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Determine the fundamental period of the following signals: (i) $x(t)=\cos (\frac \pi3t)+\sin(\frac \pi4t)$ (ii) $x[n] = \cos^2[\dfrac \pi an]$

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 05

Year : DEC 2014

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i )

Since $\sin (t) = \cos⁡(t- \dfrac π2) \\ ∴\ sin (\dfrac {πt}4)= \cos⁡(\dfrac {πt}4- \dfrac π2)\\ ∴x(t) = \cos(\dfrac π3 t) + \cos⁡(\dfrac {πt}4- \dfrac π2)$

$T= \dfrac {T_1}{T_2} = \dfrac 68= \dfrac 34$

Since T it is a rational ∴ signal is periodic

∴ Fundamental period $(T) = T_1 = \dfrac 34 T_2=6.$

ii )

$x[n] = \cos^2 \dfrac π8 n$

$\cos^2 \dfrac π8 n= \dfrac {1+ \cos(2* \dfrac {πn}8)}2 \\ = \dfrac {1+ \cos(\dfrac {πn}4)}2 \\ = 0.5 + 0.5 \cos(\dfrac {πn}4) \\ w = \dfrac π4 \\ 2πf= \dfrac π4 \\ f = 1/8$

Fundamental period T = 8.