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A set of recursive rules which generate unitary transforms with a fast algorithm (FUT) are presented. For each rule, simple relations give the number of elementary operations required by the fast algorithm. The common Fourier, Walsh-Hadamard (W-H), Haar, and Slant transforms are expressed with these rules. The framework developed allows the introduction of generalized transforms which include all common.transforms in a large class of "identical computation transforms". A systematic and unified view is provided for unitary transforms which have appeared in the literature. This approach leads to a number of new transforms of potential interest. Generalization to complex and multidimensional unitary transforms is considered and some structural relations between transforms are established. Key words, discrete transforms, fast algorithms, fast Fourier transforms, fast generalized transforms, generalized Kronecker product, Haar transform, identical computation transforms, slant transform, unitary transforms and matrices, Walsh-Hadamard transform *

LETTERS 653 pulse, i t is possible to produce a pseudorandom signal coinciding with the clocking rate of Mi.'This fact makes i t possible to achieve output rates of the order of 1 C P numbers/s. ACKNOWLEDGMENTThe author wishes to thank Prof. J. Kawata and M. Ohyama for their useful suggestions during this work. REFERENCES111 S. Ayanaga and M. Sugiura. Algebra. Japan: 1-mi. 1971. I21 C. M. Rader el ol.. "A fast method of generating digital m d o m numbers.' (31 B. R. Gaines. "Stochastic computing," in Proc. Sprirg J o i d Comgvkr Caf,. Abstracf-The slant Haar transform (SHT) is defined and related to .the slant Walsh-Hadamard transform (SWHT). A fast algorithm for the SHT is presented and its computational complexity computed. In most applications, the SHT is faster and performs as well as the SWHT. INTRODUCTION The slant \Valsh-Hadamard transform (SWHT) (originally called slant transform) has been proposed by Enomoto and Shibata [ l ] for the order 8 and used in TV image encoding. Pratt et al. (21 and Chen [3] have generalized this transform to any order 2" and compared its performance with other transforms. In [4], we have given a simpler definition of the SLVHT as a particular case of a unified treatment of fast unitary transforms and computed the number of elementary operations required by its fast algorithm.' The interesting feature of the S\VHT is the presence of a slant vector with linearly decreasing components in its basis. On the other hand, n e have found that locally dependent basis vectors, such as in the Haar transform (HT), .are of interest [SI.In this letter, we define a composite fast unitary transform: the slant Haar transform (SHT). We show that its relations to the SWHT parallel the relations between the HT and the LValsh-Hadamard transform (WHT) 161. This previous work leads us to expect that the SHT has an advantage over the SWHT because of its speed and comparable performance. DEFINITION The generalized Kronecker product of the set {a> of n matrices [.4j](j=0, . . . , n -1 ) of order m and the set {a} of m matrices [ B k ] (k = 0 , . . . , m -1) of order n is the matrix [C] of order mn such that Cum+r.u,m+r, =A..,mBm~u' when u, u'

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SThis correspondence discusses the parallel and pipeline organization = of fast unitary transforms algorithms such as the Fast Fourier Transform and points out the efficiency of a combined parallel-pipeline processor a of a transform such as the Haar transform in which (2-1) hardware-'ri H4 M4-I. U