05-May-2003 Project: IEEE P802.15 Working Group for Wireless Personal Area Networks (WPANs) Submission Title: [The ParthusCeva Ultra Wideband PHY proposal] Date Submitted: [05 May, 2003] Source: [Michael Mc Laughlin, Vincent Ashe] Company [ParthusCeva Inc.] Address [32-34 Harcourt Street, Dublin 2, Ireland.] Voice:[+353-1-402-5809], FAX: [-], E-Mail:[michael.mclaughlin@parthusceva.com] Re: [IEEE P802.15 Alternate PHY Call For Proposals. 17 Jan 2003] Abstract: [Proposal for a 802.15.3a PHY] Purpose: [To allow the Task Group to evaluate the PHY proposed] Notice: This document has been prepared to assist the IEEE P802.15. It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein. Release: The contributor acknowledges and accepts that this contribution becomes the property of IEEE and may be made publicly available by P802.15. Michael Mc Laughlin, ParthusCeva

The ParthusCeva PHY Proposal 05-May-2003 The ParthusCeva PHY Proposal Michael Mc Laughlin, ParthusCeva

Overview of Presentation 05-May-2003 Overview of Presentation PHY packet contents Coding DSSS Coding scheme - biorthogonal coding Ternary spreading codes FEC scheme - rate 4/6, 16 state convolutional coding Preamble Implementation Overview Performance Link margin Test results Data Throughput Complexity Michael Mc Laughlin, ParthusCeva

05-May-2003 Packet Contents Michael Mc Laughlin, ParthusCeva

doc.: IEEE 802.15-<doc#> <month year> doc.: IEEE 802.15-<doc#> 05-May-2003 The coding scheme 64 biorthogonal signals [Proakis1] 64 signals from 32 orthogonal sequences Ternary sequences chosen for their auto-correlation properties Code constructed from binary Golay-Hadamard sequences Michael Mc Laughlin, ParthusCeva <author>, <company>

Creating Orthogonal Ternary Sequences 05-May-2003 Creating Orthogonal Ternary Sequences Take a matrix of binary orthogonal sequences, in our case we used Golay-Hadamard sequences Add any two rows to get a ternary sequence Sum of any other two rows is orthogonal to this Continue till all the rows are used Repeat but subtracting instead of adding Michael Mc Laughlin, ParthusCeva

Finding good Ternary Golay Hadarmard codes 05-May-2003 Finding good Ternary Golay Hadarmard codes Large superset of orthogonal sequence sets to test Define aperiodic autocorrelation merit factor (aamf) as the ratio of the peak power of the autocorrelation function to the mean power of the offpeak values divided by the length of the code. Random walk used to find set with best aamf Michael Mc Laughlin, ParthusCeva

05-May-2003 Code comparison Length 32 code chosen for aamf and best matching with bit rates. Michael Mc Laughlin, ParthusCeva

Sample rate and pulse repetition frequency 05-May-2003 Sample rate and pulse repetition frequency Signal bandwidth chosen is 3.8GHz to 7.7GHz Sampling rate chosen is 7.7Ghz 32 chips per codeword, 4 bits / symbol (6 bits less 2 for convolutional code) Michael Mc Laughlin, ParthusCeva

05-May-2003 FEC scheme A 0.667 rate (rate 4/6) convolutional code was chosen for the FEC. [Proakis2] Very low complexity 16 state code, constraint length 2, Octal generators 27, 75, 72. Each of 16 states can transition to any other state, outputting 16 of 64 possible codewords. Provides 3dB of gain over uncoded errors at a cost of 50% higher bit rate Michael Mc Laughlin, ParthusCeva

Convolutional coder + + + 05-May-2003 Map every 6 bits to one of 64 biorthogonal codewords + + 2 bits in Michael Mc Laughlin, ParthusCeva

05-May-2003 Preamble Sequence Michael Mc Laughlin, ParthusCeva

05-May-2003 PAC properties Because of the perfect autocorrelation property, the channel impulse response can be obtained in the receiver by correlating with the sequence and averaging the results. Because the sequence consists of mostly 1, -1 with a small number of zerosm correlation can be economically implemented. (a length 553 PAC has 24 0’s) Michael Mc Laughlin, ParthusCeva

