### P25 vocoder FEC

Following a discussion with Carlos Cabezas EB4FBZ over on the Spanish telegram group Radiofrikis about using Codec2 with DMR, I set out to study the error correction used in DMR, since it quickly caught my eye as something rather interesting. As some may know, I’m not a fan of using DMR for Amateur Radio, so I don’t know much about its technical details. On the other hand, Carlos knows a lot about DMR, so I’ve learned much with this discussion.

In principle, DMR is codec agnostic, but all the existing implementations use a 2450bps AMBE codec. The details of the encoding and FEC are taken directly from the P25 Half Rate Vocoder specification, which encodes a 2450bps MBE stream as a 3600bps stream. Here I look at some interesting details regarding the FEC in this specification.

### A first look at 1KUNS-PF telemetry

Last Friday, three Amateur cubesats were deployed from the ISS as part of the KiboCUBE program. These were Irazú, a 1U cubesat from Costa Rica which is the first satellite in orbit from a Central American country; UBAKUSAT, a 3U cubesat from Istanbul Technical University, Turkey; and 1KUNS-PF, a 1U cubesat from University of Nairobi, Kenya, developed jointly with University of Rome La Sapienza, Italy.

Irazú and UBAKUSAT both use standard 9k6 FSK packet radio (AX.25 with G3RUH scrambler), so they can be decoded with direwolf and many other packet radio decoders. However, no one has been able to decode 1KUNS-PF yet, due to the lack of information about the modulation and coding used. Mike Rupprecht DK3WN has some information about 1KUNS-PF, including a recording of some packets. I’ve taken a look at Mike’s recording and here are my findings.

### Using a Golay(24,12) decoder for Golay(23,12)

Yesterday I explained an algebraic decoding algorithm for Golay(24,12) and commented that it was not easy to adapt it to decode Golay(23,12). Today I’ve thought of a simple way to use any Golay(24,12) decoder to decode Golay(23,12).

Recall that a systematic Golay(23,12) code is obtained from a systematic Golay(24,12) by omitting the last component of each codeword (i.e., the codeword $$(c_1,\ldots,c_{24})$$ from the Golay(24,12) code gives the codeword $$(c_1,\ldots,c_{23})$$ from the Golay(23,12) code). Conversely, one can obtain a systematic Golay(24,12) code from a systematic Golay(23,12) code by adding a parity bit at the end. This means that $$c_{24} = \sum_{j=1}^{23} c_j$$, since $$\sum_{j=1}^{24} c_j = 0$$ for all words in a Golay(24,12) code.

The idea to decode a Golay(23,12) code with a Golay(24,12) decoder is first to restore the parity bit $$c_{24}$$ and then apply the Golay(24,12) decoder. However, if there are errors in the received codeword, the restored parity bit can also be in error, increasing the number of errors in one.

The key remark is that both Golay(23,12) and Golay(24,12) are able to correct up to 3 errors. Therefore, we only care about restoring the parity bit correctly in the case when there are exactly 3 errors. If there are 2 or less errors, adding another error still gives a word decodable by the Golay(24,12) decoder.

Now note that if there are exactly 3 errors in $$(c_1,\ldots,c_{23})$$, then $$\sum_{j=1}^{23} c_j$$ gives the opposite from the parity of the original codeword. Therefore, we should restore $$c_{24}$$ as$c_{24} = 1 + \sum_{j=1}^{23} c_j$and then apply the Golay(24,12) decoder.

### Algebraic decoding of Golay(24,12)

A couple years ago, I implemented a Golay(24,12) decoder to be used in the GOMX-1 decoder in gr-satellites. The implementation can be seen here. I followed the algorithm in the book The Art of Error Correction Coding, Section 2.2.3, without taking much care to understand why the algorithm worked. Now I am doing some experiments with Golay(24,12) and Golay(23,12) codes, so I have needed to revisit that algorithm and understand it well to adapt it to my needs. Here I explain how this algebraic decoder works.

### STARcon 2018 aftermath

Please read the title of this post in a light-hearted way: the STARcon 2018 Scientific and Technical Amateur Radio congress that took place last weekend in Murcia turned out to be a huge success. Personally, the weekend has been an intense but extremely rewarding experience.

