Glossary

10.1 Fourier series for periodic functions

2 min readโ€ขaugust 7, 2024

are powerful tools for breaking down periodic functions into simpler sine and cosine waves. They help us understand complex signals by representing them as sums of simple oscillations with different frequencies and amplitudes.

This mathematical technique is crucial in many fields, from to physics. By analyzing a function's Fourier series, we can gain insights into its frequency content and energy distribution, making it easier to manipulate and study complex periodic phenomena.

Fourier Series Basics

Periodic Functions and Fourier Series

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  • A f(x)f(x) repeats its values at regular intervals PP such that f(x)=f(x+P)f(x) = f(x + P) for all xx
  • The interval length PP is called the period of the function
  • Fourier series represent a periodic function as an infinite sum of sine and cosine functions with different frequencies and amplitudes
  • The general form of a Fourier series for a function f(x)f(x) with period 2L2L is: f(x)=a02+โˆ‘n=1โˆž(ancosโก(nฯ€xL)+bnsinโก(nฯ€xL))f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(\frac{n\pi x}{L}) + b_n \sin(\frac{n\pi x}{L}))

Sine, Cosine, and Harmonic Components

  • Sine and cosine functions form the basis for Fourier series due to their periodic nature and properties
  • The fundamental frequency ฯ‰0\omega_0 is the lowest frequency component in the Fourier series and is related to the period PP by ฯ‰0=2ฯ€P\omega_0 = \frac{2\pi}{P}
  • are integer multiples of the fundamental frequency (2ฯ‰02\omega_0, 3ฯ‰03\omega_0, etc.) that contribute to the overall shape of the periodic function
  • Higher harmonics correspond to faster oscillations and finer details in the function's shape

Fourier Series Properties

Coefficients and Convergence

  • The ana_n and bnb_n in the Fourier series determine the amplitude of each sine and cosine component

  • a0a_0 represents the average value (DC component) of the function over one period

  • The coefficients can be calculated using the following integrals: an=1Lโˆซโˆ’LLf(x)cosโก(nฯ€xL)dxa_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos(\frac{n\pi x}{L}) dx

    bn=1Lโˆซโˆ’LLf(x)sinโก(nฯ€xL)dxb_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin(\frac{n\pi x}{L}) dx

  • The of a Fourier series refers to how well the series approximates the original function as the number of terms increases

  • Fourier series converge pointwise for piecewise continuous functions and converge uniformly for continuous functions with a finite number of discontinuities

Gibbs Phenomenon and Parseval's Theorem

  • occurs when a Fourier series approximates a function with a jump discontinuity, resulting in overshoots and oscillations near the discontinuity

  • The overshoots do not disappear as more terms are added to the series, but their width decreases

  • Parseval's theorem states that the total energy of a function is equal to the sum of the energies of its Fourier series components: โˆซโˆ’LLโˆฃf(x)โˆฃ2dx=a022+โˆ‘n=1โˆž(an2+bn2)\int_{-L}^{L} |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)

  • This theorem provides a way to calculate the energy content of a signal in the frequency domain and is useful in signal processing applications (filtering, compression)

Key Terms to Review (16)

