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Related Concept Videos

Trigonometric Fourier series01:17

Trigonometric Fourier series

Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
Convergence of Fourier Series01:21

Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
Discrete-Time Fourier Series01:20

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
Exponential Fourier series01:24

Exponential Fourier series

In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...

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Related Experiment Video

Updated: Jul 7, 2026

A Multimodal Wide-Field Fourier-Transform Raman Microscope
06:48

A Multimodal Wide-Field Fourier-Transform Raman Microscope

Published on: December 30, 2025

An efficient algorithm for automatically generating multivariable fuzzy systems by Fourier series method.

Liang Chen1, N Tokuda

  • 1Dept. of Math & Comput. Sci., Northern British Columbia Univ., Prince George, BC, Canada.

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|February 5, 2008
PubMed
Summary

A novel Fourier series method automatically generates multivariable fuzzy inference systems. This approach decomposes complex data, constructs independent fuzzy systems, and integrates them for precise function approximation, enhancing stability and application range.

Related Experiment Videos

Last Updated: Jul 7, 2026

A Multimodal Wide-Field Fourier-Transform Raman Microscope
06:48

A Multimodal Wide-Field Fourier-Transform Raman Microscope

Published on: December 30, 2025

Area of Science:

  • Computational Intelligence
  • Fuzzy Systems Engineering
  • Applied Mathematics

Background:

  • Multivariable fuzzy inference systems (MFIS) are crucial for complex system modeling.
  • Existing methods for constructing MFIS can be computationally intensive and lack precision.
  • The need for robust and accurate automated MFIS generation is significant.

Purpose of the Study:

  • To introduce a new constructive method for automatically generating MFIS using Fourier series expansion.
  • To achieve precise function approximation of any given sample set.
  • To develop a more stable and robust MFIS compared to existing methods.

Main Methods:

  • Decomposition of sample sets into simpler, single-input fuzzy system (SIFS) compatible clusters.
  • Independent construction of fuzzy rules and membership functions for each variable.
  • Integration of decomposed SIFS into a comprehensive MFIS using Fourier series properties.
  • Validation through two fundamental theorems ensuring decomposition and composition integrity.

Main Results:

  • A constructive algorithm for automated MFIS generation is demonstrated.
  • The method achieves specified precision for function approximation.
  • Implicit error bound analysis ensures stability, outperforming power series expansion methods (ParNeuFuz, PolyNeuFuz).
  • The developed MFIS exhibits enhanced robustness for a wider application scope.

Conclusions:

  • The Fourier series expansion offers an effective and stable method for constructing MFIS.
  • This automated approach simplifies MFIS generation while maintaining high precision.
  • The enhanced stability and broader applicability position this method as a significant advancement in fuzzy systems engineering.