Messages to be sent are restricted to a 6 bit character set consisting of the upper case letters, digits, and a selection of space, punctuation and other symbols. Lower case letters are mapped to upper case before encoding. A total of 52 character codes are defined which gives an information content of 5.7 bits per character. The table below shows the translation of ASCII characters into 6-bit source codes.

ASCII source character | Source coded value |
---|---|

Decimal 42 (*) to 59 (;) | Codes 0 to 17 |

Decimal 33 (!) | Code 18 |

Decimal 61 (=) | Code 19 |

Decimal 38 (&) | Code 20 |

Decimal 63 (?) to 90 (Z) | Codes 21 to 48 |

Decimal 10 (Newline, LF) | Code 49 |

Decimal 9 (Tab) and 32 (Space) | Code 50 |

Decimal 95 (Underscore) | Code 51 |

The 6 bit codes from the source encoding are
concatenated (lowest significant bit first) into a binary source block.
If the number of characters is `NC`, then the source block
length is `NC * 6` bits and has information content
`NC * 5.7` bits.

The outer error detection layer appends a 16 bit CRC to the source block.
The CRC is based on the CCITT standard with
polynomial hex 1021. A zero initial value is used in the CRC calculation so
that unmodulated carrier is a valid message.
The result is a message block of length `NC * 6 + 16` bits.

The message block is encoded using a standard encoder which convolves the
block with one of the polynomial sets listed in the tables
here. For constraint length `K` the encoder
maintains `K-1` memory bits.
The output bits are serialised
in the order in which the polynomials are listed in each set. The least
significant bit of each polynomial is combined with the current input bit and
the higher order bits combine with successively older memory bits.

After the last bit of the message block is encoded, the encoder memory
is flushed
with `K-1` zero input bits.
This terminates the code and ensures that the final bits of the message block
benefit from the same error correction strength as the earlier bits.

The total number of output bits from the encoder is given by
`(NC * 6 + 16 + K - 1)/R` where `R` is the rate
of the convolutional code.
The total amount of information being encoded is `NC * 5.7` bits.
Therefore the overall effective code rate is
`R * (NC * 5.7)/(NC * 6 + 16 + K - 1)`

The encoded block passes through an interleaver stage before transmission.
Encoded bits are loaded column-wise into a rectangular array, then read out
row-wise. If the size of the encoded block is `NB` bits, the
array has width given by `W = ceil(sqrt(NB))` and height given
by `H = ceil(NB/W)`. Any unpopulated array cells in a final
incomplete row are skipped during the readout.

The interleaved encoded block is transmitted by toggling the phase of the carrier. Transmission begins at a precise time which is pre-arranged with recipients.

VLF signals from the antennas are amplified, then digitised and timestamped with reference to a pulse-per-second from a GPS. Up to three channels may be received simultaneously, ideally from a pair of orthogonal magnetic loops and a vertical electric field probe. All of the following processing takes place in software.

The received signal is band limited to 3kHz width and passed through a sferic blanker with threshold set to about 10 times the mean rectified noise floor amplitude. With normal VLF conditions this results in between 25% to 35% of the signal being blanked but the gain in signal/noise ratio is around 20dB at 8kHz.

The sferic blanked signal is multiplied by a local oscillator at the carrier frequency to produce a complex baseband signal. This is further band limited and resampled to a sample rate of the order of 100 Hz. This is much wider than the signal bandwidth but the extra spectrum is useful later for measuring the background noise.

The resulting stream of complex baseband timestamped samples are stored to disk for later processing. The recording begins some 10 or 20 seconds before the pre-arranged start time of the transmission. This allows the sferic blanker time to establish a moving mean for its dynamic threshold setting. The recording is stopped approximately one symbol period after the expected end of the transmission.

The remainder of the signal processing is done by post-processing of the recorded baseband signals.

The first step of post-processing is to mix the baseband signals to synthesise a desirable antenna response. This involves correction for the known phase and amplitude response of each channel, and for the effective orientation of the loops. The antenna response is chosen to maximise the S/N ratio and the optimum beam orientation may well be different to the great circle bearing in order to steer away from the prevailing VLF background noise. The received signal is invisible at this stage, so it is desirable to have available some measurements of recent carrier transmissions as a guide.

The result of the antenna mixing is a single complex baseband signal. This
is coherently integrated per symbol period to produce an array of size
`NB` complex symbol amplitudes.

