Relation between power of the baseband signal and the low pass equivalent Let $x(t)$ be a real bandpass signal and $x_l(t) = (x(t) + j\hat{x}(t))e^{-j2\pi f_0t}$ be the lowpass equivalent. Is there any relation between $P_x$ and $P_{x_l}$ where $P_x$ refers to the average power of $x(t)$ and $P_{x_l}$ refers to the average power of $x_l(t)$? I tried to use the definition $$P_x=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}|x_l(t)|^2dt = \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}(x^2(t) + \hat{x}^2(t))dt \tag{1}$$
And I got stuck here. Maybe we should add some assumptions on $x(t)$ in order to be able to simplify $(1)$.
Since the Hilbert transform is an allpass transformation, the powers of $x(t)$ and its Hilbert transform $\hat{x}(t)$ are the same. Consequently, the complex baseband signal as defined in your question has twice the power of the real-valued bandpass signal. That's why in some textbooks the bandpass signal is defined as
$$x(t)=\sqrt{2}\textrm{Re}\big\{s(t)e^{j\omega_ct}\big\}\tag{1}$$
where $s(t)$ is the complex baseband signal.
The scaling in $(1)$ ensures that $x(t)$ and $s(t)$ have the same power.