# Fourier Transformation - Convolution quick question

Okay the question is to find the fourier transform of:

rect($\frac{x}{5})$$\otimes$($\delta$(x+3)-$\delta$(x-3))

=F$^{\infty}_{\infty}$ $\int$rect($\frac{x'}{5}$)($\delta$(x+3-x')-$\delta$(x-3-x')) dx' 

- where F represents a fourier transform.
My Issue
Okay I am fine doing this using the convolution theorem, that the fourier transform of a convultion is given by the product of the two individual fourier transforms, but I am having trouble doing it explicitly

So from  integrating over each delta function, I deduce that the first term collapses everywhere except x'=x+3, and the second everywhere except x'=x-3, . So I get:

F(rect$\frac{x+3}{5}$-rect$\frac{x-3}{5})$
= (5sinc$\frac{5k}{2}$exp$^{\frac{3ik}{5}}$exp$^{\frac{-3ik}{5}}$)
using the properties that F(rect($\frac{x}{1}$))=asinc($\frac{ka}{2}$) and that F(f(x+a))=F(f(x))exp$^{ika}$

Which does not agree with the convultion theorem were I get :

5sinc$\frac{5k}{2}$exp$^{3ik}$exp$^{-3ik}$

Thanks alot in advance for any assistance !

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vela
Staff Emeritus
$$F\left[\mathrm{rect }\left(\frac{x+3}{5}\right) -\mathrm{rect }\left(\frac{x-3}{5}\right)\right] = 5\mathrm{sinc }\left(\frac{5k}{2}\right)e^{3ik/5}e^{-3ik/5}$$ using the properties that ##F\left[\mathrm{rect }\left(\frac{x}{a}\right)\right]=a\mathrm{sinc }\left(\frac{ka}{2}\right)## and that ##F[f(x+a)] = F[f(x)]e^{ika}##. This does not agree with the convolution theorem where I get:
$$5\mathrm{sinc }\left(\frac{5k}{2}\right)e^{3ik}e^{-3ik}$$