# Fourier Transform of Shifted Triangular Pulse

Suppose you are given the following triangular pulse signal and you are asked to write it's Fourier representation.

I would like to know if I have proceeded correctly.

Since the signal is aperiodic, we will represent it with it's Fourier transform. I know this problem can be solved through rectangles but I thought it would be faster using directly the FT of the $$\Delta$$ pulse.

From an FT table we know that: $$\Delta \left(\frac{t}{\tau} \right) \xrightarrow{\mathscr{F}} \frac{\tau}{2} sinc^2 \left(\frac{\omega \tau}{4}\right)$$

where $$\tau$$ is the width of the triangular pulse.

and we can use the time shift property: $$x(t-t_0) \xrightarrow{\mathscr{F}} e^{-j \omega t_0} X(\omega)$$

The signal is a triangular pulse with doubled amplitude and shifted one unit to the right, that is, $$x(t) = 2 \Delta \left(\frac{(t-1)}{4} \right)$$

We apply the FT to $$x(t)$$: $$\mathscr{F}\{2 \Delta \left(\frac{(t-1)}{4} \right)\} = 2 \mathscr{F}\{ \Delta \left(\frac{t}{4} - \frac{1}{4} \right)\}$$

So that's the same as solving $$\mathscr{F} \{ \Delta(\frac{t}{\tau}) \}$$ with a time shift of $$\frac{1}{4}$$. Therefore,

$$X(\omega) = 2 e^{-j \omega \frac{1}{4}} \frac{4}{2} sinc^2 \left( \frac{\omega 4}{2} \right)$$

Is that correct?

$$\therefore \boxed{X(\omega) = 4 sinc^2 (2 \omega) e^{-j \frac{\omega}{4}}}$$