Proceedings of the International Symposium on Musical Acoustics, March 31st to April 3rd 2004 (ISMA2004), Nara, Japan Topics in Nonabelian Harmonic Analysis and DSP Applications William J. DeMeo Textron Systems; Hawaii, USA williamdemeo@yahoo.com Abstract Underlying most digital signal processing (DSP) algo- rithms is the group Z/N of integers modulo N , which is taken as the data indexing set. Translations are defined using addition modulo N , and DSP operations, including convolutions and Fourier expansions, are then developed relative to these translations. Recently, An and Tolim- ieri [1] considered a different class of index set mappings, which arise when the underlying group is nonabelian, and successfully apply them to 2D image data. Advantages of indexing signals with nonabelian groups are not limited to image data, but extend to audio signals as well. The present work provides an overview of DSP on finite groups and group algebras. I present the basic nonabelian group theory relevant to DSP, and define a “generalized (nonabelian) translation,” and its consequence, “generalized convolution.” Thereafter I de- scribe some specific examples of nonabelian-group in- dexing sets which are simple yet revealing, as well as useful for applications. 1. Introduction The translation-invariance of most classical signal pro- cessing transforms and filtering operations is largely re- sponsible for their widespread use, and is crucial for ef- ficient algorithmic implementation and interpretation of results [1]. Underlying most digital signal processing (DSP) algorithms is the group Z/N of integers modulo N , which serves as the data indexing set. Translations are defined using addition modulo N , and basic opera- tions, including convolutions and Fourier expansions, are developed relative to these translations. DSP on finite abelian groups such as Z/N is well understood and has great practical utility. An excel- lent treatment that is applications oriented while remain- ing fairly abstract and general, is provided by Tolim- ieri and An in [2]. Recently, however, interest in the practical utility of finite nonabelian groups has grown significantly. Although the theoretical foundations of nonabelian groups is well established, application of the theory to DSP has yet to become common-place; cf. the NATO ASI “Computational Non-commutative Al- gebras,” Italy, 2003. Another notable exception is the book [1], by An and Tolimieri (2003), which develops theory and algorithms for indexing data with nonabelian groups, defining translations with a (non-commutative) group multiply operation, and performing typical DSP operations relative to these translations. The work demonstrates that including nonabelian groups among the possible data indexing strategies significantly broadens the range of useful signal processing techniques. This paper describes the use of nonabelian groups for indexing 1-dimensional signals, and discusses the com- putational advantages and insights to be gained from this approach. We examine a simple but instructive class of nonabelian groups – the semidirect product groups – and show that, when elements of such groups are used to in- dex the data and standard DSP operations are defined with respect to special group binary operators, interest- ing and powerful signal transformations are possible. 2. Notation and Background This section summarizes the notations, definitions, and important facts needed below. The presentation style is terse since the goal of this section is to distill from the more general literature only those results that are most relevant for DSP applications. The books [1] and [2] treat similar material in a more thorough and rigorous manner. Throughout, C denotes complex numbers, G an ar- bitrary (nonabelian) group, and L(G) the collection of complex valued functions on G. 2.1. Cyclic groups A group C is called a cyclic group if there exists x ∈ C such that every y ∈ C has the form y = x n for some integer n. In this case, we call x a generator of C. Cyclic groups are frequently constructed as special subgroups of arbitrary groups. Throughout the following discussion, G is an arbi- trary group, not necessarily abelian. For x ∈ G, the set of powers of x gp G (x)= {x n : n ∈ Z} (1) is a cyclic subgroup of G called the group generated by x in G. When G is understood, we simply write gp(x). It will be convenient to have notation for a cyclic group of order N without reference to a particular un-