VLF Transmission Using Soundcard

[sru2]

I know I can make a VLF transmitter by running a software signal generator through a soundcard and into an antenna.

What happens if I play two signals at once, do I get two carrier waves?
If I use a WAV file (LPCM) and merge two sine waves, will this also create two carrier waves?

 

[sru2]

I thought I would give everyone an example. In the following diagram we have two stereo Wav files (LPCM). If we send a signal from any of these channels to the soundcard, we can produce a weak VLF wave at the same frequency.

If we then mix the two Wav files, we get the output below. Rather than two waves superimposed, we have combination of the two waves.

The question is after passing this through the DAC of the soundcard, do we get an electrical signal that looks like the Wav file, or do we get the two frequencies as above?

 

[sru2]

I have been researching this a bit and hopefully someone can tell me this is correct.

1. The above pictures are the result of additive synthesis. For the frequencies to be restored after passing through the DAC, it would need to perform some form of subtractive synthesis and this just is not required for driving a speaker?

2. Polyphony - or playing to Wav files on separate threads will result in additive synthesis during DSP with normalization to prevent clipping. Is this accurate?

3. Channel separation - This is the only way to get true independent frequencies output in the electrical signal? (such as 440Hz on left, 261.626 on right) Is this accurate?

 

[vk6kro]

At any point on the wire, there can only be one voltage at a time. So you can't get two waveforms independently. You will get a additive sum of the two.

This doesn't mean the carriers are lost and you could still recover each one with suitable filters. A suitable filter would be a radio receiver.

Note that this is not a mixer and you would not produce sums and differences of the two waveforms.

 

[sru2]

If I get you right the use two signal generators, one at 2Hz and the other at 2Hz, outputted to a cable will create a signal on the cable that will be a combination of both? ...and this can be separated by a bandpass filter at the receiver.

What happens in the case of a PCM? The following waveform is the product of combining a 2Hz, 1Hz and 1Hz sine waves. (ignore the clipping for now)

Each frequency would not be delivered separately to the cable. Does this matter? Will it still separate in the bandpass filter at the receiver?

 

[vk6kro]

You would have to exclude inputs of the same frequency as these would possibly cancel each other out if they are out of phase and of the same amplitude.

Once they were cancelled, there would be nothing left and you couldn't recover the original components.

Near where I live, there are two AM broadcast stations that use the same antenna. They just use filters to stop the other signal coming back into their transmitter.
The antenna is a 100 ft high tower on swampy ground near a river, so this is valuable enough to make the elaborate filtering worth while.

 

[sru2]

How about PCM? As I said earlier:

Each frequency would not be delivered separately to the cable. Does this matter? Will it still separate in the bandpass filter at the receiver?

 

[vk6kro]

PCM means Pulse Code Modulation?

No, it only applies to sinewaves. PCM is made up of square waves which have a complex structure of harmonics. So, it would not be possible to filter them and recover the original signals.

Such signals could be sent on the same wire, though, if they are used to modulate carriers of different frequencies.

 

[sru2]

PCM means Pulse Code Modulation?

No, it only applies to sinewaves. PCM is made up of square waves which have a complex structure of harmonics. So, it would not be possible to filter them and recover the original signals.

Such signals could be sent on the same wire, though, if they are used to modulate carriers of different frequencies.

I don't think a PCM is composed of square waves. A PCM takes a sample of the electrical signal as periodic intervals. I can use the sine wave above, in PCM format, to create a carrier wave for a radio signal by pushing directly to an amplifier and antenna.

As you said earlier, only one voltage can be present on the cable, so rather than getting multiple signals we get complex waveform that can be filtered later.

A PCM file that uses additive synthesis to merge different sine waves, should be the same as the complex waveform we would get on the cable. Thus, it should separate the same way at the receiver.

 

[davenn]

I don't think a PCM is composed of square waves. A PCM takes a sample of the electrical signal as periodic intervals. .....

a PCM signal is a digital representation of an analog signal
so by definition its a square wave

Dave

 

[sru2]

a PCM signal is a digital representation of an analog signal
so by definition its a square wave

Dave

Yeah, I was just thinking about it, but I was wondering about the slew rate on the output, does the signal have time to drop to 0?

 

[sru2]

Ok, I did some digging on pulse chain carrier waves, which is exactly what the PCM would be, and according to the following book as long as the frequency of pulses is twice that of the frequency of the signal, it will work.