Coefficients: Coefficients are numerical factors that multiply variables in mathematical expressions, often seen in algebra and calculus. In the context of Fourier series, coefficients play a crucial role in representing periodic functions as sums of sine and cosine terms, allowing for analysis and reconstruction of these functions in different domains.
Convergence: Convergence refers to the property of a sequence, series, or function approaching a specific value as its input or index approaches a limit. This concept is crucial in various fields, where understanding how and when functions or numerical methods stabilize is essential for ensuring accurate results.
Cosine function: The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It plays a crucial role in the analysis of periodic functions, especially in representing waveforms and oscillatory motion. In Fourier series, the cosine function is used to express periodic signals as sums of sine and cosine terms, enabling the decomposition of complex waveforms into simpler components.
Fourier Cosine Series: A Fourier cosine series is a way to represent a periodic function using only cosine functions, which are even functions. This series is particularly useful for analyzing functions that exhibit symmetry and can be expressed as a sum of cosines, simplifying the process of Fourier analysis. By focusing on the cosine components, this series can help in solving problems related to heat conduction, vibrations, and signal processing.
Fourier series: A Fourier series is a way to represent a periodic function as a sum of sine and cosine functions. This mathematical tool allows complex periodic signals to be decomposed into simpler components, making it useful in analyzing various physical phenomena.
Fourier Transform: The Fourier Transform is a mathematical operation that transforms a function of time or space into a function of frequency. This technique allows for the analysis of the frequency components within a signal, making it essential in various fields such as physics and engineering, where understanding the behavior of functions in terms of their frequency content is crucial. It serves as a bridge between time-domain and frequency-domain representations, enabling problem-solving in areas such as signal processing, differential equations, and wave analysis.
Gibbs phenomenon: The Gibbs phenomenon refers to the peculiar behavior of Fourier series approximations of a function, particularly at points of discontinuity, where the overshoot in the approximation can be observed. This overshoot does not disappear as the number of terms in the series increases, leading to a fixed ratio of about 9% overshoot relative to the jump size of the discontinuity. It highlights the limitations of using Fourier series for representing functions with abrupt changes and is a significant consideration in both orthogonal functions and series expansions and Fourier series for periodic functions.
Harmonics: Harmonics are the integer multiples of a fundamental frequency in a wave or signal. In physical systems, harmonics play a critical role in the analysis of periodic functions, breaking down complex waveforms into simpler components, which helps in understanding their behavior and interactions. They are essential in many applications across physics, where they describe the natural frequencies of systems and their responses to external forces.
Heat transfer: Heat transfer is the process by which thermal energy moves from one physical system to another, typically occurring due to a temperature difference. This transfer can occur via conduction, convection, or radiation and is fundamental in various scientific applications, including thermodynamics and engineering. Understanding heat transfer is crucial for analyzing and solving problems related to thermal systems, such as those involving periodic functions and their behavior under changing temperatures.
Integration: Integration is a fundamental mathematical process that involves finding the accumulation of quantities, which can be thought of as the 'reverse' operation of differentiation. It allows us to calculate areas under curves, volumes, and other accumulated values by summing infinitesimally small parts. In the context of Fourier series, integration plays a key role in determining the coefficients that represent periodic functions as sums of sine and cosine terms.
Orthogonality: Orthogonality refers to the concept where two vectors are perpendicular to each other, meaning their dot product equals zero. This fundamental idea extends beyond simple vector operations and plays a crucial role in various mathematical and physical contexts, including the behavior of functions, the nature of coordinate systems, and the analysis of differential equations.
Periodic Function: A periodic function is a function that repeats its values in regular intervals or periods. This characteristic means that there exists a positive number $T$ such that for all values of $x$, the equality $f(x + T) = f(x)$ holds. Periodic functions are fundamental in analyzing waveforms and oscillations, making them vital for understanding Fourier series, which express periodic functions as sums of sine and cosine functions.
Signal processing: Signal processing is a technique used to analyze, modify, and synthesize signals to improve their quality or extract useful information. This field is essential for applications like audio and image processing, where it helps in filtering noise, enhancing clarity, and enabling data compression. Signal processing heavily relies on mathematical tools such as Fourier analysis, which allows signals to be represented in the frequency domain for easier manipulation and interpretation.
Sine function: The sine function is a mathematical function that relates an angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. This function is periodic, oscillating between -1 and 1, and is fundamental in various applications, particularly in representing periodic phenomena like sound waves and alternating currents. In the context of Fourier series, sine functions play a crucial role in decomposing periodic functions into their constituent frequencies.
Summation: Summation is the process of adding a sequence of numbers or terms together to produce a total. In mathematical contexts, it is often represented using the sigma notation, which allows for a compact expression of sums over a specified range. This concept is crucial for analyzing periodic functions through Fourier series, where infinite sums of sine and cosine terms are used to represent complex waveforms.
Trigonometric expansion: Trigonometric expansion involves expressing functions as sums of sine and cosine functions, particularly in the context of Fourier series. This technique is essential for analyzing periodic functions, as it allows them to be represented in a form that highlights their frequency components. By decomposing a function into its trigonometric parts, one can better understand its behavior and properties over defined intervals.
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