In Mathematical Analysis, Signal processing, and Physics, the Fourier Series and the Fourier transform are crucial tools that offer reliable methods for evaluating and expressing functions, especially periodic and non-periodic signals. These ideas, which were first presented by Jean-Baptiste Joseph Fourier in the early 1800s, are the basis of modern harmonic analysis. A periodic function can be represented as an infinite summation of sine and cosine waves using the Fourier series. Because of this dissection, complex periodic waveforms can be broken down into a set of more straightforward trigonometric elements, each of which has its own distinct frequency, amplitude, and phase. The Fourier series is particularly helpful in domains like electrical engineering, acoustics, vibration analysis, and image processing because of its ability to differentiate between various components. By establishing a link between time-domain and frequency-domain representations, experts may examine signal behaviours, remove extraneous elements, and develop effective systems for signal analysis and transmission. The Fourier Transform expands the frequency analysis to periodic functions by applying the Fourier series' restriction to periodic functions to non-repeating signals. By converting a time-domain signal into a continuous array of frequency components, the Fourier Transform helps to reveal signal properties that could otherwise be obscured. In fields like quantum mechanics, telecommunications, and control systems, this transformation is essential because it makes tasks like modulation, spectrum analysis, and system identification easier. Both approaches have the major benefit of being linear and having the ability to simplify differential equations, which makes them essential for resolving boundary value issues in both engineering and physics.
Author
(s) Details
S.K. Mohapatra
Department of Mathematics, Kalinga Institute of Social Sciences (KISS)
Deemed to be University, Bhubaneswar-751024, Odisha, India.
Prashna Naik
Department of Mathematics, Kalinga Institute of Social Sciences (KISS)
Deemed to be University, Bhubaneswar-751024, Odisha, India.
Dhaneswar Beherdalai
Department of Mathematics, Kalinga Institute of Social Sciences (KISS)
Deemed to be University, Bhubaneswar-751024, Odisha, India.
M.R. Mohapatra
Department of Mathematics, KIIT Deemed to be University,
Bhubaneswar-751024, Odisha, India.
M.R. Mohapatra
Department of
Mathematics, KIIT Deemed to be University, Bhubaneswar-751024, Odisha, India.
Please see the book here:-
https://doi.org/10.9734/bpi/mcsru/v5/5624
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