The complex symbol amplitudes are then multiplied by a complex phase factor which is a guess of the signal phase corresponding to a transmitted '1' bit. If the guess is successful, all the signal will end up in the real channel. The quadrature channel will contain only noise and is discarded to obtain the 3dB S/N gain of coherent demodulation. An initial guess is obtained by summing the squared complex symbol amplitudes which can extract a usable reference phase from strong signals, and on weaker signals can bias the guess in the right direction.

The interleaving applied by the encoder is reversed at this point.

The resulting array of `NB` de-interleaved real signal
amplitudes is
available to the Viterbi decoder. This employs a completely conventional
forward pass. The trellis is of size `NB + 1` columns by
`2^(K-1)` rows and this dominates the memory required by the decoder.
The codes used are all of rate `1/n`, therefore each node of the
trellis has two incoming branches and two outgoing branches. The branch
metric of each is calculated from the inner product of a vector of 1/R
consecutive input symbol amplitudes with a vector of 1/R expected symbol values with
amplitudes +1 or -1. This is equivalent to calculating the squared
Euclidean distance
in the code space. This soft decision detection gives a 3dB advantage over
the alternative of hard decisions based on a threshold of zero.
After the earlier sferic blanking and band limiting,
the noise is approximately Gaussian and therefore the inner product produces
log-likelihood branch metrics. The logarithmic branch metrics are
summed in the normal way to produce a pair of accumulated branch metrics at each
node of the trellis. The forward pass is easily parallelised to make
good use of multiple CPU cores.

The single backward pass of conventional Viterbi decoding is replaced by the list decoding algorithm due to Soong and Huang which is very well described by Roeder and Hamzaoui in this paper. The serial algorithm iteratively extracts backward paths through the trellis in order of non-increasing path metric. The first pass is exactly equivalent to the normal Viterbi backward pass. As the backward pass proceeds, branches which diverge from the current best path are placed into a sorted stack for later consideration. Subsequent backward passes consult the top of the stack and examine paths which diverged from a previously found best path. The backward passes cannot be parallelised and can sometimes dominate the running time of the decoder.

The output from the list decoder is a series of decodes which each represent a valid codeword of the convolutional code but with increasingly less likelihood. A typical list length considers up to 50,000 of the most likely decodes but lists of up to several million can be effective.

Each decode of the list output is tested for a valid CRC applied by the outer error detection code. Those that fail are discarded. This typically leaves the correct decode in the list, along with sometimes one or two false decodes which happen by chance to have a correct CRC.

The surviving decodes in the list output are then tested for valid source encodings. Each character is represented by 6 bits, but only 52 of the possible 64 codes are valid. If any of the 6 bit codes matches an unassigned source code, the decode is discarded. By this simple scheme, the spare 0.3 bits per character contribute to the strength of the outer code and allow longer decode lists to be used, for greater overall performance.

If there are no surviving decodes, a new guess of the reference phase is applied to the symbol amplitudes and the whole decode process is repeated. Uniform phases are tried in either 30 degree or 15 degree steps, moving gradually further away from the initial guess of the phase which was derived from the signal. Usually at VLF the decode is successful with a uniform reference phase. If the full circle of reference phase has been tried, the message is split into two parts and slightly different reference phases are applied to each. Once all of these have been tried, the message is split into four pieces and permutations of four different reference phases are tried. The decoder finishes when all 108 variations of reference phase have been tried.

When a successful decode is achieved, the trialing of reference phases continues in the hope of obtaining a better decode (higher log-likelihood) at some other nearby phase.

After each decode, some further processing is applied to measure the signal.

The decoded message is re-encoded and used to reverse the original phase modulation. The result is an unmodulated carrier lasting the duration of the whole message. This is coherently integrated over the whole transmission to obtain an accurate measurement of average signal strength in a very narrow bandwidth. The noise in the full bandwidth of the recorded baseband signal is also measured to determine the noise spectral density, and from this the carrier S/N and Eb/N0 are reported.

With a successful decode the received symbols are examined to count how many of the symbols were demodulated incorrectly. From this symbol error rate the S/N of the signal can be estimated. This usually agrees within a dB or so with the average carrier S/N measured above.

From the carrier S/N, the actual value of `Eb/N0` is calculated.
This calculation uses the information content `NC * 5.7` bits, of the
message, and therefore is based on the overall effective code rate of the
cascaded codes, not the rate `R` of the inner convolutional code.
This `Eb/N0` is reported along with the calculated
channel capacity.

