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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

(E.1)

where

(E.2)

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

(E.3)

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

(E.4)

Using (E.2) and the identities

(E.5)

for odd k

(E.6)

and, for even k

(E.7)

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

(E.8)

and

(E.9)

Let k = 2

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