Question

Is the signal

$ x(t) = \sum_{k = -\infty}^{\infty} \frac {1}{(t +2k)^2 +1} $


periodic? Answer yes/no and justify your answer mathematically.

Answer

Yes because:

$ x(t+2) = \sum_{k = -\infty}^{\infty}\frac {1}{(t+2+2k)^2+1} = \sum_{k = -\infty}^{\infty}\frac {1}{(t+2(k+1))^2 + 1} $

Change of variable, let r = k+1

$ => \sum_{k = -\infty}^{\infty}\frac {1}{(t+2r)^2+1} $

This equation is of the original form, therefore it is periodic.

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