When you decompose a periodic signal into harmonics, each term has two important pieces: its amplitude and its phase. Together, they tell you both how big a wave is and where it sits in the cycle.
fn(t) = An cos(nωt + φn)
The coefficient An sets the strength of the n-th harmonic. The phase φn shifts the wave left or right in time. A change in phase does not make the wave bigger or smaller; it changes when the oscillation reaches its peaks and zeros.
Why phase matters
Two harmonic components may have the same amplitude but different phases, and the sum will look different. In signal reconstruction, this matters because the waveform depends on both the magnitude and the alignment of each component.
For a square wave, the odd harmonics are important, but their signs and phases determine whether the waveform rises or falls at a given point in time. For a shifted signal, the same frequency content can appear with different phase values even when the shape is not otherwise changed.
- Amplitude decides how much of that frequency is present.
- Phase decides where that frequency lands within the cycle.
- Sum of terms determines the final time waveform.
In practical systems, phase is often critical. In EEG, audio, control, and communications, phase relationships can affect interference, synchronization, and recovery quality. In other words, a signal is not only about frequency content; it is also about timing.
That is why the Fourier playground provides both coefficient control and phase adjustment for each harmonic. You are not just choosing whether a frequency exists—you are deciding how it contributes to the whole reconstruction.