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

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

I got the following as the solution:

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

However, in the book it is given as:

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

Wolfram alpha shows this:

https://www.wolframalpha.com/input/?i=Fourier+transform+calculator&assumption=%7B%22F%22%2C+%22FourierTransformCalculator%22%2C+%22transformfunction%22%7D+-%3E%22+rect%282Bt%29cos%28pt%29%22&assumption=%7B%22F%22%2C+%22FourierTransformCalculator%22%2C+%22variable1%22%7D+-%3E%22t%22&assumption=%7B%22F%22%2C+%22FourierTransformCalculator%22%2C+%22variable2%22%7D+-%3E%22w%22

• you should have $2B$ outside of the $\sinc$, so I'm pretty sure your book can't be right, unless $B$ is defined somewhere else. Apr 19 '20 at 17:50

If you define the $$\textrm{rect}()$$ function according to this definition, then your result is correct. However, there might be a problem with the definition of the $$\textrm{rect}()$$ function. If in your book they define that function to have unit area, then the result of the book would be correct.