Difference Between Fourier Series and Fourier Transform

Fourier Series vs Fourier Transform

Fourier series decomposes a periodic function into a sum of sines and cosines with different frequencies and amplitudes. Fourier series is a branch of Fourier analysis and it was introduced by Joseph Fourier. Fourier Transform is a mathematical operation that breaks a signal in to its constituent frequencies. The original signal that changed over time is called the time domain representation of the signal. The Fourier transform is called the frequency domain representation of a signal since it depends on the frequency. Both the frequency domain representation of a signal and the process used to transform that signal in to the frequency domain are referred to as the Fourier transform.

What is Fourier Series?

如前所述,傅里叶级数是expansion of a periodic function using infinite sum of sines and cosines. Fourier series was initially developed when solving heat equations but later it was found out that the same technique can be used to solve a large set of mathematical problems specially the problems that involve linear differential equations with constant coefficients. Now, Fourier series has applications in large number of fields including electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics and econometrics. Fourier series use the orthogonality relationships of sine and cosine functions. The calculation and the study of Fourier series is known as the harmonic analysis and is very useful when working with arbitrary periodic functions, since it allows to break the function in to simple terms that can be used to obtain a solution to the original problem.

What is Fourier transform?

Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain. The Fourier transform decomposes a function into oscillatory functions. Since this is a transformation, the original signal can be obtained from knowing the transformation, thus no information is created or lost in the process. Study of Fourier series actually provides motivation for the Fourier transform. Because of the properties of sines and cosines it is possible to recover the amount of each wave contributes to the sum using an integral. Fourier transform has some basic properties such as linearity, translation, modulation, scaling, conjugation, duality and convolution. Fourier transform is applied in solving differential equations since the Fourier transform is closely related to Laplace transformation. Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy.

Fourier series is an expansion of periodic signal as a linear combination of sines and cosines while Fourier transform is the process or function used to convert signals from time domain in to frequency domain. Fourier series is defined for periodic signals and the Fourier transform can be applied to aperiodic (occurring without periodicity) signals. As mentioned above, the study of Fourier series actually provides motivation for the Fourier transform.