Question

Consider the following three signals: a) X(t)= e 104 b) x2(t)=sin(2net)+sin(20ạt) (i.e. a combination of 1Hz and 10 Hz frequencies); c) xz(t)=e'sin(at)u(t). Calculate analytically (or derive from the tables of standard transforms) their Fourier transforms and unilateral Laplace transforms. Compare the Fourier and Laplace transforms and comment on relations between the Fourier transform and the unilateral Laplace transform. Page 1 ECCE 302 Signals and Systems Laboratory Transforms d) Fourier transform YY(6) of some unknown signal xx(6) is given as follows: i los 10% YY(0) = F{xx(t)} = { 0 |0>10 Find analytically xx(t) as the inverse Fourier transform of YY(o), i.e. find the symbolic expression F-'{YY(0)} (use the Fourier transform table of standard functions).

Answer 1

Answer: @ of LIT, we have Given that : x (6): e-ott, definition 44 $Ct) ja fest f(t)dt 44x8(t)} 5* est srca de By the SX e-10141 dt - sterst_lot at sto (590) - [o-carosty to 1 (iedo) (6+10) Esti) Stto SHO

© Xat)= sin(2017) + sin(2016) Ciea combination of fHZ and 10 Hz frequnces); x2 (tt sin (2 t) + sino (2014) taking Laplace transforms' are both side, 4{ x2Ct)y=4* sin(2016) + Sir (2014) - Lf sin (20)+-{sincont)} we have Lf sinat} = 2 = 2T s²+(21)2 + 201 824 (2011)2 © xz(t)-et synt) u(t) By the first shifting theorem we have Le-ats Ctly-f(sta) het f(1) = sin (Tt) u(t) => L{FCE)} = {{sin rie)u(t)}

we have hq f(t-a) ut-a)} = e-as f(s) - 4 sin (174-0) u (t-o), selos. L{5% (+1-ty - $2412p=f60) :.4let sincht) UCH)) = $(5+1) (Sti272,