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 Dec 13 comment Averaging the values of $\cos x$ over one period It should be included twice because it belongs to the interval $[0,2\pi]$ (not $[0,2\pi)$), doesn't it? Dec 13 asked Averaging the values of $\cos x$ over one period Oct 24 comment $f(t)f'(t)$ where $f$ is part of a Gaussian CORRECTION: edit time expired so I couldn't remove the $\frac{1}{4.5}$'s and the $\frac{1}{10}$ typo. Oct 24 comment $f(t)f'(t)$ where $f$ is part of a Gaussian then, every time the signal is applied (periodically or aperiodically) we get $$\frac{1}{T} \int\limits_{0.5}^{5} f(t)f'(t)dt = \frac{1}{4.5} \frac{1}{2} \int\limits_{0.5}^{5} d (f(t))^2 = \frac{1}{10} (f(t))^2|_{0.5}^{5} =$$ $$\frac{1}{4.5} \frac{1}{2}(-e^{-\frac{(0.5t-1)^2}{2}} -e^{-\frac{(t-1)^2}{2}})^2|_{0.5}^{5} = \frac{1}{4.5} (-1.28763) < 0$$ Oct 24 comment $f(t)f'(t)$ where $f$ is part of a Gaussian @Gerry, is this better: Let the form of the signal which is applied periodically or aperiodically on the system be a part of a Gaussian, such as $$f(t) = \begin{cases} \frac{1}{2}(-e^{-\frac{(0.5t-1)^2}{2}} -e^{-\frac{(t-1)^2}{2}})^2, & \mbox{for } t =\mbox{ 0.5