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

• kolycholy
In summary, if you want to find the inverse Fourier transform of 1/(1+8e^3jw), you can try muscle-through the contour integration.
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?

kolycholy said:
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

kolycholy said:
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.

Last edited:

What is the inverse Fourier transform?

The inverse Fourier transform is a mathematical operation that takes a function in the frequency domain and converts it back into a function in the time domain. It is the reverse process of the Fourier transform, which converts a function in the time domain into the frequency domain.

Why is the inverse Fourier transform important?

The inverse Fourier transform is important because it allows us to analyze signals and data in both the time and frequency domains. This is useful in many fields, including signal processing, image processing, and communication systems.

What is the formula for the inverse Fourier transform?

The formula for the inverse Fourier transform is: f(x) = (1/2π)∫F(ω)e^(iωx)dω, where f(x) is the function in the time domain, F(ω) is the function in the frequency domain, and ω is the frequency variable.

What is the relationship between the Fourier transform and the inverse Fourier transform?

The Fourier transform and the inverse Fourier transform are inverse operations of each other. This means that applying the Fourier transform to a function and then applying the inverse Fourier transform to the result will give back the original function.

How is the inverse Fourier transform used in real-world applications?

The inverse Fourier transform is used in a variety of real-world applications, such as audio and image compression, speech recognition, and medical imaging. It is also used in fields such as physics, engineering, and economics for data analysis and modeling.

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