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

Question

Can the intelligence message of an FM signal be cross-modulated to a a different frequency by non-linearities, but with the FM modulating message remaining (relatively) intact on the new frequency?

This isn't really trying to solve a particular problem, but I'm trying to understand if this is possible, and if the math checks out. In FM two-way radio, sometimes users experience "bleed over" from another channel. I've heard intermodulation or cross-modulation being cited as the cause. Reading the literature however I can't find this phenomenon discussed. I'm Using Leon Couch, Digital and Analog Communications Systems by the way.

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

$$V_{in} = A_{1}\sin(\omega_1t) + A_{2}\sin(\omega_2t)$$

Then part of the third order outputs are:

\begin{align}3K_{3}A_{1}^2A_{2}\sin(\omega_1t)^2\sin(\omega_2t) = \frac{3}{2}&K_{3}A_{1}^2A_{2}\{[\sin(\omega_2t) \\&- \frac{1}{2}\big[\sin(2\omega_1 + \omega_2)t + \sin(2\omega_1 - \omega_2)t\big]\}\end{align}

and another

Generally it's explained that term $\sin(2\omega_1 - \omega_2)t$ 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

$$\frac{3}{2}K_{3}A_{1}^2A_{2}\sin(\omega_2t)$$

But 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\big((\omega_2 \pm \Delta\omega) t\big) $. then the cross modulating term:

\begin{align} \sin(2 \omega_1t - \omega_2t) &= \sin\big(2\omega_1t - (\omega_2 \pm \Delta\omega)t\big)\\ &= \sin\bigg(\big(2\omega_1 - (\omega_2 \pm \Delta \omega)\big)t\bigg)\\ & = \sin\bigg(\big(2\omega_1 - \omega_2 \mp \Delta \omega\big)t\bigg)\\ & = \sin\big((2\omega_1 - \omega_2)t \mp \Delta \omega t\big) \end{align}

Which looks to me like the modulating signal would be impressed on the intermodulation product $\sin((2\omega_1 - \omega_2)t$

It seems to me about the same as super-heterodyning the message signal, which would be the $ \Delta\omega 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

I don't see any obvious issue with the OP's reasoning except to clarify that the cross-modulation is is not onto another FM signal but onto another FM channel with caveats on intelligibility. Below I add more detailed intuitive explanation as to how this occurs:

If a non-linearity creates a harmonic at $n f_c$ of a single-tone carrier signal at $f_c$ for some integer $n$, if the frequency $f_c$ is varied by $\Delta f$ over a certain duration of time $T$, then the harmonic must vary at $n\Delta f$ over that same time duration $T$: the harmonics of a single tone can only exist at integer multiples of the frequency of that tone. Thus it is clear if the carrier is FM modulated, the harmonic will also be FM modulated. But more specifically for FM signals consider the modulation index given as the ratio $\beta = \Delta_f/f_m$ where $\Delta_f$ is the frequency deviation and $f_m$ is the modulation rate. $\beta$ is the instantaneous angle of the modulated waveform (relative to the carrier). Therefore the FM signal at the harmonic will have an associated $\beta = n\Delta_f/f_m$. Increasing $\beta$ would serve to increase the amplitude of the demodulated signal to the point where it exceeds the range of the frequency discriminator resulting in an additional distortion source, but for most signals and low $n$ it would likely be completely intelligible.

With intermodulation products specifically we would have this same effect with some interesting caveats. Intermodulation products are clear to understand from two tone measurements (which are used as a consistent characterization of linearity and of primary concern with 3rd order intermodulation due to proximity in frequency of the resulting products, and 2nd order products specifically for Zero-IF Receivers). In general for any two tones, the intermodulation products are given as $n f_1 \pm m f_2$ where $n$ and $m$ are integers and $f_1$ and $f_2$ represent the frequencies that are being intermodulated. The sum $n+m$ is the order of the intermod and this effect is easily seen by carrying out the taylor series expansion for any non-linearity (such as $e^x$) and multiplying out the cosines and sines to see all the products (just as $cos(\omega t)^2$ is a frequency doubler as a 2nd order product: $\cos(\alpha)\cos(\beta) = 0.5 \cos(\alpha+\beta) + 0.5 \cos(\alpha-\beta)$ thus we get the sum and the difference, in this case the sum is a doubled frequency). So the third order products of primary concern are $2 f_1- f_2$ and $2f_2-f_1$ given these products are within $|f2-f1|$ of the signals. So closely spaced tones create third order intermodulation products that are also closely spaced to the original signals.

That said consider what would happen if $f_1$ is modulated and $f_2$ is a single tone, then clearly we can see from the first explanation that $2f_1- f_2$ would be an FM signal with $\beta$ expanded by two (the subtraction of the frequency $f_2$ is just a frequency translation which doesn't expand the frequency deviation further). But what if $f_2$ is also modulating? The result would be the combined waveform of the FM signal at $f_1$ with twice the $\beta$ together with the FM signal at $f_2$ with its original $\beta$ intact. Assuming these two signals were separated sufficiently to begin with such that the third order product is still spectrally distinguishable (not overlapping the others), although one of the two will be certainly stronger out of the demodulator if not saturating it, I doubt it will be easily intelligible when both channels are actively modulating (for voice we do have sufficient periods of silence so would be interesting how this actually would be perceived). However even in this same circumstance, the doubled $f_1$ signal will also be present (at the much higher carrier frequency so perhaps out of any band of current reception or more likely easily filtered) and intelligible, regardless of $f_2$. (And a doubled $f_2$ signal will also be present if not filtered).

Answer 2

Yes and no, it can't be intermodded onto another SIGNAL, but it can be intermodded onto another CHANNEL. For example, if you are monitoring 144 MHz, an FM signal on 145 MHz and on 146 MHz CAN, due to non-linearity, end up being heard on 144 MHz. But it can't be modulated onto an existing signal on 144 MHz. Mark

Answer 3

yes you are correct, the second harmonic will have 2x the deviation. In the case of 2F1 +/- F2, the modulation of both signals will be on the result and the 2F1 will have 2x the deviation. So yes the FM will be imparted onto the intermodulation products and can be received with the caveat that the deviation will be increased by the term.