The Split-Radix Fast Fourier Transform has the same low arithmetic complexity as the related Conjugate Pair Fast Fourier Transform. Both transforms have an irregular datapath structure which is straightforwardly expressed only in recursive forms. Furthermore, the conjugate pair variant has a complicated input indexing pattern which requires existing iterative implementations to rely on precomputed tables. It however allows optimization of the memory bandwidth as it requires a single twiddle factor load per radix-4 butterfly. In existing algorithms, this comes at the cost of using additional precomputed tables or performing recursive function calls. In this paper we present two novel approaches that handle both the butterfly scheduling and the input index generation of the Conjugate Pair Fast Fourier Transform. The proposed algorithm is cache-friendly because it is depth-first, non-recursive and does not rely on precomputed index tables. In order to achieve this, we relate the butterfly execution pattern of the Split-Radix and Conjugate Pair FFTs to the binary carry sequence . Based on this finding, we describe how common integer arithmetic and bitwise operations can be used to perform input reordering and depth-first traversal of the transform datapath with $\mathcal {O}(1)$ space complexity.

中文翻译：

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共轭对快速傅立叶变换的深度优先迭代算法

分裂基数快速傅立叶变换与相关的共轭对快速傅立叶变换具有相同的低算法复杂度。两种转换都具有不规则的数据路径结构，该结构仅以递归形式直接表达。此外，共轭对变量具有复杂的输入索引模式，这需要现有的迭代实现方式依赖于预先计算的表。但是，由于需要每根radix-4蝶形单个旋转因子负载，因此它可以优化内存带宽。在现有算法中，这是以使用其他预计算表或执行递归函数调用为代价的。在本文中，我们提出了两种新颖的方法，可同时处理蝶形调度和共轭对快速傅立叶变换的输入索引生成。所提出的算法是缓存友好的，因为它是深度优先的，非递归的并且不依赖于预先计算的索引表。为了实现这一目标，我们将分割基数和共轭对FFT的蝶形执行模式与二进制进位序列 。基于此发现，我们描述了如何使用常见的整数算术和按位运算来执行输入重排序和变换数据路径的深度优先遍历，包括：$ \数学{O}（1）$ 空间复杂度。