Modern Dictionary of Electronics
By Rudolf F. Graf

http://books.google.co.uk/books?id=o2I1JWPpdusC&pg=PA480&lpg=PA480&dq=%22pulse+chain%22+%22carrier+wave%22&source=bl&ots=ATYnO8rUb7&sig=bcE2PD-VH7MaB2ZVo5UWsy_7TAg&hl=en&sa=X&ei=1CWuT5CFM4W2hAea-YjUCA&redir_esc=y#v=onepage&q=%22pulse%20chain%22%20%22carrier%20wave%22&f=false

That means with a standard sound card, with a sampling rate of 44100, we should be able to transmit signals up to 22.5Khz without issue.

Using OFDM, we can get around the issue of harmonics and extract our frequencies with a bandpass filter.

https://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing

The PCM file acts as the mixer in the following document:

http://books.google.co.uk/books?id=...cXPhAfp9I3uCA&redir_esc=y#v=onepage&q&f=false

That is confirmed on page 32 of this document:

https://docs.google.com/viewer?a=v&...szW-9o&sig=AHIEtbTaZBvw1LrEJD2y8DqQiGYQi5OlQw

At this point, I am not too concerned with sidebands, just multiple sine waves.

 

[sophiecentaur]

a PCM signal is a digital representation of an analog signal
so by definition its a square wave

Dave

That's a bit of an over-simplification. Very few properly engineered PCM signals are square waves. A digital signal that actually has 'square edges' is grossly under-using the available bandwidth.
The Symbols on a digital signal carry information about Discrete digital values but, for example, a '1' could possibly have a whole range of analogue values from 0.5V to 1.4V and a '0' could have analogue values from -0.5V to 0.49V, depending on the filtering used and the earlier and later binary values in the stream. It is always up to the demodulating circuit to filter and 'slice' to find the actual digital value of the binary data. Google Digital Eye Patterns.

 

[sru2]

I suppose these questions need to be asked:

1. Is the output of the DAC a continuous sine wave, or a quantized representation of a sine wave?

2. If quantized, does this carry the same properties of a continuous sine wave, in that, it will produce a radio wave at the frequency of the sine wave?

 

[sru2]

Well, I found two articles on this:

http://mwrf.com/Articles/ArticleID/22873/22873.html
http://www.wirelessdesignmag.com/ShowPR.aspx?PUBCODE=055&ACCT=0000100&ISSUE=1203&RELTYPE=blog&PRODCODE=000000&PRODLETT=GL&CommonCount=0

Both appear to suggest that the resolution provided by a modern DAC is sufficient to be pumped directly into an amplifier. I'm sure at these low frequencies, harmonics won't be an issue and would be well above the frequencies of interest.

This document on SDR transmitters appears to suggest the same:

https://docs.google.com/viewer?a=v&...7XNaa9&sig=AHIEtbQGwD7Qz8n6WAIsOdwrLdQWyUSqkg

Anyone see a reason this would not apply to DACs in a sound card?

 

[sophiecentaur]

Well, I found two articles on this:

http://mwrf.com/Articles/ArticleID/22873/22873.html
http://www.wirelessdesignmag.com/ShowPR.aspx?PUBCODE=055&ACCT=0000100&ISSUE=1203&RELTYPE=blog&PRODCODE=000000&PRODLETT=GL&CommonCount=0

Both appear to suggest that the resolution provided by a modern DAC is sufficient to be pumped directly into an amplifier. I'm sure at these low frequencies, harmonics won't be an issue and would be well above the frequencies of interest.

This document on SDR transmitters appears to suggest the same:

https://docs.google.com/viewer?a=v&...7XNaa9&sig=AHIEtbQGwD7Qz8n6WAIsOdwrLdQWyUSqkg

Anyone see a reason this would not apply to DACs in a sound card?

However you choose to produce your electrical signal, the situation is exactly the same if the signal is exactly the same. You would, of course, need to filter your DAC output to eliminate harmonics.

Your main problem will be in building a suitable antenna to operate efficiently at your VLF frequency. You would also have a problem with Matching the antenna well at the two frequencies you plan to operate with because the antenna will be a tiny fraction of a wavelength (loop or long wire). The interference levels at VLF can be very high unless you are operating at a remote location - not a problem for submarines etc..
Did you have an idea of the sort of range your link would be working over?

 

[sru2]

However you choose to produce your electrical signal, the situation is exactly the same if the signal is exactly the same. You would, of course, need to filter your DAC output to eliminate harmonics.

Would this be required with a DAC from a sound card? Would it not already be filtered to audio frequencies?

