# Find fourier transform given the graph of a function

Cheers, I am given the following graph for $$x(t)$$:

and I am asked to find the fourier tranform by using the integral property and knowing that $$F(Π(t)) = sinc (\frac{\omega}{2\pi})$$

The solution I saw stated that we can see that $$x(t) = \int_{- \infty}^tΠ(ρ)dρ$$, but I just can't see why that is, could that be explained?

Also I tried to go a different route, by finding it's equation. So I saw that : $$x(t) = \begin{cases} 0, & \text{if t < 0.5 } \\ t + 0.5, & \text{if t \in [-0.5,0.5]} \\ 1, & \text{if t>0.5} \end{cases}$$

Now I try using the differentiation property of the fourier transform. So I see that: $$\frac{dx(t)}{dt} = \begin{cases} 0, & \text{if t < 0.5 } \\ 1, & \text{if t \in [-0.5,0.5]} \\ 0, & \text{if t>0.5} \end{cases}$$ which I rewrite as: $$u(t+0.5) - u(t-0.5)$$, and then I differentiate again, so I get that $$x''(t) = \delta(t + 0.5) - \delta(t-0.5)$$ and now I get that:

$$(j\omega)^2X(\omega) = e^{j\omega\frac{1}{2}} - e^{-j\omega\frac{1}{2}} \rightarrow X(\omega) = \frac{2j}{-\omega^2} \frac{e^{j\omega\frac{1}{2}} - e^{-j\omega\frac{1}{2}}}{2j} = \frac{2j}{-\omega^2} \sin(\frac{\omega}{2})$$ which is not the answer that I should get which is:

Any help on If I am doing anything right? Thanks =)

• Does your answer equate to the first part of the answer you should get? And if $f'(t) = \frac{d}{dt}f(t)$, then doesn't $\int f'(t) = f(t) + C$, where $C$ is a constant? Jan 4 at 20:15
• I suppose you are talking about the way I tried to solve it. Well I did a bit more work on it and I reach $\frac{1}{jw} \frac{ \sin (\frac{\omega}{2}) } {\frac{\omega}{2}}$. But there is no $\pi \delta(\omega)$. Is that what you mean by saying that the C is constant? That it doesn't matter? I think I am saying something very wrong here, but I do not have much experience with this type of equations, and got lost In what you are exactly implying Jan 4 at 20:32

Your method correctly produces the term $$\frac{1}{j\omega}\frac{\sin(\omega/2)}{\omega/2}$$ of the given solution, but you forget that by taking the derivative you lose information about any constant terms in $$x(t)$$. However, looking at the graph it's easy to figure out that the constant term in $$x(t)$$ is $$\frac12$$. The Fourier transform of that constant term is $$\frac12\cdot 2\pi\delta(\omega)$$, which gives you the missing term.
Concerning your first question, note that differentiating the given function resulted in a rectangle, so it shouldn't surprise you that $$x(t)$$ can be written as the integral of a rectangular function.