What is the Fourier Transform of $ rect(2Bt)cos[{\omega}_Ct + k_fm(t_k)t] $?

Question

What is the Fourier Transform of $ \text{rect}(2Bt)\text{cos}\left[{\omega}_Ct + k_fm(t_k)t\right] $?

I got the following as the solution:

$$ \frac{1}{2} \frac{1}{2B} \text{sinc}\left(\frac{\omega+{\omega}_C+k_fm(t_k) +}{4B}\right) + \frac{1}{2} \frac{1}{2B} \text{sinc}\left(\frac{\omega-{\omega}_C-k_fm(t_k) +}{4B}\right)$$

However, in the book it is given as:

$$ \frac{1}{2} \text{sinc}\left(\frac{\omega+{\omega}_C+k_fm(t_k) +}{4B}\right) + \frac{1}{2} \text{sinc}\left(\frac{\omega-{\omega}_C-k_fm(t_k) +}{4B}\right)$$

Wolfram alpha shows this:Here

Answer 1

As garyp said, there is not a single definition of the Fourier transform. People use different definitions, e.g. I use $$ {\mathscr{F}}\{f\}(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t} dt $$ Using your function $ f(t) = rect(2 B t) \cos(p t)$ yields $$ {\mathscr{F}}\{f\}(\omega) = \frac{sinc(\frac{p - \omega}{4 B}) + sinc(\frac{p + \omega}{4 B})}{4 |B|} $$ In order to obtain this result, we use:

• the convolution theorem, $ {\mathscr{F}}\{f \cdot g\}(\omega) = {\mathscr{F}}\{f \}(\omega) * {\mathscr{F}}\{g\}(\omega) $,
• the "scaling" theorem, $ {\mathscr{F}}\{f(at)\}(\omega) = \frac{1}{|a|}{\mathscr{F}}\{f\}(\omega/a) $,
• the fact that $cos$ transforms to two $\delta$-distributions,
• and the fact that the convolution of a function with $\delta$-distribution is the function evaluated at the point of the $\delta$-distribution.

Finally, supposing that $t$ has the unit $s$ than $B$ and $p$ have the unit $1/s=Hz$. So, the relation from your book has the wrong unit.