Fourier series is a powerful mathematical technique that decomposes any periodic function into a sum of simple sine and cosine waves. Here we see how a square wave can be approximated by adding together different frequency components. The more terms we include, the better our approximation becomes.
The Fourier series formula shows how any periodic function can be expressed as a sum of sines and cosines. The coefficients a-zero, a-n, and b-n determine the amplitude of each frequency component. Here we see a sawtooth wave being approximated by its Fourier series. As we add more terms, the blue approximation gets closer to the black original function.
Fourier series breaks down complex signals into simple frequency components. Each bar represents the amplitude of a specific frequency. The fundamental frequency has the highest amplitude, while higher harmonics typically have smaller amplitudes. This frequency domain representation helps us understand which frequencies are most important in the original signal.
Here we see how the Fourier series approximation is built step by step. We start with the fundamental frequency, then add the third harmonic, fifth harmonic, and so on. The red curve shows the current term being added, while the blue curve shows the cumulative sum. Notice how each additional term brings us closer to the original square wave.