Fast Compressed-Domain N-Point Discrete Fourier Transform: The "Twiddless" FFT Algorithm

Abstract: In this work, we present the \emph{twiddless fast Fourier transform (TFFT)}, a novel algorithm for computing the $N$-point discrete Fourier transform (DFT). The TFFT's divide strategy builds on recent results that decimate an $N$-point signal (by a factor of $p$) into an $N/p$-point compressed signal whose DFT readily yields $N/p$ coefficients of the original signal. However, existing compression-domain DFT analyses have been limited to computing only the even-indexed DFT coefficients. With TFFT, we overcome this limitation by efficiently computing both \emph{even- and odd-indexed} DFT coefficients in the compressed domain with $O(N \log N)$ complexity. TFFT introduces a new recursive decomposition of the DFT problem, wherein $N/2^i$ coefficients of the original input are computed at recursion level $i$, with no need for twiddle factor multiplications or butterfly structures. Additionally, TFFT generalizes the input length to $N = c \cdot 2^k$ (for $k \geq 0$ and non-power-of-two $c > 0$), reducing the need for zero-padding and potentially improving efficiency and stability over classical FFTs. We believe TFFT represents a \emph{novel paradigm} for DFT computation, opening new directions for research in optimized implementations, hardware design, parallel computation, and sparse transforms.