Glossary

7.1 Fourier Series: Periodic Functions and Convergence

3 min read•july 22, 2024

Periodic functions repeat their values at regular intervals. break these functions down into sums of sines and cosines, making them easier to analyze and manipulate. This powerful tool helps us understand complex waveforms in many areas of physics.

Fourier series converge to the original function under certain conditions. They're incredibly useful for calculating power and energy in periodic signals. This mathematical technique finds applications in everything from to quantum mechanics.

Periodic Functions and Fourier Series

Properties of periodic functions

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  • Repeat their values at regular intervals defined by the period TT
    • f(x)=f(x+T)f(x) = f(x + T) holds true for all values of xx
    • Examples: with periods of 2π2\pi
  • Can be represented as a sum of sine and cosine functions
    • Each sine and cosine term has a frequency that is an integer multiple of the fundamental frequency ω0=2πT\omega_0 = \frac{2\pi}{T}
    • The fundamental frequency is the reciprocal of the period TT

Representations in Fourier series

  • General form of a Fourier series: f(x)=a02+∑n=1∞(ancos⁡(nω0x)+bnsin⁡(nω0x))f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 x) + b_n \sin(n\omega_0 x))
    • a0a_0, ana_n, and bnb_n are that determine the amplitude of each sine and cosine term
    • nn represents the integer multiples of the fundamental frequency ω0\omega_0
  • Fourier coefficients are calculated using integrals over one period of the function
    • a0=2T∫−T/2T/2f(x)dxa_0 = \frac{2}{T} \int_{-T/2}^{T/2} f(x) dx represents the average value of the function over one period
    • an=2T∫−T/2T/2f(x)cos⁡(nω0x)dxa_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos(n\omega_0 x) dx determines the amplitude of the cosine terms
    • bn=2T∫−T/2T/2f(x)sin⁡(nω0x)dxb_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin(n\omega_0 x) dx determines the amplitude of the sine terms
  • Common periodic functions with Fourier series representations
    • : odd harmonics with amplitudes decreasing as 1n\frac{1}{n}
    • : all harmonics with amplitudes decreasing as 1n\frac{1}{n}
    • : odd harmonics with amplitudes decreasing as 1n2\frac{1}{n^2}

Convergence and Applications of Fourier Series

Convergence of Fourier series

  • occurs when the Fourier series converges to the function value at each point of continuity
    • At discontinuities, the series converges to the average of the left and right limits of the function
    • : overshoots near discontinuities that do not diminish with increasing terms
  • occurs when the Fourier series converges uniformly to the function on the entire interval
    • Requires the function to be continuous and have a finite number of maxima and minima
    • Ensures that the series can be integrated or differentiated term by term
  • for convergence
    1. The function must be periodic with period TT
    2. The function must be piecewise continuous on the interval [−T2,T2][-\frac{T}{2}, \frac{T}{2}]
    3. The function must have a finite number of maxima and minima on the interval [−T2,T2][-\frac{T}{2}, \frac{T}{2}]

Power calculation with Parseval's theorem

  • relates the energy of a function to the energy of its Fourier coefficients
    • ∫−T/2T/2∣f(x)∣2dx=T2(a022+∑n=1∞(an2+bn2))\int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{T}{2} \left(\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\right)
    • The left side represents the energy of the function over one period
    • The right side represents the energy contribution of each Fourier coefficient
  • Power of a periodic signal is the average energy per unit time
    • P=1T∫−T/2T/2∣f(x)∣2dx=a022+∑n=1∞(an2+bn2)P = \frac{1}{T} \int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)
    • Obtained by dividing Parseval's theorem by the period TT
  • Energy of a periodic signal is the total energy over one period
    • E=∫−T/2T/2∣f(x)∣2dx=T2(a022+∑n=1∞(an2+bn2))E = \int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{T}{2} \left(\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\right)
    • Directly obtained from Parseval's theorem

Key Terms to Review (17)

