I am trying to solve a question of Fourier transform in which I am given two signals:$$X(jω)=δ(ω)+δ(ω-π)+δ(ω-5)\\h(t)=u(t)-u(t-2)=\begin{cases}1&0<x<2\\0&\text{otherwise}\end{cases}$$ I am asked to find the whether the convolution of $x(t)$ and $h(t)$ is periodic or not. Now using properties I have founded the signal $x(t)$ but I am stuck at the Fourier transform of $h(t)$.
Now I need to find its Fourier Transform using the properties and the one I got is: $$\\x(t)=\begin{cases}1&|t|<T _1\\0&\text{otherwise}\end{cases} \rightarrow \frac{2sin\omega T_1}{\omega} $$ Using this property we know that the time period of the wave is 2, hence: $$H(j\omega) = \frac{2\sin\omega2}{\omega},$$ but the one mentioned in book is $$H(j\omega) = e^{-j\omega} \frac{2\sin\omega}{\omega}.$$ Can somebody explain that why my expression is wrong?