In der Regel enthalten die einzelnen frames Checksummen oder CRC-Werte. Einige Radiosonden senden frames mit Codewoertern, die eine Fehlerkorrektur ermoeglichen. Graw ==== BCH/Hamming code (N, N-R) = (2^m - 1, N-m), d.h. R=mt mit t=1, korrigiert t=1 Fehler pro Codewort der Laenge N, die Codewoerter haben den Abstand d=3 (bei 2 Fehlern wird falsch dekodiert). Fuer die Sonden DFM-06/09 ist m=3, d.h. Hamming (7, 4), jedoch mit einem Paritaetsbit zu (8, 4) erweitert. Der erweiterte Hammingcode (8, 4) kann zudem 2 Fehler als nicht-korrigierbar erkennen (wird nicht falsch decodiert). Meisei ====== BCH(63,51)-Code (m=6, t=2) mit Generatorpolynom (x^6+x+1)(x^6+x^4+x^2+x+1)=x^12+x^10+x^8+x^5+x^4+x^3+1. gekuerzt auf (46,34), die letzten 12 bit sind die BCH-Kontrollbits. Das Generatorpolynom kann man finden, wenn man geeignete Codewoerter addiert (linearer Code), um das kleinste Codewort ungleich 0 zu finden, z.B.: 0000000000000000 1 0000000000000000 1 000010111011 + 0000000000000000 1 0000000000000010 0 010001011111 = 0000000000000000 0 0000000000000010 1 010011100100 -> 101010011100100 = (x^12+x^10+x^8+x^5+x^4+x^3+1)x^2 Die 34 Nachrichtenbit sind aufgeteilt in 16+1+16+1, d.h. nach einem 16 bit Block kommt ein Paritaetsbit, dass 1 ist, wenn die Anzahl 1en in den 16 bit davor gerade ist und sonst 0. Pro Codewort koennen t=2 Fehler korrigiert werden. Ist die Anzahl Fehler #E groesser t, dann ist fuer t < #E <= 2t die Wahrscheinlichkeit, dass Fehler falsch erkannt/decodiert werden, etwa 1/t!, somit fuer t=2 garnicht gering. Da hier gekuerzte Codewoerter und 2 zusaetzliche Paritaetsbits verwendet werden, sollte man bei decodierten Codewoertern pruefen, ob die Paritaetsbits stimmen und ob bei der Decodierung Werte im Null-padding veraendert wurden. Um nur die BCH-Codewoerter zu pruefen, kann man Polynomdivision durchfuehren (CRC): 1) 0000000000000000 1 0000000000000000 1 000010111011 : 0000 0000 0BB -> 0000000000000000100000000000000001000010111011 ./crc_polymod --bch 0000000000000000100000000000000001000010111011 100000000000000001000010111011 1010100111001 1010011100100001000010111011 1010100111001 111011101001000010111011 1010100111001 10001110101100010111011 1010100111001 100111011110010111011 1010100111001 1101000010110111011 1010100111001 111100101111111011 1010100111001 10110110011011011 1010100111001 11111101001011 1010100111001 1010100111001 1010100111001 0 2) 0000000000000000 1 1010111100100010 1 000110000001 : 0000 AF22 181 -> 0000000000000000110101111001000101000110000001 ./crc_polymod --bch 0000000000000000110101111001000101000110000001 110101111001000101000110000001 1010100111001 11111100101100101000110000001 1010100111001 1010101011110101000110000001 1010100111001 1100111101000110000001 1010100111001 110011010001110000001 1010100111001 11001001101010000001 1010100111001 1100000011000000001 1010100111001 110100100001000001 1010100111001 11110111101100001 1010100111001 1011110011110001 1010100111001 1010100111001 1010100111001 0 Decoder: bch_ecc.c Vaisala ======= http://www.vaisala.com/Vaisala%20Documents/Vaisala%20News%20Articles/VN164/VN164_State-of-the-Art_Radiosonde_Telemetry.pdf CRC16: (poly=0x(1)1021, init=0xFFFF) Reed-Solomon: RS(255,231) GF(2^8)=GF(2)[x]/(f): f=x^8+x^4+x^3+x^2+1 = 0x11D Generator von GF(2^8)*: alpha=0x02 (primitive Einheitswurzel) Generatorpolynom fuer RS-Code: (X-alpha^0)...