Impulse response from frequency response

[aguntuk]

Homework Statement

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.

Homework 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)

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?

 

[donpacino]

I'm having a little bit of trouble following your work, so I am going to make some quick comments.

for part a. typically it is easier to use Laplace (at least for me). If you don't know laplace then i guess your method is ok. note: i did not check your work for part a. the method is correct. the work may not be.

for part b. You are mixing the frequency and time domain, so your method is invalid.

y(jw)=r(jw).H(jw)
or
y(t)=r(t)*H(t)

H(t)= impulse response of system.
if you are working in the time domain you use convolution to combine the r(t) and H(t) signal. if you are working in the frequency domain you can simply use multiplication.

not r(t)=A*(u(t)-u(t-T))