Sideband-asymmetry in the Luxembourg effect

The Luxembourg effect is the effect that a long/mediumwave radio signal propagating through the ionosphere can be influenced by another strong long/mediumwave transmitter. A more scientific name for it is "ionospheric cross-modulation".

Nowadays, one can easily hear the effect on 162 kHz AM via the Twente WebSDR, where one hears the modulation of RTL (234 kHz) and/or Europe1 (183 kHz) transferred to the unmodulated 162 kHz carrier from central France. [Update 2023: unfortunately, both the 183 and 234 kHz transmitters have been switched off since I wrote the above, so it's no longer so easy to observe the effect.]

This effect was first discovered and explained in the 1930s, and studied further in the 1940s using test signals broadcast outside the radio transmitters' regular operating hours.

Using modern SDR techniques, I've been studying this effect in a way that wasn't possible in the 1940s: 24h per day, and for all modulation frequencies simultaneously.

In May 2016 I found that the phase of the sidebands is often asymmetric, After that, I did many more measurements both via my Twente receiver and using portable equipment while travelling through Europe. Together with DF6NM, who had already seen a case of amplitude asymmetry in January 2013, we came up with an explanation for these asymmetries.

This work is documented in a joint publication in the scientific journal Radio Science, which can be found at https://doi.org/10.1002/2017RS006525, published in May 2018.

This diagram shows a typical measurement run, in this case of the signals from RTL's 234 kHz transmitter transferred to Radio Monte Carlo's 216 kHz signal. The horizontal axis represents time (24 hours, UTC). The vertical axis is the modulation frequency, taken positive for the upper sideband and negative for the lower sideband.
The upper graph shows the magnitude of the effect, i.e., how much the modulation from the RTL transmitter was also present on the RMC signal.
The lower graph shows the phase of the effect, i.e., the phase difference of the modulation received directly from the RTL transmitter and received on the RMC carrier.
In this example one clearly sees that the phase is asymmetric (the color bands are wider in the upper than in the lower sideband), while the amplitude picture is largely symmetric.
Note that the effect disappears at night simply because the RMC transmitter is switched off then.
Many more such images can be found in the journal article.

In order to explain this phenomenon, one needs to realize that the diagram at the top of this page is too simple. One tends to think of radio signals as propagating in a single straight line from transmitter to receiver, possibly with a reflection or a bending in the ionosphere. However, radio signals are waves, and waves tend to propagate in all directions (just imagine what happens when you throw a stone in a pond). In principle, one should take into account all possible paths from transmitter to receiver, e.g., also paths that reflect off the ionosphere at other places than right in the middle. Normally those other paths (the so-called higher Fresnel zones) don't play a role: they tend to cancel each other due to path length differences and thus phase differences. However, this changes when the reflectivity of the ionosphere is changing rapidly under the influence of (in this case) the Luxembourg transmitter. In case of ionospheric cross modulation, the result is that the different modulation frequencies are received via different paths. (For the reasoning and calculations behind that, see the journal article.) The adjacent diagram illustrates this for the case of the RMC and RTL transmitters received in Twente.
Now that we know that the sideband frequencies are received from different places in the ionosphere, we can understand the asymmetry: phase asymmetry is caused by the fact that the sidebands are received via paths with different lengths; and amplitude asymmetry can e.g. be caused by the the radiation diagrams of the antennas or differing ionization state of the ionosphere at the different reflection places.