Echo State Network / Reservoir Computing Explorer

How the fingerprint forms. Each reservoir neuron responds to the input with its own oscillation. During the steady-state window (shaded) the pattern is stable and repeating — it encodes the input's character, not its phase. Taking the RMS of each neuron's output collapses the time dimension into a single number per neuron: the oscillation amplitude. Different input signals produce distinct bar-height signatures. Try the phase shift slider — drag Signal B's phase from 0° to 360°. Watch the waveform slide in panel ①, the traces shift in panel ②, but the orange bars in panel ③ stay locked in place. Why is this? For periodic signals (sine, square wave), the averaging window spans many complete cycles and mean(x²) over any integer number of cycles equals amplitude²/2 exactly — the phase cancels algebraically. For the chirp, the instantaneous frequency at each time step is fixed by t (not by the starting phase), so different phases trace the same frequency sweep in the window; the RMS averages over many frequencies and nearly cancels the phase. The slight jitter you see with the chirp is the residual from this approximate (not exact) cancellation — the chirp never completes full cycles at any single frequency, so there is no perfect algebraic cancellation, only statistical averaging.

Each dot = one noisy trial (random phase + noise). Left — raw x(T=end): uniformly distributed on [−1, 1] regardless of frequency — zero information. Right — LDA of response-amplitude fingerprint: for each trial the ESN is driven for 100 steps; then for each of the N neurons we compute sqrt(mean(hᵢ(t)²)) over the steady-state window. This per-neuron oscillation amplitude is phase-invariant (mean(x²) over a full cycle of any sinusoid equals amplitude²/2, regardless of starting phase), so random phases no longer scatter the feature vectors. The two classes form compact, well-separated clusters along the Fisher discriminant axis; the vertical line is the LDA decision boundary. This is why Tab 3 achieves high accuracy with a simple Linear Classifier: the reservoir converts a hard temporal classification problem into an easy spatial one.

Why raw fails: x(T=end) is uniformly distributed regardless of frequency — no better than a coin flip. Why linear classification of the ESN fingerprint works at ~100%: the per-neuron response amplitude — sqrt(mean(hᵢ(t)²)) over the steady-state window — is phase-invariant: for a periodic input, mean(x²) over a full cycle equals amplitude²/2 regardless of starting phase. Each neuron’s response amplitude is frequency-specific, landing different signal types in distinct regions of ℝᴺ. In that high-dimensional space, a single hyperplane separates the classes. This is a general property of reservoir computing: the reservoir's random nonlinear dynamics expand the input into ℝᴺ, making the problem linearly solvable. High accuracy from a simple one-layer readout is expected — the reservoir did the hard work.