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I am trying to write mathematically how to calculate the power of a finite signal. I know the general formula $P_{\mathrm{x}}=\lim_{T\to\infty} \frac{1}{T} \int_{t=0}^{T} x^{2}(t) \mathrm{d} t$ used to calculate the power of a signal which has infinite Energy and finite power. But how do you express it when the does not go to infinity, is not period and does not have infinite energy?

I know its a basic but i have searched all over and have not been able to find a proper answer.

Thanks in advance.

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First of all, note that your formula for signal power is only valid for real-valued $x(t)$ that satisfy $x(t)=0$ for $t<0$.

A more general formula is

$$P_x=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2dt\tag{1}$$

For signals with finite energy

$$E_x=\int_{-\infty}^{\infty}|x(t)|^2dt\lt\infty\tag{2}$$

the power $P_x$ defined by $(1)$ is clearly zero.

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