Signal Controls
Harmonics
Filter
Display Options
Time Domain - Signal Waveform
Draw on canvas to create custom signal
Frequency Domain - FFT Spectrum
Drag to set filter cutoff frequency
Individual Harmonics
Phase Spectrum
Reconstructed Signal (Inverse FFT)
Signal Statistics
RMS Value:
0.000
Peak Amplitude:
0.000
Dominant Frequency:
0 Hz
Bandwidth:
0 Hz
THD:
0%
SNR:
0 dB
Constraint Theory Perspective
Geometric Constraints in Frequency Domain
The FFT reveals that any signal is constrained by its frequency components. Each frequency acts as a geometric constraint that shapes the signal's behavior in time.
Key Insights:
- Harmonic Series: Integer multiples of fundamental frequency create geometric progressions in frequency space.
- Pythagorean Ratios: Musical intervals emerge naturally from frequency ratios (2:1 octave, 3:2 fifth, 4:3 fourth).
- Constraint Propagation: Removing frequency components (filtering) propagates constraints through the signal, altering its shape.
- Time-Frequency Duality: Sharp features in time require broad frequency support (uncertainty principle).
Interactive Exploration:
- Draw custom signals to see their frequency content
- Add harmonics to understand complex waveform construction
- Apply filters to see constraint propagation
- Watch animated decomposition to see harmonics building the signal