Fourier series:
If a signal is periodic, then the Fourier series breaks the waveform into the summation of sinusoidal functions.The common form of the Fourier series
Sinusoidal functions are periodic over 2ฯ angular distance
For a periodic function f(x), (let’s assume a function other than a sinusoidal function i.e., a square wave with a period of 2ฯ.png)
For electronics communication, during DSB, SSB, or VSB generation we usually find f, 2f, 3f, etc. – such types of frequencies for the generation of only f. Such terms as 2f, 3f, etc. are called harmonics. The Fourier transform is the same way. A rectangular waveform of period T, for example, can be described as the sum of sine and cosine waves. Then there must be a variety of frequencies with periodic times of T, 2T, 3T, and so on, or simply multiples of fundamental period T. Where 1/T is the fundamental frequency and 1/2T, 1/3T, and so on are harmonic frequencies.
For a periodic function f(x), (let’s assume a function other than a sinusoidal function i.e., a square wave with a period of 2ฯ
.png)
For electronics communication, during DSB, SSB, or VSB generation we usually find f, 2f, 3f, etc. – such types of frequencies for the generation of only f. Such terms as 2f, 3f, etc. are called harmonics. The Fourier transform is the same way. A rectangular waveform of period T, for example, can be described as the sum of sine and cosine waves. Then there must be a variety of frequencies with periodic times of T, 2T, 3T, and so on, or simply multiples of fundamental period T. Where 1/T is the fundamental frequency and 1/2T, 1/3T, and so on are harmonic frequencies.
Applications of Fourier Series
The problems that engineers study usually contain periodic functions. Fourier series result from their representation as basic periodic functions like sine and cosine (FS). When it comes to a variety of partial differential equations difficulties, the Fourier series is a particularly effective instrument.The linear operations are carried out in one domain (frequency or time), and corresponding operations are present in the other domain, which are sometimes simpler to carry out. Some specific differential equations are considerably simpler to evaluate in the domain of frequency because the differentiation operation present in the time domain corresponds to multiplication by frequency. The frequency domain's standard multiplication corresponds to the convolution in the time domain. The transformation of the result can be traced back to the time domain after performing the necessary actions. In this case, harmonic analysis is the methodical investigation of the relationship between time scales and frequencies, including the kinds of operations or functions that are comparatively straightforward and have close ties to numerous branches of contemporary mathematics.