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Matrix Expansions for Computing the Discrete Hartley Transform

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2010, IEEE/SBrT International Telecommunication Symposium, ITS 2010

https://doi.org/10.14209/SBRT.2010.89
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Abstract

A new fast algorithm for computing the discrete Hartley transform (DHT) is presented, which is based on the expansion of the transform matrix. The algorithm presents a better performance, in terms of multiplicative complexity, than previously known fast Hartley transform algorithms. A detailed description of the computation of DHTs with blocklengths 8 and 12 is shown. The algorithm is very attractive for blocklengths N ≥ 128.

Figures (5)
Figure 1. Scheme for the computation of a DHT with blocklength N=8. The small circles into the X-box denote subtraction. The pseudo-code presented below describes the calculations, showing the floating-point multiplications. Moreover, we can see that the linear combination of adders involves only multiplications by 1 and -1, which are trivial.
Figure 1. Scheme for the computation of a DHT with blocklength N=8. The small circles into the X-box denote subtraction. The pseudo-code presented below describes the calculations, showing the floating-point multiplications. Moreover, we can see that the linear combination of adders involves only multiplications by 1 and -1, which are trivial.
Ll so rank(C,(M2)) =2. The SEF of this matrix is the same as that of C,(M,)- C.(M ,) = B, (A) —By(A),
Ll so rank(C,(M2)) =2. The SEF of this matrix is the same as that of C,(M,)- C.(M ,) = B, (A) —By(A),
Table 1. Complexity of the power series-based FHT algorithm in terms of the number of real non-trivial floating-point multiplications, compared to the radix-2 and radix-4 FHT algorithms. Complexity results for the matrix expansion FHT are shown in Table 1, in comparison with Heideman lower bound and standard radix-2, radix-4 and Split-Radix FHT algorithms [4]. Figure 2 shows the multiplicative complexity of the algorithms in Table 1 as a function of N. We must emphasize that the complexity for blocklengths N = (8, 32, 128, 512, 2048), for the Radix-4 algorithm, are not reported in [4].
Table 1. Complexity of the power series-based FHT algorithm in terms of the number of real non-trivial floating-point multiplications, compared to the radix-2 and radix-4 FHT algorithms. Complexity results for the matrix expansion FHT are shown in Table 1, in comparison with Heideman lower bound and standard radix-2, radix-4 and Split-Radix FHT algorithms [4]. Figure 2 shows the multiplicative complexity of the algorithms in Table 1 as a function of N. We must emphasize that the complexity for blocklengths N = (8, 32, 128, 512, 2048), for the Radix-4 algorithm, are not reported in [4].
spectrum. Detailed examples to illustrate the technique were presented for N = 8 and 12, but the entire procedure is systematic. The fast Hartley transform presented here is also easy to implement using DSP or low-cost high-speed Integrated Circuits. The algorithm presents a_ better performance, in terms of multiplicative complexity, than standard radix-2, radix-4 and Split-Radix Cooley-Tukey FHT algorithms. Figure 2. Multiplicative complexity for the radix-2, Split-Radix and matrix expansion FHT algorithms and Heideman lower bound, as a function of transform blocklength.
spectrum. Detailed examples to illustrate the technique were presented for N = 8 and 12, but the entire procedure is systematic. The fast Hartley transform presented here is also easy to implement using DSP or low-cost high-speed Integrated Circuits. The algorithm presents a_ better performance, in terms of multiplicative complexity, than standard radix-2, radix-4 and Split-Radix Cooley-Tukey FHT algorithms. Figure 2. Multiplicative complexity for the radix-2, Split-Radix and matrix expansion FHT algorithms and Heideman lower bound, as a function of transform blocklength.

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References (8)

  1. Olejniczak, K. J. and Heydt, G. T. (eds.), Special Section on the Hartley Transform, Proceedings of the IEEE, vol. 82, No. 3, pp. 372-447, March 1994.
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  4. Sorensen, S. V., Jones, D. L., Burrus, C. S. and Heideman, M. T., On Computing the Discrete Hartley Transform, IEEE Trans. Acoust., Speech and Signal Processing, vol. ASSP 33, No. 4, pp. 1231-1238, 1985.
  5. de Oliveira, H.M., Cintra, R.J.S., Campello de Souza, R. M., Multilevel Hadamard Decomposition of Discrete Hartley Transforms, XVIII Brazilian Telecommunications Symposium, SBrT'00, Gramado, RS, September, 2000.
  6. de Oliveira, H.M., Campello de Souza, R. M., A Fast Algorithm for Computing the Hartley/Fourier Spectrum, Proceeedings of the Brazilian Academy of Science, Rio de Janeiro, vol. 73, pp.468-468, 2001.
  7. Liu, J. G. et alli, Moment-based fast discrete Hartley transform, Signal Processing. vol. 83, No. 8, pp. 1749-1757, August 2003.
  8. Heideman, M.T. Multiplicative Complexity, Convolution and the DFT, Springer-Verlag, 1988.

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Related topics

  • Signal Processing
  • DFT calculation
  • Hartley Transform
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