# Are there amplitude and phase characteristics of harmonic signal components?

Nonlinear amplifiers are characterized by their AM/AM and AM/PM. That's what I thought. But now I have a question: The amplitude and phase distortion characteristics, do they only belong to the fundamental signal component?

I mean, to fully characterize my amplifier, are there AM/AM and AM/PM characteristics for the harmonic components as well?

• For what purpose are you using this nonlinear amplifier ? – Fat32 Feb 16 '17 at 19:26
• I am not using any amplifier. I am writing on a paper about amplifier and I want to give an introduction to how it is characterized. There are various nonlinear magnitudes, such as gain compression, harmonic distortion, and so on. But usually the nonlinear behavior is completely characterized by the AM/AM and AM/PM characteristics. That's what I thought. But I am confused right now. I am not sure anymore, if the AM/AM and AM/PM characterize the output of a single-tone input completely, or only the fundamental component? – user25356 Feb 16 '17 at 20:14
• ok in short this is an audio amplifier, at first I thought it was an RF transmiter amplifier anyway lets watch out answers. – Fat32 Feb 16 '17 at 20:32
• @Luk I'd again, as done several times now, ask you to properly cite your sources, especially if they're sources of confusion. (1 2 ) – Marcus Müller Feb 16 '17 at 21:16

If the amplifier is memoryless (i.e. there is only AM/AM conversion and no AM/PM conversion) and the input signal is bandpass (centered around some carrier frequency $F$), for input signal:

$$x(t) = a(t)\cos(2\pi Ft + \alpha(t))$$ the output signal $$y(t) = v(x(t)) = v(a\cos(2\pi F t)) = v(a \cos \theta)$$

can be expanded using a Fourier series: $$v(a\cos\theta) = v_0(a) + v_1(a)\cos(\theta) + v_2(a)\cos(2\theta) \ldots$$

If the amplifier is memoryless, the coefficients $v_m(a)$ are real and determine the harmonic amplitudes, and there is only AM/AM conversion and no phase shift on the harmonics.

The coefficients are known as the "Chebyshev Transform" of the nonlinearity $v(x)$ and are given by: $$v_m(a) = \frac{2}{\pi} \int_0^\pi v(a \cos\theta)\cos(m\theta)d\theta$$ If the amplifier has some memory, but "not much", it is known as a "quasi memoryless" system, and the coefficients are complex (giving AM/PM conversion).

To characterize the amplifier, you would need to measure $v_m(a)$ for each harmonic $m$. However, since usually all you need is the first harmonic because others will get filtered out later, you can measure $v_1(a)$ from standard power out/power in curves. (The same ones used to measure 1dB compression and 3rd order intercept). The power gain measured in this curve is just $(v_1(a)/a)^2$.

• thx Carlos! Also for the paper! So is the following correct: " Nonlinear characteristics are commonly generated from measurements with single-tone excitation. The resulting characteristics belong to the fundamental component of the output signal." – user25356 Feb 16 '17 at 21:18
• If you are talking about the first harmonic, then yes that statement is correct. In math terms, the amplitude of the first harmonic is $v_1(a)$. (If the excitation is just a single tone with no modulation, then $\alpha=0$) – Robert L. Feb 16 '17 at 22:16
• @Luk P.S. if I answered the question, marking it would be appreciated :) – Robert L. Feb 16 '17 at 22:18