05-May-2003 Preamble properties Very good detect rate and false alarm probability. Pfa and Pmd < 10-4 for CM1 to CM4 test suite at 10 metres. Different length sequences means other piconets won’t trigger detection i.e. Pfa still < 10-3 for a different piconets PACn, even at 0.3m separation. Preamble length varies from ~5s to ~15s depending on the bit rate. Lower bitrates use longer preambles (Longer distances need more training time) Michael Mc Laughlin, ParthusCeva

05-May-2003 PHY Header The PHY header is sent at an uncoded 45Mbps rate, but with no convolutional coding. It is repeated 3 times. The PHY header contents are the same as 802.15.3 i.e. Two octets with the Data rate, number of payload bits and scrambler seed. Michael Mc Laughlin, ParthusCeva

Scrambler/Descrambler 05-May-2003 Scrambler/Descrambler It is proposed that the PHY uses the same scrambler and descrambler as used by IEEE 802.15.3 Michael Mc Laughlin, ParthusCeva

Typical Tx/Rx configuration 05-May-2003 Typical Tx/Rx configuration Antenna Single Chip Possible Output data at 30 - 960 Mbps Band Reject Filter Channel Matched filter (Rake Receiver) Band Pass Filter* A/D 7.7GHz, 1 bit Switch / Hybrid LNA Correlator Bank Viterbi Decoder Descramble 256 - 3800 Mchips/sec 8-240M symbols/sec Input data at 30- 960 Mbps Band Pass Filter Band Reject Filter Chip to Pulse Generator Code Generator Convolutional encoder Scramble * Can be avoided with good LNA dynamic range Michael Mc Laughlin, ParthusCeva

Possible RF front end configuration 05-May-2003 Possible RF front end configuration Total Noise Figure = 7.0dB NF= 0.2dB (input referred) NF= 2.0dB NF= 4.0dB BP Filter* Fine Filter LNA To Rx NF= 0.8dB Tx/Rx switch / hybrid Filter From Tx * Can be avoided with good LNA dynamic range Michael Mc Laughlin, ParthusCeva

Matched Filter configuration 05-May-2003 Matched Filter configuration Cn 4 1 Di Cn+N Di-N Cn+1 4 1 Di-1 Cn+N+1 Di-N-1 ….. ….. 4x 4x 4x 4x 4 4 4 4 ….. 4 bit adder 4x 4x + 5 bit adder + ….. ….. Michael Mc Laughlin, ParthusCeva

Matched Filter configuration 05-May-2003 Matched Filter configuration Structure repeated 16 times e.g. a 500 tap filter with 4 bit coefficients would have 500 x 16 x 4 AND gates in first stage Calculates 16 outputs in parallel, each runs at 480MHz. Multiplier is 4 AND gates. First adder stage is 4 OR gates. Very little performance loss. (0dB for CM1-3, 0.23dB for CM4). Coefficients are pre-processed to remove smallest if two clash. Michael Mc Laughlin, ParthusCeva

05-May-2003 Matched filter 560 tap filter takes 150k gates or 0.9 sq mm in 0.13 standard cell CMOS Power consumption = 220mW Matched filter acts as correlator during training phase. All simulations were carried out with this filter/correlator structure Michael Mc Laughlin, ParthusCeva

05-May-2003 Michael Mc Laughlin, ParthusCeva

Distance achieved for mean packet error rate = 8% 05-May-2003 Distance achieved for mean packet error rate = 8% Michael Mc Laughlin, ParthusCeva

Distance achieved for at worst packet error rate = 8% 05-May-2003 Distance achieved for at worst packet error rate = 8% Michael Mc Laughlin, ParthusCeva

Average distance at 8% PER 05-May-2003 Average distance at 8% PER Michael Mc Laughlin, ParthusCeva

05-May-2003 480 Mbps average PER Michael Mc Laughlin, ParthusCeva

05-May-2003 240 Mbps average PER Michael Mc Laughlin, ParthusCeva

05-May-2003 120 Mbps average PER Michael Mc Laughlin, ParthusCeva

Adjacent Channel Interferers: Single uncoordinated piconet 05-May-2003 Adjacent Channel Interferers: Single uncoordinated piconet Tests were done with reference links using channel models 1-4, channels 1-5 with shadowing removed. Interferers used channels 6-10 of Channel models 1 to 4. To allow some error margin, the distances to the reference receivers were 5m, 2m and 1.5m. For each channel model, at each distance, the mean PER for all 100 tests was calculated. (5 x 4 x 5) Michael Mc Laughlin, ParthusCeva