Your main problem will be in building a suitable antenna to operate efficiently at your VLF frequency. You would also have a problem with Matching the antenna well at the two frequencies you plan to operate with because the antenna will be a tiny fraction of a wavelength (loop or long wire).

Agreed, do you have any solutions that may help? Something that may extend the frequency range.

I've been reading about how highly sensitive electrically short receivers can be made and I'm wondering can this be adapted for transmission. See here:

http://www.longrangelocators.com/forums/showpost.php?p=142280&postcount=16

The interference levels at VLF can be very high unless you are operating at a remote location - not a problem for submarines etc..

Hopefully it won't be much of an issue.

Did you have an idea of the sort of range your link would be working over?

Initially, less than 20m. The idea is to create a slow radio link between computers.

 

[sophiecentaur]

Would this be required with a DAC from a sound card? Would it not already be filtered to audio frequencies?
Agreed, do you have any solutions that may help? Something that may extend the frequency range.

I've been reading about how highly sensitive electrically short receivers can be made and I'm wondering can this be adapted for transmission. See here:

http://www.longrangelocators.com/forums/showpost.php?p=142280&postcount=16
Hopefully it won't be much of an issue.
Initially, less than 20m. The idea is to create a slow radio link between computers.

The DAC card should be fine, followed by a fairly straightforward power amp. Presumably you would be using a duplex system with separate TX and RX channels each way. A good notch filter in each receiver should allow you to use the same antenna hardware each end.

A small receiving antenna is not so much of a problem - a ferrite rod works very well on all domestic lf/mf receivers. A transmitting antenna is more of a problem because of the incredibly low radiation resistance of short radiators. A dipole of length λ/100 has a radiation resistance of around 0.02Ω, for instance.
That article on small antennas is interesting but it doesn't seem to be practically based, dealing with reception. The basic message is the same as for any thin wire antenna - its effective cross section is massive compared with two skinny bits of wire because of what happens in the near field energy flow. The high Q of a ferrite rod coil is what makes it such a good energy collector. But ferrite would saturate at very much lower powers than you would want for your transmitter.
The way to go would probably be with a large many-turn loop antenna and there are many publications about those, although I haven't any particularly in mind.

I have just read your 20m operating distance and this makes things quite a bit different - this is extremely 'near field' for the frequencies you are planning to use and two tuned loops could work with no trouble. You don't need to be considering radiated power - just the coupling between two coils.

Interference will only be a problem when you try for longer link distances.

 

[sru2]

Sorry for the late reply, I've been caught up writing some DSP software for this project. I paid attention to both vk6kro and sophiecentaur in relation to harmonics and band pass filters and decided to run some tests.

My sound card provides an internal loopback that allows me to listen to the output without connecting a cable between the speaker output to the line in. This allows me to evaluate the signal quality without introducing noise or artifacts external to my machine.

In this first diagram, I am playing a PCM at 440Hz. Without any sound, this spectrum would be entirely black. As we can see, the harmonics are extensive and propagate all the way up to 20Khz.

In this second diagram, we narrow in on our area of interest which is below 1Khz. We can see how the carrier wave at 440Hz is clearly defined and that the signal is well above the surrounding harmonics. What is clear is that sophiecentaur was correct and a band pass filter is required on the output stage. The source of the harmonics is the card's circuitry and internal crosstalk. If we look at the signal stability, we can see that the steep roll-off begins after 10-20 Hertz and a shallower roll-off of about 100-150Hz. This means that we can do FDM with about 25-30Hz separation from the main carrier frequency.

In this final diagram I test vk6kro's statement that due to the harmonics, the original frequencies would be unrecoverable. This signal is from a PCM that uses additive synthesis of two sine waves, 440Hz and 261.626Hz, which were each reduced by -6db to prevent clipping. As we can see, the harmonics are dreadful, but the two original carriers are clearly stronger by a detectable amount.

This analysis reveals that there are two ways to approach this problem. The easy route would be to ignore the harmonics and focus on selecting the strongest signals at the receiver. Its not a great solution, but it will work and more importantly will continue to function even if we change frequency. That said, we would need to inform the receiver of the number of frequencies to lock on to. In the case above, selecting the two strongest signals would achieve a link on our carriers of interest.

The second solution is a variable bandpass filter. This is a complex setup as it must function across the entire range of frequencies and be capable of controlled by computer. The upshot is that we can eliminate the harmonics and even narrow the bandwidth of the output signal. This provides more channels in our FDM setup.

Anyone got comments, good ideas, or schematics for a good variable band pass filter?

