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?
Best Answer
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.