Trigonometric Fourier Series

While studying heat flow, Fourier discovered that a non-sinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions. Recall that a periodic function is one that repeats every $ T $ seconds. In other words, a periodic function $ f(t) $ satisfies where $ n $ is an integer and $ T $ is the period of the function. According to the Fourier theorem, any practical periodic function of frequency $ \omega_{0} $ can be expressed as an infinite sum of sine or cosine functions that are integral multiples of $ \omega_{0} $. Thus, $ f(t) $ can be expressed as or where $ \omega_{0}=2 \pi / T $ is called the fundamental frequency in radians per second. The sinusoid $ \sin n \omega_{0} t $ or $ \cos n \omega_{0} t $ is called the $ n $th harmonic of $ f(t) $; it is an odd harmonic if $ n $ is odd and an even harmonic if $ n $ is even. Equation $ 3 $ is called the trigonometric Fourier series of $ f(t) $. The constants $ a_{n} $ and $ b_{n} $ are the Fourier coefficients. The coefficient $ a_{0} $ is the dc component or the average value of $ f(t) $ (Recall that sinusoids have zero average values.) The coefficients $ a_{n} $ and $ b_{n} $ (for $ n \neq 0 $ ) are the amplitudes of the sinusoids in the ac component. Thus, A function that can be represented by a Fourier series as in Eq. (3) must meet certain requirements, because the infinite series in Eq. (3) may or may not converge. These conditions on $ f(t) $ to yield a convergent. Fourier series are as follows:

• $ f(t) $ is single-valued everywhere.
• $ f(t) $ has a finite number of finite discontinuities in any one period.
• $ f(t) $ has a finite number of maxima and minima in any one period.
• The integral $ \int_{t_{0}}^{t_{0}+T}|f(t)| d t<\infty $ for any $ t_{0} $.

These conditions are called Dirichlet conditions. Although they are not necessary conditions, they are sufficient conditions for a Fourier series to exist. A major task in Fourier series is the determination of the Fourier the coefficients $ a_{0}, a_{n} $, and $ b_{n} $. The process of determining the coefficients is called Fourier analysis. The following trigonometric integrals are very helpful in Fourier analysis. For any integers $ m $ and $ n $, Let us use these identities to evaluate the Fourier coefficients. We begin by finding $ a_{0} $. We integrate both sides of Eq. (3) over one period and obtain Invoking the identities of Eqs. (4.a) and (4.b), the two integrals involving the ac terms vanish. Hence, or showing that $ a_{0} $ is the average value of $ f(t) $.To evaluate $ a_{n} $, we multiply both sides of Eq. (3) by $ \cos m \omega_{0} t $ and integrate over one period: The integral containing $ a_{0} $ is zero in view of Eq. (4.b), while the integral containing $ b_{n} $ vanishes according to Eq. (4.c). The integral containing $ a_{n} $ will be zero except when $ m=n $, in which case it is $ T / 2 $, according to Eqs. (4.e) and (4.g). Thus, or In a similar vein, we obtain $ b_{n} $ by multiplying both sides of Eq. (3) by $ \sin m \omega_{0} t $ and integrating over the period. The result is Be aware that since $ f(t) $ is periodic, it may be more convenient to carry the integrations above from $ -T / 2 $ to $ T / 2 $ or generally from $ t_{0} $ to $ t_{0}+T $ instead of 0 to $ T $. The result will be the same. An alternative form of Eq. (3) is the amplitude-phase form we can apply the trigonometric identity to the ac terms in Eq. (10) so that Equating the coefficients of the series expansions in Eqs. (3) and (12) shows that or To avoid any confusion in determining $ \phi_{n} $, it may be better to relate the terms in complex form as The plot of the amplitude $ A_{n} $ of the harmonics versus $ n \omega_{0} $ is called the amplitude spectrum of $ f(t) $; the plot of the phase $ \phi_{n} $ versus $ n \omega_{0} $ is the phase spectrum of $ f(t) $. Both the amplitude and phase spectra form the frequency spectrum of $ f(t) $. Thus, the Fourier analysis is also a mathematical tool for finding the spectrum of a periodic signal. To evaluate the Fourier coefficients $ a_{0}, a_{n} $, and $ b_{n} $, we often need to apply the following integrals: It is also useful to know the values of the cosine, sine, and exponential functions for integral multiples of $ \pi $. These are given in Table $ 1 $, where $ n $ is an integer.