The exploration of the properties of the Fourier series begins with linearity.

When considering two periodic signals and forming a third by their linear combination, the Fourier coefficients of this third signal are simply a linear combination of the coefficients of the original signals.

When a periodic signal is shifted in time, the magnitude of its Fourier coefficients remains unchanged, keeping the signal preserved despite the time shift.

When a continuous-time signal undergoes time reversal, the sequence of its Fourier series coefficients also experiences a time reversal.

If a signal demonstrates even symmetry, its corresponding Fourier series coefficients will also be even symmetric. Similarly, the Fourier series coefficients for an odd signal will also exhibit odd symmetry.

In radio broadcasting, the time-shifting property of the Fourier series ensures signal quality during frequency modulation.

The linearity property allows multiple signals to be transmitted over the same channel without interference in FM radio.

The time-reversal property is utilized in digital signal processing, aiding operations such as the convolution of signals.