Fast algorithms for complex-valued discrete Fourier transform with separate real and imaginary inputs/outputs
Aleksandr Cariow
TL;DR
The study tackles computing the discrete Fourier transform when inputs and outputs are represented as separate real and imaginary parts rather than as single complex samples. It develops a fast, Cooley–Tukey–style FFT built from a vector–matrix formulation that uses a Kronecker-product block structure and a bit-reversal permutation, culminating in a forward transform described by ${f y}_{2N} = {f W}^{(m)}_{2^{m+1}} {f D}^{(m)}_{2^{m+1}} dots {f W}^{(1)}_{2^{m+1}} {f D}^{(1)}_{2^{m+1}} {f S}_{2^{m+1}} {f x}_{2N}$. The key contribution is an explicit construction that interleaves real and imaginary parts into two channels, enabling independent real processing while preserving the transform’s overall complexity, effectively unifying the I/O representation without altering the fundamental FFT complexity. Although not introducing a new transform principle, the approach provides a practical data-path reorganization that can simplify post-processing across independent channels. This has practical implications for signal processing pipelines where separate real/imag spectral channels are desirable or required.
Abstract
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of complex-valued data when each element of the sequence represents a single whole. However, in some cases, it is more advantageous to represent each element of the input and output sequences by a pair of real numbers. Such a need arises, for example, when further post-processing of spectral coefficients is carried out through two independent channels. Taking into account the noted need, the article proposes an algorithm for fast complex-valued discrete Fourier transform with separate real and imaginary inputs/outputs. A vector-matrix computational procedure is given that allows one to adequately describe and formalize the sequence of calculations when implementing the proposed algorithm.