# Inverse Fourier transforms

Luongo
1. find the inverse fourier transform of f(w)=e-i5wsinc(2w)

2. I set up the integral to be from defn of sinc: 1/2pi*integral from -infinity to infinity (sin(2w)/2w)*e^-5w

3. i have no idea how to solve this integral, is there a better way to do this?
i know that rect(t) has a F.T. of sinc(w/2) but how do i go the other way if it's 2w, not w/2?

## Answers and Replies

Homework Helper
The way that one normally does this sort of thing it to use countour integration.

Homework Helper
Gold Member
1. find the inverse fourier transform of f(w)=e-i5wsinc(2w)

2. I set up the integral to be from defn of sinc: 1/2pi*integral from -infinity to infinity (sin(2w)/2w)*e^-5w

3. i have no idea how to solve this integral, is there a better way to do this?
i know that rect(t) has a F.T. of sinc(w/2) but how do i go the other way if it's 2w, not w/2?

Yes, there is a better way to do it. Use the shifting property of FT. If we denote the transform of f(t) by F(ω), one of the shifting properties gives:

$$f(t-t_0) \leftrightarrow e^{-i\omega t_0}F(\omega)$$

Staff Emeritus