120 Mbps with single adjacent interferer 05-May-2003 120 Mbps with single adjacent interferer Single uncoordinated piconet, Reference Link 120Mbps at 5m, cm1-4 -0.5 -1 -1.5 0 average PER 8% PER -2 channel model 1 channel model 2 1 channel model 3 log -2.5 channel model 4 -3 -3.5 -4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Interferer Distance (m) Michael Mc Laughlin, ParthusCeva

240Mbps with single adjacent interferer 05-May-2003 240Mbps with single adjacent interferer Single uncoordinated piconet, Reference Link 240Mbps at 2m, cm1-4 -0.5 -1 -1.5 8% PER log10 Average PER channel model 1 channel model 2 -2 channel model 3 channel model 4 -2.5 -3 -3.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Interferer Distance (m) Michael Mc Laughlin, ParthusCeva

480Mbps with single adjacent interferer 05-May-2003 480Mbps with single adjacent interferer Michael Mc Laughlin, ParthusCeva

Two adjacent channel interferers 05-May-2003 Two adjacent channel interferers Tests were done with reference links using channel models 1-4, channels 1-5 with shadowing removed. Adjacent channel interferers used a freespace channel To allow some error margin, the distances to the reference receivers were 5m, 2m and 1.5m. For each channel model, at each distance, the mean PER over the 5 channels is plotted. Michael Mc Laughlin, ParthusCeva

120Mbps - Two free space interferers 05-May-2003 120Mbps - Two free space interferers Michael Mc Laughlin, ParthusCeva

240Mbps - Two free space interferers 05-May-2003 240Mbps - Two free space interferers Michael Mc Laughlin, ParthusCeva

480Mbps - Two free space interferers 05-May-2003 480Mbps - Two free space interferers Michael Mc Laughlin, ParthusCeva

120Mbps - Three free space interferers 05-May-2003 120Mbps - Three free space interferers Michael Mc Laughlin, ParthusCeva

05-May-2003 More interferers What matters is the total interference power, very little effect due to delay spread of the interfering channels. 2 interferers have 3dB more power than 1 which translates to 50% worse distance performance. 3 interferers have 1.76dB more power than 2 which translates to 22% worse distance performance. 4 interferers have 1.76dB more power than 3 which translates to 15% worse distance performance. Michael Mc Laughlin, ParthusCeva

Co-channel interference 05-May-2003 Co-channel interference Different piconets use exactly the same data mode codes as each other. Separation is achieved mainly because a different piconet will have a different impulse response and thus will not correlate with the matched filter which has been trained for the piconet of interest. For this reason, co-channel data mode interference has the same effect as adjacent channel interference. Training to the preamble will be affected more markedly by co-channel interference. Difficult to simulate. Michael Mc Laughlin, ParthusCeva

05-May-2003 Co existence Out of band signal (< 3.85GHz and >10.6GHz) are always filtered out. Any desired in band energy can be filtered out, with minimal effect on performance because the whole band is used to transfer data. Only adverse effect is the transmit power reduction (e.g. Dropping 400MHz for 802.11a loses <0.5dB) Michael Mc Laughlin, ParthusCeva

Interference and susceptibility 05-May-2003 Interference and susceptibility As for co existence, out of band signal are always filtered out. Again, any desired in band energy can be filtered out, with minimal effect on performance because the whole band is used to transfer data. Only adverse effect is the receive power reduction (e.g. Dropping 400MHz for 802.11a loses <0.5dB), its just a part of the channel. Michael Mc Laughlin, ParthusCeva

PHY-SAP Data Throughput 05-May-2003 PHY-SAP Data Throughput At higher bit rates, a 1024 byte frame is very short. The channel will be stationery for more than one frame so it is possible to send multiple frames for each preamble. Michael Mc Laughlin, ParthusCeva

05-May-2003 Scalable solution Possible to improve distance achieved by increasing coder complexity. (Decoder used here <15k gates) 30 - 960 Mbps. PHY power consumption/gate count changes little over range 110 - 480. Short range solution with much smaller matched filter for smaller delay spread Michael Mc Laughlin, ParthusCeva

Complexity - Area/Gate count, Power consumption 05-May-2003 Complexity - Area/Gate count, Power consumption Michael Mc Laughlin, ParthusCeva