(X-alpha^23) = X^24 + 0x7A X^23 + 0x76 X^22 + 0xA9 X^21 + 0x46 X^20 + 0xB2 X^19 + 0xED X^18 + 0xD8 X^17 + 0x66 X^16 + 0x73 X^15 + 0x96 X^14 + 0xe5 X^13 + 0x49 X^12 + 0x82 X^11 + 0x48 X^10 + 0x3D X^9 + 0x2B X^8 + 0xCE X^7 + 0x01 X^6 + 0xED X^5 + 0xF7 X^4 + 0x7F X^3 + 0xD9 X^2 + 0x90 X + 0x75 Es koennen t=12 Fehler pro Codewort korrigiert werden. Fehler der Anzahl t < #E <= 2t werden zuverlaessig erkannt. Decoder: bch_ecc.c http://www.ka9q.net/code/fec/ https://en.wikiversity.org/wiki/Reed%E2%80%93Solomon_codes_for_coders RS92 ---- HEADER 6 DATA 210 RS-PARITY 24 2a2a2a2a2a10 (header) 6510 (L=2*0x10=32) e81820204b34393533393334006100083d3d07b342bb3e9809d3bc3f754c963b ebfb 690c (L=2*0x0c=24) ca2b0fd9670f00670f8c0f11a458111e8810da410d30430d dc84 673d (L=2*0x3d=122) b01c852167ab07298b3cb05a536e0fffcf4faf7f4f3fff8fcfffffffff7fe5918aef20e8110175879900d0e0e900061f8c048aa393009edea1fe2557dc0019549f04b5160d00ae998b00e12a4200370b8608cacc1900b3e08905bd60040126da9303e1d4b4006d04a00573469700d6a98b00699f120195828f04 6ed7 6805 (L=2*0x05=10) 03030000000000000000 b27d ff02 (L=2*0x02=4) 02000200 f0be2a40a7cd69b9ed0668ec12182e8560ea6dd0733612a1 (Reed-Solomon parity) ecc-rs_vaisala.c: 1) ./ecc-rs_vaisala 2a2a2a2a2a106510e81820204b34393533393334006100083d3d07b342bb3e9809d3bc3f754c963bebfb690cca2b0fd9670f00670f8c0f11a458111e8810da410d30430ddc84673db01c852167ab07298b3cb05a536e0fffcf4faf7f4f3fff8fcfffffffff7fe5918aef20e8110175879900d0e0e900061f8c048aa393009edea1fe2557dc0019549f04b5160d00ae998b00e12a4200370b8608cacc1900b3e08905bd60040126da9303e1d4b4006d04a00573469700d6a98b00699f120195828f046ed7680503030000000000000000b27dff0202000200f0be2a40a7cd69b9ed0668ec12182e8560ea6dd0733612a1 RS92 codeword errors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frame: 2a2a2a2a2a106510e81820204b34393533393334006100083d3d07b342bb3e9809d3bc3f754c963bebfb690cca2b0fd9670f00670f8c0f11a458111e8810da410d30430ddc84673db01c852167ab07298b3cb05a536e0fffcf4faf7f4f3fff8fcfffffffff7fe5918aef20e8110175879900d0e0e900061f8c048aa393009edea1fe2557dc0019549f04b5160d00ae998b00e12a4200370b8608cacc1900b3e08905bd60040126da9303e1d4b4006d04a00573469700d6a98b00699f120195828f046ed7680503030000000000000000b27dff0202000200f0be2a40a7cd69b9ed0668ec12182e8560ea6dd0733612a1 2a) ./rs92gps -r rs92.wav > rs92raw.txt cat rs92raw.txt | ./ecc-rs_vaisala > rs92raw_ecc.txt ./rs92gps --rawin1 -a almanac.txt rs92raw_ecc.txt 2b) cat rs92raw.txt | ./ecc-rs_vaisala | ./rs92gps --rawin1 -a almanac.txt 2c) ./rs92gps -r rs92.wav | ./ecc-rs_vaisala | ./rs92gps --rawin1 -a almanac.txt 3) ./rs92ecc -b --ecc -e rinex.txt rs92.wav ./rs92ecc -b --ecc --crc -e rinex.txt rs92.wav ./rs92ecc -b --ecc -r