Name: fourier series truncation and gibbs ripples
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 21 s
Channel: JoVE

The Fourier series of a signal is an infinite sum of complex exponentials. The infinite sum is often truncated to a finite partial sum to make it practical.
Increasing terms in a partial sum should make the approximation converge to the signal. Yet near discontinuities, persistent ripples occur, getting compressed towards the discontinuity - a phenomenon known as the Gibbs phenomenon.
Gibbs observed these high-frequency ripples and overshoots near discontinuities in truncated Fourier series approximations.
To mitigate this, choose a large number of terms so that the ripple&#39;s total energy is negligible.
Despite ripples, the energy in the approximation error decreases with more terms, allowing the Fourier series to represent discontinuous signals.
Truncating the Fourier series to a desired number of terms provides the best approximation, minimizing the error. The error reduces with more terms and eventually becomes zero if the signal can be represented by a Fourier series.
For instance, in image processing, reducing the error is crucial when approximating an image signal using a truncated Fourier series to avoid visual artifacts.

convergence of fourier series

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 in the signal when approximated by a truncated Fourier series. These high-frequency ripples do not vanish with an increasing number of terms; instead, they become compressed towards discontinuity. Gibbs first observed this effect, noting the characteristic ripples and overshoots that persist regardless of how many terms are included in the partial sum.
One strategy to mitigate the impact of the Gibbs phenomenon is to increase the number of terms in the partial sum. While the ripples&#39; amplitude near the discontinuity remains, their total energy becomes negligible with a sufficiently large number of terms. Consequently, the overall energy in the approximation error decreases, allowing the Fourier series to effectively represent discontinuous signals. Truncating the Fourier series to a specific number of terms provides the best possible approximation under the given constraints, minimizing the error. As the number of terms increases, the error reduces, approaching zero if the signal is ideally represented by a Fourier series. This characteristic is particularly important in applications such as image processing, where minimizing error is crucial to avoid visual artifacts. In image signal approximation, reducing the Fourier series truncation error ensures higher fidelity and better visual quality.
As a result, while the Gibbs phenomenon presents a challenge in signal approximation using the Fourier series, increasing the number of terms and understanding the energy distribution in the approximation error can significantly mitigate its effects, allowing for accurate representations of even discontinuous signals.

The Fourier series of a signal is an infinite sum of complex exponentials. The infinite sum is often truncated to a finite partial sum to make it ...

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Fourier Series

KEY TERMS AND CONCEPTS

fourier series truncation and gibbs ripples

Fourier series can represent a periodic signal as an infinite sum of complex exponentials. For practical use, that infinite series is cut off after a finite number of terms. The result is a partial sum that gives a workable approximation of the original signal.

Truncating the series creates a challenge near discontinuities. The approximation develops the Gibbs phenomenon, which is the set of persistent oscillations and overshoots around a jump in the signal. These high-frequency ripples do not disappear when more terms are added. Instead, they become squeezed closer to the discontinuity.

Gibbs first noticed this effect, and the same ripples and overshoots can still appear no matter how many terms are used in the partial sum. One way to reduce the problem is to increase the number of terms. The ripple size near the jump stays, but the total energy in that error becomes very small when enough terms are included.

As the approximation error loses energy, the overall error also drops. That means a truncated Fourier series can still represent a discontinuous signal well under the limits of a finite sum. The chosen number of terms gives the best approximation available for that setup, so the error is minimized.

This idea matters in applications such as image processing. If Fourier series truncation error is too large, it can leave visible artifacts in an image. Understanding how the error energy behaves helps improve image fidelity and visual quality, even when the original signal has discontinuities.

Fourier Series Truncation and Gibbs Ripples

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Fourier series can represent a periodic signal as an infinite sum of complex exponentials. For practical use, that infinite series is cut off after a finite number of terms. The result is a partial sum that gives a workable approximation of the original signal. Truncating the series creates a challenge near discontinuities. The approximation develops the Gibbs phenomenon, which is the set of persistent oscillations and overshoots around a jump in the signal. These high-frequency ripples do not...

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key terms and concepts

Learn how truncating a Fourier series creates Gibbs ripples near discontinuities, why the error persists, and how more terms reduce total error energy.

Fourier Series Truncation and Gibbs Ripples | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

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Fourier Series Truncation and Gibbs Ripples

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