Difference between Fourier Series and Fourier Transform

Fourier Series

The exponential form of Fourier series of a continuous-time periodic signal x(t) is given by,

$$\mathrm{x(t)= \sum_{n=-\infty}^{\infty}C_{n}\:e^{jn\omega_{0}t}}$$

The set of coefficients $[C_{n}]$ is called the set of the Fourier series coefficients or the spectral coefficients of signal x(t).

$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\:e^{-jn\omega_{0}t}dt}$$

The complex coefficients $[C_{n}]$ measure the portion of the signal x(t),that is at each harmonic of the fundamental component.

$$\mathrm{}$$

The coefficient $[C_{0}]$ is the DC component of the function x(t), which is the average value of the signal over one period, i.e.,

$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)dt}$$

The Fourier series coefficients of the function x(t) are discrete in nature and hence we obtain a discrete spectrum.

The Fourier transform of a continuous-time non-periodic signal x(t) is defined as,

$$\mathrm{X(\omega)=\int_{-\infty }^{\infty}x(t)e^{-j\omega t}dt}$$

$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)e^{j\omega t}d\omega}$$

$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$

The equation of inverse Fourier transform$[i.e.,x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)e^{j\omega t}d\omega]$ plays a role for non-periodic signals similar to the equation of Fourier series $[i.e.,x(t)=\sum_{n=-\infty}^{\infty}= C_{n}\:e^{jn\omega_{o}t}]$ for periodic signals. Because both the equations represent the linear combination of complex exponentials.
For periodic signals, the spectral coefficients have amplitudes C_n and occur at discrete set of harmonically related frequencies $(n\omega_{0})$, where,$(n=0,\pm 1,\pm 2...)$ These amplitudes are given by,

$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\:e^{-jn\omega_{0}}dt}$$

For non-periodic signals, the complex exponentials occur at continuous frequencies with magnitude $[X(\omega)d\omega/2\pi]$.
The Fourier transform $X(\omega)$ of an aperiodic signal x(t) is called the spectrum of the signal x(t).

Updated on 15-Dec-2021 08:49:46