The Hartley transform produces real output for a real input, and is its own inverse. It therefore can
have computational advantages over the discrete
Fourier transform, although analytic expressions are usually more complicated
for the Hartley transform.

In the discrete case, the kernel is multiplied by

(4)

instead of

(5)

The discrete version of the Hartley transform--using an alternate convention with
the plus sign replaced by a minus sine can be written explicitly as