While the discrete Fourier transform (DFT) is defined in the field of complex numbers, number theoretic transforms (NTT's) operate in finite rings and fields. Some of these NTT's have a fast-transform structure similar to that of the fast Fourier transform (FFT) and can be used for fast digital signal processing. Both the computational effort and the signal-to-noise ratio (SNR) performance of transform-domain signal processing with NTT's are investigated in this paper. In particular, the effect of limited word lengths, i.e., b \leq 16 , and long transform lengths on the SNR of NTT filtering is analyzed. For small word lengths and/or moderate to large transform lengths, NTT filtering is shown to achieve a better SNR than FFT filtering with fixed-point arithmetic. Finally, new NTT's with a single- or mixed-radix fast-transform structure are presented. While these NTT's require efficient implementations of modulo arithmetic operations, their transform length is optimum for any given work length b in the range 8 \leq b \leq 16 .