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:

The addition theorem:

And

The derivation is given as: