🌊 WAVE & SIGNAL EXPLORER

interactive experiments · click any canvas to pause/resume · explore frequencies, brain waves & decomposition
fundamental

📐 Sine & Cosine Waves

The building blocks of all waveforms. sine cosine phase
sine cosine unit circle
🔑 Key concept: Sine and cosine are phase-shifted by 90° (π/2). The unit circle shows how they emerge from circular motion — the x-coordinate gives cosine, the y-coordinate gives sine. Phase shift controls the horizontal offset. This is the foundation of all signal processing.
fourier

🌀 Frequency Decomposition

Break down a complex wave into pure sine components. fourier spectrum harmonics
composite f1 f2 spectrum
🧠 Fourier's insight: Any periodic wave can be built from sine waves of different frequencies. The spectrum shows which frequencies are present and their amplitudes. This is how EEG, audio, and radio signals are analyzed in the frequency domain.
neuroscience

🧠 Brain Wave Decomposition

EEG signals combine different frequency bands. δ 0.5-4Hz θ 4-8Hz α 8-13Hz β 13-30Hz
EEG signal delta (δ) theta (θ) alpha (α) beta (β)
🧠 EEG frequency bands: Delta (δ) — deep sleep · Theta (θ) — drowsiness, meditation · Alpha (α) — relaxed, eyes closed · Beta (β) — active thinking, concentration. Adjust sliders to see how each band contributes.
interference

🌊 Superposition & Interference

Two waves traveling in opposite directions create standing waves. constructive destructive standing wave
wave 1 → wave 2 ← sum envelope
🌈 Superposition principle: When waves meet, they add together. Constructive interference (in-phase) creates larger waves, while destructive interference (out-of-phase) cancels them. The envelope shows the beating pattern.
modulation

📻 Amplitude Modulation (AM)

How information is encoded in wave amplitude. carrier modulation envelope
carrier wave modulated signal envelope
📻 AM explained: The carrier wave (high frequency) has its amplitude varied by the modulation signal (low frequency). This is how radio broadcasts transmit audio — the envelope contains the actual information.
spectrogram

📊 Spectrogram View

Watch the frequency spectrum evolve over time. time-frequency spectrogram waterfall
time domain spectrogram energy
📊 Spectrogram: A time-frequency representation showing how frequency content changes over time. The vertical axis shows frequency, horizontal shows time, and color indicates energy/intensity. Used in audio analysis, speech recognition, and EEG.
EMD

📉 Empirical Mode Decomposition (EMD)

Step-by-step sifting process to decompose a brain signal into Intrinsic Mode Functions (IMFs). IMF sifting adaptive
Step: 0/6
original signal current IMF residue envelope (max/min) extracted IMFs
🔬 EMD Sifting Process: Step 1: Identify local maxima/minima · Step 2: Create upper/lower envelopes · Step 3: Calculate mean envelope · Step 4: Subtract mean to get IMF candidate · Step 5: Repeat until IMF criteria met · Step 6: Subtract IMF from signal, repeat on residue. EMD is data-adaptive — no fixed basis functions, making it ideal for non-stationary EEG signals.
wavelet

🔍 Discrete Wavelet Transform (DWT)

Multilevel decomposition using wavelets — revealing both time and frequency information. approximation detail multiresolution
Level: 2
original signal approximation (A) detail 1 (D1) detail 2 (D2) detail 3 (D3)
🔍 Wavelet Transform: Unlike Fourier, wavelets provide time-frequency localization. Approximation (A) captures low-frequency content (trend). Details (D1, D2, D3) capture high-frequency components at different scales. This is the foundation of EEG artifact removal, image compression (JPEG2000), and denoising. Multiresolution analysis reveals both global trends and local details simultaneously.
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