Can the intelligence message of an FM signal be cross-modulated onto another FM signal due to nonlinearities?

Question

Can the intelligence message of an FM signal be cross-modulated onto another FM signal due to non-linearities?

Where I work, we have several FM two way repeater radio sites.

Over the years I've heard complaints of reception of a message on the wrong channel. Sometimes intermodulation or cross-modulation has been cited as the cause. Reading the literature however, I can't find this phenomenon discussed. I'm using Leon Couch's Digital and Analog Communications Systems.

Studying non-linearities, intermodulation and cross modulation, I see that the 3rd order output is:

$$ V_{out} = K_{0} + K_{1}(V_{in})+ K_{2}(V_{in})^2 + K_{3}(V_{in})^3 $$

If you put signals into the system where

$$ V_{in} = (A_{1})\sin(ω_1t) + (A_{2})\sin(ω_2t) $$

Then part of the third order outputs are:

$$ 3K_{3} (A_{1})^2 (A_{2}) \sin(ω_1t)^2 \sin(ω_2t) = \\ = \frac{3}{2} K_{3} (A_{1})^2 (A_{2}) \left\{ \sin(ω_2t) - \frac{1}{2} [ \sin(2ω_1 + ω_2)t + \sin(2ω_1 - ω_2)t ] \right\} $$

Generally it's explained that the term \$ \sin(2ω_1 - ω_2)t \$ accounts for intermodulation distortion that is difficult to filter because it is close to the desired signal. Also they show that if there is any amplitude variation on signal 1, that it can amplitude modulate signal 2, because

$$ \frac{3}{2} K_{3} (A_{1})^2 (A_{2}) \sin(ω_2t). $$

I never see a discussion of an FM signal being cross-modulated onto another FM signal. It seems to me that if signal 2 is an FM signal as: \$ \sin((ω_2 \pm \Deltaω) t) \$ then the cross modulating term:

$$\begin{aligned} &\sin(2ω_1t - ω_2t) = \\ &= \sin\left[ 2ω_1t - (ω_2 \pm \Deltaω)t \right] \\ &= \sin\left[ (2ω_1 - (ω_2 \pm \Deltaω))t \right] \\ &= \sin\left[ (2ω_1 - ω_2 \mp \Deltaω)t \right] \\ &= \sin\left[ (2ω_1 - ω_2)t \mp \Deltaω t \right] \\ \end{aligned}$$

It looks to me like the modulating signal would be impressed on the intermoduation product \$ \sin((2ω_1 - ω_2)t \$.

Of course my reasoning might be totally incorrect.

Like I said, I've never been able to find a mathematical or physics treatment of the phenomenon, but I've heard people argue that it happens, and others argue that in the FM domain cross modulation will only cause noise, not the imparting of an intelligent signal on another signal like in AM.

Answer 1

Can the intelligence message of an FM signal be cross-modulated onto another FM signal due to non-linearities?

Yes.

I haven't rigorously checked your trig analysis but you are going in the correct direction with that. The frequencies of intermodulation products depends on the frequencies of the intermodulating signals so FM on the latter will be transferred to the intermodulation products.

I'm afraid I must respectfully disagree with @JRE's suggestion that this is "wild, new physics". I've been using amateur radio FM voice repeaters (G8 call) for nearly 40 years and been involved with managing my local ones for around half that time. Intermod interference is an entirely standard feature. However, the other causes listed by JRE are also possible. If you are investigating these complaints you'll need to work through all possible causes.

Reading the literature however I can't find this phenomenon discussed.

I'm not familiar with Couch but university-style textbooks tend to major on topics that can be analysed theoretically but are thin on practical issues. So expect plenty of coverage of e.g. AGWN (additive Gaussian white noise) but no material on how large your tower supports need to be or what to do if the tower owner won't tell you who you are sharing the tower with.

If you are investigating this it would help to get some logs of the apparent interference. Are you the only radio user on your repeater sites or are the sites shared with other users? Are you using cavity filters on shared sites?

As your maths analysis implies, receiving intermod signals depends on having local transmitters with the correct frequencies to generate products on your receive frequency. Can you correlate interference products to specific repeater sites? You would be unlucky to get intermod signals on all your sites so reports of interference across your sites would imply problems in the common infrastructure or operator error as suggested by JRE.

others argue that in the FM domain cross modulation will only cause noise

You'll only get an intelligible signal out if the interfering signal is itself FM modulated. If it's not then you'll get noise. On this topic have your radio operators been able to recognise the interfering signals? Can they identify the users who seem to be interfering with your radio system?

The rusty bolt effect means that intermodulation can be generated in radio masts, etc not just in electronic equipment.