Quantization errors in the computation of the discrete Hartley transform

The principal objective of this paper is the study of the arithmetic roundoff error characteristics of several discrete Hartley transform (DHT) algorithms. We first summarize a variety of efficient DHT algorithms including Bracewell's original decimation-in-time radix-2 algorithm. Statistical models for fixed- and floating-point arithmetic roundoff errors are then used as the basis for a theoretical study of roundoff noise characteristics of a number of the DHT algorithms. The results of a detailed experimental study of roundoff noise are compared to the theoretical predictions. In fixed-point implementation of the decimation-in-time and frequency radix-2 algorithms, it is found that the noise-to-signal ratio increases approximately 1.1 bits per stage. For the floating-point implementation, the number of bits of rms noise-to-signal ratio for all the algorithms increase as
\sqrt{\log_{2}N}
, so that doubling the number of points produced a mild increase in the output noise.