Convolution with the Hartley transform

The Hartley transform can be computed by summing the real and imaginary parts of the Fourier transform.

$$\begin{array}{}\text{(1)}& \begin{array}{rl}\mathcal{F}\mathbf{a}& =\mathbf{x}+i\mathbf{y}\\ \mathcal{H}\mathbf{a}& =\mathbf{x}+\mathbf{y},\end{array}\end{array}$$

where $\mathbf{a}$, $\mathbf{x}$, and $\mathbf{y}$ are real-valued vectors, $\mathcal{F}$ is the Fourier transform, and $\mathcal{H}$ is the Hartley transform. It has several useful properties.

• It is unitary, and also an involution: it is its own inverse.
• Its output is real-valued, so it can be used with numerical routines that cannot handle complex numbers.
• It can be computed in $\mathcal{O}(n\log (n))$ time using standard Fast Fourier Transform (FFT) libraries.