
	ECE4253 Digital Communications

	Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

Linear Recursive Sequence Generator

Shift registers with feedback essentially divide polynomials to create distinctive binary sequences.

This online tool draws and analyzes digital circuits which generate Linear Recursive Sequences (LRS) based on a defining polynomial P(x). The circuit shown below is traced through all possible states. Maximum length sequences are identified. The autocorrelation of each sequence can also be checked (maximum 1023 bits).

Fibonacci Implementation

alternate configuration Fibonacci

Circuit based on P(x) = x⁸+x⁷+x⁶+x⁴+x³+x²+1

The circuit taps correspond to P(x) = (111011101).

Taps: (111011101) (prime)

Sequence #1 (Starting with 0)

States: 0 0 forever...

Sequence #2 (Starting with 1)

States: 1 128 192 96 176 88 172 214 107 181 90 173 86 171 213 234 245 250 125 190 95 175 87 43 21 138 69 162 209 104 52 26 13 134 67 33 144 72 36 146 73 164 82 41 20 10 133 194 97 48 152 204 102 51 25 140 198 227 241 120 188 94 47 151 75 165 210 233 116 186 221 110 183 91 45 150 203 101 178 89 44 22 11 5 2 1

Period = 85 (autocorrelation)
Output =
1000000011010110101011111010100010110000100100101000011001100011
110100101110110100110...

Sequence #3 (Starting with 3)

States: 3 129 64 160 208 232 244 122 189 222 239 247 251 253 126 63 31 15 135 195 225 240 248 124 62 159 207 231 115 185 92 46 23 139 197 98 177 216 108 182 219 237 246 123 61 30 143 71 163 81 168 84 170 85 42 149 74 37 18 137 196 226 113 184 220 238 119 59 157 206 103 179 217 236 118 187 93 174 215 235 117 58 29 14 7 3

Period = 85 (autocorrelation)
Output =
1100000010111101111110000111110011101000110110111100010101010010
001110111001101110101...

Sequence #4 (Starting with 4)

States: 4 130 193 224 112 56 28 142 199 99 49 24 12 6 131 65 32 16 136 68 34 17 8 132 66 161 80 40 148 202 229 114 57 156 78 167 211 105 180 218 109 54 27 141 70 35 145 200 228 242 249 252 254 255 127 191 223 111 55 155 77 38 147 201 100 50 153 76 166 83 169 212 106 53 154 205 230 243 121 60 158 79 39 19 9 4

Period = 85 (autocorrelation)
Output =
0010000011100011000001000100001010011100101101100010011111111011
001001100101011001111...

See a detailed analysis and State Table for this circuit.

Specify the taps for your sequence

Modulo 2 addition is shown schematically equivalent to Exclusive-OR gates.


2025-06-23 20:39:07 ADT Last Updated: 2014-01-13	Richard Tervo [ tervo@unb.ca ]	Back to the course homepage...