Impulse response of LTI Systems

I am trying to solve the attached question. Please I would appreciate if some one can check my approach and tell me if I am correct and how can I proceed to find h0(t)  [![enter image description here]]

• Your calculation is correct, but I think it will not lead to a simple solution (You'd need to calculate inverse FT of (1/jw), which does converge in the common sense, I think.) Instead, I give you two hints: 1) Check out the time-shift property of the FT and apply it to your equation. 2) Remember the what the Fourier transform of a rectangular function is and try to apply it here. Let us know, if you can solve on your own now or need more help, explaining your problems with my hints. – Maximilian Matthé Mar 31 '17 at 11:31
• Maximilian, thanks very much for the tips. I am trying to apply the Inverse FT but I am confuse on how to simply it. – Soso Mar 31 '17 at 12:02
• I didn't check your calculations, but $\frac{2}{j\omega}$ is the FT of sgn function. So you actually have a difference of a sgn and a shifted sgn by $T$ which is a box function (width of $T$). However, a DC value is missing (compared to the IFT of the sinc). So some delta should also be there in your result... – msm Mar 31 '17 at 12:10
• I have tried to do some maths manipulation which I have attached above, i don't know if I am on the right track. – Soso Mar 31 '17 at 12:39
• @msm: You are right with the signs, but I think actually solving this integral is not easly. Further, I dont think there is a DC missing, the IFT of a sinc is also just a box. – Maximilian Matthé Mar 31 '17 at 12:50

$h_0(t)$ is inverse Fourier transform of $H(jw)$. Your formula is OK, you can continue your calculation to practice your math manipulation, why not. To check the result, you can remark that $H(jw)$ is rotated $sinc$ function. And $sinc$ must be the Fourier transform of a rectangular box function.