Question

Consider a periodic, continous-time signal x(t) = sin(21500t) a) Find the Fourier series form of x(t) b) Find the Fourier transform of (t) using the Fourier series form of x(t) you found on (a) c) Throughly explain the formulation between Fourier series coefficients and Fourier transform.

Please write neatly, readable and explain the steps please. Thank you.

Answer 1

Given; x(t): Sin (29. soot) Time period of xlt) - 27 . We can write act) = eiust - Just & W. = 1000 TT rad/see Y 500(201) 500 By Definition of Exponential Fourier series, Any signal #tt Ticta Š cheokass to acti con be written as on on comparing 0 We get @ = -1 & c. = 1 and remaining Coefficients are Zero. suot ilire to Fourier series of act) = c. elust &

- 6 By the concept of Duality we get TF{l.cvut ? = .5w-w) * XW) - e, the Just } + c, asetusty Xu = 5, [20-646+, (2n-66w.cz] en] s[wtw.)+7 (20) 56w-w) X(w) - 1660-0) - scw+wa])69 = lovit c) from Fourier series, we got 1 x(t) = 24 kust where = + salt) e ka de oet If we consider, one Cycle of this periodic signed alt) ! By Definition of Fourier Transform, - flow) = facts. ejet at (w) - Cacts ējut.dt. periodie Signad alt) le; âlt) 94 T is such a large value . Let us consider t o - and If we sampled that it is 0. the above Expression is w w = kW & Taking limits on both sides 660.no, #$ 44). estostert

6 T700 T han f fw auto, e t avec lest det As 7-700 ; ált) rölt) * * xwolwewe = . szelle) e strast det 4 x kw) - € 194 Suc4.eitas dit ika" xlt) e de LE LE to T ) t W = kw. THOOT T+00 Tro TT lei ffrom ég 0 Station fx|ke) - 4. ( THO * TCH - X (Mad) tok Cr= 4xcam..