Self evaluation : General Criteria 05-May-2003 Self evaluation : General Criteria Michael Mc Laughlin, ParthusCeva

Self evaluation : PHY protocol 05-May-2003 Self evaluation : PHY protocol Michael Mc Laughlin, ParthusCeva

Self evaluation : MAC enhancements 05-May-2003 Self evaluation : MAC enhancements Michael Mc Laughlin, ParthusCeva

} Summary of advantages Ternary spreading codes 05-May-2003 Summary of advantages Ternary spreading codes Better auto-correlation properties Perfect PAC training sequence Simple RF section 1 bit A/D converter No AGC required No mixers required Long matched filter possible 4 bit coefficients 1 bit data no multipliers } Low cost Low power consumption Low Noise figure Michael Mc Laughlin, ParthusCeva

05-May-2003 Backup Slides Michael Mc Laughlin, ParthusCeva

Ternary orthogonal sequences 05-May-2003 Ternary orthogonal sequences From any base set of 32 orthogonal binary signals, can generate 32C16 sets of 32 orthogonal ternary sequences. Generate by adding and subtracting any 16 pairs. Generally, if the base set has good correlation properties, so will a generated set. Michael Mc Laughlin, ParthusCeva

05-May-2003 Good base binary set Base set is a set of binary Golay-Hadamard sequences Take a binary Golay complementary pair. s116=[1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1]; s216=[1 1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1]; if A=circulant(s116) and B=circulant(s216) and G32= A B BT -AT then G32 is a Hadamard matrix. [Seberry] This type has particularly good correlation properties[Seberry] Detector can use the Fast Hadamard Transform Michael Mc Laughlin, ParthusCeva

Creating Orthogonal Ternary Sequences 05-May-2003 Creating Orthogonal Ternary Sequences Take a matrix of binary orthogonal sequences Add any two rows to get a ternary sequence Sum of any other two rows is orthogonal to this Continue till all the rows are used Repeat but subtracting instead of adding Michael Mc Laughlin, ParthusCeva

Orthogonal Ternary Example 05-May-2003 Orthogonal Ternary Example E.g. 1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 pairing 1 with 3 and 2 with 4 gives this orthogonal matrix 2 0 0 2 2 0 0 -2 0 2 2 0 0 -2 2 0 Michael Mc Laughlin, ParthusCeva

Finding good Ternary Golay Hadarmard codes 05-May-2003 Finding good Ternary Golay Hadarmard codes Large superset of orthogonal sequence sets to test Define aperiodic autocorrelation merit factor (aamf) as the ratio of the peak power of the autocorrelation function to the mean power of the offpeak values divided by the length of the code. Random walk used to find set with best aamf Michael Mc Laughlin, ParthusCeva

05-May-2003 Code comparison Length 32 code chosen for aamf and best matching with bit rates. Michael Mc Laughlin, ParthusCeva

