Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial
Abstract
:1. Introduction
- We prove that the Roache approach (i.e., using simple polynomials derived from the residual error framework) is spectrally equivalent to the Lanczos approach (i.e., using quasi-Bernoulli polynomials derived from an integration-by-parts framework) for any harmonic different from zero (i.e., ), where simple polynomial coefficients are determined by a low-cost backward algorithm. Although both approaches have the same complexity when using linear operators based on integrals or derivatives, simple polynomials are easier to manipulate using these operators, and they reduce rounding errors by lowering addition and multiplication operations.
- We propose a reprojection method that allows the transformation from Fourier series coefficients to mixed Fourier series coefficients. We introduce the term “mixed Fourier series” to designate the summation of series derived from a smooth periodic residual error, wherein one of the series is the Fourier series. This method allows for recovering convergence using standard Fourier coefficients obtained from any smooth function. The proposed method has the advantage of avoiding the temporal information of g:. Therefore, it has the potential to be particularly useful for native spectral applications (e.g., solving differential equations with spectral methods).
- By employing the Maliev–Lanczos approach and leveraging the residual error framework, we introduce and evaluate a novel sub-harmonic mixed Fourier series. This new series demonstrates enhanced performance and versatility in approximating wide-band or pass-band functions compared to the quasi-Bernoulli series. It is worth noting that the Maliev–Lanczos approach presents a set of continuity-based constraints that can be applied to any series complementing the Fourier series. Moreover, the conditions for achieving accelerated convergence can be readily obtained using the residual error framework.
- We discuss several examples of common smooth functions whose approximations using polynomials and trigonometric series exhibit several well-known adverse phenomena, such as the Gibbs phenomenon, Runge’s phenomenon, spectral leakage, and non-convergence by a non-analytic point or a limited region of convergence (using the Taylor series), which are successfully represented by the mixed Fourier series. The results demonstrate the potential of the Maliev–Lanczos approach in the approximation of the usual smooth functions in applied problems, even outperforming, in several scenarios, the Taylor series, orthogonal polynomials, and Chebyshev polynomials using nonuniform sampling.
- We illustrate the application of the mixed Fourier series with linear operators. In particular, we solve a common direct problem in applied mathematics (Numerical Riemann Integration) and a common inverse problem in fluid dynamics (Poisson’s equation). In both examples, we show the benefit of employing simple polynomials, and we illustrate fast convergence without the Gibbs phenomenon.
- We evaluate the use of a mixed evaluation (i.e., a combination of closed-form derivatives and the DFT approach) to find the mixed Fourier series of functions without closed-form Fourier coefficients. In that case, we show that the DFT reflects the property for smooth functions, which allows accelerated discrete Fourier processing. Therefore, this approach has a huge potential for a wide range of practical situations.
- Finally, we show in detail the methodology used to define a new mixed Fourier series using the residual error framework. Additionally, the versatility of this new series is demonstrated through several examples.
2. Continuous-Time Theory
2.1. Fourier Series Fundamentals
2.2. Mixed Fourier Series
2.3. Polynomial Coefficients in Closed Form
2.3.1. Case
2.3.2. Case
2.3.3. Case
2.3.4. General Case: Arbitrary Such That
2.4. Fourier Coefficients in Closed Form
2.5. Enhanced Continuous-Time Processing
2.6. Relation with the Maliev–Lanczos Approach
2.7. A Simple Reprojection Method: Using Standard Closed-Form Fourier Coefficients to Define a Mixed Fourier Series
3. Continuous-Time Examples and Applications
3.1. A Different Perspective for Convergent Series of Functions
3.2. Canonical Examples of Approximation Using Closed-Form Smooth Functions
3.2.1. Generic Sawtooth Function
3.2.2. Power Function
3.2.3. Exponential Function
3.2.4. Base-Band Cosine Function
3.3. Comparison with Selected State-of-the-Art Techniques
3.4. A Canonical Direct Problem: Numerical Riemann Integration of Closed-Form Smooth Functions
3.5. A Canonical Inverse Problem: Solution of a Boundary Value Problem (BVP) Using Standard Closed-Form Fourier Coefficients
3.6. A Canonical Inverse Problem: Solution of a Boundary Value Problem (BVP) Using the DFT
3.7. Toward an Ideal Sampling Theorem for Truncated Continuous-Time Functions
- 1.
- If , then such that , where can be as small as desired.
- 2.
- If , then such that , where can be as small as desired. The bandwidth of with this approach is .
- 3.
- Conclusively, if both previous limits converge to zero, then , where can be as small as desired.
3.8. Canonical Example of a Non-Polynomial Mixed Fourier Series: The Sub-Harmonic Case
4. Open Challenges and Future Work
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Páez-Rueda, C.-I.; Fajardo, A.; Pérez, M.; Yamhure, G.; Perilla, G. Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial. Math. Comput. Appl. 2023, 28, 93. https://doi.org/10.3390/mca28050093
Páez-Rueda C-I, Fajardo A, Pérez M, Yamhure G, Perilla G. Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial. Mathematical and Computational Applications. 2023; 28(5):93. https://doi.org/10.3390/mca28050093
Chicago/Turabian StylePáez-Rueda, Carlos-Iván, Arturo Fajardo, Manuel Pérez, German Yamhure, and Gabriel Perilla. 2023. "Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial" Mathematical and Computational Applications 28, no. 5: 93. https://doi.org/10.3390/mca28050093