active-noise-control

name	active-noise-control
description	Active noise control (ANC) — feedforward/feedback ANC, LMS filtered-x algorithm, secondary path, coherence, ANC headphones, duct noise cancellation, ANSI S1.11.
metadata	{"priority":7,"promptSignals":{"phrases":["active noise control","ANC","noise cancellation","active noise cancellation","filtered-x LMS","feedforward noise control"],"minScore":3}}

Active Noise Control (ANC) — Complete Skill

Principle

Destructive interference: secondary source generates anti-noise (180° out of phase) to cancel primary noise Effective for: low-frequency tonal noise (< 500 Hz) where passive treatment ineffective without large mass

Feedforward ANC: reference sensor upstream of primary source; digital filter generates anti-noise Feedback ANC: error microphone only; no reference sensor; models primary noise from error

System Components

Reference microphone (x): upstream; captures primary noise before cancellation zone Controller (digital): adaptive filter W(z); processes reference to generate anti-noise signal Secondary source (loudspeaker/actuator): produces anti-noise Error microphone (e): downstream; measures residual noise; feedback to adapt filter Secondary path S(z): transfer function from secondary source to error microphone (must be modeled)

Filtered-X LMS Algorithm (FXLMS)

Algorithm: most widely used ANC algorithm

Secondary path estimate: Ŝ(z) ≈ S(z) (measured offline or online identification)

Filtered reference signal: x'(n) = Ŝ(z) * x(n) [* = convolution; filter reference through secondary path model]

Weight update: w(n+1) = w(n) - μ × e(n) × x'(n)

μ = step size [controls convergence speed and stability] e(n) = error microphone signal at time step n x'(n) = filtered reference vector

Stability condition: μ < 1 / (N × P_x × ‖Ŝ(z)‖_max) N = filter length; P_x = average power of reference; ‖Ŝ‖ = max gain of secondary path

Convergence rate: τ ≈ 1 / (4 × μ × f_s × P_x × |S(f)|²) [time constant in samples; f_s = sample rate]

Filter length N: N ≥ 4 × f_max / Δf_resolution [longer N = better attenuation but slower convergence] Typical: N = 64–512 for 0–1 kHz ANC

Coherence and ANC Potential

Coherence γ²(f) between reference and error: γ²(f) = |G_xe(f)|² / (G_xx(f) × G_ee(f)) [0 ≤ γ² ≤ 1]

Maximum achievable attenuation: ΔL_max(f) = -10 log₁₀(1 - γ²(f)) [dB] γ² = 0.9 → ΔL = 10 dB; γ² = 0.99 → ΔL = 20 dB; γ² = 0.999 → ΔL = 30 dB

Required coherence for target attenuation A [dB]: γ²_min = 1 - 10^(-A/10)

Coherence improves with:

• Single dominant source (tonal noise)
• Reference sensor close to primary source
• Minimum reflections/reverberations between source and reference

ANC for Ducts (1D Propagation)

Duct ANC effective below cut-on frequency: f_cut-on = c / (2 × D_duct) [Hz; D = duct diameter; c = 343 m/s] Circular duct 600 mm: f_cut-on = 343/(2 × 0.6) ≈ 286 Hz → ANC effective < 286 Hz

Attenuation in duct: ΔL ≈ 10–30 dB for tonal HVAC noise (fans, pumps) at 50–250 Hz

Loudspeaker placement: Secondary source > λ/4 upstream of error microphone Reference microphone > 2D upstream of secondary source (avoid acoustic feedback)

Feedback ANC (No Reference Sensor)

Applicable to: random or tonal noise without accessible reference Controller: predict primary noise from error signal; generates anti-noise

IMC (Internal Model Control) structure: d_hat(n) = [1 - S(z)W(z)] / [1 - S(z)W(z)] × e(n) [internal model of disturbance]

Performance: ~6–15 dB attenuation at resonant frequencies Limitation: cannot achieve high coherence; better for narrowband tonal noise

ANC Headphones/Earphones

Feedforward: external microphone on cup; cancels external noise before reaching ear Feedback: internal microphone inside cup; cancels noise at ear canal entrance

Performance: Low frequency (20–500 Hz): 20–35 dB attenuation (feedforward) Mid frequency (500 Hz–1 kHz): 5–15 dB attenuation High frequency (> 1 kHz): passive attenuation dominates

Analog feedback ANC (simple low-cost version): Feedback gain G: |G × S| < 1 for all frequencies (stability criterion) Attenuation: 1/(1 + G × S) → G large → attenuation → G unstable above crossover

ANC for Engine Orders (Automotive)

Engine noise → ANC in exhaust/cabin: Reference: crankshaft tachometer (tacho); extracts engine order harmonics Frequency tracks with speed → linear prediction algorithms

Engine order attenuation: 1st–4th orders: 10–25 dB with adaptive FXLMS

ASEAN/road noise ANC (RANC): Reference: accelerometers on body; predict road-excitation noise in cabin More complex (multiple references, multiple error sensors)

Standards and References

Standard	Scope
ANSI S1.11	Octave-band filters for ANC evaluation
ISO 11820	Measurements on silencers including ANC
IEC 60268-7	Headphones; includes ANC testing
SAE J1477	Interior automotive noise measurement

Output

Provide: ANC type (feedforward/feedback), reference coherence γ²(f) and achievable attenuation [dB by octave band], filter length N, step size μ, secondary path delay [ms], sampling rate [kHz], loudspeaker/actuator specifications (frequency range, power [W]), expected attenuation vs. frequency [Hz → dB], coherence requirement for target attenuation, and applicable standard (ANSI S1.11, ISO 11820).