Fourier Series representation of Periodic Signals

Consider a periodic signal x(t) with fundamental period T, i.e.

Then the fundamental frequency of this signal is defined as the reciprocal of the fundamental period, so that

Under certain conditions, a periodic signal x(t) with period T can be expressed as a linear combination of sinusoidal signals of discrete frequencies, which are multiples of the fundamental frequency of x(t). Further, sinusoidal signals are conveniently represented in terms of complex exponential signals. Hence, we can express the periodic signal in terms of complex exponentials, i.e.

Such a representation of a periodic signal as a combination of complex exponentials of discrete frequencies, which are multiples of the fundamental frequency of the signal, is known as the Fourier Series Representation of the signal.