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from my DSP book,i am reading that exponential signal $x(t)=e^{-a*t}$ has some time constant value $c$ let say if $e^{-a*t}=e^{-t/c}$ which means that $a=1/c$,but i want to understand what is general formula to find time constant for other signals and also what does it express?i meant what is real mining of time constant in signal?clearly it cant be value where signal repeats itself,because in this case we call this number period,thanks in advance

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There is no general formula to calculate time constants for all signals. Time constants are defined for some signals. For your exponential signal

\begin{equation} e^{-a \cdot t} = e^{-t / c} \end{equation}

it just means that always after the time $c$ has passed, the value of the function has been reduced by $1 / e$. A more common interpretation is half-life defined as

$$t_{1/2} = \frac{ln(2)}{a}$$

which is the time after the function has gone from 1 (at time zero) to 1/2 (at time $t_{1/2}$) and so on.

However, there is no general time constant that can be applied to any signal, because, as you already mentioned, it is unclear what it represents.

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