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Signal Processing operators

37 operators in the signal_processing category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
ECHO10Echo signal operator for sixth-harmonic decay analysis, extracting spectral envelope from echo responses.D_univ = κ_d · ℜ[∫ I(t) · e^(-i2π·1.287·t) dt] · φ
ECHO11Echo damped oscillation operator modeling seventh-order resonance decay in multi-path echo environments.D_c(ϕ) = ∬ ρ(x,t) · δ(∫ϕ dx - Φ₀) · e^(i·1.287·t) dx dt
ECHO12Echo signal processing operator for eighth-harmonic attenuation modeling in echo return analysis.I_h = κ_i · Re[ Σ ϕ_n · e^(i·n·2π·1.287·t) / √(1+n²) ]
ECHO13Echo decay operator for ninth-order damped oscillation extraction in time-frequency echo processing.B_ψ = lim_(T→∞) (1/T) · ∫₀ᵀ ψ₁(t)·ψ₂*(t) · sin(2π·1.287·t) dt
ECHO14Echo signal operator for tenth-harmonic damped resonance in propagation channel characterization.Q_λ = Σ |λ_i⟩⟨λ_i| ⊗ ϕ(λ_i) · pulse(1.287)
ECHO15Echo processing operator analyzing eleventh-order decay modes for echo pulse compression.G_ms = ∂²ϕ/∂t² - v²∇²ϕ + α·sin(2π·1.287·t)·ϕ³
ECHO16Echo signal operator for twelfth-harmonic damped oscillation in broadband echo spectral analysis.K_rm(τ) = ∫ ϕ(t)·ϕ(t-τ)·e^(-γ|τ|)·cos(2π·1.287·τ) dτ
ECHO17Echo decay operator modeling thirteenth-order attenuation in multi-bounce echo environments.C_x = ∬ ϕ_phys(x)·ϕ_info(y)·δ(|x-y|-v·1.287⁻¹) dx dy
ECHO18Echo signal processing operator for fourteenth-harmonic resonance in dispersive echo channel analysis.∇_tel ϕ = (∂ϕ/∂t) / |∇ϕ| · Ĥ_1.287(ϕ)
ECHO19Echo damped oscillation operator for fifteenth-order decay analysis in echo signal deconvolution.A_sf = β · (ϕ₁ * ϕ₂)(t) · comb_{1.287}(t)
ECHO20Phase extraction: the argument (mod 2π) of the HulyaPulse-demodulated signal integral ∫ϕ·e^(−i·2π·1.287·t) dt.Φ_c = arg[ ∫ ϕ e^(-i·2π·1.287·t) dt ] mod 2π
ECHO21Echo decay operator modeling seventeenth-order attenuation for final-stage echo signal characterization.χ_τ = ∫∂ϕ/∂t · e^(-i·2π·1.287·(t-τ)) · |∇S_ϕ| dτ
ECHO5Echo damped oscillation operator A*e^(-t/tau)cos(omegat) modeling exponentially decaying resonant echo signals.E_eth = -∇[φ² · log(φ²)] · H(1.287·t)
ECHO6Echo signal processing operator for second-harmonic damped oscillation analysis in reverberant environments.P_h(t+Δt) = (1/2π) ∫ φ(ω) · e^(iωΔt) · δ(ω-2π·1.287) dω
ECHO7Echo decay operator modeling third-order temporal attenuation in acoustic or electromagnetic echo returns.S_sov = (∂φ/∂t) · (1 - e^(-‖φ - φ_coll‖²))
ECHO8Echo signal operator for fourth-harmonic damped resonance extraction from time-domain echo data.Q_filt = φ · exp(-‖∇²φ‖/φ_c) · cos(2π·1.287·t)
ECHO9Echo processing operator analyzing fifth-order damped oscillatory modes in echo signal decomposition.H_r = ∫ φ_injured · φ_whole · e^(-γt) · sin(2π·1.287·t) dt
SP1Discrete Fourier Transform X[k] = sum of x[n]e^(-j2pikn/N) decomposing a discrete signal into frequency components.X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}
SP10Nyquist-Shannon sampling theorem requiring f_s >= 2*f_max to perfectly reconstruct a bandlimited signal from samples.f_s \geq 2 f_{max}
