Noise Cancellation

Adaptive noise cancellation is one of the most important practical applications of adaptive filters. It uses a reference input that carries a correlated version of the noise but not the desired signal, allowing the adaptive filter to model the noise path and subtract it from the primary input. The result is a signal with significantly reduced noise content, even when the noise characteristics are unknown or time-varying.

Core Concept of Noise Cancellation

The basic assumption in adaptive noise cancellation is that the desired signal s(n) and the reference noise n1(n) are uncorrelated with each other. The primary microphone or sensor picks up the desired signal s(n) plus a noise component n0(n), which is a filtered version of the original noise source. The reference sensor is placed near the noise source to pick up n1(n), which is correlated with n0(n) but not with s(n).

The adaptive filter W(z) takes n1(n) as input and tries to produce an output y(n) that matches n0(n) as closely as possible. When subtracted from the primary input, the noise component largely cancels out, leaving an estimate of s(n) as the error signal e(n). The clever insight is that the algorithm does not need to know the transfer function of the noise path; the adaptive filter learns it automatically over time.

The LMS algorithm is typically used to update the filter weights because the desired signal s(n) serves implicitly as a training signal. Since s(n) and n1(n) are uncorrelated, the LMS algorithm drives W(z) toward the noise path transfer function and not toward s(n).

Mathematical Expression

Let the primary input be d(n) = s(n) + n0(n) and the reference input be n1(n). The adaptive filter output is y(n) = W^T(n) * n1_vec(n). The system output is e(n) = d(n) - y(n) = s(n) + n0(n) - y(n). When W(z) perfectly models the noise path, y(n) = n0(n), and e(n) = s(n), achieving perfect cancellation.

The signal-to-noise ratio improvement depends on the coherence between n0(n) and n1(n). Higher coherence means better cancellation. The coherence function gamma^2(f) at frequency f is defined as |G_n0n1(f)|^2 / (G_n0n0(f) * G_n1n1(f)), where G denotes power spectral density. Coherence close to 1 across the noise bandwidth enables near-perfect cancellation.

Practical Understanding

A classic application is active headphone noise cancellation, where an outer microphone captures ambient noise and an adaptive filter models the acoustic path to the ear. The filtered noise is subtracted from the audio stream, attenuating unwanted sounds without affecting music content.

Another application is electrocardiography (ECG), where power-line interference at 50 or 60 Hz corrupts the cardiac signal. A reference sensor picks up the power-line signal directly, the adaptive filter models how it couples into the electrode, and the filtered version is subtracted from the ECG. This approach is superior to fixed notch filters because it adapts to variations in coupling over time.

Given:
Desired signal power P_s = 1 W, Noise power P_n0 = 4 W
After noise cancellation, residual noise power P_res = 0.2 W

Why this formula applies:
SNR improvement = 10 * log10(SNR_out / SNR_in)

Formula:
SNR_in = P_s / P_n0
SNR_out = P_s / P_res
SNR_improvement = 10 * log10(SNR_out / SNR_in)

Substitution:
SNR_in = 1/4 = 0.25
SNR_out = 1/0.2 = 5
SNR_improvement = 10 * log10(5/0.25) = 10 * log10(20)

Calculation:
log10(20) = 1.301
SNR_improvement = 10 * 1.301 = 13.01 dB

Final Answer:
Noise cancellation improves SNR by approximately 13 dB.

Exam Tip: In GATE problems on noise cancellation, the error signal e(n) converges to the desired signal s(n), NOT zero. Zero error would mean the noise is zero, but the desired signal is always present. Also, correlation between reference and desired signal will cause signal cancellation, a critical design constraint.

• Requires reference sensor correlated with primary noise but uncorrelated with desired signal.
• LMS or RLS can drive the adaptive filter; LMS is most common due to simplicity.
• Cancellation quality is limited by coherence between reference and primary noise components.
• Applications include active noise cancellation headphones, ECG denoising, and sonar.
• If the reference input leaks desired signal, the algorithm will also cancel part of s(n).

Quick Revision

• Primary input: d(n) = s(n) + n0(n); reference input: n1(n) correlated with n0(n).
• e(n) converges to s(n), not to zero, at convergence.
• SNR improvement depends on coherence between n0(n) and n1(n).
• GATE trap: reference must not contain desired signal, or desired signal is degraded.
• Works for non-stationary noise because adaptive filter tracks changes in noise path.
• Higher filter order allows modeling more complex noise transfer paths.