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