Frequency Difference Keying (FDK) - An Experimental Modulation Method

I am not sure that this is a new idea (although I have been told NASA use a similar mode for status monitoring of  deep-space craft).  If it is a new idea, you heard it first here - if not, don't be too unkind!!

General Philosophy of FDK: - The idea for FDK came from looking at many LF signals (some live, but mostly pictures posted to the 'Net) using Spectrogram , a very useful FFT display especially for receiving slow CW.   In order to maximise the range of their stations many European stations use slow CW (QRSS) with dot durations sometimes over 10 seconds.   Looking at these QRSS pictures, I noticed a couple of things.

Firstly, at poor S/Ns or in the presence of heavy QRN, because the decoding requires some fuzzy logic (deciding if the carrier is there or not), the full benefit of using narrowband reception is not realised.  To receive a character using 10 second dot durations you would need in the order of a minute to receive the character.   To make sure that the dots and dashes are not blurred, the narrowest FFT BW would be about 0.3Hz.   However, if you only needed to detect the presence of a carrier (not dots and dashes) in that minute then you could use an FFT BW of around 0.017Hz (1/60).  You can readily detect the presence of carriers as traces on a FFT spectrum display at S/Ns well below those levels needed to visually decode QRSS.

Secondly, for the decoding of the signals a visual display is necessary, ie, an operator needs to be watching the display as no more than about a dozen characters could be displayed at any one time. For slow data rates which might take tens of minutes to pass relevant data, this can be a problem.

These are the reasons for investigating other narrowband methods for LF.

Why Frequency Difference Keying: - Obviously different characters cannot be encoded by the the presence of a steady carrier alone.  Some unique characteristic needs to be able to be identified to enable decoding of the characters at the other end.

Some possibilities are:-

  • Absolute frequency encoding.  Here the characters are identified by the frequency of a tone sent.  For SSB reception this is difficult as it would require very high RF frequency accuracy for transmitters and receivers.    This could be minimised by using wide frequency spacings between different encoded characters but would still require sending of a reference tone to establish the position of the tones.  The required reference could be eliminated by using AM modulation.  The demodulated tone frequencies are in this case independent from relative offsets (within reason) between the transmitted and received RF signals.   However, AM suffers from a rapid falling off of the output to input S/N at low S/N ratios. This is counter to the goals here.

  • Absolute phase encoding. Here the characters are identified by the phase of a signal frequency tone sent.  Would require sending of a reference phase (perhaps 30 seconds reference, 30 seconds character phase).  Not sure about the effectiveness of this method.

  • Frequency difference encoding.  The method investigated here. The characters are encoded as the difference between two tones.   A reasonable mistuning between transmitter and receiver can be tolerated as the information is encoded in the difference between the two tones sent.  The transmitted signals are similar to the two-tone PEP SSB test signals so familiar to amateur operators.  Two reception modes are possible:-

    An obvious linear mode where the received audio signal from a receiver in SSB mode is processed by an FFT algorithm and two tones are identified.   The frequency difference is the encoding for each character.  This method gives the maximum possible advantage from narrowband reception for FDK.

    A square-law mode where the received audio signal from a receiver in SSB mode is mixed with itself (producing sum and difference products) and then processed by the FFT algorithm to find the difference frequency directly.  Mistuning and drift are largely irrelevant here, but it suffers from the same low S/N dropout as AM above in the paragraph headed 'Absolute frequency encoding'.

    These FDK two modes are described in more detail below.

Note: The character duration for FDK has been initially chosen to be one minute per character.

FDK Transmission Mode:- When transmitting a character, the difference frequency assigned to that character is calculated (say 5Hz for the space character ' ').  Two tones are generated with that frequency difference (5Hz) spaced equally around the centre frequency (nominally 1000Hz), i.e., (1000 - 2.5) = 997.5Hz and (1000+2.5)=1002.5Hz - giving a difference of the required 5Hz. Each burst of the tones for each character sent lasts for 60secs and is synchronised with the transmitting PC clock time.  The 'channel spacing' for each character has been initially set to 0.1Hz. A beacon mode can be provided for repeating a set message if necessary.

FDK Reception Mode:- The receiver synchronises to the receiving PC clock time and acquires data for 47.6 seconds giving a record length of 524288 samples.  This is because FFT raw data should have a length which is a power of  2.   The nearest block time to one minute using 11025Hz sampling is (524288 / 11025)= 47.6 seconds.

