# How to isolate overlapping FM signals?

Analogue radio stations often have a range of frequencies that they can use. Different transmission towers will transmit on different carrier frequencies in this range, because the signal takes time to propagate through air and if they used the same carrier then the received signal would be... Well, it wouldn't be well-formed FM, and it would almost certainly be echoey and distorted.

But what if they did all use the same carrier? You'd need multiple aerials placed some distance apart (would 3 be enough?) and would have to do something clever with them... But would it be possible to extract a signal from that?

If it's not too different a question, would it be possible to extract the individual signals if the transmission towers were transmitting something different to each other on the same carrier frequency?

• Could you explain in detail a method to isolate two overlapping FM signals using multiple antennas? I don't think it can be done. – MBaz Jul 7 '18 at 16:23
• @MBaz well, beamforming. See the last part of my answer. But it's really not a viable choice for broadcast FM. – Marcus Müller Jul 7 '18 at 17:44
• @MarcusMüller Right. The thought of beamforming for FM broadcasting is so outlandish and unfeasible that it didn't even cross my mind :) – MBaz Jul 7 '18 at 18:55
• @MBaz The one I was thinking of was measuring a pattern of high | of mid | of low using multiple antennas, and seeing how that shifted between the antennas. Now that you have an approximate idea of the distances-in-time between them (and perhaps their amplitudes), use a clever echo cancelling thing to get a vague idea of what the signal should be. Then use this vague idea to refine the idea of what the signal should be. That should give you a reasonably correct analogue signal. Then simply decode the FM. – wizzwizz4 Jul 7 '18 at 19:59
• dude, I wrote that answer... – Marcus Müller Jul 8 '18 at 7:04

## 4 Answers

What if they did all use the same carrier?

What you're describing is a single-frequency network (SFN). These are in common use for things that are not stupid FM broadcast.

The whole truth is that reception from different senders in an SFN just look like heavy multipath propagation, where the sent signal just takes multiple paths of different length to the receiver, and thus the receiver sees the signal as what would come out of a tapped delay-line.

If you consider it like that, the effect is clear: the different delays with which the signal can reach the receiver lead to something that for all practical purposes is an FIR filter. If you look at it from the viewpoint of digital communications, you get Inter-Symbol Interference (ISI) as delayed versions the last sent symbol still reach the receiver while the shorter paths already brought on the next symbol.

If you look at the same thing from the perspective of spectrum analysis, it is a frequency-selective channel.

Either way you look at it, you counter that with an equalizer that requires knowledge of how the channel looks (i.e. the different propagation delays, or the frequency-dependent behaviour of your channel, which are just two descriptions of the same thing, linked by the Fourier transform) and can then do its best to reverse that effect.

Now, sadly, FM radio is not digital communications. You don't get symbols, and by the very nature of FM being a modulated carrier, you don't even get any kind of "known, not very repetitive" input with which you can estimate what your channel looks like.

This needn't be a problem – if the resulting delays don't make the channel too harsh, you don't lose much SNR at the receiver. We can measure that by comparing the delay spread (difference between shortest and longest path) to the inverse of the channel bandwidth. If the channel bandwidth is "sufficiently" smaller than the inverse delay spread, we're probably in luck and can still consider the channel "sufficiently" flat. "Sufficiency" here depending on your needs, so let's just do a order-of-magnitudes calculation.

An FM channel is roundabout 150 kHz wide. Invert that, and you get $6\frac23$ µs.

Electromagnetic waves travel with the speed of light in vacuum, and for all practical purposes here, in air, too. So those microseconds correspond to a distance difference to two transmitters of $c_0\cdot \frac{20}3\cdot10^{-6}\,\text{s}=3\cdot 10^{8}\frac{\text{m}}{\text{s}}\cdot \frac{2}3\cdot10^{-5}\,\text{s}=2\,\text{km}$.

Since we can be pretty sure that different FM radio transmitters aren't only 2 km apart, we can directly see that the resulting radio channel will be wildly non-flat and frequency selective. In fact, since the distance between these will be orders of magnitudes larger, we can assume that even given good channel information, we might have frequency-dependent deep fades that we simply can't properly "equalize away" (you get nothing through at those frequencies, so you can't even amplify anything to make it "flat" again).

