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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?

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  • $\begingroup$ One could explain why the book's expression is right.... $\endgroup$ Jan 6 '19 at 10:00
  • $\begingroup$ @LordSharktheUnknown can you please point out my mistake? $\endgroup$ Jan 6 '19 at 10:03
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    $\begingroup$ I cannot point out your mistake, and nor can anyone else apart from yourself, since you have not provided details of your calculation. $\endgroup$ Jan 6 '19 at 10:04
  • $\begingroup$ @LordSharktheUnknown can you point out my mistake now? $\endgroup$ Jan 6 '19 at 11:56
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    $\begingroup$ Your formula works for $x(T)$ which is $1$ on the interval $(-T_1,T_1)$ and zero outside. But your $h$ is not of this form. $\endgroup$ Jan 6 '19 at 14:07

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