ON THE AUTO AND CROSSCORRELATION OF PN SEQUENCES

Sep 2, 2026

Uploaded by Rhys

This paper analyzes the autocorrelation and crosscorrelation properties of pseudorandom (PN) sequences using key PN-sequence properties. The discussion avoids heavy linear-algebra methods by building intuition from structural behavior of linear maximal PN sequences generated by m-stage feedback shift registers. It defines maximal sequences via primitive generating polynomials and derives essential characteristics such as period occupancy, balance, and a shift-and-add transformation that simplifies correlation evaluation. Experimental results and sequence sets with good crosscorrelation properties are included.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages48

Reading Time2 h 1 min

AccessFree to download

Contents

Abstract

Auto-Correlation of a Linear Maximal PN Sequence

Cross-Correlation of Linear Maximal PN Sequences

Sequences of m=7

The Correlation as a Random Variable

Sets of Sequences with Good Crosscorrelation Properties

References

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

ON THE AUTO AND CROSSCORRELATIONOF PN SEQUENCES James C.MorakisDecember 1969Goddard Space Flight CenterGreenbelt,Maryland CONTENTS 名……… ABSTRACT………………………………………………………………………………… Auto-Correlation of a Linear Maximal PN Sequence………………………………………………4 Cross-Correlation of Linear Maximal PN Sequences ……………6 Sequences of m=7……………………………………………………………………8 The Correlation as a Random Variable…………………………………………11 Sets of Sequences with Good Crosscorrelation Properties……………………15 REFERENCES……………………………………………………………………………………………………………………21 APPENDIX A………………………………………………………………………………………………………………………22 APPENDIX B…………………………………………………………………………………………………………… APPENDIX C……………………………………………………… APPENDIX D………………………………………………………………………………:· APPENDIXE…………… Preceding page blank Page V Experimental Results on the Crosscorrelation of Linear Maximal PN The Correlation of PN Sequences Without Limitation on the Length.. 16 25 29 33 41 APPENDIX F. 43 # ON THE AUTO AND CROSSCORRELATIONOF PN SEQUENCES James C.Morakis # ABSTRACT The autocorrelation and crosscorrelationproper-ties of pseudorandom(PN)sequences are analyzed byusing some important properties of PN sequences.These properties make this discussionunderstandablewithout the need of linear algebraic approach.Theanalysis is followed by some experimental.results, # ON THE AUTO AND CROSSCORRELATIONOF PN SEQUENCES ·· Due to increasing interest in PN codes for applications in multiple user andor combat of multipath and others the author has decided to compile some of thealready known and other recent developments on the correlation properties ofPN coding schemes and their intercomparison.The basic linear PN code gen-eration is discussed extensively in[1]and some of the important properties ofPN codes will be repeated for convenience. A maximal linear PN Sequence is defined as a sequence generated by alinear*feedback shift register m stages long such that the length of the sequenceis the maximum possible for a constant m。It can be easily shown that if such nectionalways).One can consider the contents of the shift registeras the state of the shift register.If the state is known at time t,then the nextstate is given by a relationship of the original state and the transition matrixA**。 The characteristic equation of the matrix A turns out to be[1] withh;=0,1 and h₀=h=1. Figure 1-A Linear Feedback Shift Register This polynomial is also called the generating polynomial of the sequence.Ifthis polynomial is primitive then the sequence is maximal.This polynomial,just like any other polynomial of degree m,has m roots which might be desig-nated as a,a₁,a₂ am-rThe knowledge of any one of the above mroots is sufficient because all the others can be represented as powers of theone chosen.Thus we can say that PA(x)has a root a.If a polynomial has aroot β such that β=ak,k any integer,then that polynomial is either polynomialor another primitive polynomial.The number of distinct maximal se-quences that can be generated by an m-stage shift register is given by[1] where φ(j)is the Euler phi function and it is equal to the number of integersprime to j but less than j.So ifjis a primeφ(j)=j-1.Table 1 gives the num~ber of maximal sequences the length L and other information for each m.Table1 also gives a listing of the first few Mersenne primes which are defined as them's which are prime such that 2m-1is also a prime.Obviously Mersenneprimes result in large numbers of maximal sequences.To investigate the cor-relation properties of shift register generated sequences we must first discusssome pertinent properties。A sequence of period Lcan be viewed as &sequenceofL states(or Ln-dimensional vectors;L=2n-1 for binary)。 Property A.A maximal binary sequence will occupy all 2\#-1 non-zero statesbefore it starts repeating. Property B.(Balance property)The number of 1's in a maximal sequence ex-ceeds the number of 0's by 1. Thus the number of zeros of a maxima

FAQs

What properties are used to analyze autocorrelation and crosscorrelation of PN sequences?

The analysis relies on important properties of PN sequences, including definitions and transformation concepts that make correlation behavior understandable without linear-algebra techniques.

How is a maximal linear PN sequence defined in this document?

A maximal linear PN sequence is generated by an m-stage linear feedback shift register, producing the maximum possible length for a given m with period L based on that stage count.

Which transformation simplifies correlation calculations in binary waveforms?

The shift-and-add transformation converts the correlation computation by turning multiplications into mod-2 additions, enabling direct bitwise construction of related sequences.

Sep 2, 2026

Uploaded by Rhys