Fast Hartley Transform - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 486

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

N -

Â

1

2 p nk

N

Ê
Ë

ˆ
¯

() = () Hk xn cas , k = 01 , ,..., N - 1 (F.1)

n = 0

where

cas u = cos u + sin u

(F.2)

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

( ) - N 21 N - 1

2 p nk

N

2 p nk

N

Ê
Ë

ˆ
¯ +

Ê
Ë

ˆ
¯

 Â

() = () ()

Hk xn cas xn cas (F.3)

n = 0 nN = 2

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