[Math] Periodic and Aperiodic signal properties.

periodic functionssignal processing

I have three question on periodic and aperiodic signal that I have doubt about also I couldn't find the answer on searching so I am asking here.

  1. Sum of a periodic and aperiodic signal always be a aperiodic signal?
  2. If a signal is periodic will its derivative always be a periodic?
  3. If on signal $x(t)$ applying $\lim_{t\to\infty}x'(t)=0$,can we say its aperiodic?Why?(this is question because of this answer )

Please answer with a example

Best Answer

  1. No, the sum of a periodic and an aperiodic signal can be periodic. Consider for instance:

    • $f(x)=\sin(x)$, which is a periodic function with period $2\pi$,
    • $g(x)=-\sin(x)+\sin(\sqrt{2}x)$, which is an aperiodic function as $\displaystyle \inf_{u,v\in\mathbb{Z}}\{\left|u\sqrt{2}+v\right|\}=0$,
    • $f(x)+g(x) = \sin(\sqrt{2}x)$ is a periodic function of period $\frac{2\pi}{\sqrt{2}}$.

  2. Yes, if the derivative of a function $f(x)$ exists, and $f(x)$ is periodic with period $T$, then $f'(x)$ is also periodic with the same period $T$ because, for all $x$ and all integers $k$, $$f^\prime(x+kT)=\lim_{h\to0}\frac{f(x+kT+h)-f(x+kT)}{h}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=f^\prime(x).$$

  3. If a periodic function $f(x)$ satisfies $\lim_{x\to\infty}f(x)=0$ then $f(x)=0$ for all $x$. You can prove it using $$\left|f(x)\right|=\left|f(x+kT)\right|<\varepsilon_k$$ with $\lim_{k\to\infty}\varepsilon_k = 0$ by the definition of the convergence. Therefore, $f(x)=0$ for all $x$.
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