Parseval's Theorem - Average Power of the difference of functions

[Master1022]

Homework Statement

Given a function $e(t) = f(t) - g(t)$, find the average power of the function.

Relevant Equations

Parseval's theorem

Hi,

So I have a quick conceptual question about Parseval's theorem in this application. In previous parts of this question, we have found the average powers of both $f(t)$ and $g(t)$ by integration and using the complex Fourier series respectively (not sure if this is relevant to my question). I wanted to know how do we use these to calculate the average power of e(t)?

The answer simply does subtract the two aforementioned expressions to get the answer. However, I was wondering why we wouldn't need to consider the 'cross-terms'? So in the average power integral, we are dealing with $|e(t)|^2 = |e(t)||e^*(t)| = |f(t) - g(t)||g^*(t) - f^*(t)|$ and wouldn't this contain terms that look like $f(t) g^*(t)$ and $f^*(t) g(t)$ (omitting the signs for now)? The calculation that they have done does not include any of these and I was wondering whether we are allowed to combine average powers like that?

Thanks in advance.

(N.B. I do understand that this might be question-specific, but just want to understand the concept first as this is the last sub-part of a longer question and typing that out would include seemingly redundant information - I can include it if necessary)

 

[Valkarie]

Yes, you are allowed to combine average powers like that. This is because of Parseval's Theorem, which states that the sum of the squared magnitudes of the Fourier coefficients is equal to the integral of the squared magnitude of the original signal. Thus, the average power of e(t) can be calculated by subtracting the average powers of f(t) and g(t), without taking into account the cross-terms.