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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 1 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

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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):
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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Information TechnologyTechnologyImage ProcessingKronecker productsFast AlgorithmsDiscrete Hartley Transform
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