I want to know whether the following is periodic or not periodic

Question

I have a question about system properties of the following function whether it is periodic or aperiodic.

With an insight, I'd determine the function is aperiodic since the unit-step term looks implying that jump discontinuities occur at odd number times but don't have a detailed solution in mathematical terms.

Many thanks in advance for your help.

$$f(t)=\sum_{n=-\infty}^{+\infty}e^{-(2t-n)}u(2t-n)$$

Yes, I also saw the problem in that way at first but was doubtful if there were errors, especially regarding u(t). Thank you for your advice. — MathEW, Sep 19 '14 at 08:52

Chris · Accepted Answer · 2014-09-19T08:46:57.863

1

The function is periodic with period $1/2$ since you have $$ f(t+1/2)=\sum_{n=-\infty}^\infty e^{-(2t-n+1)}u(2t-n+1)=f(t), $$ as you can easily see by an index shift in the sum.

Hope this helps...

edited Sep 19 '14 at 08:46

answered Sep 19 '14 at 08:38

Chris

398

First of all, thank you Chris for your quick answer. But can you assure me if jump discontinuities don't take place despite the equation u(t)=u(2t)? – MathEW Sep 19 '14 at 08:56
You're welcome. You can have as many discontinuities as you want. Just define $u$ to be $0$ outside the interval $I=[0,1/2]$ and discontinuous inside. Then you will find that $f(t)=e^{-2t+1}u(2t+1)$ for $t\in I$. If $u$ was discontinuous, then $f$ is also. – Chris Sep 19 '14 at 09:23

score 1 · Answer 2 · edited Sep 19 '14 at 08:53

1

Hint: Let $a\gt0$. Every function $f$ defined as $$f(t)=\sum_{n=-\infty}^\infty g(at-n),$$ is periodic with period $____$.

edited Sep 19 '14 at 08:53

Jean-Claude Arbaut

23,277

answered Sep 19 '14 at 08:50

Did

279,727

1

?? "Next time", you will choose the answer which suits you best, no? – Did Sep 19 '14 at 09:07

I want to know whether the following is periodic or not periodic

2 Answers2