Signal processing with number theoretic transforms and limited word lengths

Number Theoretic Transforms (NTT's), unlike the Discrete Fourier Transform (DFT), are defined in finite rings and fields rather than in the field of complex numbers. Some NTT's have a transform structure like the Fast Fourier Transform (FFT) and can be used for fast digital signal processing. The computational effort and the signal-to-noise ratio (SNR) performance of linear filtering in finite rings and fields are investigated. In particular, the effect of limited word lengths, i.e.,
b \leq 16
, and long transform lengths on the SNR is analyzed. It is shown that for small word lengths and/or moderate to large transform lengths NTT filtering achieves a better SNR than FFT filtering with fixed-point arithmetic. Some new NTT's with a single- or mixed-radix fast transform structure are presented. While these NTT's may require special modulo arithmetic they achieve optimum transform length for any given word length b in the range
8 \leq b \leq 16
.