 

[sru2]

Fixed the images above...

 

[sophiecentaur]

What actual frequencies are you intending to use for the link? 'VLF' usually refers to at least a few tens of kHz. The problem of achieving coupling between two coils would be greater as the frequency is reduced.
I realize that the availability of a sound card makes high audio frequencies attractive but there are several other factors to take into account if you want a working system.

What data rate were you expecting?

 

[sru2]

What actual frequencies are you intending to use for the link?

The frequency response of my sound card is 10-20000Hz. The idea is to use as many frequencies as possible by employing frequency division multiplexing. To do this, I need to avoid the complex harmonics. Given that the harmonics will vary between sound cards, OFDM is unsuitable as a solution. I was thinking of an adaptive form of OFDM (AOFDM). I could write an application that detects the harmonics, similar to images above, then inserts a new carrier in a blank space until the spectrum is full.

I would need to compare that result, with harmonics induced in the receiver and eliminate carriers causing problems. I could do this at the sync stage.

Let's leave this as an advanced step for now. Just assume a manual setup with as many carriers as possible between 10-20000Hz.

'VLF' usually refers to at least a few tens of kHz. The problem of achieving coupling between two coils would be greater as the frequency is reduced.

This is an area I would be weak on. Any suggestions would be much appreciated. I would like to understand the differences between working in the 'near field' and the 'far field'. Does this introduce noise? Increased complexity? What the difference between coupling and radiating? etc...

I realize that the availability of a sound card makes high audio frequencies attractive but there are several other factors to take into account if you want a working system.

Any factors that may cause issues I would be glad to hear about.

What data rate were you expecting?

With FDM I'm sure we can aim for 56K as a good target.

 

[sru2]

I wanted to check the harmonics from the additive synthesis of two signals at higher frequencies. For this test I increased the resolution of the spectrum analyzer to get a better picture of signal stability. This process is slower, but shows long term trends a lot better.

In the picture below, the signal is the product of an 18Khz and 19Khz signals. The harmonics are extensive but with digital filtering a threshold can be set allowing carriers to occupy frequencies occupied by weak harmonics. See the last image for more on this.

I also found some noise below 200Hz, around 1KHz and a similar noise pattern around 11Khz. Noise suppression under 200Hz is probably the result of filters rolling off. The noise at 1Khz and 11Khz is a little more difficult to explain and has no clear source.

Finally, the last diagram demonstrates the power of digital filtering. Achieving the picture below at a receiver with electronics would be both expensive and time consuming. Its a very complex design. In software, rejecting signals below a threshold is simple and you don't lose any power from the signal in the process. Using this threshold, it is possible to squelch weak signals on the same frequency (such as harmonics) and focus only on the main carrier of interest.

 

[vk6kro]

The harmonics of a data signal are much more than an inconvenience. They are there as a result of the shape of the original signal.

So, to reconstruct the shape of the original signal, you need to recover all the harmonics and make sure they are not mixed in with the harmonics of any other signal.

This is why you won't be able to do it if there are different signals on the same wire..

If you reduced all signals back to their sinewave fundamental, there will be data errors when a previous signal does not go away before a new one is introduced. Instead of nice sharp rises and falls, the signal will be a confusing mess of sinewaves in various stages of rising and falling.

Radiation from low frequency AC signals on wires does occur but at a very low level. Some loss is experienced in power line transmission and various methods are used to minimise it, but the percentage loss is very small.

So, you will probably not be able to detect the signal more than a few inches away.

 

[sophiecentaur]

If you are expecting such a high data rate within such a narrow channel then you are going to need a pretty high signal to noise ratio. (Basic Shannon information theory) This means that you will need plenty of coupling between transmitter and receiver. Before going much further in your experiments with the DAC, I might recommend trying some coils, fed with audio frequencies, and see just how much signal will couple from one to the other over a range of different separations. You could then see how increasing your operating frequency might help. I think that you'll find it necessary to operate at a frequency that your DAC will not produce. However, you can always 'mix up' the signal the DAC has produced to a frequency that is more amenable to RF transmission. This is how most comms systems work.
You will also appreciate that a system, operating at 'baseband' can only operate in the absence of any other similar nearby systems. This may not matter for a one-off experiment but it's very relevant if you want to take things further. WiFi (and all other wireless systems) have a choice of a number of different channels for this reason.

