Fourier Series

18MAT31 - ENGINEERING MATHEMATICS 3 | Module-2 VTU Notes

VTU | 18MAT31 -MODULE-2

Fourier series is a mathematical technique used to represent a periodic function as a sum of sinusoidal functions (sines and cosines). This representation allows for the analysis and approximation of various types of periodic phenomena, such as waveforms, vibrations, and oscillations. Fourier series is a fundamental tool in signal processing, harmonic analysis, and various areas of science and engineering.

Key Points:

1. Periodic Functions: Fourier series is applicable to functions that repeat themselves over a specific interval, known as the period. The function's behavior outside this interval is assumed to be periodic as well.

2. Representation: A periodic function f(t) with period T can be represented as a sum of sines and cosines (harmonics) of varying frequencies and amplitudes:

f(t) = a₀/2 + ∑ [aₙ * cos(nωt) + bₙ * sin(nωt)]

where ω = 2π/T is the angular frequency and a₀, aₙ, and bₙ are coefficients determined by the function's properties.

3. Calculation of Coefficients: The coefficients a₀, aₙ, and bₙ are computed using integration techniques that involve multiplying the function by sines and cosines and then integrating over one period.

4. Orthogonality and Completeness: The success of Fourier series relies on the orthogonality and completeness properties of sines and cosines. This property ensures that any periodic function can be approximated by its Fourier series.

5. Convergence: The convergence of Fourier series depends on the properties of the function being approximated. Smooth and well-behaved functions tend to have better convergence, while functions with discontinuities or abrupt changes may exhibit convergence issues at those points.

6. Applications: Fourier series is widely used in various fields, including physics (quantum mechanics, waves), engineering (signal processing, communication systems), mathematics (harmonic analysis), and even music (sound synthesis).

7. Fourier Transform: Fourier series is closely related to the Fourier transform, which extends the concept to non-periodic functions and provides their frequency-domain representation. The Fourier transform is particularly useful for analyzing signals with continuous spectra.

8. Fast Fourier Transform (FFT): For practical computations, the FFT algorithm is used to efficiently compute the coefficients of the Fourier series and transform between the time and frequency domains.

In summary, Fourier series is a powerful mathematical tool that allows periodic functions to be represented as combinations of sinusoidal components. It plays a pivotal role in understanding and analyzing periodic phenomena, making it a cornerstone in fields that involve signal analysis, wave behavior, and oscillatory systems.