Arikan and Alamouti matrices based on fast block-wise inverse Jacket transform

EURASIP Journal on Advances in Signal Processing

Table 1 Computational complexity of the proposed fast algorithms for the block Jacket transforms compared with its direct computation (DC)

DC	Proposed
DC	N= 2^k	N= 3^k	N= 5^k	N= p ^k	N= p ^m q ⁿ	Two-dimensional: *p = q, m = n*
ADD
N(N – 1)	N log₂ N	2N log₃ N	4N log₅ N	(p – 1)N log_p N	n(p – 1)p ⁿ + m(q – 1)q ^m	2m(p – 1)p ^m
MUL
N × N	1/2N log₂ N	4/3N log₃ N	16/5N log₅ N	$\frac{{(p - 1)}^{2}}{P} N {log}_{p} N$	n(p – 1)² p ^n–1 + 2 m(q – 1)q ^m–1	m(p ² – 1)p ^m–1

In Table 1, ADD and MUL, respectively, abbreviate real additions and real multiplications.