 

[sru2]

I think it may be best just to bite the bullet and try it. I feel the best approach is to get a functional receiver built first. Nothing fancy, just an average functional antenna. I have been doing some reading and the recommendation is to add some form of surge protection. Even though it is unlikely that the antenna will encounter lightning indoors, it may encounter signals from Medium and Long wave transmitters that could damage the sound card.

I also need to convert between the high impedance of the antenna and the low impedance of the sound card. I think I found a good circuit here:

http://gkircher.stormloader.com/sbhiz/

Although, I will need to check if all my inputs support this, otherwise I will need to move to the line in.

Any ideas on how to construct the surge protection? Or adapt that circuit linked above for use with line in?

 

[sophiecentaur]

Oh yes. Matching is essential but could involve no more than a suitable 'tap' onto your transmit and receive coils. I am assuming that your antennae will consist of large(ish) coils - not pretty but necessary for such low frequency operation. That link talks about a whip antenna for VLF transmissions - which have already been launched with an enormous transmitting array, covering literally a couple of fields. Your situation is different in that you need to transmit and receive over a short distance. This would be best done, I feel, by magnetic coupling and not electric coupling. Think of a very large transformer rather than two radio antennae! Google LF frame or loop antenna if you want a laugh at just how big these things can get.

Can you obtain a signal generator and an oscilloscope for the initial work on the antennae? You will soon see what I mean about the distance being a problem. Close up, two coils will talk to each other fine but once you get them apart, the signal will drop off alarmingly and a 'scope display will give you a good feel for things.

 

[sophiecentaur]

I foundhttp://www.rac.ca/tca/RF_Coil_Design.html which tells you the inductance of an air cored coil if you put in the dimensions. You may find it useful.

 

[sru2]

I've been brushing up on my antenna theory and I understand the difference between near field and far field now. In the near field we must deal with a mixture of electrical and magnetic currents from a variety of sources, in the far field we must only deal with electric and magnetic fields that are self-generated (i.e. an EM wave).

sophiecentaur, you suggested that I should focus on magnetic coupling, rather like near-field communications in contactless payments or RFID. A loose coupling transformer in short.

The following article suggests that the electric field is stronger in the near field than the magnetic field:

Loop Aerial Theory

A major problem in receiving VLF and LF signals is the high level of local noise generated from noisy power lines and consumer electrical equipment. In the presence of this type of noise, the received signal-to-noise ratio can be improved with the use of a loop aerial.

To explain this, it is necessary to briefly discuss the fields around a radiating element. At distances up to around half a wavelength, the induction or near field is prominent but it falls away at a greater rate with distance than the radiation field. At distances greater than one half wavelength, the radiation field is prominent. The relationship between field strength and distance is as follows:

1. The electric component of the induction field decreases with the cube of the distance and dB = 60log (d2/d1) where d2 and d1 are the relative distances.

2. The magnetic component of the induction field decreases with the square of the distance and dB = 40log (d2/d1)

3. Both the electric and magnetic components of the radiation field decrease directly with distance and dB = 20log (d2/d1)

The effect of all this is that, in the near field, the electric component is much stronger than the magnetic component. This is illustrated graphically in Figure 1.

http://users.tpg.com.au/users/ldbutler/VLF-LFLoopAerial.htm

I've taken a look at various antenna designs:

Magnetic
http://users.tpg.com.au/users/ldbutler/VLF-LFLoopAerial.htm
http://users.tpg.com.au/users/ldbutler/Ferrite_Loop_Ant.pdf
http://sidstation.loudet.org/antenna-en.xhtml
http://www.vlf.it/bikeloop/bykeloop.htm

Electric
http://www.home.pon.net/785/equipment/build_your_own.htm
http://www.vlf.it/torsten/_B3CKS-ANTENNA.htm
http://www.home.pon.net/785/equipment/antenna/index.htm
http://www.vlf.it/FSR/FSR.html
http://www.techlib.com/electronics/VLFwhistle.htm#Super-Tiny
http://gkircher.stormloader.com/sbhiz/

In most designs an amplifier is also included in the design. My sound card has a Signal-to-Noise Ratio of 109dB and I'm nearly positive that the other sound cards I have a fairly similar (maybe 90-100dB). The question is, could I perform amplification in the digital domain? Or would the signal be out of this range after broadcast??

I like the idea of the super-tiny. Its sensitivity seems perfect for this, as well as its size. What are the implications of this receiver picking up the signal broadcast from the same sound card? Just harmonics, or is there a possibility of inducing currents?

 

[sru2]

Can you obtain a signal generator and an oscilloscope for the initial work on the antennae?