Ternary Orthogonal Length 32 Code Set 05-May-2003 Ternary Orthogonal Length 32 Code Set + 0 - 0 - 0 - 0 + 0 + 0 - 0 + 0 + 0 - 0 - 0 - 0 - 0 - 0 + 0 - 0 - 0 + - 0 - 0 - + + + 0 0 0 0 0 0 0 0 0 - 0 + 0 0 - 0 - - + - - 0 0 0 0 - - 0 0 0 0 0 0 + + 0 0 - + 0 0 - - - + - + 0 0 - - + - 0 0 0 + + - - 0 0 - 0 0 + 0 + - 0 0 0 0 + - - 0 0 - 0 - - 0 - + - + + 0 0 0 0 - - 0 - + 0 + 0 0 + - - + 0 0 0 - - 0 - 0 0 - 0 0 0 0 0 + - 0 0 0 0 0 0 - + 0 0 0 - 0 0 - + + + - - 0 0 - + - - - 0 + - 0 0 0 + + - - 0 0 - 0 0 + 0 - + 0 0 0 0 + - - 0 0 - 0 - - 0 0 0 0 + + - 0 + - - - 0 0 0 + 0 0 0 0 - 0 0 0 + - - - 0 - + - 0 0 0 0 0 0 + - 0 0 0 0 0 0 - + - - - - 0 0 - + + + - - 0 0 - + 0 0 0 + 0 0 0 0 - + + 0 - - + - 0 - - - 0 0 0 0 + 0 0 0 - - + - 0 0 0 0 + + 0 0 0 0 0 0 - - 0 0 - + 0 0 - - + - - + 0 0 - - - + + - 0 0 0 + 0 0 0 0 - + + 0 - - + - 0 - - - 0 0 0 0 + 0 0 0 - - 0 0 0 0 0 0 + + 0 0 0 0 0 0 - - - + - + 0 0 - - + - - + 0 0 - - - - + - 0 0 0 + 0 0 0 0 - + + 0 - - + - 0 - - - 0 0 0 0 + 0 0 0 0 0 0 0 - + 0 0 0 0 0 0 + - 0 0 - - 0 0 - + - - - - 0 0 - + + + + 0 - - + - 0 0 0 + 0 0 0 0 - + 0 0 - - + - 0 - - - 0 0 0 0 + 0 0 + 0 - 0 + 0 + 0 + 0 + 0 + 0 - 0 + 0 - 0 + 0 + 0 - 0 - 0 - 0 + 0 + 0 0 - 0 + 0 0 0 0 + + - + + + + + - 0 - 0 - + 0 - 0 0 0 0 0 + - + + 0 0 - + - + + + 0 0 - + 0 0 + + 0 0 0 0 0 0 - - 0 0 0 0 + + - 0 0 0 0 - + 0 + + 0 + 0 0 - - + + 0 0 0 - + 0 + 0 0 - 0 0 0 0 0 - + + - 0 0 + 0 0 + 0 + + 0 0 0 0 + + - 0 0 + 0 + - 0 - - + - + 0 0 + + - - - + 0 0 + + + 0 + - 0 0 0 0 0 0 - + 0 0 0 0 0 + 0 0 + + - 0 0 0 0 - + 0 + + 0 - 0 0 - - + + 0 0 0 - + 0 + 0 0 + + + - 0 0 0 + 0 0 0 0 + - + 0 - - - + 0 - + + 0 0 0 0 + 0 0 0 + + + + - + 0 0 + + - - - + 0 0 0 0 0 0 + - 0 0 0 0 0 0 - + 0 0 + + + 0 + - + + 0 0 0 - 0 0 0 0 + 0 0 0 - + - - 0 + + - 0 0 0 0 - + + + 0 0 - + + - + + 0 0 - + 0 0 - - 0 0 0 0 0 0 + + 0 0 0 0 0 0 + + + 0 + - + + 0 0 0 - 0 0 0 0 + 0 0 0 - + - - 0 + + - 0 0 - + - + + + 0 0 - + + - + + 0 0 0 0 0 0 - - 0 0 0 0 0 0 + + 0 0 0 0 0 0 + + + 0 + - + + 0 0 0 - 0 0 0 0 + 0 0 0 - + - - 0 + + - - - - + 0 0 + + + + - + 0 0 + + 0 0 - + 0 0 0 0 0 0 + - 0 0 0 0 0 - 0 0 0 0 + + + 0 + - + + 0 0 + - 0 0 0 0 + 0 0 0 - + - - 0 + Michael Mc Laughlin, ParthusCeva

