# CTFT of a Sinusoid Convolved with a Sinc

• May 25th 2011, 02:38 AM
rebghb
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!
• May 25th 2011, 10:55 PM
CaptainBlack
Quote:

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