Two FDK modes of analysis of this data are possible:-

  • Square Law Mode:-  Here each sample is multiplied by itself (squared) before running the FFT.   This implements a square law detection process which produces sums and products.  For our space character (frequency difference = 5Hz) this produces a difference signal at 5Hz (1002.5 - 997.5) and a sum signal at 2000Hz (1002.5 + 997.5).  Note that the sum product will always be 2000Hz for a centre frequency of 1000Hz.   The spectrum is scanned for a maximum between 5Hz and 12Hz.  The maximum is taken to be the signal for the character and is displayed.

    The biggest advantage for this mode is that it is immune to both mistuning (within reason - as long as the two tones are in the audio passband obviously) and short term drift.    The BIG disadvantage is that it has poor performance at low S/N ratios.   This is because the signal presented to the FFT process is the result of the mixing of the received signal with itself.   This has implications for output S/N (after multiplying) to input S/N (before multiplying).   Assuming a signal with, say, a -3dB S/N ratio.  After mixing with itself the resultant output S/N is -6dB.   Therefore, compared to the linear process described below, this approach has an output S/N which is only 3dB worse.   However, looking at an input signal which has a S/N of -20dB, after mixing, the output S/N is -40dB, or now 20dB worse than the linear method.  So you see at low S/Ns the performance of the square law method falls off rapidly.

    Note the above analysis is a rough one and is meant to illustrate relative performance not absolute.   Using the square law reception mode I have found that it is roughly equivalent to PSK31 in performance but with a much reduced data rate.  The main advantage is that no accurate tuning is required.  It is really mostly an academic exercise - but interesting nonetheless.  The performance of this mode is greatly improved by filtering the audio before acquiring the data.  Using a narrow RF filter or audio output DSP filter of say 200Hz (instead of the normal voice bandwidth) raises the performance by about 6dB or more.

    Another disadvantage to this mode is that ANY two signals in the acquired audio passband which produce a substantial difference product in the range of of about 5Hz to 12Hz have the potential to interfere with the decoding process.

  • Linear Mode:-  Here the audio data is fed straight to the FFT algorithm.   The output spectrum is scanned for the maximum amplitude frequency and then re-scanned for the next highest amplitude frequency.  These two frequencies are taken to be the two transmitted tones.  The difference frequency between the two tones is calculated and the character corresponding to that difference is displayed.  This method is largely immune to mistuning (once again within reason - as long as the two tones are in the audio passband) but is susceptible to drift in frequency over the one minute acquisition period.

    This method requires short-term drift to be less than 0.017Hz over one minute to maintain the S/N advantage and about 0.1Hz over one minute to minimise decoding errors.  At LF frequencies of say 190KHz this translates to about 0.1ppm and about 0.5ppm respectively, over one minute.  In these days of modern receivers with absolute frequency specs. of +/- 5ppm over a wide temperature range, this is not too tough.   Sorry, but narrowband work requires good stability - there is no way around this.   Using an FRG-100 with +/-2ppm spec. TCXO and an FT-847 with +/-5ppm spec. plus mixing down from 160m to LF (which makes things worse), I have had no noticeable problems.

Some Notes on the Linear Mode:-    Observing the butterfly display of the linear FDK reception mode brings to mind some possibilities:-

  1. Even with errors rates of 50% or more it is easy to see the symmetrical spacing of the two tones.  Either by manual or automatic means it looks possible to identify the centre frequency and thereby adjusting the centre and the range of the scan for the maximums.  This would improve the error rate by eliminating from the scan tones outside the character tone difference range.

  2. Following on from above - after identifying the centre frequency, the error rate could further reduced by doing a SmartScan algorithm.  Starting from the strongest tones find the strongest pair which are symmetrically placed around the centre frequency.   Kind of like the slow tuning lock found in PSK31.

Test Results:-  Ok, so that's the theory, what about the practice?  As luck would have it (in this case good luck) there was a very high level of QRN when the first live tests were done.  Hear a 10 second WAV file of the audio here (110Kbytes) or a smaller 5 second file here (55Kbytes).

The pictures below are low-resolution to reduce their size (the FDK screenshots were taken in 1024x768 mode), but hopefully you 'get the picture' :-)

A Spectrogram of the audio with the FDK signal just visible.

Note that while the signal is discernible, trying to use QRSS under these conditions would be well-nigh impossible.

Linear Mode Screenshot

A screenshot of the FDK software in Linear Mode using the same signal as the Spectrogram above.

