Dyadic symmetry and Walsh matrices W.K. Cham, PhD R.J. Clarke, PhD, CEng, MIEE Indexing terms: Image processing, Matrix algebra, Mathematical techniques Abstract: A unified matrix treatment which is defined for binary Walsh matrices is presented. This unified treatment, based on the concept of dyadic symmetry, defines Walsh matrices of differ- ent orderings using a simple equation. This equa- tion in turn provides a straightforward derivation of various reordering schemes and Walsh matrix properties. Various fast computational algorithms can also be derived within the same framework using dyadic decompositions. It is hoped that this unified treatment of Walsh matrices using dyadic symmetry will provide a better understanding of Walsh transforms, and will lead to the discovery of more useful transforms. 1 Introduction The simplicity and ease of implementation of the Walsh transform have resulted in a wide range of applications and investigations of its properties. Throughout the development of the Walsh transform, different definitions and different generation methods have been adopted, leading to a certain degree of confusion. Attempts have been made, therefore, to unify the nomenclatures, and Fino and Algazi produced a unified matrix treatment to provide a common framework for all areas of interest [1]. They defined Walsh matrices having different order- ings using the Kronecker matrix product and various permutation matrices. In this paper we propose an alternative unified matrix treatment which is defined for binary Walsh matrices [2]. This unified treatment, based on the concept of dyadic symmetry, defines Walsh matrices of different orderings using a simple equation. This equation in turn provides a straightforward derivation of various reordering schemes and Walsh matrix properties. Various fast computational algorithms can also be derived within the same frame- work using dyadic decomposition. The concept of dyadic symmetry has also recently been utilised to generate two new transforms, the high correlation transform (HCT) and the low correlation transform (LCT) [3]. It is hoped that the unified treatment of Walsh matrices using dyadic symmetry will provide a better understanding of Walsh transforms, and will lead to the discovery of other useful transforms. In this paper the nomenclature standards Paper 5202F (E4), first received 23rd September and in revised form 26th November 1986 Dr. Cham is with the Department of Electronics, Chinese University of Hong Kong, Shatin, NT, Hong Kong Dr. Clarke is with the Department of Electrical & Electronic Engineer- ing, Heriot-Watt University, 31-35 Grassmarket, Edinburgh EH1 2HT, United Kingdom proposed by Ahmed et al. [4] for Walsh matrices will be adopted. In Section 2 dyadic symmetry within a vector is first defined, and then the properties of dyadic symmetry are derived. These results, together with a simple equation which defines a Walsh matrix, are then used in Section 3 to derive properties of Walsh matrices. In Section 4 dyadic symmetry decomposition is defined and its rela- tion to various fast computational algorithms is given. 2 Dyadic symmetry Definition 1: A vector of 2 m elements [a 0 a^ ••• a 2m -i]' is said to have the Sth dyadic symmetry if a J = C<*j<BS (1) where © is 'exclusive-OR',; lies in the range [0, 2 m — 1] and S in the range [1, 2 W — 1], and c is a constant which determines the type of the dyadic symmetry. If c = 1 then the symmetry is said to be 'even', and if c = — 1 then the symmetry is said to be 'odd'. Theorem 1: If a 2 m -vector has dyadic symmetries S lt S 2 , ..., S r , this vector also has dyadic symmetry S k , where "fc — "1 VI7 (2) Proof: Let vector A be [a 0 a t ••• a 2 m-i'\ t having dyadic symmetry S u S 2 ,..., S r . From the definition of dyadic symmetry, we have a j = C X a J = C r Oj for all) within the range [0, 2 m — 1]. Therefore, we have ®j = C k QjQSk where c k = c Y c 2 •• c r and S k = S t © S 2 © • • • © S r . Let F be a binary field which has 0 and 1 as its elements, 'logical AND' as the multiplication operation and 'exclusive-OR' as the addition operation. A dyadic symmetry S = [s x s 2 ••• s m ] is a vector in F. [s t s 2 •• • s m ] is also the binary representation of S, where s t is the most significant and s m is the least signifi- cant bit. Unless stated otherwise, all vectors in F will be row vectors and all vectors in the number field will be column vectors. Definition 2: The dyadic symmetries S lt S 2 , ..., S m are said to be dependent if there exist m elements k lt k 2 , ..., k m in F, not all zero, such that Otherwise, the m symmetries are said to be linearly inde- pendent. IEE PROCEEDINGS, Vol. 134, Pt. F, No. 2, APRIL 1987 141