What is the book all about?

This is Euclid's definition for “perpendicular”, which is synonymous with the word “orthogonal” used in the title of this book. Although the meaning of this word has been generalized since Euclid's time to describe the relationship between two functions as well as two vectors, as what we will be mostly concerned with in this book, they are essentially no different from two perpendicular straight lines, as discussed by Euclid some 23 centuries ago.

Orthogonality is of important significance not only in geometry and mathematics, but also in science and engineering in general, and in data processing and analysis in particular. This book is about a set of mathematical and computational methods, known collectively as the orthogonal transforms, that enables us to take advantage of the orthogonal axes of the space in which the data reside. As we will see throughout the book, such orthogonality is a much desired property that can keep things untangled and nicely separated for ease of manipulation, and an orthogonal transform can rotate a signal, represented as a vector in a Euclidean space, or more generally Hilbert space, in such a way that the signal components tend to become, approximately or accurately, orthogonal to each other.

This is the most general, but also the most abstract chapter on sequence generators in the book. In this chapter we describe the theory of algebraic feedback shift registers (AFSRs). This is a “shift register” setting for the generation of pseudo-random sequences, which includes as special cases the theory of linear feedback shift register (LFSR) sequences, feedback with carry shift register (FCSR) sequences, function field sequences, and many others. The general framework presented here first appeared in 1999 [120] (following an intermediate generalization of FCSRs [105]). It “explains” the similarities between these different kinds of sequences. Many of the interesting properties of these sequences simply reflect general properties of any AFSR sequence. However, because this chapter is fairly abstract, the rest of the book has been written so as to be largely independent of this chapter, even at the expense of sometimes having to repeat, in a special case, a theorem or proof which is fully described in this chapter.

Definitions

The ingredients used to define an algebraic shift register are the following.

1. An integral domain R (see Definition A.2.1).

2. An element π ∈ R.

3. A complete set of representatives S ⊂ R for the quotient ring R/(π). (This means that the composition S → R → R/(π) is a one to one correspondence. See Section 5.2.)

In this chapter we briefly review some of the sequences that are related to m-sequences, many of which have found applications in communications. See also Section 15.5, where the linear span of sequences derived from m-sequences is discussed. Further information on these and many other sequences may be found in Helleseth and Kumar's survey article [88], Cusick, Ding, and Renvall's monograph [35], and Golomb and Gong's textbook [63].

Welch bound

For applications to spread spectrum communications, one attempts to find a collection of shift distinct sequences with low pairwise cross-correlation values. For a given (maximal) cross-correlation, there are theoretical limitations on the number of sequences in such a collection, the simplest of which is the Welch bound [204].

Suppose we have a collection of n periodic sequences of elements in a finite field F, each with the same period T. We can expand this set to include all the shifts of these sequences. If T is the minimal period of each sequence and the sequences are pairwise shift distinct, then we obtain a set of N = Tn vectors, commonly referred to as a signal set. Let us denote these vectors by a(1), a(2), …, a(N). Let χ : F → ℂ× be a nontrivial character.