**E**

**Fast Hartley Transform**

Whereas complex additions and multiplications are required for an FFT, the Hartley transform [1–8] requires only real multiplications and additions. The FFT maps a real function of time into a complex function of frequency, whereas the fast Hartley transform (FHT) maps the same real-time function into a real function of frequency. The FHT can be particularly useful in cases where the phase is not a concern.

The discrete Hartley transform (DHT) of a time sequence *x*(*n*) is defined as

where

In a similar development to the FFT, (E.1) can be decomposed as

Let *n* = *n* + *N*/2 in the second summation of (E.3)

Using (E.2) and the identities

for odd *k*

and, for even *k*

Using (E.6) and (E.7), (E.4) becomes

and

Let *k* = 2