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This expression: $x(t)=[e^{-3t+5}] u(t-1)$.

I am trying to take the Fourier transformation of the above expression.

I know that for $x(t)=[e^{-at}] u(t) \leftrightarrow \frac1{i\omega+a}$.

But, how am I supposed to transform the above expression? It looks similar to the known transformation, but it certainly is not =/


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up vote 0 down vote accepted

Write $x(t) = e^{-3t+5}u(t-1) = e^{-3t+3+2}u(t-1) = e^2\cdot e^{-3(t-1)}u(t-1)$.

Now notice that $x(t)$ is a shifted version of $y(t) = e^2 \cdot e^{-3t}u(t)$, so apply the shift theorem to $y(t)$ and you should have your result.

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$x(t)=[e^{-3t+5}] u(t-1)=[e^{-3(t-1)+2}] u(t-1)=e^2 e^{-3\tau}u(\tau), $ where $\tau=t-1.$ Then use linearity of Fourier transform.

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