Matrix approximation methods have successfully produced efficient, low-complexity approximate transforms for the discrete cosine transforms and the discrete Fourier transforms. For the DFT case, literature archives approximations operating at small power-of-two blocklenghts, such as \{8, 16, 32\}, or at large blocklengths, such as 1024, which are obtained by means of the Cooley-Tukey-based approximation relying on the small-blocklength approximate transforms. Cooley-Tukey-based approximations inherit the intermediate multiplications by twiddled factors which are usually not approximated; otherwise the effected error propagation would prevent the overall good performance of the approximation. In this context, the prime factor algorithm can furnish the necessary framework for deriving fully multiplierless DFT approximations. We introduced an approximation method based on small prime-sized DFT approximations which entirely eliminates intermediate multiplication steps and prevents internal error propagation. To demonstrate the proposed method, we design a fully multiplierless 1023-point DFT approximation based on 3-, 11- and 31-point DFT approximations. The performance evaluation according to popular metrics showed that the proposed approximations not only presented a significantly lower arithmetic complexity but also resulted in smaller approximation error measurements when compared to competing methods.

翻译：矩阵近似方法已成功为离散余弦变换和离散傅里叶变换生成高效、低复杂度的近似变换。对于离散傅里叶变换的情况，文献中记载的近似方法主要针对较小的2的幂次方块长度（如{8, 16, 32}）或较大的块长度（如1024），后者是通过基于Cooley-Tukey算法的近似方法获得的，该方法依赖于小尺寸块长度的近似变换。基于Cooley-Tukey的近似方法继承了中间旋转因子的乘法运算，这些因子通常不被近似；否则，由此产生的误差传播将妨碍近似整体良好的性能。在此背景下，质因子算法可为推导完全无乘法器的离散傅里叶变换近似提供必要的框架。我们提出了一种基于小质数尺寸离散傅里叶变换近似的近似方法，该方法完全消除了中间乘法步骤，并防止了内部误差传播。为验证所提方法，我们基于3点、11点和31点离散傅里叶变换近似，设计了一个完全无乘法器的1023点离散傅里叶变换近似。根据常用指标的性能评估表明，与现有方法相比，所提出的近似不仅具有显著更低的算术复杂度，而且产生了更小的近似误差测量值。