Abstract: Maximal length sequences (m-sequences) are also
known as pseudo random sequences or pseudo noise sequences
for closely following Golomb-s popular randomness properties: (P1)
balance, (P2) run, and (P3) ideal autocorrelation. Apart from these,
there also exist certain other less known properties of such sequences
all of which are discussed in this tutorial paper. Comprehensive proofs
to each of these properties are provided towards better understanding
of such sequences. A simple test is also proposed at the end of
the paper in order to distinguish pseudo noise sequences from truly
random sequences such as Bernoulli sequences.
Abstract: Different pseudo-random or pseudo-noise (PN) as well as orthogonal sequences that can be used as spreading codes for code division multiple access (CDMA) cellular networks or can be used for encrypting speech signals to reduce the residual intelligence are investigated. We briefly review the theoretical background for direct sequence CDMA systems and describe the main characteristics of the maximal length, Gold, Barker, and Kasami sequences. We also discuss about variable- and fixed-length orthogonal codes like Walsh- Hadamard codes. The equivalence of PN and orthogonal codes are also derived. Finally, a new PN sequence is proposed which is shown to have certain better properties than the existing codes.
Abstract: The paper provides an in-depth tutorial of mathematical
construction of maximal length sequences (m-sequences) via primitive
polynomials and how to map the same when implemented in
shift registers. It is equally important to check whether a polynomial
is primitive or not so as to get proper m-sequences. A fast method to
identify primitive polynomials over binary fields is proposed where
the complexity is considerably less in comparison with the standard
procedures for the same purpose.