The butterfly shape can be clearly seen.  The centre frequency could be easily determined.  The error rate is around 6%.

Square Law Mode Screenshot

Here is a screenshot of the FDK display in Square Law Mode at a S/N where PSK31 has well-and-truly dropped out.

Here the display only shows the difference frequency as appropriate for the square-law mode.   0% error rate.

Some Facts About FDK

Fact #1 - FDK is NOT the same as DFCW or VFSKCW:

Please don't confuse FDK with DFCW / VFSKCW.    The DFCW mode was one (I called it VFSKCW) that I abandoned early in favour of FDK a long time ago.  FDK is NOT DFCW and is a completely different mode.

In fact, in an effort to counter the confusion I have re-named FDK as Wandjina. This is a Koori word for the Spirit People who travelled across the seas in long boats to watch over them.

Fact #2 - You Do NOT Need a Linear Amplifier to Transmit FDK (Wandjina):

For minimum bandwidth it IS best to use Linear Amplifiers.  The principle of FDK (Wandjina) is to transmit two tones simultaneously separated by a small frequency difference.   This small frequency difference encodes the character.  Each character has a unique frequency difference assigned to it.   The waveform looks identical to that of the two-tone test waveform.   Under linear conditions the spectrum space occupied is about 7Hz for the 60 second burst FDK mode.

However, by using the same method that is commonly used for BPSK/WOLF the bandwidth is about 30Hz for products less than 20dB down.  This method uses an XOR gate to phase-switch the carrier at a rate which is half the frequency difference.   Here you don't have to use a Linear Amplifier.   Considering all the resistance to advanced modes from various sections (especially from those parts where they are crammed into 1600Hz of spectrum space) it would be advisable to only use this mode either for QRP experiments or in the LowFer US band.   If you currently transmit WOLF or BPSK then you can transmit FDK (Wandjina).

Fact #3 - FDK (Wandjina) is a NOT a Complicated Mode:

Actually I believe that the simplicity of the method causes people to have difficulty understanding it.   If you understand QRSS which is standard, but slow CW, or DFCW / VFSKCW where the dots and dashes of ordinary Morse Code are represented by two different but fixed frequencies sent sequentially, then you are 90% of the way to understanding FDK (Wandjina).   Watching  QRSS on, say, Argo. shows lines across the screen which can be visually decoded into dots and dashes by the actual duration of the lines.    DFCW / VFSKCW compresses the time needed to send information by representing the dots and dashes not as different durations (1:3), but as two distinct frequencies slightly different in frequency.   Each frequency (assuming no drift) does not change.     Watching DFCW / VFSKCW shows "dots" which have no gaps (except for letter and word spaces) but are slightly different in frequency.

If you watch FDK you would see two lines at the same time on the screen.    The difference between the two lines is what encodes the character,    The duration of each pair is 60 seconds and as each character is sent the frequency spacing between the two tones changes for each different character.  You could actually roughly guess what character is being transmitted by looking visually at the space between the tones.     Certainly you could guess it within a range of 5 characters.   However, as the step in difference is of the order of 0.1Hz and there are about 49 of them spaced across the screen, electronic identification is needed for accurate decoding.

Fact #4 - FDK (Wandjina) Does Not Suffer from a Big Power Deficit Compared to Other Modes:

To get to the truth of this, more than a simplistic analysis is needed.   In particular, it needs to be taken into account what governs the output power characteristics of the intended transmitter and put this against the S/N implications of each.    One could classify this into several categories:

  1. Rules Limited - the Tx is limited by the maximum allowable input power (US Lowfer).

  2. Dissipation Limited - the Tx is limited to a maximum power level by overheating of components.

  3. Peak Limited - the Tx is limited to a maximum peak power.

Let's take the first case where we are limited to an input power and apply it to the situation for a number of modes - QRSS, DFCW and FDK (Wandjina) where we are attempting to transmit a message of seven characters plus word space in 8 minutes - say "WANJINA".

Rules Limited (e.g. US LowFer regulations):

Firstly, looking at receiver bandwidth needed :-

  • QRSS - needs 73 units in 480 seconds (8 minutes) - gives 6.6 second dots. Needs FFT equal to or shorter than about 3.3 seconds to resolve visually - equals operating BW of about 0.3Hz..

  • DFCW - needs 24 units in 480 seconds - gives 20 second dots.   Needs FFT equal or shorter than about 10 seconds to resolve visually - equals operating BW of about 0.1Hz.

