Synthesizing Signals with the Fourier Series
Periodic signals can be synthesized as a linear combination of harmonically related complex sinusoids. The theory of Fourier series provides the mathematical tools for this synthesis by starting with the analysis formula, which provides the Fourier coefficients Xn corresponding to periodic signal x(t) having period T0.
Common periodic signals include the square wave, pulse train, and triangle wave. This table shows the Fourier series analysis and synthesis formulas and coefficient formulas for Xn in terms of waveform parameters for the provided waveform sketches: