Finding Energy of an Odd Signal Given Even Part and Total Energy

[Axis001]

Homework Statement

X(t) = Xe(t) + Xo(t) where Xe(t) = (1/2){X(t)+X(-t)} is the even part and Xo(t) = (1/2){X(t)-X(-t)} is the odd part of the signal. Let X(t) be an energy signal with energy 5 Joules. Suppose the even part of X(t) is Xe(t) = exp(-|t|). Determine the energy in Xo(t).

Homework Equations

E = ∫|X(t)|^2 dt from -infinity to infinity

The Attempt at a Solution

I have attempted this problem two ways.

First I tried to derive Xo(t) by using the above energy equation and substituting {Xe(t) + Xo(t)} with E = 5 J and from there I got the result Xo(t) = -exp(-|t|). Which didn't make sense since that would mean that X(t) = 0 but somehow has energy of 5 J.

My second attempt at this I started with the base equations provided for Xe(t) and set them equal to each other which resulted in X(-t) = 2exp(-|t|) + X(t). Taking this result and plugging it into the equation for Xo(t) again resulted in Xo(t) = -exp(-|t|). This again makes no sense.

I think I am simply over complicating this problem or I am just doing something very wrong. Either way any help would be greatly appreciated.

 

[KataKoniK]

Thank you for your post. I understand your confusion and I am here to help clarify the problem.

Firstly, let's start with the given equation: X(t) = Xe(t) + Xo(t)

In this equation, X(t) represents the energy signal with energy 5 Joules. This means that the total energy of the signal is 5 Joules.

Now, let's look at Xe(t) = exp(-|t|). This is the even part of the signal, which means that it is symmetric about the y-axis. This also means that X(-t) = X(t).

Using this information, we can rewrite the equation as: X(t) = Xe(t) + Xe(-t).

Substituting Xe(-t) with Xe(t) (since they are the same), we get: X(t) = 2Xe(t).

Now, we can solve for Xe(t) by dividing both sides by 2: Xe(t) = X(t)/2.

Since we know that X(t) = 5 Joules, we can substitute this value into the equation to get: Xe(t) = 2.5 Joules.

Finally, to find the energy in Xo(t), we use the given equation: Xo(t) = (1/2){X(t)-X(-t)}.

Substituting X(t) with 5 Joules and X(-t) with X(t), we get: Xo(t) = (1/2){5-5} = 0 Joules.

Therefore, the energy in Xo(t) is 0 Joules.

I hope this helps clarify the problem for you. Let me know if you have any further questions or concerns.