
	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) = x³+x+1

The circuit taps correspond to P(x) = (1011).

Taps: (1011) (prime)

Sequence #1 (Starting with 0)

States: 0 ⇒ 0, forever...

Sequence #2 (Starting with 1)

States: 1 ⇒ 5 ⇒ 7 ⇒ 6 ⇒ 3 ⇒ 4 ⇒ 2 ⇒ 1

Period = 7 (Maximum Length Sequence) (autocorrelation)
Output = 1110100...

A maximum length sequence was found. For this polynomial of degree 3, the maximum length sequence has a period of 2³-1 states. This only happens when the characteristic polynomial is prime, as in this case. Use of a prime polynomial is a necessary but not sufficient condition for a maximum length sequence. For example, try 1010111 which is prime, but does not give a maximum length sequence.

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 01:41:38 ADT Last Updated: 04-09-25	Richard Tervo [ tervo@unb.ca ]	Back to the course homepage...