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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 73, Iss. 12 — Dec. 1, 1983
  • pp: 1832–1835

Discrete Hartley transform

R. N. Bracewell  »View Author Affiliations


JOSA, Vol. 73, Issue 12, pp. 1832-1835 (1983)
http://dx.doi.org/10.1364/JOSA.73.001832


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Abstract

The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most iniage-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of fl) times; (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [R. V. L. Hartley, Proc. IRE 30,144 (1942)] as the DFT bears to the Fourier transform.

© 1983 Optical Society of America

Citation
R. N. Bracewell, "Discrete Hartley transform," J. Opt. Soc. Am. 73, 1832-1835 (1983)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-73-12-1832


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References

  1. R. V. L. Hartley, "A more symmetrical Fourier analysis applied to transmission problems," Proc. IRE 30, 144–150 (1942).
  2. S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill, New York, 1965).
  3. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  4. Zhong-de Wang, "Harmonic analysis with a real frequency function. I. Aperiodic case," Appl. Math. Comput. 9, 53–73 (1981); "II. Periodic and bounded cases," 9, 153–163 (1981); "III. Data sequence," 9, 245–255 (1981).

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