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Chebyshev and The Fast Fourier Transform Methods for Signal Interpolation

Ishmael N. Amartey

TL;DR

This work contrasts Chebyshev interpolation, implemented via the Chebfun framework, with Fourier-based interpolation for signal reconstruction under varying node distributions and noise. It analyzes node distributions, convergence properties, and basis conditioning, showing that Chebyshev interpolation remains accurate for both evenly and unevenly spaced nodes and exhibits favorable numerical stability compared to the monomial basis. The gamma variate curve serves as a core test function, with Chebyshev interpolation reliably recovering maxima and shape even under noise, whereas Fourier interpolation struggles with uneven sampling and, in some cases, noise. The findings suggest Chebyshev methods offer practical advantages for real-world signals with irregular sampling, while FFT-based approaches retain value under ideal, evenly spaced conditions; future work should quantify these trade-offs more broadly and explore FFT-based comparisons.

Abstract

Approximation theorem is one of the most important aspects of numerical analysis that has evolved over the years with many different approaches. Some of the most popular approximation methods include the Lebesgue approximation theorem, the Weierstrass approximation, and the Fourier approximation theorem. The limitations associated with various approximation methods are too crucial to ignore, and thus, the nature of a specific dataset may require using a specific approximation method for such estimates. In this report, we shall delve into Chebyshev's polynomials interpolation in detail as an alternative approach to reconstructing signals and compare the reconstruction to that of the Fourier polynomials. We will also explore the advantages and limitations of the Chebyshev polynomials and discuss in detail their mathematical formulation and equivalence to the cosine function over a given interval [a, b].

Chebyshev and The Fast Fourier Transform Methods for Signal Interpolation

TL;DR

This work contrasts Chebyshev interpolation, implemented via the Chebfun framework, with Fourier-based interpolation for signal reconstruction under varying node distributions and noise. It analyzes node distributions, convergence properties, and basis conditioning, showing that Chebyshev interpolation remains accurate for both evenly and unevenly spaced nodes and exhibits favorable numerical stability compared to the monomial basis. The gamma variate curve serves as a core test function, with Chebyshev interpolation reliably recovering maxima and shape even under noise, whereas Fourier interpolation struggles with uneven sampling and, in some cases, noise. The findings suggest Chebyshev methods offer practical advantages for real-world signals with irregular sampling, while FFT-based approaches retain value under ideal, evenly spaced conditions; future work should quantify these trade-offs more broadly and explore FFT-based comparisons.

Abstract

Approximation theorem is one of the most important aspects of numerical analysis that has evolved over the years with many different approaches. Some of the most popular approximation methods include the Lebesgue approximation theorem, the Weierstrass approximation, and the Fourier approximation theorem. The limitations associated with various approximation methods are too crucial to ignore, and thus, the nature of a specific dataset may require using a specific approximation method for such estimates. In this report, we shall delve into Chebyshev's polynomials interpolation in detail as an alternative approach to reconstructing signals and compare the reconstruction to that of the Fourier polynomials. We will also explore the advantages and limitations of the Chebyshev polynomials and discuss in detail their mathematical formulation and equivalence to the cosine function over a given interval [a, b].
Paper Structure (32 sections, 26 equations, 37 figures)

This paper contains 32 sections, 26 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Chebyshev points and interpolants
  3. Chebyshev Interpolation through Random Data
  4. Rounding errors in computing Chebyshev points
  5. Chebyshev points of the first kind
  6. Convergence of Chebyshev interpolants
  7. Geometric mean distance between points
  8. Chebyshev Points
  9. Legendre Points
  10. Equally Spaced Points
  11. Scaled Chebyshev points to the interval [a, b]
  12. Chebyshev polynomials and series
  13. A Chebyshev coefficient
  14. Dependence on wave number
  15. Chebyshev series of a complicated function
  16. ...and 17 more sections

Figures (37)

  • Figure 1: A Chebfun interpolation through random data for 10 points. Elapsed time = 0.28 seconds, min=-0.8299, max=1.1792
  • Figure 2: A Chebfun interpolation through random data for 100 points. Elapsed time = 0.07 seconds, min=-1.4093, max = 1.3018.
  • Figure 3: A Chebfun interpolation through random data for 1000 points. Elapsed time = 0.084655 seconds, min = -1.5144, max = 1.8876
  • Figure 4: A Chebfun interpolation through random data for 10000 points. Elapsed time = 0.116381 seconds, min =-1.8462, max= 1.7696
  • Figure 5: A comparison of the Chebyshev points of the first kind and Legendre points. The maximum between the two points is 0.0084.
  • ...and 32 more figures