# APPENDIX

# J Maximal-Length Sequences

Basically, maximal-length sequences, also referred to in the literature as *m*-*sequences*, are *linear cyclic codes*, the generation of which is realized by using a linear feedback-shift register (LFSR) as discussed in Chapter 10 on error-control coding; Figure J.1 is an illustrative example of LFSR. However, from a practical perspective insofar as this book is concerned, it is the pseudo-noise (PN) characteristic that befits their use in producing spread-spectrum signals, an issue that was discussed in Section 9.13 of Chapter 9. In short, a maximal-length sequence viewed as a “carrier” may be used to spread the spectrum of an incoming message sequence in the transmitter and despread the received signal so as to recover the original message signal at the receiver output.

It is therefore apropos that we begin the discussion of maximal-length sequences in this appendix by discussing their basic properties, illustrated by the LFSR as the sequence generator.

## J.1 Properties of Maximal-Length Sequences

Maximal-length sequences^{1} have many of the properties possessed by a truly *random binary sequence*. A random binary sequence is a sequence in which the presence of binary symbol 1 or 0 is equally probable. Maximal-length sequences have the following properties.

**PROPERTY 1** ** Balance Property**

*In each period of a maximal-length sequence, the number of 1s is always one more than the number of 0s.*

**PROPERTY 2** **Run Property**

*Among the runs of 1s and of 0s in each ...*