
	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).

Galois Implementation

alternate configuration Galois

Circuit based on P(x) = x⁴+x³+x²+x+1

The circuit taps generally correspond to P(x) = (11111) in reversed order, although this specific ordering is not evident when the bit sequence is symmetric as in this example.

Taps: (11111) (prime)

Sequence #1 (Starting with 0)

States: 0 ⇒ 0, forever...

Sequence #2 (Starting with 1)

States: 1 ⇒ 15 ⇒ 8 ⇒ 4 ⇒ 2 ⇒ 1

Period = 5 (autocorrelation)
Output = 11000...

Sequence #3 (Starting with 3)

States: 3 ⇒ 14 ⇒ 7 ⇒ 12 ⇒ 6 ⇒ 3

Period = 5 (autocorrelation)
Output = 10100...

Sequence #4 (Starting with 5)

States: 5 ⇒ 13 ⇒ 9 ⇒ 11 ⇒ 10 ⇒ 5

Period = 5 (autocorrelation)
Output = 11110...

In this example, a prime polynomial failed to give a maximum length sequence. For this polynomial of degree 4, the maximum length sequence would have a period of 2⁴-1 states. Use of a prime polynomial is a necessary but not sufficient condition for a maximum length sequence. Only "primitive primes" give maximum length sequences.

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.


2024-05-04 17:29:25 ADT Last Updated: 04-09-25	Richard Tervo [ tervo@unb.ca ]	Back to the course homepage...