Why is e[n] plotted as the “cleaned” output?

In this feedforward arrangement the adaptive filter is trying to copy the noise component v that is correlated with the reference x. When ŷ ≈ v, subtracting it from the primary leaves e = p − ŷ ≈ s + v − v = s. So the error node is exactly where you would listen after cancellation.

When does LMS diverge or oscillate?

For standard LMS on FIR systems, the classic upper bound is 0 < μ < 2 / λ_max(R_xx) where R_xx is the tap-input autocorrelation matrix. Too large μ grows the eigenmodes of the weight-error recursion until the adaptation blows up — try the “LMS too large μ” preset with NLMS turned off. NLMS rescales μ by instantaneous input energy, which usually widens the stable range.

What if L is smaller than the true noise path length L_h?

The adaptive FIR only has L degrees of freedom. If the physical coupling needs more taps, the residual v − ŷ stays correlated with x and e still contains noise. Increase L until the misalignment ‖w − h‖ stops improving for a fixed random path.

How is this different from the Kalman filter simulator?

Kalman filtering assumes an explicit dynamical model for the hidden state and Gaussian noise covariances; the gain matrix comes from solving Riccati-style recursions. LMS is a stochastic-gradient descent on the instantaneous squared error with respect to FIR weights — no explicit state transition matrix, but also no optimality guarantee under non-Gaussian or nonstationary statistics.

LMS / NLMS Adaptive Noise Cancellation

Q: What if L is smaller than the true noise path length L_h?

The adaptive FIR only has L degrees of freedom. If the physical coupling needs more taps, the residual v − ŷ stays correlated with x and e still contains noise. Increase L until the misalignment ‖w − h‖ stops improving for a fixed random path.

Q: How is this different from the Kalman filter simulator?

Kalman filtering assumes an explicit dynamical model for the hidden state and Gaussian noise covariances; the gain matrix comes from solving Riccati-style recursions. LMS is a stochastic-gradient descent on the instantaneous squared error with respect to FIR weights — no explicit state transition matrix, but also no optimality guarantee under non-Gaussian or nonstationary statistics.

This page simulates a textbook feedforward active noise control (ANC) path: the primary sensor records p[n] = s[n] + v[n], where s is a deterministic desired signal (sine, two tones, or a linear chirp) and v is interference generated by passing white Gaussian reference noise x[n] through a fixed unknown FIR h of length L_h (random coefficients each time you press “new noise path”). An L-tap adaptive FIR forms ŷ[n] = Σ_j w_j x[n−j] and the error e[n] = p[n] − ŷ[n] drives the Widrow–Hoff LMS update w ← w + μ e[n] x_n, where x_n is the tap-delay vector of reference samples. Normalized LMS (NLMS) divides the step by ε + ‖x_n‖², which stabilizes adaptation when the reference power swings. With L ≥ L_h and a small enough step, w converges toward the true h (up to delay ambiguity) so ŷ tracks v and e tracks s — the same algebra used to explain adaptive line enhancers, echo cancellation training, and simplified headphone ANC, modulo causality and secondary paths in real hardware.

Who it's for: Undergraduate signals & systems or adaptive DSP courses; anyone comparing Kalman filtering (state model) with stochastic-gradient adaptive FIRs (reference regression).

Key terms

LMS adaptive filter
NLMS
Widrow–Hoff
noise cancellation
FIR identification
mean-square error
tap-delay line
step-size μ

Live graphs

How it works

Feedforward noise cancellation with an FIR adaptive filter. The primary microphone picks up p[n] = s[n] + v[n] where s is the desired signal and v is interference built as a fixed unknown FIR of a reference noise x[n] (white Gaussian here). The filter forms ŷ[n] = wᵀxₙ with xₙ = [x[n], x[n−1], …, x[n−L+1]]ᵀ and minimizes e[n] = p[n] − ŷ[n] online using the LMS update w ← w + μ e[n] xₙ (or NLMS with μ / (ε + ‖xₙ‖²) for gentler scaling). When L is large enough and x spans the noise subspace, ŷ → v so e → s — classic adaptive DSP homework made visual.

Key equations

ŷ[n] = Σⱼ wⱼ x[n−j], e[n] = p[n] − ŷ[n]

LMS: w ← w + μ e[n] xₙ

NLMS: w ← w + (μ / (ε + ‖xₙ‖²)) e[n] xₙ

Frequently asked questions

Why is e[n] plotted as the “cleaned” output?: In this feedforward arrangement the adaptive filter is trying to copy the noise component v that is correlated with the reference x. When ŷ ≈ v, subtracting it from the primary leaves e = p − ŷ ≈ s + v − v = s. So the error node is exactly where you would listen after cancellation.
When does LMS diverge or oscillate?: For standard LMS on FIR systems, the classic upper bound is 0 < μ < 2 / λ_max(R_xx) where R_xx is the tap-input autocorrelation matrix. Too large μ grows the eigenmodes of the weight-error recursion until the adaptation blows up — try the “LMS too large μ” preset with NLMS turned off. NLMS rescales μ by instantaneous input energy, which usually widens the stable range.
What if L is smaller than the true noise path length L_h?: The adaptive FIR only has L degrees of freedom. If the physical coupling needs more taps, the residual v − ŷ stays correlated with x and e still contains noise. Increase L until the misalignment ‖w − h‖ stops improving for a fixed random path.
How is this different from the Kalman filter simulator?: Kalman filtering assumes an explicit dynamical model for the hidden state and Gaussian noise covariances; the gain matrix comes from solving Riccati-style recursions. LMS is a stochastic-gradient descent on the instantaneous squared error with respect to FIR weights — no explicit state transition matrix, but also no optimality guarantee under non-Gaussian or nonstationary statistics.

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LMS / NLMS Adaptive Noise Cancellation

Live graphs

How it works

Key equations

Frequently asked questions

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Live graphs

How it works

Key equations

Frequently asked questions