Difference between Fourier Series and Fourier Transform

Fourier series is a branch of Fourier analysis of periodic signals. Fourier series splits a periodic signal into a sum of sines and cosines with different amplitudes and frequencies. Fourier series was introduced by a French mathematician Joseph Fourier. On the other hand, the Fourier Transform is a mathematical operation that decompose a signal into its constituent frequencies. The Fourier transform is also called frequency domain representation of a signal because it depends on the frequency of the signal.

What is Fourier Series?

The mathematical method of decomposing a periodic signal into a sum of sines and cosines is referred to as Fourier Series. Fourier series utilizes orthogonal relationship between sine and cosine functions. Fourier series allows us to split a periodic function into the sum of simple terms that can be used to obtain the solution of a given problem.

Fourier series was originally developed to solve heat equations, but at present, it has applications in a large number of fields including electronics, electrical, signal processing, quantum mechanics, image processing, etc. The study of Fourier series is called harmonic analysis and it is widely used to analyze periodic functions.

• The exponential form of Fourier series of a continuous-time periodic signal is given by,

• The set of coefficients [Cn] is called the set of the Fourier series coefficients or the spectral coefficients of signal x(t).

• The complex coefficients [Cn] measure the portion of the signal that is at each harmonic of the fundamental component.

• The coefficient [Co] is the DC component of the function , which is the average value of the signal over one period, i.e.,

• The Fourier series coefficients of the function are discrete in nature and hence we obtain a discrete spectrum.

What is Fourier Transform?

Fourier transform is a mathematical operation that defines the relationship between the time domain representation of a single and its frequency domain representation. It decomposes a signal or function into oscillatory functions. In the Fourier transform, we can obtain the original signal from its transformation, therefore, no information is lost or created in the transformation process.

Fourier transform is widely used to solve differential equations. Fourier transform has some mathematical properties such as linearity, scaling, duality, convolution, modulation, conjugation, etc. It is used in nuclear magnetic resonance and other types of spectroscopy.

• The Fourier transform of a continuous-time non-periodic signal is defined as,

• The inverse Fourier transform is defined as,

• x(t) and X(ω) form a Fourier transform pair, which is represented as,

• The equation of inverse Fourier transform [i.e., ] plays a role for non-periodic signals similar to the equation of Fourier series [i.e.,] for periodic signals. Because both the equations represent the linear combination of complex exponentials.

• For periodic signals, the spectral coefficients have amplitudes Cn and occur at discrete set of harmonically related frequencies (nω0), where (n = 0, ±1, ±2...). These amplitudes are given by,

• For non-periodic signals, the complex exponentials occur at continuous frequencies with magnitude [X(ω)dω/2π].

• The Fourier transform  of an aperiodic signal is called the spectrum of the signal .

Difference between Fourier Series and Fourier Transform

The following table highlights the important differences between Fourier Series and Fourier Transform −

Characteristic Fourier Series Fourier Transform
Definition Fourier series is a technique of decomposing a periodic signal into a sum of sine and cosine terms. Fourier Transform is a mathematical operation for converting a signal from time domain into its frequency domain.
Type of signal Fourier series can be applied to periodic signals only. Fourier transform can be applied to periodic signals as well as aperiodic signals.
Use Fourier series is used for harmonic analysis of arbitrary periodic functions. Fourier transformer is used to solve differential equations.
Applications The applications of Fourier series include electrical, electronics, signal processing, quantum mechanics, etc. The application of Fourier transform is in nuclear magnetic resonance, and other types of spectroscopy.

Conclusion

The concepts of Fourier series and Fourier transform are quite useful in the study of signals and systems. Fourier transform is a generalization of the Fourier series because it enables the Fourier series to extend to nonperiodic functions.