In other words, FM and SFN don't play well together. There's been attempts to still implement SFNs for FM broadcast (I think Netherlands tried that), but as far as I know, they failed – they can only work by "aligning" the interference for limited locations, and if your network can't be too large and there's no strong variation of propagation environments for different transmitters (which would explain why small and flat Netherlands was an attempted adopter).

That's one of the most important reasons the world is leaving analog broadcasting behind. DAB and DAB+ are SFN-based, as their heavily multi-carrier oriented OFDM system with channel bandwidths of 1 kHz allows for much larger delay spread, and their frame structure and channel coding allow for heavy error correction, so that even deeply faded subcarriers don't matter that much.

So, yes, you've found a (if not the most important) spectrum management reason why FM should, if you ask me, die as soon as possible: it enforces the non-reuse of spectrum, and that's a waste of a precious resource.

Now, you mention multiple antennas:

Well, yes, if you have multiple antennas, you can use phasing to make them have some directivity, i.e. you can "simulate" having a single antenna that only receives from one direction. Hm. You buy multiple antennas, build a complex and large phased array and try to operate in a SFN, and the best-case scenario is you get the same performance as in a MFN with a single antenna. Doesn't sound too tempting, and, you'd need to have an idea in which direction you want to "look". It's notoriously hard to estimate that out of, again, an analogue signal with no known non-periodic structure. In a situation where you're at a fixed position (moving obviously breaks knowledge of directions) and adjust antenna phases just to get broadcast radio: Have you considered satellite radio?

The short answer to your question is yes and the method is called beamforming which you alluded to in your question by mentioning multiple aerials, and knowing the propagation time from two distinct transmission locations broadcasting on the same frequency to each receiving aerial is key to beamforming.

The number of required aerials depends on the specific details of the actual scenario.

As mentioned by MBaz, one can use a modulation called CDMA to share a carrier frequency by multiple transmitters. The US GPS satellite constellation does this on 2 shared carriers.

The longer answers is yes but beamforming is expensive and every user will need their own beamforming system because all the relative transmission delays change with location. Commercial FM is also something that people used to enjoy listening to while driving which greatly complicates straightforward beamforming. Extraneous echos and multi path complicate beamforming and impact achievable performance.

Spectrum management is a much better solution and modulation schemes like CDMA are better solutions.

So, I would say the answer is yes, it is possible but isn’t justified in all but the most unique circumstances.

• I must admit that I'm quite conflicted about this: yes, CDMA does allow a single frequency to be shared, but a) if you want to share with $N$ senders, your bandwidth increases by at least $N$, so you gain nothing in terms of spectrum efficiency, and more importantly b) CDMA is hard to get right with large delay spread, and doppler spread, at the same time (which is the reason "the world" abandoned it for true multi-user systems like cellular data and ad-hoc high rate networking – UMTS was a CDMA system, but the impossibility to get perfect auto- and crosscorr properties led to its obsolence, – Marcus Müller Jul 8 '18 at 4:34
• so that in many places we now have 2G and 4G and soon 5G working in parallel, but 3G is already phased out. Long story short: CDMA is meh, because sync is hard, and because you can't have a set of bases that are perfectly orthogonal under both time- and frequency shifts) GPS is not really a multi-user system, so it only marginally counts here :) Long range broadcasts use OFDM these days (DAB, DAB+, DRM, HD Radio, …) – Marcus Müller Jul 8 '18 at 4:35

The use of different carrier frequencies is unrelated to propagation delay. The reason is that the use of different carriers makes the signals orthogonal to each other, meaning that they can overlap and still be told apart.

To separate signals using the same carrier frequency, it is necessary to introduce orthogonality in some other fashion. One common example is CDMA, which multipies signals by orthogonal codes, thus creating orthogonality independent of the carrier frequency.

There are plenty of questions and answers on this site related to signal orthogonality and CDMA, which you may want to browse.

I think that the answer is no, you won't be able to recovery your signals if they overlap in frequency. Only if the other signals are weak enough you could recover one of them, but it would have some residual interference from the other sources.

There are things like MIMO that are based in multiple sources and multiple recivers and spatial characteristics of the path, but if there is interference in your signal it is very unlikely that you get rid of it. And it does not seem like what you are looking for.

But there is something that sounds fishy to me in your question. I think that the speed of propagation (and the time if the transmitter antennas are at the same distance) of the wave is not related with the modulation frequency.

I hope it helps.

• Speed of propagation means that the signals don't line up in time. Also, what's MIMO? Also, tour. – wizzwizz4 Jul 7 '18 at 15:36