Computation of Fourier Transform

[ColdStart]

Homework Statement

x(t) = t*exp(a)*exp(-a*t)*u(t-1) - exp(a)*exp(-a*t)*u(t-1)

I need to find X(jw)...

Homework Equations

how to apply properties of Fourier transform to get an answer? Because i know that the only effective method for this..

The Attempt at a Solution

For example from properties i know that exp(-a*t)u(t) is 1/(a+jw) and t*exp(-a*t)*u(t) is 1/(a+jw)^2..

now... i can think of my X(jw) consisting of 2 parts.. and i need to apply properties to each of those parts... however... properties are given in case if i have u(t) in expression... but in my problem i have u(t-1).. which kind of spoils things...

How should i deal with that properly?

 

[ColdStart]

ok i'v been thinking more of this problem, now here is what i decided to do:

lets say I am concentrating my attention on the 1st part of x(t) which is: t*exp(a)*exp(-a*t)*u(t-1), and i will compute X1(jw) for it first.

i am changing t-1 to Tao, => Tao = t - 1, t=Tao+1, then i have the following:

(Tao+1) *exp(a) * exp(-a*(Tao+1)) * u(Tao) = Tao*exp(a)*exp(-a*Tao)*exp(-a)*u(Tao) + exp(a)*exp(-a*Tao)*exp(-a)*u(Tao).

Now, recognizing mentioned properties above, i see that the last expression has transform:

1/(a+jw) + 1/(a+jw)^2 ... and this will be the transform for my X1(jw)..

Was the thinking correct? if yes then simililar stuff applied to second part of x(t) and problem solved..

thanks..