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尽管离散付立叶变换(DFT)是在复数域定义的,然而数论变换(NTT)却是在有限域和环内运算的。这些NTT中的某些变换具有类似于快速付立叶变换(FFT)的快速变换结构,因而能用于快速数字信号处理。本文对采用NTT进行变换域信号处理的计算效果以及信噪此(SNR)性能作了研究。特别是,分析了有限字长(b≤16)和长变换长度对NTT滤波的影响。分析表明,对于短的字长或中到大的变换长度,NTT滤波比定点运算的FFT滤波能达到更好的SNR。最后,提出了一种具有单基或混合基快速变换结构的新的NTT。虽然这些NTT需要高效率地实现取模运算,然而对于在8≤b≤16范围内的任一给定的工作长度b,这些新的NTT的变换长度是最佳的。
Although Discrete Fourier Transform (DFT) is defined in the complex field, Number Theoretic Transformation (NTT) operates in finite fields and in loops. Some of these NTT transforms have fast transforms similar to Fast Fourier Transform (FFT) and are therefore used for fast digital signal processing. In this paper, the computational effect and signal-to-noise (SNR) performance of NTT transform domain signal processing are studied. In particular, the effects of finite word length (b ≤ 16) and long transform length on NTT filtering are analyzed. Analysis shows that NTT filtering achieves better SNR than fixed-point FFT filtering for short word sizes or medium to large transform lengths. Finally, a new NTT with a single-base or hybrid-based fast-transition structure is proposed. Although these NTTs require efficient modulo operation, the transformation length of these new NTTs is optimal for any given working length b in the range of 8≤b≤16.