7.1 Fourier Series: Periodic Functions and Convergence

Periodic functions repeat their values at regular intervals. Fourier series 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 signal processing 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 $T$
  • $f(x) = f(x + T)$ holds true for all values of $x$
  • Examples: sine and cosine functions with periods of $2\pi$
• 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 $\omega_0 = \frac{2\pi}{T}$
  • The fundamental frequency is the reciprocal of the period $T$

Representations in Fourier series

• General form of a Fourier series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 x) + b_n \sin(n\omega_0 x))$
  • $a_0$, $a_n$, and $b_n$ are Fourier coefficients that determine the amplitude of each sine and cosine term
  • $n$ represents the integer multiples of the fundamental frequency $\omega_0$
• Fourier coefficients are calculated using integrals over one period of the function
  • $a_0 = \frac{2}{T} \int_{-T/2}^{T/2} f(x) dx$ represents the average value of the function over one period
  • $a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos(n\omega_0 x) dx$ determines the amplitude of the cosine terms
  • $b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin(n\omega_0 x) dx$ determines the amplitude of the sine terms
• Common periodic functions with Fourier series representations
  • Square wave: odd harmonics with amplitudes decreasing as $\frac{1}{n}$
  • Sawtooth wave: all harmonics with amplitudes decreasing as $\frac{1}{n}$
  • Triangle wave: odd harmonics with amplitudes decreasing as $\frac{1}{n^2}$

Convergence and Applications of Fourier Series

Convergence of Fourier series

• Pointwise convergence 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
  • Gibbs phenomenon: overshoots near discontinuities that do not diminish with increasing terms
• Uniform convergence 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
• Dirichlet conditions for convergence
  1. The function must be periodic with period $T$
  2. The function must be piecewise continuous on the interval $[-\frac{T}{2}, \frac{T}{2}]$
  3. The function must have a finite number of maxima and minima on the interval $[-\frac{T}{2}, \frac{T}{2}]$

Power calculation with Parseval's theorem

• Parseval's theorem relates the energy of a function to the energy of its Fourier coefficients
  • $\int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{T}{2} \left(\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\right)$
  • 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
  • $P = \frac{1}{T} \int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$
  • Obtained by dividing Parseval's theorem by the period $T$
• Energy of a periodic signal is the total energy over one period
  • $E = \int_{-T/2}^{T/2} |f(x)|^2 dx = \frac{T}{2} \left(\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\right)$
  • Directly obtained from Parseval's theorem