Matlab Code to generate PAC sequences 05-May-2003 Matlab Code to generate PAC sequences % function phi=ipatov(nu,multiplier,mul2); % % Generate a length nu, ternary perfect periodic autocorrelation sequence % using Singer Cyclic Difference Sets e.g. (553, 24, 1) function phi=ipatov(nu,multiplier,mul2); if nargin==1 multiplier=1; mul2=-1; end % multipliers 1,-1 are most commonly good if nargin==2 mul2=-1; end % multipliers 1,-1 are most commonly good phi=0; if gcd(nu,multiplier)>1; % must not be a common divisor of nu return; end if gcd(nu,mul2)>1; % must not be a common divisor of nu switch nu case 7 % nu=7;k=3;lamda=1; D=[1 2 4 ]; case 13 nu=13;k=4;lamda=1; D=1+[ 0 1 3 9 ]; case 21 nu=21;k=5;lamda=1; D=[3,6,7,12,14]; case 31 nu=31;k=6;lamda=1; D=[1 5 11 24 25 27 ]; case 57 nu=57;k=8;lamda=1; D=[0 1 6 15 22 26 45 55 ]; case 63 % multipliers 1,5 gives perfect ternary nu=63;k=31;lamda=15; D=1+[0 1 2 3 4 6 7 8 9 12 13 14 ... 16 18 19 24 26 27 28 32 33 35 36 38 ... 41 45 48 49 52 54 56 ]; case 73 nu=73;k=9;lamda=1; D=[1, 2, 4, 8, 16, 32, 37, 55, 64]; % P73 case 91 nu=91;k=10;lamda=1; D= [1 2 4 10 28 50 57 62 78 82]; % case 133 nu=133;k=12;lamda=1; D=[ 1 10 11 13 27 31 68 75 83 110 115 121]; case 183 nu=183;k=14;lamda=1; D=[ 1 13 20 21 23 44 61 72 77 86 90 116 ... 122 169 ]; case 273 nu=272;k=17;lamda=1; D=[0 1 22 33 83 122 135 141 145 159 175 200 ... 226 229 231 238 246]; case 307 nu=307;k=18;lamda=1; D=[ 0 1 3 30 37 50 55 76 98 117 129 133 ... 157 189 199 222 293 299 ]; case 341 nu=341;k=85;lamda=21; % 1,5 gives a perfect ternary sequence D= [ 0 1 2 3 5 7 8 11 15 17 20 23 ... 24 31 32 35 40 41 42 47 49 58 63 65 ... 68 71 76 80 81 83 85 95 99 117 120 127 ... 128 130 131 132 137 142 143 153 161 163 167 170 ... 171 174 180 182 186 190 191 199 204 208 210 230 ... 234 235 236 241 255 257 260 261 263 265 272 274 ... 275 285 287 288 300 306 307 314 320 323 327 330 ... 335 ]; case 364 nu=364;k=121;lamda=40; % 1,5 gives a perfect ternary sequence D=[0 1 2 3 4 6 8 9 11 16 17 19 ... 22 24 32 35 36 41 42 46 48 50 56 73 ... 76 78 79 80 88 89 92 99 105 106 107 109 ... 110 111 114 122 123 127 128 132 133 134 142 151 ... 152 153 156 162 163 169 171 174 177 181 183 187 ... 189 190 198 201 203 207 210 212 214 218 221 222 ... 223 224 229 234 237 241 246 248 249 250 251 256 ... 260 262 264 274 281 283 285 286 289 292 299 300 ... 302 305 306 307 309 311 314 315 317 318 322 326 ... 327 330 331 336 343 348 350 351 353 354 357 358 ... 360 ]; case 381 nu=381;k=20;lamda=1; D=1+[0 1 19 28 96 118 151 153 176 202 240 254 ... 290 296 300 307 337 361 366 369 ]; case 511 % 1,3 gives a perfect ternary sequence nu=511;k=255;lamda=127; D=[ 1 2 4 7 8 13 14 16 17 21 23 26 ... 28 31 32 33 34 35 37 39 42 46 49 51 ... 52 53 55 56 59 61 62 64 66 68 70 74 ... 75 77 78 79 81 83 84 85 89 91 92 95 ... 98 99 101 102 103 104 105 106 109 110 112 113 ... 115 118 121 122 123 124 128 131 132 133 136 137 ... 140 141 143 147 148 149 150 153 154 155 156 158 ... 161 162 163 165 166 168 169 170 178 182 183 184 ... 187 190 196 198 199 201 202 203 204 206 207 208 ... 210 211 212 215 217 218 219 220 221 223 224 225 ... 226 227 230 235 236 237 242 244 246 248 249 251 ... 256 259 262 264 266 267 271 272 273 274 275 280 ... 281 282 283 285 286 293 294 295 296 297 298 300 ... 301 303 305 306 307 308 310 312 313 316 317 321 ... 322 324 326 327 329 330 332 333 336 337 338 340 ... 347 349 355 356 357 359 361 363 364 365 366 367 ... 