  • FDK (Wandjina) - needs 8 units in 480 seconds - gives 60 second dots.   Can use FFT equal to or shorter than about 60 seconds to resolve (not resolved visually) - equals operating BW of about 0.0165Hz.

Taking FDK (Wandjina) as a base then we can see that DFCW is 7.8dB down on FDK and QRSS is 12.6dB down on FDK in terms of S/N gain from the respective required receiver bandwidth.

Secondly, looking at power out when limited to, say, 1W input :-

  • QRSS - 1W input - assume 100% eff.  = 1W for the single carrier.

  • DFCW - 1W input - assume 100% eff.  = 1W for the single carrier.

  • FDK (Wandjina) - 1W input  - assume 100% eff. = 0.5W for each of the two tones.

Therefore FDK is 3dB down on both QRSS and DFCW in terms of putting power into the receiver for the needed tone(s).

So the score card now is :-

  1. FDK = 0dB

  2. DFCW = -7.8dB + 3dB = -4.8dB

  3. QRSS = -12.6db + 3dB = -9.6dB

So, FDK is 4.8dB better than DFCW and 9.6dB better than QRSS in terms of Rules Limited operation.

Dissipation Limited:

Looking at receiver bandwidth needed the same factors apply as in the Rules Limited case above :-

Taking FDK (Wandjina) as a base then DFCW is 7.8dB down on FDK and QRSS is 12.6dB down on FDK in terms of S/N gain from the respective required receiver bandwidth.

Now looking at power out when limited to, say, 1W input average and taking into account the duty cycle :-

  • QRSS - 1W input - assume 100% eff.  and approx. 48% duty cycle = 2.1W for the single carrier.

  • DFCW - 1W input - assume 100% eff.  and approx. 63% duty cycle= 1.6W for the single carrier.

  • FDK (Wandjina) - 1W input  - assume 100% eff. and approx. 100% duty cycle= 0.5W for each of the two tones.

Therefore FDK is 6.2dB down on QRSS and 5.1dB down on DFCW in terms of putting power into the receiver for the needed tone(s).

So the score card in this case is :-

  1. FDK = 0dB

  2. DFCW = -7.8dB + 5.1dB = -2.7dB

  3. QRSS = -12.6db + 6.2dB = -6.4dB

So, FDK is 2.7dB better than DFCW and 6.4dB better than QRSS in the Dissipation Limited case.

Peak Limited:

Looking at receiver bandwidth needed the same factors apply as in the Rules Limited and Dissipation Limited cases above :-

Taking FDK (Wandjina) as a base then DFCW is 7.8dB down on FDK and QRSS is 12.6dB down on FDK in terms of S/N gain from the respective required receiver bandwidth.

Now looking at power out when limited to, say, 1W input peak :-

  • QRSS - 1W peak - assume 100% eff.  = 1W for the single carrier.

  • DFCW - 1W peak - assume 100% eff.  = 1W for the single carrier.

  • FDK (Wandjina) - 1W peak  - assume 100% eff. = 0.25W for each of the two tones.

Therefore FDK is 6dB down on both QRSS and DFCW in terms of putting power into the receiver for the needed tone(s).

So the score card in this case is :-

  1. FDK = 0dB

  2. DFCW = -7.8dB + 6dB = -1.8dB

  3. QRSS = -12.6db + 6dB = -6.6dB

So, FDK is 1.8dB better than DFCW and 6.6dB better than QRSS in the Peak Limited case.

NOTE #1:  This is for the linear case.   For the non-linear case we can claw back 2.08dB, so the score card is :-

  1. FDK = 0dB

  2. DFCW = -7.8dB + 6dB - 2.08dB = -3.88dB

  3. QRSS = -12.6db + 6dB - 2.08dB = -8.68dB

So, FDK is 3.88dB better than DFCW and 8.68dB better than QRSS in the Peak Limited case using non-linear mode (XOR gate 180degrees phase-switching).

Fact #5 - The 3dB Penalty of FDK Can Be Eliminated by a Single Tone System:

It has been suggested by several that the basic 3dB deficit over single-tone systems caused by the transmission of two tones simultaneously could be overcome by sending the tones sequentially.    Unfortunately, I don't think this is the case.   In the 60 seconds, one tone would be sent for 30 seconds and then the second tone would have to be sent for the remaining 30 seconds.    This means you have half the length of FFT record for each tone and therefore twice the operating bandwidth, resulting in a 3dB loss in S/N.     So 3dB gain in power is offset by 3dB loss in S/N due to wider bandwidth.   In addition it would necessitate some kind of time synchronisation (i.e. is this tone the last one of a pair or the first one in the next ?).    Of course, time synchronisation will improve all methods (QRSS, DFCW / VFSKCW and FDK), but is not in the realm of the basic factors looked at here.

