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?

$X_I(t) = \sum(A_i(t) * \cos[\omega_i * t + \theta_i (t)],i,N)$

$X_q(t) = \sum(A_i(t) * \sin[\omega_i * t + \theta_i (t)],i,N)$

  • 1
    $\begingroup$ What does AM/AM and AM/PM mean? $\endgroup$
    – Arnfinn
    Dec 10, 2016 at 22:44
  • $\begingroup$ i think, in order to be clear about what "AM/AM" and "AM/PM" mean, you should define what the "nonlinear amplifier" does mathematically. $\endgroup$ Dec 10, 2016 at 23:21
  • $\begingroup$ Since the amp is nonlinear it's AM AM and AM PM conversions operate on the whole input. I e the equivalent amplitude and phase of the sun of two sinusoids. Not on two amplitudes and two phases. $\endgroup$
    – Nir Regev
    Dec 11, 2016 at 7:54
  • $\begingroup$ AM stands for amplitude modulation, PM stands for phase modulation. The AM/AM characteristics shows how the amplitude is changed by the amplifier, depending on the input amplitude. The AM/PM characteristic shows how the phase is changed, depending as well on the input amplitude. How can I determine the equivalent amplitude and phase of the sum of two sinusoids? $\endgroup$
    – user25356
    Dec 11, 2016 at 12:14

1 Answer 1


The sum of two sinosoids with different frequencies $f_{c1}, f_{c2}$ does not add up to a single sinosoid. Only, if both sinosoids have the same frequency, you can combine them into a single sinosoid using the equation you state at the end of your post. See also for example this post.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.