Find Fourier transform of triangular function based on a Fourier results of rectangular

Question

I have a triangular pulse given by $$x\left(\frac{t}T\right) = \begin{cases} 1-\frac {|t|}T, & \text{if $T\ge t$} \\ 0, & \text{otherwise} \end{cases}$$

Given that $F\left(\operatorname{rect}\left(\frac{t}T\right)\right)=T\operatorname{sinc}(fT)$

where $F$ denotes a Fourier transformation

$$\operatorname{rect}\left(\frac{t}T\right)=\begin{cases} 1, & \frac{T}{2} \ge t \ge \frac{-T}{2} \\ 0, & \text{otherwise} \end{cases}$$

Prove that $F\left(x\left(\frac{t}T\right)\right)=T\operatorname{sinc}^2(fT)$.

As my knowledge, I prove above equation is $T^2\operatorname{sinc}^2(fT)$, instead of $T\operatorname{sinc}^2(fT)$.

We have $x\left(\frac{t}T\right)=\operatorname{rect}\left(\frac{t}T\right)*\operatorname{rect}\left(\frac{t}T\right)$ Then \begin{align} F\left(x\left(\frac{t}T\right)\right) & =F\left(\operatorname{rect}\left(\frac{t}T\right)\right)\cdot F\left(\operatorname{rect}\left(\frac{t}T\right)\right)\\ & =T \operatorname{sinc}(fT) \cdot T \operatorname{sinc}(fT) =T^2\operatorname{sinc}^2(fT). \end{align}

What is wrong in my solution? How to correct it?

Answer 1

For $t=0$, we have $x(t/T)=1$ but $\operatorname{rect}(t/T)*\operatorname{rect}(t/T)=T$. This, and the difference between your answer and the correct one, should give you a hint of what is wrong.