I'm dealing with an echo signal from a hybrid. An echo path includes DAC, the hybrid, ADC and the line infrastructure. I can see, that after using a digital linear echo-canceller the residual echo signal is still significant and I suspect that echo-path has additional non-linear components. Now I want to estimate, how much of the incoming signal is an actual echo and not a noise, and I want to do it before actually implementing or inferring any specific non-linear function (e.g. without assumptions regarding the order of Volterra series).

I've found the ITU-T O.42 recommendation, which suggests using the 4-tone intermodulation method to measure nonlinear distortion impairments on analogue circuits. But it doesn't suit me, since my echo path includes the DAC.

So is there a way to measure or estimate the level of the non-linearity in my echo path? I would highly appreciate any references. Also, if this question is more related to electronics, please feel free to migrate it to the corresponding site.


Definition of non-linearity: based on the source, a non-linearity for an echo signal is anything that cannot be modeled with a linear filter: a dc component, a long tail and truly non-linear dependencies. So the question becomes: is there a way to measure or estimate the level of those components compared to noise? The peculiarity here is that both a dc component and a long tail are actually linear in their nature.

Signal measurements: Following are the data that I got by changing the power of transmitted signal (from 1 to 6).

Power spectral density plots (regular and normalized), where frequencies are limited by a sampling rate:

PSD PSD normalized

Left: echo mean z mean and variance z var and residual echo mean e mean and variance e var for a regular (random) transmitted signal (using some arbitrary length of a linear filter). Right: echo mean and variance and residual echo mean and variance (using filter weights from training on the regular signal) for a constant transmitted symbol.

Regular tx Constant tx


  1. The PSD plots look identical, except for unclear shift of a plot for the lowest power level (1).

  2. Mean and variance plots do not resemble any obvious non-linearity, which would have shown as a power law dependency everywhere. Based on increased e mean for a constant transmitted symbol, there is a tail in the echo.

  3. What puzzles me the most is a linear growth of a residual echo variance, which I cannot explain.

So the original question is still relevant. Additionally, can I actually use obtained plots to estimate the echo non-linearity and if so, how should I do it?


1 Answer 1


excellent, you have PAM-4!

Send a long sequence of your least-amplitude symbol, and integrate the received power to calculate the energy of your received signal.

Do the same with the three other symbols.

Now, you can solve the equation

$$E_{RX} = \alpha E_{TX} + E_N + F(P_{TX}),$$

with $\alpha$ being the channel gain, which has to be the same for all 4 of your measurements, $E_{RX},E_{TX}$ being the transmitted and received energies (hint: normalize them to the least-power symbol), $E_N$ is noise energy, i.e. noise power times number of samples (which becomes asymptotically constant if you just average enough symbols), and $F$ is the nonlinear error term.

With $E$ known, and $\alpha$ and $E_N$ overdetermined determined from four measurements, $F$ is the reason for the error.

  • $\begingroup$ Great suggestion! The main concern: doesn't this approach require that I make an assumption about the non-linearity in order to solve the equation system? E.g. I set $F(P_{TX}) = a * P_{TX}^2 + b * P_{TX}^3$, and then solve the system for $a$, $b$, $\alpha$ and $E_N$. This is somewhat limiting. What if I want to estimate all non-linear components? $\endgroup$
    – megasplash
    Jan 25, 2021 at 10:13
  • $\begingroup$ all the nonlinear components are in $F$; to determine it, you might need more than 4 measurements with different powers , though. $\endgroup$ Jan 25, 2021 at 10:38
  • $\begingroup$ So did I understand you correctly, that I have to use an explicit form of $F$? $\endgroup$
    – megasplash
    Jan 25, 2021 at 10:56
  • $\begingroup$ If that is true, it will limit the results, as function $F$ will include only the explicitly defined non-linear components, and not all of them. $\endgroup$
    – megasplash
    Jan 27, 2021 at 11:27
  • $\begingroup$ no, you only get an intergral of "all the power that doesn't fit in the linear model" as F, not a model of any kind. $\endgroup$ Jan 27, 2021 at 12:08

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