Inverse Fourier Transform of 1/(1+8e^3jw): A Contour Integration Approach

[kolycholy]

How do I find inverse Fourier transform of 1/(1+8e^3jw)??
Now, it would have been easier to find inverse of 1/(1+1/8e^jw), because that would be just (1/8)^n u[n]
i think i basically need a way to write 1/(1+8e^3jw) in a form described below:
A/(1+ae^(jw)) + B/(1+be^(jw) +C/(1+ce^(jw)
where a, b, c are less than 1.
How do I do that? partial fraction can be killing, because the process will be too long. Any other smarter methods?

 

[axax]

How do I find inverse Fourier transform of 1/(1+8e^3jw)??
Now, it would have been easier to find inverse of 1/(1+1/8e^jw), because that would be just (1/8)^n u[n]
i think i basically need a way to write 1/(1+8e^3jw) in a form described below:
A/(1+ae^(jw)) + B/(1+be^(jw) +C/(1+ce^(jw)
where a, b, c are less than 1.
How do I do that? partial fraction can be killing, because the process will be too long. Any other smarter methods?

find (2-e^(-jw))/(1+e^(-j3w)/8) inverse Fourier transform

 

[jackmell]

How do I find inverse Fourier transform of 1/(1+8e^3jw)??
Now, it would have been easier to find inverse of 1/(1+1/8e^jw), because that would be just (1/8)^n u[n]
i think i basically need a way to write 1/(1+8e^3jw) in a form described below:
A/(1+ae^(jw)) + B/(1+be^(jw) +C/(1+ce^(jw)
where a, b, c are less than 1.
How do I do that? partial fraction can be killing, because the process will be too long. Any other smarter methods?

How about just muscle-through the contour integration:

$$\frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{e^{iwx}}{1+8e^{3iw}}dw$$

I suspect that may be just a sum of residues if x>0 however I've not gone over it rigorously so I'm not sure. Just another possibility you may wish to consider.