Warren,
A good general book on coding theory, that also covers convolutional codes, is “Error Control Coding” by Lin and Costello or “Principles of Digital Communications
and Coding” by Viterbi and Omura. The latter is now, I believe available from Dover Books.
My congratulations also to Dex and Paul; a very impressive feat.
73,
David Weinreich
WA2VUJ/8
From: [email protected] [mailto:[email protected]]
On Behalf Of Warren Ziegler
Sent: Sunday, January 04, 2015 2:28 PM
To: rsgb_lf_group
Subject: Re: VLF: Transatlantic messages at 8822Hz
I did find some good online material from M.I.T. course notes:
On Sun, Jan 4, 2015 at 2:22 PM, Warren Ziegler <[email protected]> wrote:
First congratulations Dex and Paul !
Second, it has been many years since my graduate work in mathematics at Courant, and then it was oriented toward PDE's, fluid mechanics, complex variable, and mathematical methods.
What is the best text for getting up to speed on convolutional coding and Viterbi decoding?
On Sun, Jan 4, 2015 at 10:11 AM, Markus Vester <[email protected]> wrote:
Big congratulations again to Paul and Dex!
What Paul described in unpretentious and matter-of-fact words should really be regarded as a major achievement. It has been a well-deserved fruit of several months of effort, and there were
a number of difficulties to be overcome. As I was lucky to be included in the preceding email exchange, I had the chance to witness milestones and setbacks during the process.
Most members of the LF group will appreciate the challenges to Dex on the sub-9kHz transmit side, dealing with large coils and high voltage, realizing accurate GPS-derived frequencies and sub-second
bit timing, and last not least being able to leave the signal on air reliably for many hours.
The information processing side handled solely by Paul surely presented an even higher hurdle: In the first place, he searched and found a small handful of "good" FEC codes. The search involved
extensive simulations on powerful multicore computers hired from the Amazon cloud. Then the soft Viterby decoding of a potential receive signal is also computer intensive, especially for longer messages. There need to be numerous trials while optimising reference
phase evolution, bit timing, and antenna weights.
Perhaps the most challenging part was guessing appropriate parameters before a transmission, ie. how many characters should be sent in which amount of time. Although some experience had been
gained from carrier measurements during previous nights, ionospheric and atmospheric variations make it hard to predict SNR accurately enough if you want to exploit the channel capacity to the last couple of dBs. In the third round on new year's night, Dex
and Paul dared to take a bet, and won. Allow me to cite from their final email exchange on Dec 31st:
Dex:
Want to test the limits?
Paul:
Yes please.
Let's go for broke.
8 seconds is close to the limit, which is what I'd like to see.
The ultimate goal of this work has been to take decoding sensitivity close to the theroretical limit. An universal metric for this is Eb/N0, the ratio of the received signal energy per payload bit (Eb in Joules) and the noise spectral power density (N0 in Watt/Hertz,
equivalent to "noise energy" in Joules). The Shannon limit for long messages spread to infinite bandwidth is
Eb/N0 = ln(2) = -1.59 dB,
which (similar to the speed of light) cannot be surpassed by any possible encoding scheme. Paul's and Dex' experiments showed that his codes can come within about a dB of this limit in a real long-distance propagation experiment.
To put that into perspective, let's derive Eb/N0 figures for two popular digital modes:
WSPR-15 transmits 50 information bits in 15 minutes, ie one bit in 18 seconds. The decoding threshold is -38 dB in 2.5 kHz, or -4 dBHz. This gives
Eb/N0 = 10 log(18) - 4 dB = +8.5 dB,
ie. about 10 dB above the Shannon limit. Note that although different speed variants (eg WSPR-2) need different power, the minimum energy per bit has to remain the same.
Opera-32 carries 28 information bits in 32.6 minutes, ie. one bit in 70 seconds. The threshold is about -39.5 dB in 2.5 kHz, (-5.5 dBHz), referenced to the average power of the 50% dutycycle
on-off keying. This gives
Eb/N0 = 10 log(70) - 5.5 dB = +13 dB
or about 14.5 dB above Shannon. Note however that for LF / VLF transmissions, the limit will often be antenna voltage and peak power rather than average power, which can result in a further 3 dB disadvantage for Opera against frequency- or phase-modulated techniques.
The opds correlation decoder can go about 9 dB lower than Opera. But of course it can only find the best match from an a-priori defined list of callsigns, and doesn't attempt to decode any
message.
However we must recognize that the amateur modes spend a significant part of their energy to provide a reference for synchronisation, so not all of the Eb/N0 difference is due to less efficient
encoding. The "nude" FEC-PSK mode doesn't contain any such overhead. So it can only work when the link has a stable phase (like on VLF), and the decoder has been given accurate information on carrier frequency and symbol timing.
All the best,
Markus (DF6NM)
Sent: Sunday, January 04, 2015 1:13 PM
Subject: VLF: Transatlantic messages at 8822Hz
W4DEX achieved another 'first' recently by sending a series of
messages across the Atlantic at 8822 Hz which were successfully
copied at Todmorden UK, range 6194km.
Transmissions used coherent BPSK signalling with ERP of around
150uW. The modulation encoded the messages using a rate
1/16 terminated convolutional code with constraint length 25,
cascaded with an outer error detection code.
The first message was received at 2014-12-30 03:00, a 4
character message 'EM95'. Eb/N0 was -0.8dB using 9 second
symbols.
A second test the following night managed 12 characters 'PAUL
HNY DEX' using 14 second symbols giving Eb/N0 of +1.0dB.
Conditions were good and we could have used shorter symbols
and a longer message.
The third test and best result so far was a 25 character message
'8822HZ 2015 JAN 1 TA TEST' sent from 2015-01-01 00:00 using
8 second symbols. This was received with Eb/N0 = -0.1dB.
In the 0.125 Hz bandwidth of a code symbol, the S/N was -13.2dB.
That corresponds to -56dB S/N in a 2.5kHz audio bandwidth
after sferic blanking. Before the blanker the S/N would be
around -76dB.
The source encoding uses 6 bits per character to produce a
payload of 150 bits. An outer code adds a 16 bit CRC and the
convolutional encoder expands the message to 3040 signal bits.
The effective code rate is therefore 150/3040 = 1/20.27.
Of the 3040 signal bits, 1153 were demodulated incorrectly
but the FEC was able to fix them all to reveal the message.
Received signal was around 0.12 fT and it was necessary to
combine H-field and E-field receiver outputs to obtain a
sufficient S/N to decode.
The decoder is a soft Viterbi list decoder. The signals are too
weak to reveal a reference phase by the usual method of summing
the squared complex symbol amplitudes. Instead the decoder has
to do a brute force trial and error search.
The information rate in the 3rd test was 24.6 bits per hour
which is 80% of the channel capacity.
--
Paul Nicholson
http://abelian.org/
--
--
73 Warren K2ORS
WD2XGJ
WD2XSH/23
WE2XEB/2
WE2XGR/1
--
73 Warren K2ORS
WD2XGJ
WD2XSH/23
WE2XEB/2
WE2XGR/1
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