Family of unified complex Hadamard transforms

Abstract

Novel discrete orthogonal transforms are introduced in this paper, namely the unified complex Hadamard transforms. These transforms have elements confined to four elementary complex integer numbers which are generated based on the Walsh-Hadamard transform, using a single unifying mathematical formula. The generation of higher dimension transformation matrices are discussed in detail.Index Terms-Digital signal processing, discrete transforms, fast algorithms, orthogonal transforms, unified complex Hadamard transforms.

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems

“…These sequences are generated by the unified complex Hadamard transform matrix discussed in [96], the correlation properties of which are studied in [97], where it is shown that the unified complex Hadamard transform sequences possess the better autocorrelation properties in comparison with the WH sequences, which are characterrized by very poor autocorrelation properties.…”

“…They are categorized into two groups: the halfspectrum property orthogonal unified complex Hadamard transform spreading sequences and the non-half-spectrum property orthogonal unified complex Hadamard transform spreading sequences. Consider briefly how these sequences can be generated and note the main definitions and remarks discussed in [96,97].…”

“…A unified complex Hadamard transform matrix A of order n N 2  is a square matrix with elements 1  and j  , and is constructed by [96,97] [97] that the non-half-spectrum property unified complex Hadamard transform sequences have very similarly poor autocorrelation properties as WH sequences, and some of the half-spectrum property unified complex Hadamard transform sequences exhibit a reasonable compromise between the autocorrelation and cross-correlation functions. In this section, we just consider the half-spectrum property unified complex Hadamard transform sequences, i.e., j j a   or 3 .…”

A new parametric transform

“…The WHT has a wide spread use in many applications and specifically with a remarkable success in encryption [6], [7] and watermarking [8], [9] mainly because of its simple structure. Rahardja and Falkowski [10] have introduced a complex WHT by using the second-order Hadamard matrix and the Kronecker product. The entries of the resulting matrix operator are all from the set {+1, -1, + j , -j }, where…”

A new recursive formulation for 2-D WHT

“…We chose this convention to show the direct correspondence between the derived algorithms and the proposed VLSI networks. Although, as far as we know from the literature, the recursive 1-D WHT algorithm is widely presented in the literature [8], [11], neither the proposed 2-D WHT algorithm nor the modular forms were previously derived.…”