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Fourier Series Representation of Periodic Signals
What is Fourier Series?
In the domain of engineering, most of the phenomena are periodic in nature such as the alternating current and voltage. These periodic functions could be analysed by resolving into their constituent components by a process called the Fourier series.
Therefore, the Fourier series can be defined as under −
“The representation of periodic signals over a certain interval of time in terms of linear combination of orthogonal functions (i.e., sine and cosine functions) is known as Fourier series.”
The Fourier series is applicable only to the periodic signals i.e. the signals which repeat itself periodically over an interval from $(-\infty\:to\:\infty)$and it cannot be applied to non-periodic signals. Although, not all the periodic signals can be represented by Fourier series. The Fourier series analysis of signals is also known as harmonic analysis.
Representation of Fourier Series
The Fourier series representation of a signal may have the following three forms −
Trigonometric form
In the trigonometric Fourier series representation, the orthogonal functions are the trigonometric functions, i.e.,
$$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t}$$
Cosine form
The cosine representation of $x(t)$ is given by,
$$\mathrm{x(t)=A_{0}+\sum_{n=1}^{\infty}A_{n}[cos(n\omega_{0} t+\theta_{n})]}$$
Exponential form
In the exponential Fourier series representation, the orthogonal functions are the exponential functions, i.e.,
$$\mathrm{x(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{jn\omega_{0} t}}$$
Dirichlet’s Condition for Existence of Fourier Series
A German mathematician Dirichlet defined the conditions for the existence of Fourier series. A periodic signal x(t) can be represented by the Fourier series if it satisfies the conditions which are known as Dirichlet’s conditions. These conditions are as follows −
$x(t)$ must be a single valued function.
$x(t)$ has a finite number of discontinuities.
$x(t)$ has only a finite number of maxima and minima.
$x(t)$ is absolutely integrable over one period, i.e.,
$$\mathrm{\int_{0}^{T}x(t)\:dt<\infty}$$
These four conditions are sufficient but not necessary conditions for the existence of the Fourier series of the periodic function x(t). Here, the fourth condition is known as the weak Dirichlet condition.
If a function satisfies the weak Dirichlet condition, then the existence of the Fourier series of the function is guaranteed, but the Fourier series may not converge at every point.
The second and third conditions are known as strong Dirichlet conditions. If these two conditions are satisfied for the function, then the convergence of the series is also guaranteed.
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