Fourier Tranform of piece wise function

Question

I want to find the fourier transform of this input signal.

Let the unit function unit (t, a, b) have the value 1 on the interval a≤ t < b and the value 0 otherwise. f(t) = (t)unit(t, 0, 0.5) + (-t)unit(t, 0.5, 1.5) + (t)unit(t, 1.5, 2). I am lost on where to go from here. Without being able to write it using euler's formula(e^(it) = cos(t) + isin(t)), how should I proceed? I am not sure which fourier integral to use.

This is the equation I hope to use.

Answer 1

f(t) = {0 <= t < (1/2), 2t, (1/2) <= t < (3/2), 2-2t, (3/2) <= t < 2, 2t}

integral of [f(t) * e^(-iwt)] from 0 to 2 is the fourier transform of the function.

Answer 2

I think a simpler and more general way (it can be extended to any continuous piecewise linear function) is to write this function as a linear combination of triangle functions $\Lambda$, with $\Lambda(t)=1-|t|$ with support on $[-1,1]$.

$$f(t)=\Lambda(2t-1)-\Lambda(2t-3)$$

and then apply the rules about F. Transform, yielding something like

$$(e^{-i\pi u}-e^{-3i\pi u})sinc(u/2)^2 \ \ (*)$$

(where $sinc$ is the classical cardinal sine, with sinc $u = \dfrac{\sin{\pi u}}{\pi u}$ for $u \neq 0$)

(I say "something like" because there are different cousin definitions of the F.T., of the cardinal sine, of the triangle function, etc.)

Remark: formula (*) can be transformed by factorization of $e^{-2i\pi u}$.