HRATLEY TRANSFORM

The Hartley transform of s sequence { xi }, i = 0 ... N-1, is

Xk = Sumi xi cas( 2 Pi i k / N )

where cas(x)=cos(x)+sin(x). The inverse is

xi = N-1 Sumk Xk cas( 2 Pi i k / N )

The inner product of the basis functions is

Sumk [ cos(2 Pi k n / N) + sin(2 Pi k n / N) ] [ cos(2 Pi k m / N) + sin(2 Pi k m / N) ]
= Sumk [ cos(2 Pi k [n-m] / N) + sin(2 Pi k [n+m] / N) ]
= sin(Pi a)/sin(Pi a/N) cos(Pi a (1-N)/N ) + sin(Pi b)/sin(Pi b/N) sin(Pi b (1-N)/N )

where a=n-m and b=n+m. For n and m integer this is zero unless n=m, and in this case is has value N.

Given a discrete sequence { xi }, the continuous Hartley transform

X(w) = Sum xi cas( i w )

can be written

X(w) = Sum Xk [ H1(w - 2 Pi k/N) + H2(w + 2 Pi k/N) ]

where

H1(w) = N-1 sin(w N/2)/sin(w/2) cos([N-1]w/2)
H2(w) = N-1 sin(w N/2)/sin(w/2) sin([N-1]w/2)



Marco Corvi - ...