Fourier transform (integration)

[gomes.]

Got stuck in the second part, any help is appreciated, cheers.

 

[vela]

What's the definition of the Fourier transform? The problem is asking you to find the Fourier transform of that function using it.

 

[gomes.]

thanks, i tried doing that but I am not sure what to do with the u(t+4) bit.

btw is my first part correct? (question 1)

 

[vela]

The step function makes the integrand equal to 0 when t+4<0, so that let's you replace the lower limit of the integral with -4.

 

[gomes.]

thanks, but how would i integrate u(t+4)e^[t(-1-iw)-4]?

How do i deal with the u(t+4)?

 

[vela]

What is u(t+4) equal to for t>-4?

 

[gomes.]

sorry I am really stuck, u(t)?

 

[vela]

The unit step function must surely be discussed in your notes or textbook.

http://www.efunda.com/math/unit_delta/unit_delta.cfm

 

[gomes.]

thanks, i think i might have missed out on that lecture, which explains why.

Is what I've done correct?

 

[vela]

Well, the first line definitely doesn't equal the second line.

The integrals look okay.

 

[gomes.]

sorry, I am really really stuck. What should the next line be, after the first line? Could someone possibly show me how to proceed from the 1st line?

 

[vela]

Perhaps you misunderstood. I meant:

$$u(t) \ne \int_{-\infty}^\infty u(t+4) e^{t(-1-i\omega)-4}\,dt$$

The integral in the second line is fine to calculate the Fourier transform you're looking for, but it's definitely not equal to u(t) as you wrote.

Try the substitution t'=t+4 to evaluate the integral.

 

[gomes.]

thanks! got it now