Partitioning the energy of a signal between its components In physics, the total energy of a system is the summed energy of its component subsystems, and energy must be conserved, ie, adding the energy of the components gives the total energy.
In signal analysis (discrete time signals), one would expect that the energy of a signal would be defined in such a way that energy would again be conserved.  This would make it consistent with normal usage in physics.
There are many types of component analysis. In general, there is no requirement that the components are zero-mean or orthogonal, and in data-dependent component extractions, the components are not even predictable, but in all cases it is expected that they linearly sum to the original signal values.
This is incompatible with the usual definition of signal energy as the sum of the squared component values, since the component energies do not sum to the original signal energy.
This is confusing.  Is there another approach to defining the component energies that conserves energy but does not destroy the requirement that the linear sum of components must be equal to the original signal?
Say you decompose the signal $x(t)$ into two components: $$s(t) = x(t) + y(t).$$ As you say, there are myriad ways to do this. The energy of $s(t)$ is $$E_s = \int_{-\infty}^\infty s^2(t) dt.$$
If we calculate it using the decomposition, we obtain:
\begin{align}E_s &amp;= \int_{-\infty}^\infty (x(t) + y(t))^2(t) dt \\
                 &amp;= \int_{-\infty}^\infty x^2(t) dt + \int_{-\infty}^\infty y^2(t) dt + 2\int_{-\infty}^\infty x(t)y(t) dt
\end{align}
Clearly, if you require that $E_s = \int_{-\infty}^\infty x^2(t) dt + \int_{-\infty}^\infty y^2(t) dt$, then it is necessary that $$\int_{-\infty}^\infty x(t)y(t) dt = 0$$ or, in other words, $x(t)$ and $y(t)$ must be orthogonal.
By the way, I have never seen it claimed that all existing decompositions have the energy-conserving property you are looking for, not even in the context of physics rather than engineering.