The best way of overcoming the 3dB deficit is to use Piccolo Mk1.  I re-invented this over three years ago (1998) and called it AFK.  It has then been re-re-invented by several others under the name of PUA-43 and PGP-1 (or something like that).    Piccolo was invented in 1957 by some English boffin.     In Piccolo the character is encoded in an absolute frequency.    It requires high standards for both accuracy and stability.     In fact FDK was my attempt to relax the requirement on accuracy (but still requiring a high standard of stability).

NOTE:  Piccolo MK1 is very suitable for VLF (especially below 9kHz) as both accuracy and stability can more easily be satisfied.

Fact #6 - Required Tx and Rx Stability:

FDK DOES require a higher level of stability over the 60 second epochs than QRSS and DFCW / VFSKCW.

FDK Spectrum for Non-Linear Transmissions

Unmodulated RF Carrier:-  The digital square wave at the carrier frequency unmodulated contains components at the fundamental frequency (say 177.5kHz for the examples here) and odd harmonics only (assuming a perfect 50% duty cycle).    The third harmonic at 532.5kHz is 20 Log(1/3) down on the fundamental which about -10dB.  The fifth harmonic at 887.5kHz is likewise 20 Log(1/5) down which is about -14dB down - and so on for the rest of the odd harmonics.

These RF carrier harmonics in most practical systems are rejected either by deliberate low-pass filtering or by the inherently high Q tuned antenna system used on LF.

This situation will remain unchanged (i.e. RF carrier harmonics) when the signal is modulated with FDK.   However - of course the spectrum around the fundamental will contain sidebands carrying the information and these are described next.

FDK Modulated RF Carrier:-  Taking a highest difference frequency of 6.8Hz for example, which requires a modulation frequency of 3.4Hz, in a linear system a balanced modulator would produce just two sidebands spaced 3.4Hz around the 177.5kHz carrier frequency.    Note that the carrier is suppressed just leaving the sidebands.   The spacing between the sidebands is the required 6.8Hz.     Assuming a perfectly linear system no other components are produced (remember we are ignoring RF carrier harmonics) as shown below:-

The characteristic of the balanced modulator (effectively a DSB modulator) is such that it switches the phase by 180 degrees every half cycle of the modulating frequency (3.4Hz).    In a linear system also the amplitude tapers off sinusoidally to zero before the phase switch and increases sinusoidally after.    This is needed to ensure only the two sidebands are produced.

But we need a non-linear system of modulation.   This can be simply done by running an RF digital signal through a circuit which switches the phase by 180 degrees (i.e. inverts) and back at the rate of the modulating frequency.   This is commonly done by using an XOR (exclusive-OR) digital gate.   In my case I have used four AND gates to implement this as I need extra control for the other modes.     The difference here is that while we have the required 180 degrees phase shift as in the linear balanced modulator, the tapering off before switching is absent.   That is the 180 degrees phase shift is done at full amplitude.    This results in the spectrum as below:-

The green lines are the third harmonics (spaced at 10.2Hz out from the centre) of the 3.4Hz modulating frequency and (similarly to the RF carrier example above) are about -10dB down on the red pair.     The magenta lines are the fifth harmonics (17Hz at about -14dB down) while the blue lines are the ninth harmonics at 30.6Hz out and -19dB down.

I have made several attempts to reduce the level of the sidebands.    The first was to soften the phase shift by going to 180 shift in two steps.    That is instead of a 0-180-0 phase shift sequence I used a 0-90-180-90-0 sequence.    This did reduce the level of the sidebands marginally but brought up a significant carrier frequency fundamental component only -6dB down between the wanted two-tone pair.     The other approach was a stepped amplitude so that the 180 degrees phase shift occurred at half amplitude.    This reduced the third harmonics to about -15dB down instead of about -10dB but the extra complication of providing the half-amplitude switching was not considered worthwhile at this time.

Considering that the non-linear FDK modulator harmonic products are over -19dB down on the main information carrying tone pair outside +/-17Hz and that a PSK31 signal will produce full strength sidebands at about +/-15Hz I feel that this is acceptable.     Also a 12 wpm normal signal will cover about the same spectrum to maintain keying shape.

Last updated Sunday, 11 May 2008