Question

$\text{If f(t) is a real even continuous time signal, then its Fourier transform will be}$

• A. $2{\int }_0^{\infty }f\left(t\right)sin\left(\omega t\right)dt$
• B. ${\int }_0^{\infty }f\left(t\right)cos\left(2\omega t\right)dt$
• C. $2{\int }_0^{\infty }f\left(t\right)cos\left(\omega t\right)dt$
• D. ${\int }_0^{\infty }f\left(t\right)sin\left(\omega t\right)dt$

Answer 1

The correct option is C $2{\int }_0^{\infty }f\left(t\right)cos\left(\omega t\right)dt$

$F\left(\omega \right)={\int }_{-\infty }^{\infty }f\left(t\right)e^{-j\omega t}dt={\int }_{-\infty }^{\infty }\left[f\right(t)cos\omega t-jf(t\left)sin\omega t\right]dt={\int }_{-\infty }^{\infty }f\left(t\right)cos\omega tdt-j{\int }_{-\infty }^{\infty }f\left(t\right)sin\omega tdtf\left(t\right)\Rightarrow \text{even signal}f\left(t\right)cos\omega t\Rightarrow \text{even signal}f\left(t\right)sin\omega t\Rightarrow \text{odd signal}{\int }_{-\infty }^{\infty }f\left(t\right)sin\omega tdt=0{\int }_{-\infty }^{\infty }f\left(t\right)cos\omega tdt=2{\int }_0^{\infty }f\left(t\right)cos\omega tdt\therefore F\left(\omega \right)=2{\int }_0^{\infty }f\left(t\right)cos\omega tdt$