Slant Haar transform

Abstract

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'<n and w , w ' < m . [C] can be factorized and has a fast algorithm [4]. The generalized Kronecker product provides a simple way to recursively define fast unitary transforms. Consider the matrix of order 2 : Then the matrix of order 2", denoted (Tp], is obtained from the matrix of order 2"-' by {a).@ { T p -I ] , [Tp-I]) where @ denotes a generalized Kronecker product. R l t h this recursive notation, the H T is obtained for {a]. = [K(r/4)], [Iz], . * . , [I,] and the LVHT Sote that the number of operations given in (31 seems in error and that fewer operations may in fact he needed in a different organization 141.

Parallel and pipeline computation of fast unitary transforms

“…Many other transforms can have similar pipeline algorithms with reduced amount of hardware: the Modified generalized discrete transforms [11], the WFH transforms [1], the Slant Haar transforms [12] and other generalized Slant transforms [13]. In all cases, the pipeline-parallel algorithm needed to perform a transform of order 2 n in one cycle is the total number of butterflies appearing in the flow diagram of the algorithm.…”