# Math Help - CTFT of a Sinusoid Convolved with a Sinc

1. ## CTFT of a Sinusoid Convolved with a Sinc

Hello Evryone!

I've noticed this about Fourier transform.

Suppose we're convolving a single tone sinusoid with a sinc function, namely [TEX]\sin(at)/\pi t[\TEX], then, by taking their CTFT, we get a rect multiplied with 2 pure imaginary deltas. Now what's the significance of our transform being imaginary, not just in this case? Does it just come out of the math involved?

Thanks!

2. Originally Posted by rebghb
Hello Evryone!

I've noticed this about Fourier transform.

Suppose we're convolving a single tone sinusoid with a sinc function, namely [TEX]\sin(at)/\pi t[\TEX], then, by taking their CTFT, we get a rect multiplied with 2 pure imaginary deltas. Now what's the significance of our transform being imaginary, not just in this case? Does it just come out of the math involved?

Thanks!
The FT of a convolution is the pointwise product of the Fourier transforms.

Now what is the FT of sin( w t) ? (it is purely imaginary with conjugate symmetry because the original function is real).

As we also know the FT of a sinc is a top hat.

CB