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I want to get the expression of signal in time domain by using inverse fourier transform. The signal in frequency domain is a little special: H(jw) is 1/(a+jw) only when |w|< W; and when |w|> W H(jw)=0. Can we use inverse fourier formula to get h(t)? In other words, can we get the expression of h(t) by calculating the integral below? $$ h(t) = \frac{1}{2π}\int_{-W}^W \frac{e^{jwt}}{a+jw}dw\,. $$

Please help me! Thanks a lot.

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  • $\begingroup$ Assuming that $a$ is not purely imaginary, yes. You could also use some convolution result. $\endgroup$ – copper.hat May 20 '15 at 20:51
  • $\begingroup$ In fact a is positive real. Could you give me more details? $\endgroup$ – Albert May 20 '15 at 21:26
  • $\begingroup$ I'm not sure there is an elementary answer. I thought your question was about existence. $\omega \mapsto {1 \over a+ i\omega}$ is the Fourier transform of $t \mapsto e^{-at} u(t)$, and $\omega \mapsto 1_{(-W,W)}(\omega)$ is the Fourier transform of a scaled version of $\text{sinc}$, so $h$ must be a convolution of these two. $\endgroup$ – copper.hat May 20 '15 at 21:36

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