An Integral Transform which shares some features with the Fourier Transform, but which (in the discrete
case), multiplies the Kernel by

(1) |

(2) |

The discrete version of the Hartley transform can be written explicitly as

(3) | |||

(4) |

where denotes the Fourier Transform. The Hartley transform obeys the Convolution property

(5) |

(6) | |||

(7) | |||

(8) |

(Arndt). Like the Fast Fourier Transform, there is a ``fast'' version of the Hartley transform. A decimation in time algorithm makes use of

(9) | |||

(10) |

where denotes the sequence with elements

(11) |

(12) | |||

(13) |

The Discrete Fourier Transform

(14) |

(15) | |||

(16) |

so

(17) |

**References**

Arndt, J. ``The Hartley Transform (HT).'' Ch. 2 in ``Remarks on FFT Algorithms.'' http://www.jjj.de/fxt/.

Bracewell, R. N. *The Fourier Transform and Its Applications.* New York: McGraw-Hill, 1965.

Bracewell, R. N. *The Hartley Transform.* New York: Oxford University Press, 1986.

© 1996-9

1999-05-25