When we plot a graph of |ck|2 as a function of frequencies kF0, k=0,±1,±2… the following spectrum is obtained which is known as Power density spectrum.

For any signal x(t) to be represented as Fourier series, it should satisfy the Dirichlet conditions which are x(t) has a finite number of discontinuities in any period, x(t) has finite number of maxima and minima during any period and x(t) is absolutely integrable in any period.

cos(2πkF0 t+θk) = cos2πkF0 t.cosθk-sin2πkF0 t.sinθk

Since we are synthesizing the Fourier series of the signal x(t), we call it as synthesis equation, where as the equation giving the definition of Fourier series coefficients is known as analysis equation.

The average power of a periodic signal x(t) is given as

In general, Fourier coefficients ck are complex valued. Moreover, it is easily shown that if the periodic signal is real, ck and c-k are complex conjugates. As a result

If the given signal is x(t) and F0 is the reciprocal of the time period of the signal and ck is the Fourier coefficient then the Fourier series representation of x(t) is given as \\(\\sum_{k=-\\infty}^{\\infty}c_k e^{j2Ï€kF_0 t}\\).

We know that, Fourier series coefficients are complex valued, so we can represent ck in the following way.

When we apply integration to the definition of Fourier series representation, we get

We know that, for an periodic signal, the Fourier series coefficient is