Difference Between Fourier Series and Fourier Transform

Once a question always in a mind of a maths student is what is the Difference Between Fourier Series and Fourier Transform. In this article, we not only discuss the Fourier Series and Fourier Transform.

Difference Between Fourier Series and Fourier Transform

What is a Fourier series?

As the Fourier series is an extension of the periodic function that uses an infinite number of sines and cosines. The Fourier series was originally developed to solve thermal equations, but it later became clear that the same technique could be used to solve a wide variety of mathematical problems, especially those involving linear differential equations with constant coefficients.

Fourier series are used today in various fields including electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, and econometrics.

The Fourier series uses the orthogonality of the sine and cosine functions. Calculating and studying the Fourier series is called harmonic analysis and is very useful when working with any periodic function because it allows the function to be broken down into simple terms that can be used to solve the original problem obtained.

What is Fourier Transform?

The Fourier transform defines the relationship between a signal in the time domain and its representation in the frequency domain. The Fourier transform breaks down the function into an oscillating function. Since this is a transformation, the original signal can be derived from the knowledge of the transformation, where no information is created or lost.

The study of the Fourier series actually motivates the Fourier transform. Due to the nature of sine and cosine, it is possible to use the integral to recover the sum of each wave contributing to the sum.

Fourier transform has some basic properties such as linearity, translation, modulation, scaling, conjugation, duality, and convolution. The Fourier transform is used when solving differential equations because the Fourier transform is closely related to the Laplace transform. The Fourier transform is also used in magnetic resonance imaging (MRI) and other types of spectroscopy.

Difference between Fourier series and Fourier transform

Fourier series is an extension of the periodic signal as a linear combination of sine and cosine, while the Fourier transform is a process or function used to convert signals in the time domain to the frequency domain. Fourier series is defined for periodic signals and the Fourier transform can be applied to aperiodic signals (without periodicity). As noted above, studying the Fourier series actually provides the motivation for Fourier transforms.