A Novel Radix-3/9 Algorithm for Type-Iii Generalized Discrete Hartley Transform

Abstract

A novel radix-3/9 algorithm for type-III generalized discrete Hartley transform (GDHT) is proposed, which applies to length-3p sequences. This algorithm is especially efficient in the case that multiplication is much more time-consuming than addition. A comparison analysis shows that the proposed algorithm outperforms a known algorithm when one multiplication is more time-consuming than five additions. When combined with any known radix-2 type-III GDHT algorithm, the new algorithm also applies to length-2q3p sequences.

Algorithm of efficient computation of generalised discrete Hartley transform based on cyclic convolutions

“…Combination of the results of convolutions is performed horizontally (Table 4). Output data of the transform in the result of computations corresponds to the following sequence: X(1), X(3), X(9), X(13), X(11), X(5), X(8), X(10), X(2), X(6), X(4), X (12) and X (7).…”

“…The hashing array of Pc(n) specifies the order of the input data when we conduct the discrete cosine transform using cyclic convolutions. The values of elements require transition from the hashing array of elements (2n +1)→ n to the appropriate hashing array with values of elements Pc(n) (1,5,25,17,13,29,19,23,7,35,31,11,3,15,21,33,9,27) (0, 2,12,8,6,14,9,11,3,17,15,5,1,7,10,16,4,13) Performance of element-wise additions of input data will be used for analysis of identity and quasi-identity cyclic submatrices placed horizontally. Computation of cyclic convolution is performed once for combined input data for identity and quasi-identity submatrices selected for analysis vertically.…”

“…( described by the hashing array of arguments with elements of smaller period equal to 8N in the form P (16) = (1,3,9,27,17,19,25,11) (7,21,31,29,23,5,15,13) The values of elements require transition from hashing array (2n +1)→ n to the appropriate hashing array Accordingly, eight rows of hashing array P(n) for DHT IV horizontal are selected. They allow computing the eight output data through cyclic convolutions.…”

“…Combination of the results of cyclic convolutions is performed at the base coordinates of the first elements of submatrices horizontally. Output data of transform as a result of computation are scaled by two and are determined for: X(0), X(1), X(4), X(5), X(3), X(2), X (7) and X (6). Therefore horizontal 8 rows of respective hashing array P (16) are selected, allowing us to compute output data values via cyclic convolutions.…”

“…The most diverse is the study of the discrete Fourier transform class. Multivariate effective computing algorithms are divided into: size of radix two [3][4][5], split radix [6], mixed radix [7], special size [8] and prime factors composite transform size [9]. The existing approaches have advantages and features that are specified by the generalised representation and the synthesis of efficient algorithms [10][11][12].…”