﻿ Hartley transform

# Hartley transform

The Hartley transform, abbreviated as HT, is in the functional analysis, a branch of mathematics, a linear integral transformation with respect to the Fourier transform and how this frequency transformation. In contrast to the complex Fourier transform, the Hartley transform is a real transform. It is named after Ralph Hartley, who introduced them in 1942.

The Hartley transform exists in a discrete form, the discrete Hartley transform, abbreviated as DHT, which is used in digital signal processing and image processing. This form was published in 1994 by RNBracewell.

## Definition

Hartley transform of a function f (t) is defined as:

ω the angular frequency and the abbreviation:

Which is referred to as " core - Hartley ".

Exist in the literature regarding the factor different definitions, which normalize to 1 and this factor occurs in the inverse of the Hartley transform factor.

## Inverse transformation

The Hartley transform is inverse to itself according to the above definition, making it an involutive transformation:

## Relative to the Fourier transform

Fourier transformation

Differs by its complex core:

With the imaginary unit j of the purely real core cas (? t ) from the Hartley transform. With an appropriate choice of normalization factors, the Fourier transform can be calculated directly from the Hartley transform:

The real and imaginary parts of the Fourier transform is formed by the even and odd parts of the Hartley transform.

## The Hartley - core relationships

For the " Hartley - core " can be the following relationships from the trigonometric functions derived: