Fourier transform of tent signal

[Neutronium]

Hello, I'm having an issue with a given problem.

Homework Statement

Using Parseval's Equation find the energy of the signal $$z(t)=\frac{4}{4+t^{2}}$$

Homework Equations

The book solves that problem by using the tent signal CTFT and duality property (i.e ). However that properly isn't in the formula sheet, and the book derives it inversely!

The Attempt at a Solution

I tried to develop it by plugging $$z(t)$$ into the CTFT analysis equation, but it becomes a serious mess with integration by parts.

So what should I do? Take the tent property as it is and apply it to the question?

Thank you.

 

[MATLABdude]

Hello, I'm having an issue with a given problem.

Homework Statement

Using Parseval's Equation find the energy of the signal $$z(t)=\frac{4}{4+t^{2}}$$

Homework Equations

The book solves that problem by using the tent signal CTFT and duality property (i.e ). However that properly isn't in the formula sheet, and the book derives it inversely!

The Attempt at a Solution

I tried to develop it by plugging $$z(t)$$ into the CTFT analysis equation, but it becomes a serious mess with integration by parts.

So what should I do? Take the tent property as it is and apply it to the question?

Thank you.

I would suggest you try doing a partial fraction decomposition on z(t). That should simplify things a little (also keep in mind the time shifting property of the convolution). I believe this works, but I haven't done a Fourier transform in a few years...

EDIT: And welcome to PhysicsForums!

 

[Mene]

Hello,

It seems like you are trying to find the energy of the signal z(t)=\frac{4}{4+t^{2}} using Parseval's Equation. It is important to note that Parseval's Equation can only be used for signals with finite energy, meaning that the integral of the squared magnitude of the signal must converge.

In this case, the tent signal has infinite energy, so Parseval's Equation cannot be directly applied. However, you can still use the duality property to solve this problem. The duality property states that the Fourier transform of a function is equal to the Fourier transform of its time-reversed version. In other words, the Fourier transform of z(t) is equal to the Fourier transform of z(-t).

Using this property, you can rewrite z(t) as z(t)=\frac{4}{4+t^{2}}=\frac{4}{4+(-t)^{2}}=z(-t). Then, you can apply Parseval's Equation to z(-t) to find its energy. This energy will be the same as the energy of z(t) since they have the same Fourier transform.

I hope this helps! Good luck with your problem.