# Impulse response of LTI Systems

I am trying to solve these questions. I have attached my solutions. Please I would appreciate if some one can check my approach and tell me if I am correct or not. Also, how can I proceed to find $$h_0(t)$$?

Question (s) with My Attempts

1. Given the LTI-System with frequency response $$\LARGE H(j\omega)=T_1e^{-\frac{j\omega T_1}{2}}\cdot\frac{\sin(\frac{\omega T_1}{2})}{\frac{\omega T_1}{2}}$$Find
a. Impulse Response $$h_0(t)$$ of this system.
b. Reaction $$y(t)$$ of this system to the input signal $$\Gamma(t)$$ 1. Determine the Fourier Transform of the Rectangular Pulse shown in the following figure.  • 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. Mar 31, 2017 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, 2017 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, 2017 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, 2017 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. Mar 31, 2017 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.