Fourier transform of a triangular function

Question

How do I calculate Fourier Transform? I want to use the formula to calculate the Fourier transform, but I can not come to the right answer,anything wrong?

Answer 1

The function in your plot is $$f (t) = \left\{ \begin{array}{ll} 0 & t\le-1\\ 1+t & -1<t\leq 0 \\ 1-t & 0< t \leq 1 \\ 0 & 1 <t. \\ \end{array} \right. $$

Note that you can split the FT up into $4$ pieces, where two of them are zero. So $$\mathcal{F}\{f\}(\omega) = \int\limits_{-\infty}^{\infty}f(t)e^{-i\omega t}\,\mathrm{d}t = \int\limits_{-1}^{0}(1+t)e^{-i\omega t}\,\mathrm{d}t+\int\limits_{0}^{1}(1-t)e^{-i\omega t}\,\mathrm{d}t.$$ From now on this should be easy to solve. Multiply out, use linearity of the integral and solve four easy integrals of exponential functions.