### S-NET telemetry parser

Recently I have added a telemetry parser to the S-NET decoder in gr-satellites. Recall that I have talked about S-NET and its decoder in a previous post. To implement this telemetry parser I have used the information in Mike Rupprecht DK3WN’s web, some additional information shared by the S-NET team, as well as some recordings done by Mike. Many thanks to Mike and the S-NET team for all their help. Here I describe a few details about the telemetry.

### Soft Viterbi decoder for AO-40 FEC

A year ago, I made a decoder for the AO-40 FEC. While AO-40 has been dead for many years, the same FEC system is used in AO-73 and the rest of the FUNcube satellites. This decoder was later included in gr-satellites and it is currently used in the decoders for AO-73, UKube-1 and Nayif-1.

When I implemented this FEC decoder, for simplicity I used a hard decision Viterbi decoder, since my main concern was to get all the system working. I always intended to replace this by a soft decision Viterbi decoder, but it seems that I forgot completely about it.

Now, while thinking about integrating gr-aausat (my AAUSAT-4 decoder OOT module) into gr-satellites and adding a soft Viterbi decoder for AAUSAT-4, I have remembered about this. While the decoder for AAUSAT-4 will have to wait, since I have found a bug in the GNU Radio Viterbi decoder that makes it segfault, I have already added a soft Viterbi AO-40 FEC decoder to the FUNcube decoders in gr-satellites.

### Decoder for PolyITAN-2-SAU

PolyITAN-2-SAU, or QB50 UA01, is a cubesat from National Technical University of Ukraine, that was launched on May 2017 as part of the QB50 proyect. When it was launched, I made a recording of several QB50 satellites, including PolyITAN-2-SAU. Presumably, the modulation and coding used by this satellite is 9k6 BPSK AX.25, with G3RUH scrambling. Back then, I commented that although the signal was strong and I could get a clean constellation plot, I was unable to get valid AX.25 packets.

I had completely forgotten this satellite, but the other day I saw that Andy UZ7HO had added support for PolyITAN-2-SAU to his SoundModem. I asked Andy for some help, since I suspected that the coding wasn’t exactly standard G3RUH AX.25.

The trick is that this satellite uses two “layers” of NRZI encoding. The relevant part of the decoder is shown below. The BPSK symbols come in from the left of the figure and the AX.25 packets exit by the right. A standard G3RUH AX.25 decoder wouldn’t have the extra NRZI decoder on the right.

Note that NRZI decoding and G3RUH descrambling commute, since the G3RUH polynomial has an odd number of terms. Therefore, the decoder can also be organized in a different way, with both NRZI decoders at one side (either the input or output) of the descrambler.

Having two NRZI decoders in chain is a really funny concept, so it almost seems as some kind of mistake from the satellite team (most QB50 satellites use standard BPSK or FSK AX.25 packet radio for compatibility). In fact, if we write an NRZI decoder as $$y_n = 1 + x_n + x_{n-1}$$, where $$x_n$$ is the input sequence, $$y_n$$ is the output sequence and the operations are performed on the finite field $$GF(2)$$, then the effect of two NRZI decoders in chain can be written as$z_n = 1 + y_n + y_{n-1} = 1 + x_n + x_{n-2},$ which is a rather strange form of differential decoder.

Thanks to Andy for giving me the clue about the extra NRZI decoder, as I would have had a hard time in finding it by myself (although, in retrospective, it is not that difficult to guess it by looking at the descrambled stream and seeing how HDLC 0x7e flags can be obtained from it). I have now added a decoder for PolyITAN-2-SAU to gr-satellites.

### BCH decoder for S-NET

In my previous post, I talked about the coding used by the S-NET cubesats and the implementation of my decoder included in gr-satellites. The decoder was still missing a BCH decoder. I have now implemented a BCH decoder and included it in gr-satellites. Here I describe the details of this decoder.

### Decoding S-NET

S-NET is a swarm of 4 cubesats from TU-Berlin. Their mission is to test SLink, an S-band transceiver for inter-satellite communications. They were launched on February 1 this year and they use use Amateur frequencies for their telemetry downlink on the 70cm band. Several weeks ago, Mike Rupprecht DK3WN raised my attention towards these satellites. Since they use a rather particular coding, custom software would be needed to decode the telemetry. Then, I set to add support for S-NET to gr-satellites

After some really helpful communication with the S-NET team, in particular with Walter Frese, and some exchanges of ideas with Andrei Kopanchuk UZ7HO, who was also working to add an S-NET decoder to his soundmodem, I have finally added a basic S-NET decoder to gr-satellites.