Impulse response from frequency response

1. The problem statement, all variables and given/known data



I am having some problems to derive Inverse Fourier transform of sinc function & exponential function. It's actually for getting the Impulse response from the given frequency response [which comprises of both sinc function & exponential function]. Also need to know the output y(t) for the given input. Can anyone help me out?



The question is in the attachment.



2. Relevant equations



(a) The impulse response is the inverse transform [IFT] of the transfer function. I think IFT will lead me to get the impulse response.



(b)



3. The attempt at a solution



Solution:



(a)



I stepped forwards as following:



h_0 (t)= F^(-1) [H(jω)]

= 1/2π ∫_(-∞)^(+∞)▒〖T_1 . e^(-jω T_1/2) . sin⁡〖ω T_1/2〗/(ω T_1/2)〗 .〖 e〗^(-jωt) dω

= T_1/2π ∫_(-∞)^(+∞)▒〖e^(jω(t-T_1/2)) dω .∫_(-∞)^(+∞)▒sin⁡〖ω T_1/2〗/(ω T_1/2)〗 dω

= T_1/2π δ(t-T_1/2) ∫_(-∞)^(+∞)▒sin⁡〖ω T_1/2〗/(ω T_1/2) dω



I could not get the sinc function [sin⁡〖ω T_1/2〗/(ω T_1/2)] to fourier transform and also the derivation from time lapse exponential function [e^(jω(t-T_1/2))] to delta function [δ(t-T_1/2)]. Can anyone help me out the full derivatives of the problem?





(b) Is it the right equation to get the system output y (t) as follows?



y(t)= r(t)*[H(jω)]

= A T_1 . e^(-jω T_1/2) . sin⁡〖ω T_1/2〗/(ω T_1/2)

= A T_1 . e^(-jω T_1/2) . sinc(ω T_1/2)



Is the approach ok? And how can it be more simplified?

Attached Images

ImpulseResonse_Freq Response.jpg (28.3 KB)