2 typo in the first integral edited Aug 18 '14 at 9:16 Gilles 2,25633 gold badges1616 silver badges2626 bronze badges \begin{align*} X\left(j\omega \right) &= \int_0^{\tau}E\cdot t \cdot e^{-j\omega t}dt + \int_{\tau}^T E\cdot e^{-j\omega t}dt\\ &=\frac{E}{\tau} \left[ \frac{-t}{j\omega} e^{-j\omega t}\right]_0^{\tau} - \frac{E}{\tau}\int_0^{\tau}\left( \frac{-1}{j\omega}e^{-j\omega t}\right)dt + E \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{\tau}^T\\ &=\frac{jE}{\omega}e^{-j\omega \tau} + \frac{E}{\omega^2 \tau}\left( e^{-j\omega \tau} - 1\right) + \frac{jE}{\omega}\left( e^{-j\omega T} - e^{-j\omega \tau}\right)\\ &=\frac{jE}{\omega}\left[ e^{-j\omega T} + e^{-j\frac{\omega \tau}{2}}\left(\frac{e^{-j\frac{\omega \tau}{2}} - e^{j\frac{\omega \tau}{2}}}{2j\frac{\omega\tau}{2}}\right)\right]\\ \Rightarrow X\left(j\omega \right) &= \frac{jE}{\omega} \left(e^{-j\omega T} - e^{-j\frac{\omega \tau}{2}} \text{sinc}\left( \frac{\omega \tau}{2}\right) \right) \end{align*}\begin{align*} X\left(j\omega \right) &= \int_0^{\tau}\frac{E}{\tau}\cdot t \cdot e^{-j\omega t}dt + \int_{\tau}^T E\cdot e^{-j\omega t}dt\\ &=\frac{E}{\tau} \left[ \frac{-t}{j\omega} e^{-j\omega t}\right]_0^{\tau} - \frac{E}{\tau}\int_0^{\tau}\left( \frac{-1}{j\omega}e^{-j\omega t}\right)dt + E \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{\tau}^T\\ &=\frac{jE}{\omega}e^{-j\omega \tau} + \frac{E}{\omega^2 \tau}\left( e^{-j\omega \tau} - 1\right) + \frac{jE}{\omega}\left( e^{-j\omega T} - e^{-j\omega \tau}\right)\\ &=\frac{jE}{\omega}\left[ e^{-j\omega T} + e^{-j\frac{\omega \tau}{2}}\left(\frac{e^{-j\frac{\omega \tau}{2}} - e^{j\frac{\omega \tau}{2}}}{2j\frac{\omega\tau}{2}}\right)\right]\\ \Rightarrow X\left(j\omega \right) &= \frac{jE}{\omega} \left(e^{-j\omega T} - e^{-j\frac{\omega \tau}{2}} \text{sinc}\left( \frac{\omega \tau}{2}\right) \right) \end{align*} \begin{align*} X\left(j\omega \right) &= \int_0^{\tau}E\cdot t \cdot e^{-j\omega t}dt + \int_{\tau}^T E\cdot e^{-j\omega t}dt\\ &=\frac{E}{\tau} \left[ \frac{-t}{j\omega} e^{-j\omega t}\right]_0^{\tau} - \frac{E}{\tau}\int_0^{\tau}\left( \frac{-1}{j\omega}e^{-j\omega t}\right)dt + E \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{\tau}^T\\ &=\frac{jE}{\omega}e^{-j\omega \tau} + \frac{E}{\omega^2 \tau}\left( e^{-j\omega \tau} - 1\right) + \frac{jE}{\omega}\left( e^{-j\omega T} - e^{-j\omega \tau}\right)\\ &=\frac{jE}{\omega}\left[ e^{-j\omega T} + e^{-j\frac{\omega \tau}{2}}\left(\frac{e^{-j\frac{\omega \tau}{2}} - e^{j\frac{\omega \tau}{2}}}{2j\frac{\omega\tau}{2}}\right)\right]\\ \Rightarrow X\left(j\omega \right) &= \frac{jE}{\omega} \left(e^{-j\omega T} - e^{-j\frac{\omega \tau}{2}} \text{sinc}\left( \frac{\omega \tau}{2}\right) \right) \end{align*} \begin{align*} X\left(j\omega \right) &= \int_0^{\tau}\frac{E}{\tau}\cdot t \cdot e^{-j\omega t}dt + \int_{\tau}^T E\cdot e^{-j\omega t}dt\\ &=\frac{E}{\tau} \left[ \frac{-t}{j\omega} e^{-j\omega t}\right]_0^{\tau} - \frac{E}{\tau}\int_0^{\tau}\left( \frac{-1}{j\omega}e^{-j\omega t}\right)dt + E \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{\tau}^T\\ &=\frac{jE}{\omega}e^{-j\omega \tau} + \frac{E}{\omega^2 \tau}\left( e^{-j\omega \tau} - 1\right) + \frac{jE}{\omega}\left( e^{-j\omega T} - e^{-j\omega \tau}\right)\\ &=\frac{jE}{\omega}\left[ e^{-j\omega T} + e^{-j\frac{\omega \tau}{2}}\left(\frac{e^{-j\frac{\omega \tau}{2}} - e^{j\frac{\omega \tau}{2}}}{2j\frac{\omega\tau}{2}}\right)\right]\\ \Rightarrow X\left(j\omega \right) &= \frac{jE}{\omega} \left(e^{-j\omega T} - e^{-j\frac{\omega \tau}{2}} \text{sinc}\left( \frac{\omega \tau}{2}\right) \right) \end{align*} 1 answered Aug 17 '14 at 23:05 Gilles 2,25633 gold badges1616 silver badges2626 bronze badges \begin{align*} X\left(j\omega \right) &= \int_0^{\tau}E\cdot t \cdot e^{-j\omega t}dt + \int_{\tau}^T E\cdot e^{-j\omega t}dt\\ &=\frac{E}{\tau} \left[ \frac{-t}{j\omega} e^{-j\omega t}\right]_0^{\tau} - \frac{E}{\tau}\int_0^{\tau}\left( \frac{-1}{j\omega}e^{-j\omega t}\right)dt + E \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{\tau}^T\\ &=\frac{jE}{\omega}e^{-j\omega \tau} + \frac{E}{\omega^2 \tau}\left( e^{-j\omega \tau} - 1\right) + \frac{jE}{\omega}\left( e^{-j\omega T} - e^{-j\omega \tau}\right)\\ &=\frac{jE}{\omega}\left[ e^{-j\omega T} + e^{-j\frac{\omega \tau}{2}}\left(\frac{e^{-j\frac{\omega \tau}{2}} - e^{j\frac{\omega \tau}{2}}}{2j\frac{\omega\tau}{2}}\right)\right]\\ \Rightarrow X\left(j\omega \right) &= \frac{jE}{\omega} \left(e^{-j\omega T} - e^{-j\frac{\omega \tau}{2}} \text{sinc}\left( \frac{\omega \tau}{2}\right) \right) \end{align*}