Adaptive function approximation based on the Discrete Cosine Transform (DCT)

TL;DR

The paper tackles univariate, memoryless function approximation by adopting a cosine-based basis and learning the coefficients through supervised training rather than directly computing the DCT. By exploiting the finite dynamics and orthogonality of the cosine basis, the authors derive a simple NLMS gradient-based update with a diagonal autocorrelation matrix, enabling predictable convergence and low misadjustment. Theoretical results link the misadjustment to a tunable parameter $\alpha$, and empirical evaluations on linear and $\sqrt{x}$-type functions validate the approach, showing convergence times in line with theory and favorable learning-quality-to-complexity trade-offs. The work suggests that DCT-inspired adaptive function approximation is a practical, scalable component for more complex learning systems and potential wireless computing applications.

Abstract

This paper studies the cosine as basis function for the approximation of univariate and continuous functions without memory. This work studies a supervised learning to obtain the approximation coefficients, instead of using the Discrete Cosine Transform (DCT). Due to the finite dynamics and orthogonality of the cosine basis functions, simple gradient algorithms, such as the Normalized Least Mean Squares (NLMS), can benefit from it and present a controlled and predictable convergence time and error misadjustment. Due to its simplicity, the proposed technique ranks as the best in terms of learning quality versus complexity, and it is presented as an attractive technique to be used in more complex supervised learning systems. Simulations illustrate the performance of the approach. This paper celebrates the 50th anniversary of the publication of the DCT by Nasir Ahmed in 1973.

Figures (15)

• Figure 1: Mirror image that is assumed by the DCT in order to represent $y(x)$ in exchange for doubling the length of its periodicity.
• Figure 2: DCT approximation with 6 coefficients (Q=12) of a logarithmic compander.
• Figure 3: Linear function to be approximated.
• Figure 4: Optimal coefficients of the cosine transform. The DCT has been computed with $N$=512 points
• Figure 5: Linear function versus its approximation with the cosine transform. The approximation error is 0.01$\%$ because only 5 coefficients of \ref{['eq4']} have been considered.