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Determine whether or not the following signals are periodic. If periodic specify its fundamental period.

Mumbai University > Computer Engineering > Sem 7 > Digital Signal Processing

Marks: 10 Marks

Year: May 2016

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  1. $x(n) = \sin(0.25\pi n + 0.4)$

    Solution:

    Consider Standard equation $x(n) = A \cos(\pi n+\theta)$

    Comparing given signal we get, $\omega n = 0.25 \pi n \\ \therefore 2\pi fn = \dfrac {\pi n}{4} \\ \therefore f = \dfrac18$

Result : Given signal, x(n) is periodic with Fundamental Period,N = 4

  1. $x(n) = cos(0.5n\pi) + sin(0.25n\pi)$

    Solution:

    Consider x(n) to be combination of $x_1(n)$ & $x_2(n)$

    Where, $x_1(n)= \cos(0.5 \pi n_1) \& \ x_2(n)=\sin(0.25 \pi n_2)$

    $\therefore, \omega_1n_1 = 0.5\pi n1 \ \ \ \ \& \omega_2n_2 = 0.25 \pi n2$

    $\therefore, 2 \omega f_1n_1 = \dfrac{1}{2} \pi n1 \ \ \ \& 2 \pi f_2n_2 = \dfrac{1}{4} \pi n_2$

    $\therefore, f_1 = \dfrac{1}{4} \ \ \ \& f_2 = \dfrac{1}{8}$

    $\therefore, N_1 = 4 \ \ \ \ \& N_2 = 8$

    $\therefore, N = LCM of N_1 \ \ \& N_2 = 8$

Result : Given signal, $X_2 (n)$ is periodic with Fundamental Period, N = 8

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