Question

1. (a) You have seen that the Fourier transform of cos(wt) and sin(wt) func- tions results in even and odd combinations of delta functions in the frequency domain. Prove the opposite. That is, find the combination of delta functions in the time domain that give cosine and sine functions in the frequency domain. (b) Use the signum function to relate these two combinations of delta functions and use the convolution theorem to show that sin (wt) = cos (wt) * - cos (wt) = – sin (wt) * 3

Answer 1

Tourile transform F 1P (4) e 1204 fourier bransform. X(w) * ct) é just at x ) = S(t -to) X(w) - I See-to & Jutat hat petto dp-dt xiwla Is(p) & Jew (pt to ap- & -jw to x4 )= Sletto) X justo ( w e wto -jw to jwto ti je to Cos(w to d = Acording 8 (Etto) + 8(t-to) to superposition will give Cos(wlo) sltto) - 8(E-tol and give sin(wto). will

i setto) + 3u-to) Cosc w to) FT of 84 4to) - 867-tol) sin(who) Fot og {sgnus) jú FoT of {.f@g} = {f) •F{g} - Cowolution Theorem. Coswt) ® z afity scetto) + sutto] 2 irongisgncer? - j Fre Settlo? +S4-fo) Sgn (t)} FTS sittto) Sgnl) + sit-to sguit) to) sonu) tsu-tol sgult) =fot s sgnto) f ( t tto) + sit-to) sznu a we know sgnl-x) = -Sgnix) » Foti { sgn.(to) [- sattb) + 8 + ] 2 -fath S(t+16) - Sct - to?) sgu (tos? = F-Ts Sutto)-s4-to) = . sin(wto)

-Sinlwt) Z – – F.Th St to) - Sto F.T & Spelti & 23 – F.Th 14+to) -8 (t to) - sgnlt, } FT { Sutto) Sgnit) - 14t-to) Sgn H) 3. = -foT { Sgn (3 to seto) - 84 to) Sgri (to) } 2 tol sittto) + 8 it -foT los (wit)
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