368 369 373 374 380 381 385 389 391 392 393 396 ... 397 398 402 403 404 406 407 408 409 412 414 416 ... 419 420 422 424 429 430 433 434 435 436 437 438 ... 439 440 442 446 448 450 451 452 454 457 459 460 ... 465 470 472 473 474 475 481 484 485 488 492 493 ... 496 498 502]; case 553 nu=553;k=24;lamda=1; D=[ 1 23 52 90 108 120 152 163 173 178 186 223 ... 232 272 359 407 411 431 438 512 513 515 529 548]; case 651 nu=651;k=26;lamda=1; D=[ 0 1 3 43 64 73 92 161 169 175 214 251 ... 268 309 396 421 453 471 500 505 515 527 531 538 ... 551 586 ]; case 757 nu=757;k=28;lamda=1; D=[ 1 2 63 103 112 114 119 158 171 199 242 264 ... 333 345 363 371 405 408 437 556 591 644 661 680 ... 711 734 738 744]; case 781 nu=781;k=156;lamda=31; % 1,2 gives a perfect ternary sequence D=1+[0 1 2 3 5 6 7 10 11 15 25 26 ... 27 30 31 35 38 41 48 50 51 55 57 64 ... 67 75 83 86 94 96 112 113 117 125 126 127 ... 130 131 135 137 143 150 151 153 155 169 175 190 ... 197 198 202 204 205 209 222 229 237 239 240 244 ... 250 251 255 258 264 266 272 275 285 301 303 313 ... 320 322 329 335 341 343 352 364 373 375 381 387 ... 402 404 414 415 417 419 430 439 448 451 457 458 ... 462 469 470 474 480 482 491 492 494 496 508 509 ... 513 516 523 533 539 546 549 552 560 561 565 567 ... 579 582 585 588 594 597 616 625 626 627 630 631 ... 633 635 642 644 650 651 655 675 678 685 693 701 ... 715 723 724 728 734 737 748 750 751 755 765 775]; case 871 nu=871;k=30;lamda=1; D=[ 1 24 29 69 151 167 216 234 259 263 295 321 ... 329 414 488 543 582 599 645 659 683 689 696 716 ... 731 819 820 822 831 841 ]; case 993 nu=993;k=32;lamda=1; D=1+[ 0 1 3 13 101 127 154 169 204 210 226 235 ... 259 289 297 317 356 434 474 478 495 538 570 584 ... 589 607 618 654 749 756 801 920 ]; case 1057 nu=1057;k=33;lamda=1; D=[ 1 2 32 36 41 150 156 172 192 370 397 426 ... 451 509 522 559 614 628 652 675 701 716 718 775 ... 778 786 796 885 913 961 1007 1014 1026]; case 1407 nu=1407;k=38;lamda=1; D=1+[ 0 1 37 63 205 274 289 302 314 316 321 362 ... 414 420 436 465 469 486 550 621 644 652 655 711 ... 731 844 854 924 981 1098 1122 1152 1187 1230 1248 1316 ... 1325 1369 ]; case 1723 nu=1723;k=42;lamda=1; D=[ 0 1 3 107 125 216 224 239 245 291 295 422 ... 435 444 471 499 770 807 840 909 935 952 982 992 ... 1050 1103 1138 1183 1222 1264 1296 1312 1372 1459 1545 1564 ... 1569 1589 1623 1630 1661 1712]; otherwise return end % end switch D=sort(mod(D*multiplier,nu)); % transform to new difference set. while any(D==0) D=mod(D+1,nu); Dhat=sort(mod(D*mul2,nu)); % transform to another new difference set while any(Dhat==0) Dhat=mod(Dhat+1,nu); ; Xd=zeros(1,nu); Xdhat=Xd; Xd(D)=1; Xdhat(Dhat)=1; phi=xcorr([Xd Xd],[Xdhat])-lamda; phi=round(phi(2*nu+1:2*nu+nu)); % phi is the ternary sequence if nargout==0 plot(xcorr(phi,[phi phi phi])) Michael Mc Laughlin, ParthusCeva

05-May-2003 References [Proakis1] John G. Proakis, Digital Communications 2nd edition. McGraw Hill. pp 224-225. [Proakis2] John G. Proakis, Digital Communications 2nd edition. McGraw Hill. pp 466-470. [Seberry et al] J. Seberry, B.J. Wysocki and T.A. Wysocki, Golay Sequences for DS CDMA Applications, University of Wollongong [Ipatov] V. P. Ipatov, “Ternary sequences with ideal autocorrelation properties” Radio Eng. Electron. Phys., vol. 24, pp. 75-79, Oct. 1979. [Høholdt et al] Tom Høholdt and Jørn Justesen, “Ternary sequences with Perfect Periodic Autocorrelation”, IEEE Transactions on information theory. Michael Mc Laughlin, ParthusCeva