Natural-ordered complex Hadamard transform

This paper presents a new transform known as natural-ordered complex Hadamard transform (NCHT) which is derived from the Walsh–Hadamard transform (WHT) through the direct block matrix operation. Some of its properties, including the exponential property of the NCHT and the shift invariant property of the NCHT power spectrum, are presented. The relationship of the NCHT with the sequency-ordered complex Hadamard transform (SCHT) is discussed. In fact, it is shown that NCHT is a natural-ordered version of complex Hadamard transform whereas SCHT shows the sequency ordering. This is parallel to their real-valued counterparts, the WHT and the sequency-ordered Walsh transform (SOWT). A fast algorithm for computing the NCHT is also developed using sparse matrix factorization and its computational complexity is examined.