Calculate Fourier Transform of x(t): \frac{1}{(1+t^2)} & abs(t)

[razorwind]

Determine the Fourier transform of the following signals: $$x(t) = \frac{1}{(1+t^2)}$$

I start off by doing $$X(f) = \int{x(t)*e^{-j2\pi{}ft} dt}$$

So i plug in x(t) into that equation, but I'm lost as to how to integrate it. Am i going in the right direction?

Also: x(t)=abs(t) for $$-2\leq{}t\leq{}2$$

I am splitting it to -t for $$-2\leq{}t\leq{}0$$ and t for $$0\leq{}t\leq{}2$$ then doing the transform with the equation above. Is it allowed to split it into 2 integrals? So it ends up being $$X(f) = \int{-t*e^{-j2\pi{}ft} dt}$$ + $$\int{t*e^{-j2\pi{}ft} dt}$$ and I simplify from there.

 

[LightHero]

Is this the right approach?Yes, this is the correct approach. You will need to use integration by parts to solve the integral for the first signal. For the second signal, splitting it into two integrals is the way to go.

 

[Mene]

Yes, you are on the right track. To integrate x(t) = 1/(1+t^2), you can use the substitution method. Let u = t^2 + 1, then du = 2t dt. So the integral becomes:

X(f) = \frac{1}{2}\int{\frac{1}{u}e^{-j2\pi{}fu} du}

Using the fact that \int{\frac{1}{u} e^{-j2\pi{}fu} du} = \frac{e^{-j2\pi{}fu}}{-j2\pi{}f}, we can write:

X(f) = \frac{1}{2}\cdot \frac{e^{-j2\pi{}fu}}{-j2\pi{}f} = -\frac{1}{2j\pi{}f}\cdot e^{-j2\pi{}fu}

To calculate the Fourier transform of x(t) = abs(t), you are correct in splitting it into two integrals. This is allowed because the Fourier transform is a linear operation. So your final answer would be:

X(f) = -\frac{1}{2j\pi{}f}\cdot e^{-j2\pi{}fu} + \frac{1}{2j\pi{}f}\cdot e^{j2\pi{}fu}

= \frac{1}{j\pi{}f}\cdot sin(2\pi{}fu)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot cos(\pi{}fu)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot \frac{1}{2}\cdot \left(e^{j\pi{}fu} + e^{-j\pi{}fu}\right)

= \frac{1}{j\pi{}f}\cdot sin(\pi{}fu) \cdot \left(e^{j\pi{}fu} - e^{-j\pi{}fu}\right)

= \frac{2}{j\pi{}f}\cdot sin(\pi{}fu) \cdot sin(\pi{}fu)

= \frac{4}{j\pi{}f}\cdot sin^2(\pi{}fu)

= \frac{4}{j\pi{}f}\cdot \frac{1}{2}\cdot \left(1 - cos(2\