Difference between Fourier Series and Fourier Transform
Fourier Series
$$\mathrm{x(t)= \sum_{n=-\infty}^{\infty}C_{n}\:e^{jn\omega_{0}t}}$$
$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)\:e^{-jn\omega_{0}t}dt}$$
$$\mathrm{}$$
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.,
$$\mathrm{C_{n}=\frac{1}{T}\int_{-T/2}^{T/2}x(t)dt}$$
Fourier Transform
$$\mathrm{X(\omega)=\int_{-\infty }^{\infty}x(t)e^{-j\omega t}dt}$$
$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)e^{j\omega t}d\omega}$$
$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$
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 Cn 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).
Published on 15-Dec-2021 06:57:57
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