The Fourier series representation of a periodic signal, with period T=1/fo, is defined by
where the complex Fourier series coefficients, also expressed in polar
are computed via the integral formula
where T is the fundamental period of the signal. The DC component of the signal is equal to the first Fourier series coefficient and is simply the average value of the signal over one period.
The sinusoidal components of the signal that occur at multiples of the
fundamental frequency are called harmonics.
In general, for well-behaved (continuous) periodic signals, a sufficiently large number of harmonics can be used to approximate the signal reasonably well. For periodic signals with discontinuities, however, such as a periodic square wave, even a large number of harmonics will not be sufficient to reproduce the square wave exactly. This effect is known as Gibbs phenomenon and it manifests itself in the form of ripples of increasing frequency and closer to the transitions of the square signal.
An illustration of Gibbs phenomenon is shown in the figure below. The figure shows the result of adding one, three, five, seven, and nine harmonics. In all cases, and regardless of the number of harmonics, it is observed that the overshoot of the ripples has a constant magnitude (around 18%).