rs92.wav RS41 ---- HEADER 8 RS-PARITY 48 = 2*24 DATA 264-462 (max 2*231=462) "interleaved" Die 2 Codewoerter setzen sich zusammen aus den DATA-Bytes an den geraden bzw. ungeraden Positionen und ersten bzw. zweiten RS-PARITY-Teil. RS(255,231) RS(255,231) L=40 CRC L=42 CRC L=30 CRC L=89 CRC L=21 CRC L=17 CRC 8635f44093df1a60 2c87e0fa0521e8943d9cef4c7a67393f6d39fb546461f2111b6447ab79a746c80350cda5344157f8c0c12234f4690222 0f 7928 16174b313933303239331a00000300000a00002f0007322ce53e31991abf12dada3eb68468c16755 d51c 7a2a 15310216060245f302000d08a31607821e08bb210219060243f302000000000000000000000000000000 220d 7c1e 0807d03cdc071fd81ddb19d70a8d0eb602b60cb518d40692ff00ff00ff00 1c27 7d59 b8d83301ff0f881f0f38f4fe18b283038735ff000000003eb8ff4947201e6e3aff55415f13fc6e005440440cf100009e9f7406f85800832b631719d70010bebc172a8b00000000000000000000000000000000000000000000 a48b 7b15 366181193ef05d07e1245b1be0f721f801f6080410 7b0b 7611 0000000000000000000000000000000000 ecc7 1) 8635f44093df1a60 (HDR) 2c87e0fa0521e8943d9cef4c7a67393f6d39fb546461f2111b6447ab79a746c80350cda5344157f8c0c12234f4690222 (2xRS) 0f 7928 (L=0x28=40) 16174b313933303239331a00000300000a00002f0007322ce53e31991abf12dada3eb68468c16755 d51c (CRC-OK) 7a2a (L=0x2a=42) 15310216060245f302000d08a31607821e08bb210219060243f302000000000000000000000000000000 220d (CRC-OK) 7c1e (L=0x1e=30) 0807d03cdc071fd81ddb19d70a8d0eb602b60cb518d40692ff00ff00ff00 1c27 (CRC-OK) 7d59 (L=0x59=89) b8d83301ff0f881f0f38f4fe18b283038735ff000000003eb8ff4947201e6e3aff55415f13fc6e005440440cf100009e9f7406f85800832b631719d70010bebc172a8b00000000000000000000000000000000000000000000 a48b (CRC-OK) 7b15 (L=0x15=21) 366181193ef05d07e1245b1be0f721f801f6080410 7b0b (CRC-OK) 7611 (L=0x11=17) 0000000000000000000000000000000000 ecc7 (CRC-OK) ./ecc-rs_vaisala 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 RS41 codeword1 errors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codeword2 errors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frame: 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 2) ./ecc-rs_vaisala 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RS41 codeword1 errors: 1 pos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codeword2 errors: 2 pos: 106 176 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3) ./rs41ecc -b --ecc rs41.wav ./rs41ecc -b --ecc --crc rs41.wav ./rs41ecc -b --ecc -r rs41.wav CRC16 ----- ./crc_polymod --crc 0000000000000000000000000000000000 ecc7 C7EC 11111111111111110000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100011111101100 10001000000100001 1110111111011111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100011111101100 10001000000100001 110011111001111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100011111101100 10001000000100001 10001111000111110000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100011111101100 10001000000100001 111000011111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100011111101100 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