Finding Fourier Transform for x(t): A Math Student's Query

[cepheid]

In an assignment, I've been given a function:

$$x(t) = \theta(t-t_1) - \theta(t-t_2)$$

Assume $t_2 > t_1$

and we are asked to find the Fourier transform. So I wrote down:

$$x(\omega) = \int_{-\infty}^{\infty}{e^{-i\omega t} [\theta(t-t_1) - \theta(t-t_2)] dt}$$

I know that the function given is the heaviside step function. Its derivative is the dirac delta function, and it is itself the derivative of the ramp function. But I just found this stuff out by looking online. We've learned neither convolution nor Fourier transform in math class, yet somehow this physics prof expects us to do it. Can someone at least point me in the right direction?

Thanks.

 

[Tide]

Your integrand is zero everywhere except in the interval from t1 to t2 - integrate it directly!

 

[cepheid]

Ok, I feel stupid! :rofl:

Just to see if I'm understanding you correctly, x(t) can be broken down as follows:

x(t) = 0 - 0 for t < t1

x(t) = 1 - 0 for t1 < t < t2

x(t) = 1 - 1 for t > t2

This is just the first statement in your post.

So the integral reduces to:

$$x(\omega) = \int_{t_1}^{t_2}{e^{-i\omega t}dt}$$

Am I right? The help was much appreciated!

 

[Tide]

You got it - WTG!