I consider a sinusoidal signal with fixed amplitude $A_{\rm in}$ and frequency $f_c$.
This signal is sent into a nonlinear amplifier, whose AM/AM and AM/PM characteristics are known. Using these characteristics for the given amplitude, I can determine the output amplitude $A_{\rm out}$ and the output phase $\phi_{\rm out}$.
Now however, I consider a sum of two sinusoids with different amplitudes $A_{\rm in_1}$ and $A_{\rm in_2}$ and different frequencies $f_{\rm c_1}$ and $f_{\rm c_2}$. First, I want to describe this input signal by its equivalent lowpass signal. I want to do this in MATLAB. I know the hilbert() command gives the analytic signal. So I suppose when my first sinusoid is called $x_1(t)$, the equivalent lowpass signal can be described by $\mathcal H(x_1(t)) \cdot \exp(j2\pi f_{c_1} t)$. The same is valid for the second sinusoid. But I don't know how to represent the sum by an equivalent lowpass signal. In a second step, I want to apply the AM/AM and AM/PM characteristics to the signal. But now, I have two different amplitudes.
I found this: Consider a multiple carrier input: $x(t) = \sum(A_i(t)*\cos[(\omega_c + \omega_i)* t + \theta_i (t)],i,N)$
This can be written as $x(t) = A(t)*\cos[\omega_c *t + \theta(t)]$
where $A(t) = [x_I^2(t) + x_q^2(t)]^{1/2}$
$\theta(t) = tan^{-1} [x_I(t) / x_q(t)]$
I really don't understand this equation given for the resulting phase. Why is it valid? I mean X_I and X_Q are not just real and imaginary parts of x(t), are they? I know by Euler's formula, that the cos corresponds to the real part of the exponential and the sin corresponds to the imaginary part. But (see below) X_I and X_q are both sums of cos or sin waves of different amplitudes and different phases. How can I just add them?
$X_I(t) = \sum(A_i(t) * \cos[\omega_i * t + \theta_i (t)],i,N)$
$X_q(t) = \sum(A_i(t) * \sin[\omega_i * t + \theta_i (t)],i,N)$
