Discrete Fourier Transform Approximations Based on the Cooley-Tukey Radix-2 Algorithm

TL;DR

This work introduces a family of DFT approximations that replace exact Cooley–Tukey twiddle factors with low-complexity, α-scaled rounded values to form tilde{F}_N recursively. It establishes rigorous bounds and convergence results showing tilde{F}_N → F_N in Frobenius norm as α grows, and proves invertibility for all nonzero α via bounds on the approximate twiddle factors. The study also analyzes arithmetic complexity, provides error analyses (recursive Frobenius bounds and Fourier-series-based estimates), and demonstrates practical viability through architecture illustrations and multi-beam-forming applications, along with an estimation-theory framework based on approximate periodograms. Collectively, the results offer a principled, hardware-friendly approach to near-orthogonal, invertible, low-complexity DFT approximations suitable for waveform analysis and array-processing tasks.

Abstract

This report elaborates on approximations for the discrete Fourier transform by means of replacing the exact Cooley-Tukey algorithm twiddle-factors by low-complexity integers, such as $0, \pm \frac{1}{2}, \pm 1$.

Figures (17)

• Figure 1: Curves for scaled rounded sine and cosine waves for scale parameters $\alpha = 2$ and $4$. Dashed lines represent original sine and cosine waves.
• Figure 2: Curves for estimated and computed relative $N$-point DFT approximation error in terms of Frobenius norm for precision parameters $\alpha = 2, 4, 8$ and $16$. Circle mark represents estimated relative $N$-point DFT approximation error and squared mark represents computed relative $N$-point DFT approximation error.
• Figure 3: Curves for estimated and computed $N$-point DFT approximation relative error in terms of Frobenius norm for precision parameters $\alpha = 2, 4, 8$ and $16$. Circle mark represents estimated $N$-point DFT approximation error and squared mark represents computed $N$-point DFT approximation relative error.
• Figure 4: Curves for Frobenius norm of exact and approximated $N$-point DFT and its Frobenius norm approximation. Solid lines represents the curve for exact $N$-point DFT Frobenius norm, dashed lines represents the Frobenius norm for the $N$-point DFT approximation and dashed-dotted lines represents the approximating curve for Frobenius norm of $N$-point DFT approximation.
• Figure 5: Solid lines represents the curve for exact $N$-point DFT Frobenius norm, dashed lines represents the estimated Frobenius norm for the $N$-point DFT approximation error matrix and dashed-dotted lines represents the curve for Frobenius norm of computed $N$-point DFT approximation error matrix.

Theorems & Definitions (32)

• Definition 1: Scaled Rounding Operator
• Theorem 1
• proof
• Theorem 2: Asymptotic Convergence of Approximate Twiddle Factor
• proof
• Theorem 3
• proof
• Definition 2
• Example 1
• Theorem 4