Fast Computation of the Discrete Fourier Transform Square Index Coefficients

TL;DR

It is shown that the number of points of some DFT coefficients can be dramatically reduced by means of elementary mathematical properties, and any regular DFT algorithm can be straightforwardly applied to compute the SICs with a reduced number of complex multiplications.

Abstract

The $N$-point discrete Fourier transform (DFT) is a cornerstone for several signal processing applications. Many of these applications operate in real-time, making the computational complexity of the DFT a critical performance indicator to be optimized. Unfortunately, whether the $\mathcal{O}(N\log_2 N)$ time complexity of the fast Fourier transform (FFT) can be outperformed remains an unresolved question in the theory of computation. However, in many applications of the DFT -- such as compressive sensing, image processing, and wideband spectral analysis -- only a small fraction of the output signal needs to be computed because the signal is sparse. This motivates the development of algorithms that compute specific DFT coefficients more efficiently than the FFT algorithm. In this article, we show that the number of points of some DFT coefficients can be dramatically reduced by means of elementary mathematical properties. We present an algorithm that compacts the square index coefficients (SICs) of DFT (i.e., $X_{k\sqrt{N}}$, $k=0,1,\cdots, \sqrt{N}-1$, for a square number $N$) from $N$ to $\sqrt{N}$ points at the expense of $N-1$ complex sums and no multiplication. Based on this, any regular DFT algorithm can be straightforwardly applied to compute the SICs with a reduced number of complex multiplications. If $N$ is a power of two, one can combine our algorithm with the FFT to calculate all SICs in $\mathcal{O}(\sqrt{N}\log_2\sqrt{N})$ time complexity.

Figures (2)

• Figure 1: Butterfly diagram for the computation of the $9$-point DFT coefficients $X_{0\sqrt{9}}$, $X_{1\sqrt{9}}$, and $X_{2\sqrt{9}}$ of the input signal ${\mathbf{x}}=\{x_0,\cdots,x_8\}$. Firstly, ${\mathbf{x}}$ is compressed into the signal $\mathbf{\hat{x}}=\{\hat{x}_0, \hat{x}_1, \hat{x}_2\}$ according to (\ref{['eqn:xhat']}). Then, a $\sqrt{9}$-point DFT on $\mathbf{\hat{x}}$ results in the coefficients $\hat{\mathbf{X}}=\{\hat{X}_0,\hat{X}_1,\hat{X}_2\}$ such that $X_{k\sqrt{9}}=\hat{X}_k$ ($k=0,1,2$).
• Figure 2: FFT vs. proposed SIC DFT algorithm: 1st to 6th harmonics of the A440 piano key with fundamental frequency $440$ Hz.