw.)? Cos hwo (w.)? 2 Sinnwo Wo & Sinnwo an=2 ( 2 Cos hwo 2 hwo z + 2 (hwo)? (hw.)?

=)=212 (2 sino & Sinnwo Coshwo + 2 2 hoo hwo (hwo)? (hwo)? f(t) = 9+ E an cos (hot) + { bn sinchat) n=1 IF(A) = 3 + Coshod +2 § 28 sinnwo + Sin nood nwo 2 2 h=1 (hwest hwo (hw.)? Coshwall Solution - (b 4 5 2 3 4 6 - 4 we calculate T the This is odd signal then only 9. and bo - anno for old signal

Calculation of go iTo =) 971 + f(A) dt To 2 79. ។ 5 dtt odt t 6 (4 + +16)dt U 12 6 3 -4dt --- +(2-4) 4 8 = 4 3 6 =) 9 = 1 (uli

To - bn= bn=2 27 f(A) Sinnwordt 2 bn= 2 5 sinnwot + (-44 +16) Sin hooldt 6 0 6 s -4 Sin noot et 2 bn=15 -Coshwoh 3 hwo +16) (-Coshwol 4 6 + 4 4 Coshwat - 4 Sinnoot (hw.)? 2. nwo 3 5 bna 5 (cosa nwo - Jeft 3 howo y sin unwo (nw.j² Sinhwa (how.)? + Coshoof + 4. ( hwo + 4 coschwo Cos 5hwo hwo

= bn Cosznworl - 4 Sinunwo (cos zowo-1) 3 hwo ty Cos3hoo (w.)? Sin 3hwo i nowoje hwo + 4 (Costnewo - Cos5 new) hwo f(t) = 90+ Eran cosnw, t & br Sinnoot :) h= es (cos (2hw.)-1) - 4 sinunwo If (A) = 4 -5 htoo 2 + olm (hu) n=1 + 4 cos3nw. & sin 3n400 +4 (Cosso-lossnew hwo Thew.)? hwo & sinnwot