SP11Hamming window function reducing spectral leakage in DFT analysis with optimized sidelobe suppression.W(n) = 0.54 - 0.46\cos\frac{2\pi n}{N-1}
SP12Hann (Hanning) window function applying a raised-cosine taper to reduce spectral leakage in frequency analysis.W(n) = \frac{1}{2}\left(1 - \cos\frac{2\pi n}{N-1}\right)
SP13Bandpass filter transfer function passing frequencies within a specified range while attenuating those outside.H_{BP}(s) = \frac{\omega_0/Q \cdot s}{s^2 + \omega_0/Q \cdot s + \omega_0^2}
SP14Butterworth lowpass filter with maximally flat magnitude response in the passband, no ripple characteristics.H_{LP}(s) = \frac{1}{\prod_{k}(s - p_k)}
SP15Hilbert transform computing the analytic companion of a real signal via principal-value convolution with 1/(pi*t).\mathcal{H}\{x(t)\} = \frac{1}{\pi}\text{P.V.}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau
SP16Analytic signal z(t) = x(t) + j*H{x(t)} combining a real signal with its Hilbert transform for envelope detection.x_a(t) = x(t) + j\mathcal{H}\{x(t)\}
SP17Wigner-Ville distribution providing high-resolution time-frequency energy density for non-stationary signal analysis.W_x(t,\omega) = \int x(t+\tau/2)x^*(t-\tau/2)e^{-j\omega\tau}d\tau
SP18Short-time Fourier transform windowing a signal in time before applying DFT for localized spectral analysis.STFT\{x\}(t,\omega) = \int x(\tau)w(\tau-t)e^{-j\omega\tau}d\tau
SP19Continuous wavelet transform decomposing a signal using scaled and shifted wavelets for multi-resolution analysis.CWT\{x\}(a,b) = \frac{1}{\sqrt{|a|}}\int x(t)\psi^*\left(\frac{t-b}{a}\right)dt
SP2Inverse Discrete Fourier Transform reconstructing a time-domain signal from its frequency-domain representation.x[n] = \frac{1}{N}\sum_{k=0}^{N-1} X[k] e^{j2\pi kn/N}
SP20Time-frequency uncertainty principle: a signal cannot be simultaneously localized in both time and frequency beyond a fundamental limit.\sigma_t \cdot \sigma_\omega \geq \frac{1}{2}
SP3Z-transform transfer function H(z) characterizing a discrete-time linear system in the complex frequency domain.H(z) = \frac{Y(z)}{X(z)} = \frac{\sum b_k z^{-k}}{\sum a_k z^{-k}}
SP4Infinite impulse response (IIR) filter difference equation with feedback coefficients enabling recursive signal filtering.y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k]
SP5Autocorrelation function R_xx[m] measuring the similarity of a signal with a time-shifted version of itself.R_{xx}[m] = \sum_{n} x[n] x[n+m]
SP6Power spectral density via Wiener-Khinchin theorem relating autocorrelation to frequency-domain power distribution.S_{xx}(\omega) = \sum_{m} R_{xx}[m] e^{-j\omega m}
SP7Signal-to-noise ratio SNR = 10*log10(P_signal/P_noise) quantifying signal quality in decibels.SNR = 10\log_{10}\frac{P_{signal}}{P_{noise}}
SP8Discrete convolution y[n] = sum of x[k]*h[n-k], the fundamental operation for linear time-invariant system response.y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]
SP9Frequency response magnitude and phase |H(e^(j*omega))| and angle(H), characterizing filter gain and delay across frequencies.H(\omega) = |H(\omega)| e^{j\phi(\omega)}

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["ECHO10"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​