Basis functions: Basis functions are a set of functions that can be combined to represent other functions in a specific space, often used in mathematical analysis and signal processing. They serve as building blocks for constructing more complex functions, enabling the representation of periodic functions through series expansions. Understanding basis functions is crucial for analyzing convergence properties in Fourier series and applying them to practical problems in various fields, such as particle physics and condensed matter.
Dirichlet Conditions: Dirichlet conditions are a set of criteria that ensure the convergence of a Fourier series to a function at certain points. These conditions state that a function must be periodic, piecewise continuous, and have a finite number of discontinuities in any given interval. Understanding these conditions is crucial for analyzing the convergence behavior of Fourier series and determining how well they can represent various periodic functions.
Fourier Coefficients: Fourier coefficients are the numerical constants that arise when a periodic function is expressed as a Fourier series. These coefficients represent the amplitudes of the sine and cosine functions in the series, capturing how much of each frequency component is present in the original function. They play a crucial role in reconstructing the periodic function through its series representation, helping us understand its behavior in terms of simpler trigonometric functions.
Fourier Series: A Fourier series is a way to represent a periodic function as a sum of sine and cosine functions. This powerful tool allows complex periodic signals to be broken down into simpler components, making it easier to analyze their frequency content and behavior. Fourier series play a crucial role in various fields, such as signal processing and quantum mechanics, where understanding waveforms and oscillations is essential.
Gibbs Phenomenon: The Gibbs phenomenon refers to the peculiar behavior observed when approximating a discontinuous function with its Fourier series, where overshoots occur at the points of discontinuity. This overshoot approaches a constant value, specifically about 9% above the function's actual value at the jump, regardless of how many terms are included in the series. It highlights the limitations of Fourier series in accurately representing functions with sharp transitions and emphasizes the convergence characteristics of these series.
Harmonic analysis: Harmonic analysis is a branch of mathematics focused on the representation of functions or signals as the superposition of basic waves, typically sines and cosines. It deals with the decomposition of functions into their frequency components, providing powerful tools for understanding periodic phenomena and signal processing. This concept plays a crucial role in the study of Fourier series and transforms, which allow for the analysis of functions in both continuous and discrete domains.
Jean-Baptiste Joseph Fourier: Jean-Baptiste Joseph Fourier was a French mathematician and physicist known for introducing the concept of Fourier series and Fourier transforms, which are crucial tools for analyzing periodic functions and signal processing. His work laid the foundation for understanding how complex waveforms can be represented as sums of simpler sine and cosine functions, thereby connecting mathematical theory to practical applications in physics and engineering.
Orthogonality: Orthogonality is a concept that describes the perpendicularity of vectors or functions in a given space, meaning that their inner product is zero. This property is crucial for various mathematical and physical applications, allowing different functions or vectors to maintain independence from one another. It plays a significant role in simplifying complex problems, facilitating analysis in different coordinate systems, and optimizing solutions in series expansions and special functions.
Parseval's Theorem: Parseval's Theorem states that the sum of the squares of a function is equal to the sum of the squares of its Fourier coefficients. This concept establishes a critical connection between the time domain and frequency domain representations of signals, ensuring energy conservation in both domains. It is particularly useful in analyzing periodic functions and is foundational in understanding the relationships within discrete Fourier transforms.
Periodicity: Periodicity refers to the property of a function that repeats its values at regular intervals, known as periods. This concept is crucial in various areas of mathematics and physics, as it allows for the analysis and synthesis of functions that exhibit cyclic behavior. Recognizing periodicity can lead to simplified calculations and deeper insights when working with Fourier series, which break down complex periodic functions into simpler sine and cosine components, and in transformations that convert discrete signals for efficient computation.
Pointwise convergence: Pointwise convergence refers to a type of convergence for a sequence of functions where, at each point in the domain, the sequence converges to a limit function. In this context, understanding pointwise convergence is essential for analyzing how series and transforms behave as they approximate functions. This concept helps us determine if a sequence of approximations captures the desired properties of the original function over its domain.
Sawtooth Wave: A sawtooth wave is a type of non-sinusoidal waveform that rises upward linearly and then sharply drops, resembling the teeth of a saw. This waveform is characterized by its distinctive linear rise and abrupt fall, which gives it a unique harmonic content that can be decomposed into a series of sinusoidal components using Fourier series. Its periodic nature makes it important in the study of signal processing and various applications in electronics and acoustics.
Signal processing: Signal processing refers to the techniques used to analyze, manipulate, and synthesize signals to extract useful information or improve signal quality. This field is essential for converting real-world signals like sound, images, and sensor data into formats that can be efficiently processed and understood, making it a vital tool in various applications, including communications and data analysis.
Sine and cosine functions: Sine and cosine functions are fundamental trigonometric functions that describe the relationship between the angles and lengths of a right triangle. These functions are periodic, meaning they repeat their values in regular intervals, which makes them essential in representing waveforms and oscillatory motion. In the context of Fourier series, these functions serve as building blocks for expressing periodic functions as infinite sums of sine and cosine terms.
Square Wave: A square wave is a non-sinusoidal waveform that alternates between two fixed values, typically high and low, with a rapid transition between these levels. This type of waveform is characterized by its sharp transitions and a duty cycle, which defines the proportion of the period that the wave is at its high level compared to its low level. Square waves are significant in the study of periodic functions and their representation through Fourier series, as they can be decomposed into a sum of sine and cosine functions, revealing the harmonics present in the signal.
Triangle wave: A triangle wave is a non-sinusoidal waveform that resembles a series of triangular shapes, characterized by its linear rise and fall over time, making it a periodic function. Its unique shape makes it an important signal in various applications, including signal processing and music synthesis, and it can be represented as a Fourier series, highlighting its significance in the study of periodic functions and convergence.
Uniform Convergence: Uniform convergence occurs when a sequence of functions converges to a limit function in such a way that the rate of convergence is uniform across the domain. This concept is significant because it guarantees that certain properties of functions, such as continuity and integrability, are preserved in the limit. Understanding uniform convergence is essential when dealing with Fourier series and transforms, as it ensures that the series will converge to the function uniformly, allowing for the interchange of limits and integration.
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