Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 05

Year : DEC 2014

(i) $x(t) = t^3+3t$

$$x(t) = t^3+3t$$

$x(-t) = -t^3-3t$

Even component $= 0.5[ x(t) + x(-t)] \\ = 0.5[t^3+3t-t^3-3t] \\ = 0$

Odd component $= 0.5[ x(t) - x(-t)] \\ = 0.5[t^3+3t+t^3+3t] \\ =0.5[2t^3+6t] \\ = t^3+3t.$

(ii) $ x[n] = \cos[n] +\sin[n] +\sin[n]*\cos[n]$

$$x[n] = \cos[n] +\sin[n] +\sin[n]*\cos[n]$$

$x[-n] = \cos[-n] +\sin[-n] +\sin [-n]*\cos[-n] \\ = \cos[n] – \sin[n] – \sin[n]*\cos[n] $

Even component $= 0.5[x[n] + x[-n]] \\ =0.5[\cos[n] +\sin[n] +\sin[n]*\cos[n] + \cos[n] – \sin[n] – \sin[n]*\cos n] \\ = 0.5[2\cos[n]] \\ = \cos[n] $

Odd component $= 0.5[x[n] – x[-n] ]\\ =0.5[\cos[n] +\sin[n] +\sin[n]*\cos[n] – \cos[n] + \sin[n] + \sin[n]*\cos[n] \\ =0.5[2\sin[n] + 2\sin[n]*\cos[n]] \\ =\sin[n] + \sin[n]*\cos[n] \\ =\sin[n] [1+ \cos[n]].$