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Let the function be

$$ f(t) = e^{-3t} u(t -1) $$ The criterion is set at 70% of the total signal energy.

I know how to find total energy, but not how to find the the $W$ (or frequency) at which the signal is a certain amount.

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Find the Fourier Transform $F(\omega)$ and perform the following integration.$\frac{1}{2\pi}\int_{-W}^{W}|F(\omega)|^2\mathrm{d}\omega=0.7E$, where $E$ is the total Energy of the function given by: $E=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|F(\omega)|^{2}\mathrm{d}\omega$.

The value of $W$ is the 70% Bandwidth.

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  • $\begingroup$ Isn't the first integral you said multiplied by 1/pi? $\endgroup$ – bluejamesbond Sep 24 '13 at 2:48
  • $\begingroup$ Why do you want to multiply by $\frac{1}{\pi}$? According to Parseval's theorem, $\int_{\infty}^{+\infty}|x(t)|^{2}\mathrm{d}t=\int_{\infty}^{+\infty}|X(f)|^{2} \mathrm{d} f$. Simply stated, the Energy in the signal can be found out directly by summing the squared magnitudes of the Fourier Transform across all the frequencies. $\endgroup$ – Sudarsan Sep 24 '13 at 3:42
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    $\begingroup$ A minor correction: the Fourier transform is $X(f)$, not $|X(f)|$. Since OP @mk1 is using $f$ for the function and distinguishing between "W" (presumably $\omega$ or radian frequency is meant) and "frequency", he/she is undoubtedly using Fourier transforms w.r.t radian frequency in which case factors of $1/2\pi$ (or $1/\pi$ for those who do not believe in negative frequencies) do show up in Parseval's theorem. $\endgroup$ – Dilip Sarwate Sep 24 '13 at 13:42
  • $\begingroup$ Modified for the Fourier Transform relationships in Angular frequency. $\endgroup$ – Sudarsan Sep 24 '13 at 17:06
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    $\begingroup$ @mk1 What is meant is, calculate the value of the integral $\frac{1}{2\pi}\int_{-W}^{W} \cdots $ which will give a **function** of $W$, say $g(W)$. Set $g(W) = 0.7E$ where $E$ is a known number, and solve for $W$. $\endgroup$ – Dilip Sarwate Sep 24 '13 at 18:47

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