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Fourier Series – Representation and Properties
Jean Baptiste Joseph Fourier developed a technique to analysing non-sinusoidal waveforms applicable to a wide range of engineering problems. Many times all the information available in time domain is not sufficient for the analysis of the circuits, for this reason we have to transform the signal into frequency domain for extracting more information about the signal. The Fourier series is a tool used for transforming a signal from time domain to the frequency domain. In the Fourier series the signal is decomposed into harmonically related sinusoidal functions.
Frequency Domain Analysis
A periodic signal can be decomposed into a linear weighted sum of harmonically related sinusoidal functions or complex exponential functions, which are known as Fourier series. The analysis of signal by decomposing it into its frequency related components is known as Frequency Domain Analysis.
The Fourier series can be used to represent those periodic signals only, which satisfies the Dirichlet’s conditions.
Dirichlet’s Condition
A function f(t) can be absolutely integrated over any period t, if
f(t) has a finite number of maxima and minima within the any finite interval of period (t).
f(t) has a finite number of discontinuities within any finite interval of period (t), and each of these discontinuities are finite.
Fourier Series Representation
There are two types of Fourier series representations, both are equivalent to each other. Depending on the type of signal, most convenient representation is chosen.
Exponential Form of Fourier Series
Trigonometric Form of Fourier Series
Exponential Form of Fourier Series
J. B. J. Fourier demonstrated that a periodic function f (t) can be expressed as a sum of sinusoidal functions. According Fourier representation,
$$f(t)=a_{0}+\displaystyle\sum\limits_{n=1}^\infty M_{n}\cos(n\omega_{0}t+\theta_{n})$$
Where $\omega_{0}=\frac{2\Pi}{T_{0}^{\prime}}$
T0 is the time period, when n = 1, one cycle covers T0 seconds while $M_{1}\cos(\omega_{0}t+\theta_{1})$is called as fundamental. When n = 2, T0 represents two cycles within T0 seconds while $M_{2}\cos(2\omega_{0}t+\theta_{2})$is termed as 2nd harmonic. In the same manner, for n = K, K cycles falls within T0 seconds and $M_{K}\cos(K\omega_{0}t+\theta_{K})$ is the Kth harmonic term.
Hence by using Euler identity,
$$f(t)=a_{0}+\displaystyle\sum\limits_{n=-\infty\
eq\:0}^\infty C_{n}e^{jn\omega_{0}t}$$
Where, Cn = Complex Fourier Coefficient,
$$f(t)=a_{0}+\displaystyle\sum\limits_{n=1}^\infty a_{n}\cos(n\omega_{0}t)+b_{n}\sin(nw_{0}t)$$
The Fourier coefficient is defined by the expression,
$$C_{n}=\frac{1}{T_{0}}\int_{t_{1}}^{t_{1}+T_{0}}f(t)e^{-jn\omega_{0}t}dt\:\:\:...(1)$$
The expression (1) represents the exponential form of Fourier series.
Trigonometric Form of Fourier Series
The trigonometric form of Fourier series can be easily derived from the exponential form. The trigonometric Fourier series representation of a periodic signal f (t) with fundamental time period T0 is given by,
$$f(t)=a_{0}+\displaystyle\sum\limits_{n=1}^\infty (a_{n}\cos\:n\omega_{0}t+b_{n}\sin\:n\omega_{0}t)$$
Where, $\omega_{0}=\frac{2\Pi}{T_{0}}$ and $a_{n}$ and $b_{n}$ are Fourier Coefficients given by,
$$a_{n}=\frac{2}{T_{0}}\int_{t_{1}}^{t_{1}+T_{0}}f(t)\cos(n\omega_{0}t)dt$$
$$a_{n}=\frac{2}{T_{0}}\int_{t_{1}}^{t_{1}+T_{0}}f(t)\sin(n\omega_{0}t)dt$$
The a0 is the average value of the waveform that can be directly evaluated from the waveform and is given by,
$$a_{0}=\frac{1}{T_{0}}\int_{t_{1}}^{t_{1}+T_{0}}f(t)dt$$
Properties of Fourier Series
If f(t) is an even function i.e. f(-t) = f(t), then
$$a_{0}=\frac{2}{T_{0}}\int_{0}^{T_{0}/2}f(t)dt\:,$$
$$a_{n}=\frac{4}{T_{0}}\int_{0}^{T_{0}/2}f(t)dt\cos(n\omega_{0}t)dt\:,$$
$$b_{n}=0$$
If f(t) is an odd function i.e. f(-t) = - f(t), then
$$a_{0}=0$$,
$$a_{n}=0$$,
$$b_{n}=\frac{4}{T_{0}}\int_{0}^{T_{0}/2}f(t)\sin(n\omega_{0}t)dt\:,$$
If f(t) is a half wave symmetric function i.e. $f(t)==f(t-\frac{T_{0}}{2})$, then
When n is even,
$$a_{0}=0$$,
$$a_{n}=b_{n}=0$$,
When n is odd,
$$a_{n}=\frac{4}{T_{0}}\int_{0}^{T_{0}/2}f(t)\cos(n\omega_{0}t)dt\:,$$,
$$b_{n}=\frac{4}{T_{0}}\int_{0}^{T_{0}/2}f(t)\sin(n\omega_{0}t)dt$$
Summary of Properties of Fourier Series
For an even function, all terms of its Fourier series are cosine terms. No sine terms are present. However, the function does have an average value a0
For an odd function, the series contains only sine terms. There is no average value and no cosine terms.
If the given function is of half wave symmetry, only odd harmonics are present in the series and the series would contain both sine and cosine terms when n is odd. The average value is zero.
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