A message transmitted on 8822Hz by Dex McIntyre W4DEX in North Carolina which was received and decoded in the UK provides a good example.

Transmission | |

Transmit ERP | Estimated 150uW |

Message length | 25 characters |

Symbol period | 8 seconds |

Duration | 24320 seconds: 6 hrs, 45 mins, 20 seconds |

Encoding | Rate 1/16 K=25 |

Information content | 25 * 5.7 = 142.5 bits |

Inner code block size | 25 * 6 + 16 = 166 bits |

Encoder output | (25 * 6 + 16 + 25 - 1) * 16 = 3040 symbols |

Overall code rate | 142.5 / 3040 = 1/21.33 |

Bandwidth | 0.125 Hz |

Reception | |

Es/N0 (S/N in 0.125Hz) | -13.2 dB |

Eb/N0 | -0.1 dB |

S/N in 1Hz | -22.2dB |

S/N in 2500 Hz | -56.2 dB, approx -76dB before sferic blanking |

Symbol error rate | 0.38 |

Rank in Viterbi list output | 89 |

Viterbi list length | 50,000 |

Channel capacity | 30.34 bits per hour |

Capacity achieved | 21.1 bits per hour, 70% of capacity |

Decoder memory | 49 Gbyte |

Decoding PC | Xeon E5-2680 v2 32 core 2.8GHz 64 Gbyte RAM |

Decode time | 2.5 minutes |

The spectrogram on the right shows the signal from W4DEX after the decoder has reversed the phase modulation to reconstruct the carrier. It is remarkable that 140 bits of information can be reliably decoded from the signal. Without removing the phase modulation, the signal is completely invisible in a spectrogram beyond about 300km range.

The Eb/N0 of -0.1dB and the ranking 89th out of 50,000 in the decoder's output list suggests that there was about 0.5dB or so of spare signal. A slightly shorter symbol period would also have decoded and achieved a higher channel capacity. The decoder still has a reasonable success rate with Eb/N0 of -1dB but it is better to leave a bit in hand to allow for phase variations. Several carrier transmission by W4DEX in the days leading up to this test gave S/N variations of around +/-2 dB and the average signal during the message was about in the center of that range.

The Viterbi algorithm produces a true maximum likelihood decode of the signal, is simple to implement, and is very efficient for short constraint length codes. It works very naturally with soft-decision BPSK and is easy to extend to list decoding. The only problem with the Viterbi decoder is that its memory requirements increase exponentially with increasing constraint length. For this reason it is normally used only up to constraint lengths of 15 or so. Its use up to constraint lengths of 25 is unusual because of the very large amount of memory required for the trellis, of the order of 2 Gbyte per message character. However, the latest generation of Intel based PCs provide 32 or more CPU cores and up to 240 Gbytes of RAM which makes it possible to experiment with long constraint lengths without having to use a less-than-optimum serial decoding algorithm. Furthermore, such computers are readily available to rent at low cost by the hour, typically less than 50 cents per hour for a 36 core 240 Gbyte machine. Considerable use has been made of this class of machine via the AWS EC2 service to develop and test the convolutional codes used by this mode. It only takes a few minutes to start one or several of these up and install and run the decoder, and the whole process can be scripted.

A feature of convolutional codes is a diminishing return for increasing constraint length. Dropping from constraint length 25 down to 21 for example only incurs a fraction of a dB loss of coding gain. It takes hundreds of hours of CPU time to actually measure accurately the difference in performance. Consequently, not much is lost by using constraint length 21, or even lower. This brings the decoder memory requirement within reach of domestic computers. A machine with 4 GByte of spare RAM can handle a 12 character message with constraint length 21 and a 30 character message with constraint length 19.

In any case, if PC developments continue at their recent pace - which they will have to do to keep up with increasingly bloated web browsers and email clients, it will probably not be many years before the average home computer has 32 cores and 64 GByte of RAM.

It is crucial for this mode of operation that the PC clocks at both ends are synchronised accurately to universal time. The decoder relies on the signal timestamps to know when each transmitted symbol begins and it is desirable for the timing to be accurate to within 1/10th of a symbol period. Running the ntpd daemon is sufficient for symbol periods of 1 second and longer. If ntpd is able to use a GPS directly connected to the receiver PC then even shorter symbol periods may be usable.