Forgot to answer this one. I have two DSS signal generators on order, but I don't know if they will be useful without amplification. I also have a couple of of scopes, so I would have issues there.

 

[sophiecentaur]

There is a bit of confusion here. The article on the loop aerial that talks of the relative drop off of fields is, I think, referring to the characteristics of a transmitting monopole element (which is what is normally used for broadcasting). In that case the E field will dominate up close.
But I have to ask you why, if the near E field is inherently so much higher than the H field, why are magnetic circuits used for RF transformers? You could use electric coupling between two plates or wires for your system but it's easy to calculate the relative capacity between two distant wires and compare this with their capacity to ground. This corresponds to a considerable Capacitative 'pot-down'. The magnetic field around a coil surely dominates over the E field in the near field. Each turn of the coil adds to the H field but only one turn contributes to the E field - the others will be cancelling each other out.
I seriously suggest that you try a simple experiment with two coils and two wires and compare the effectiveness of the two systems. Easy done in an afternoon and it would point you in the right direction - and convince you, too!

Also, the SNR of your DAC is not what I refer to. I am referring to the ratio of your wanted, received signal (highly attenuated by the space between the terminals) and the receiver noise and external interference. That's what your receiver / demodulator has to deal with when it tries to extract the data from the received signal (as with all signal receiving equipment). The original DAC 'noise' (or rather distortion) is still 109dB below your wanted signal and is not a problem.

 

[sophiecentaur]

Forgot to answer this one. I have two DSS signal generators on order, but I don't know if they will be useful without amplification. I also have a couple of of scopes, so I would have issues there.

You are sure to need some amplification in your final system but, even with just 20 mW of signal generator output, you should be able to get an idea of the problems. You will certainly have a lot of trouble matching 20kHz into any length of wire monopole antenna you may be able to string up. In fact, 20m would take you the whole way from Tx to Rx, wouldn't it? As I wrote earlier, you should be able to tap into a multiple turn coil at somewhere around the 50Ω value and Bob would be your proverbial uncle.
Needless to say, this would only work at all at the higher audio frequencies. You need Ferromagnetic cores or fat capacitors to couple circuits at a few kHz.

 

[sru2]

You are sure to need some amplification in your final system but, even with just 20 mW of signal generator output, you should be able to get an idea of the problems. You will certainly have a lot of trouble matching 20kHz into any length of wire monopole antenna you may be able to string up. In fact, 20m would take you the whole way from Tx to Rx, wouldn't it? As I wrote earlier, you should be able to tap into a multiple turn coil at somewhere around the 50Ω value and Bob would be your proverbial uncle.

Are there any quick formulas for coupling and radiated power? Also, let's say I use additive synthesis to combine two sine waves. What effect does suppressing the harmonics have, on both transmission and signal reconstruction?

Needless to say, this would only work at all at the higher audio frequencies. You need Ferromagnetic cores or fat capacitors to couple circuits at a few kHz.

Any good circuit designs?

 

[sophiecentaur]

The linearity of the amplfiers and synthesiser won't affect the coupling.
More important than 'harmonic' performance would be intermodulation performance. Non linearity can produce intermodulation products that can be at sum and difference frequencies of the fundamentals and can be the result of mixing three of more of the input frequencies. These products can fall 'in band' so can't be eliminated by simple band pass filtering. Any coupling between antennae will be strictly linear but, over the massive range of frequencies you seem to be discussing (several octaves), you can expect a huge frequency tilt in the system response. For such a wide band system, you will not be able to gain any advantage from 'resonance', which is what many of the wireless systems you can read about rely on in order to get coupling.
There are a number of crucial factors involved with this project which you will need to get to grips with so that you can make an informed 'go-no go' decision as to its viability / or its information handling capacity.

Are there any quick formulas for coupling and radiated power?

You will not be concerned with "radiated power"; ideally, there would be none. What you need is a good coupling coefficient between coils. I have no data about that but I'm sure Google could help you there. In the distant past, I have used formulae for impedance matrices for arrays of dipoles but that wouldn't be of any use for your purpose.

 

[sru2]

The linearity of the amplfiers and synthesiser won't affect the coupling.
More important than 'harmonic' performance would be intermodulation performance. Non linearity can produce intermodulation products that can be at sum and difference frequencies of the fundamentals and can be the result of mixing three of more of the input frequencies. These products can fall 'in band' so can't be eliminated by simple band pass filtering.

I intend to deal with this in the digital domain. I could look at the phase differences, test for modulation and eliminate the intermodulation that way.

Do you see any issues with this?