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New fast Walsh-Hadamard-Hartley transform algorithm

Profile image of International Journal of Electrical and Computer Engineering (IJECE)International Journal of Electrical and Computer Engineering (IJECE)

2023, International Journal of Electrical and Computer Engineering (IJECE)

https://doi.org/10.11591/IJECE.V13I2.PP1533-1540
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Abstract

This paper presents an efficient fast Walsh-Hadamard-Hartley transform (FWHT) algorithm that incorporates the computation of the Walsh-Hadamard transform (WHT) with the discrete Hartley transform (DHT) into an orthogonal, unitary single fast transform possesses the block diagonal structure. The proposed algorithm is implemented in an integrated butterfly structure utilizing the sparse matrices factorization approach and the Kronecker (tensor) product technique, which proved a valuable and fast tool for developing and analyzing the proposed algorithm. The proposed approach was distinguished by ease of implementation and reduced computational complexity compared to previous algorithms, which were based on the concatenation of WHT and FHT by saving up to 3N-4 of real multiplication and 7.5N-10 of real addition.

Figures (5)
The factorization shown in (4) can be stated in terms of tensor (Kronecker) product as (5). as shown in (2) may be factorized into the (4). Using similar approach, WHy can be written as (6):
The factorization shown in (4) can be stated in terms of tensor (Kronecker) product as (5). as shown in (2) may be factorized into the (4). Using similar approach, WHy can be written as (6):
As an example, and without losing the generality, let us assume the transform length N=16; the Hy matrix can be represented as (18). Int J Elec & Comp Eng, Vol. 13, No. 2, April 2023: 1533-1540 Since N is a power of two, after that (16) can be written in compact form as (17):
As an example, and without losing the generality, let us assume the transform length N=16; the Hy matrix can be represented as (18). Int J Elec & Comp Eng, Vol. 13, No. 2, April 2023: 1533-1540 Since N is a power of two, after that (16) can be written in compact form as (17):
Therefore, it is feasible to calculate transformations using the matrices provided in (18) using the butter! structure, as illustrated in Figure 1. Figure 1. Radix-2 FWHT signal flow diagram when N=16, with (20) multiplications and (50) additions where the solid and dotted lines denote additions and subtractions, respectively
Therefore, it is feasible to calculate transformations using the matrices provided in (18) using the butter! structure, as illustrated in Figure 1. Figure 1. Radix-2 FWHT signal flow diagram when N=16, with (20) multiplications and (50) additions where the solid and dotted lines denote additions and subtractions, respectively
Figure 2. An in-place butterfly of radix-2 FWHT algorithm An and My denote the overall number of real additions and multiplications. The comparison in the total number of operations number between the FWHT and radix-2 WHT followed by the radix-2 FHT is shown in Table 1. The comparison shows that the FWHT algorithm requires (3N-4) real multiplications and (7.5N-10) real additions less than the existing algorithm. Furthermore, the proposed algorithm was also applied on MATLAB (R2021b), loaded on a laptop computer processor (Intel Core 17), and Windows-10 system to validate and confirm the results of the mathematical operations, as shown in Figures 3(a) and 3(b). Table 1. Comparison of real arithmetic operations
Figure 2. An in-place butterfly of radix-2 FWHT algorithm An and My denote the overall number of real additions and multiplications. The comparison in the total number of operations number between the FWHT and radix-2 WHT followed by the radix-2 FHT is shown in Table 1. The comparison shows that the FWHT algorithm requires (3N-4) real multiplications and (7.5N-10) real additions less than the existing algorithm. Furthermore, the proposed algorithm was also applied on MATLAB (R2021b), loaded on a laptop computer processor (Intel Core 17), and Windows-10 system to validate and confirm the results of the mathematical operations, as shown in Figures 3(a) and 3(b). Table 1. Comparison of real arithmetic operations
Figure 3. Shows the overall number of real operations (additions and multiplications) for the proposed radix-2 FWHT and WHT+FHT (a) real additions and (b) real multiplications
Figure 3. Shows the overall number of real operations (additions and multiplications) for the proposed radix-2 FWHT and WHT+FHT (a) real additions and (b) real multiplications

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