The received signal should be timestamped at the point where it is digitised, and at the same time the sample rate must be continuously corrected for A/D conversion rate drift. Spectrum Lab for Windows and vlfrx-tools for unix-like systems will perform the GPS timestamping, sample rate drift correction, and storage of signal to disk.

The decode success rate of short block codes has an upper bound given by the sphere packing limit. This refers to the probability that a received noisy message will lie within the correct non-intersecting hamming sphere which surrounds each valid code word in the code space. As the length of a code block increases to infinity, the slope of the sphere packing limit steepens and tends towards a vertical line at the Shannon limit of Eb/N0 = -1.59 dB.

The sphere packing limit for the rate 1/8 K=25 code with 5 character messages is plotted in the graph along with measured performance points of the code. The limit is calculated using the overall rate 1/19.65 of the cascaded combination of inner convolutional code and outer error detection layer.

It is clear from the graph that list decoding increases the chance of
successful decode by a significant factor, in this case a factor of more
than 10
with the weakest signal. Without list decoding, a 50% success rate
is obtained at

The performance can be brought closer to the limit by increasing the list length. As the sphere packing limit is approached, the outer error detection layer begins to make errors in the form of false detections as the capability of the CRC (and the redundant 0.3 bits per character of the source coding) becomes exhausted. This can be corrected by increasing the CRC size enough to restore error-free operation. However, the resulting increase in overhead of the outer code produces a reduction of the overall code rate which in turn shifts the sphere packing limit further away. Consequently the sphere packing limit can be approached by longer and longer list lengths but it remains just out of reach.

The number of bits used for error detection is `nc * 0.3 + 16` where
`nc` is the number of characters. Therefore, for any given
list size, false detections are more likely when using short messages and each
additional three characters reduces the false detection rate by nearly 50%.

Although in principle the limit to list decoding is set by the error detecting strength of the outer code, in practice the operator can provide a third layer of error detection. In most cases a genuine decode will be immediately recognised as a valid callsign, QRA locator, or greeting message. By contrast a false decode caused by a chance match with the CRC will produce a string of random characters which the operator will ignore.

An unmodulated carrier is a valid message and decodes to a string of '*' characters. This is useful for testing a path to see how short a symbol period it will support. A long unmodulated carrier can be recorded and presented to the decoder to test the decode performance using different code strengths, message lengths and symbol periods - up to a maximum total duration limited by the length of the carrier sample.

The cascaded code is linear, therefore the performance of a carrier 'message' is representative of any other message.

After a successful decode, a data file is produced which gives
information about the received message. On the Windows version of the
decoder this file is named by replacing the `.wav` of the signal
filename with `-symbols.csv`.

If a better decode (greater overall log-likelihood) is found later in the phase search, the file will be over-written.

The file has one row per symbol in the order in which the symbols were received. Each row has comma separated fields as follows:

Column | Contents |
---|---|

1 | Symbol position, counting from zero |

2 | Symbol amplitude - in-phase component |

3 | Symbol amplitude - quadrature component |

4 | Symbol error: 1 if the symbol was demodulated incorrectly, otherwise 0 |

5 | The transmitted symbol, 1 or 0 |

It can be useful to plot column 4 (using impulses) against column 1 to see if the symbol errors were distributed uniformly, or whether parts of the message were received worse than others. If the signal is strong, columns 2 and 3 can be plotted against each other (using points) to display a constellation diagram. However the $E_b/N_0$ needs to be greater than +3 dB or so to see anything more than an amorphous central blob. Most signals are much weaker than this (they should be, to make full use of the channel capacity), therefore the constellation plot is rarely useful except for local testing of your phase modulator and carrier stability.

Some example decode times on an average quad core domestic PC. The times given are for a full search through all the reference phase patterns. If the message is going to decode, it usually does so within about the first 10% of the full search time, during which uniform phase patterns are tried. It may take more of a search if propagation was disturbed or if one end of the link has oscillator problems.

Code and message length | Run time |
---|---|

Rate 1/8 K=19, 31 characters, list 1000 | 9 minutes |

Rate 1/8 K=19, 31 characters, list 50000 | 12 minutes |

Rate 1/8 K=19, 31 characters, list 200000 | 16 minutes |

Rate 1/8 K=21, 12 characters, list 50000 | 50 minutes |

The running time of the forward pass is proportional to message length and also will double for each increment of K, so K=21 will take four times longer than K=19. The backward pass running time is roughly proportional to $N \log_2(N)$ where $N$ is the message length. This can dominate when using low K and long lists with a long message.