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A Family of Adaptive Filter Algorithms in Noise Cancellation for Speech Enhancement
Sayed. A. Hadei, M. lotfizad
TL;DR
Rapidly changing noise-cancellation signals motivate adaptive algorithms that converge quickly, while established methods trade off convergence and computational complexity. This paper applies FAP and FEDS to speech-noise cancellation; both converge faster than LMS, NLMS, and AP and comparably to RLS.
Problem
Adaptive noise cancellation needs rapid convergence as signal characteristics change, but established algorithms present a tradeoff between convergence speed and computational complexity.
Source: §I · p. 1Method
The paper applies fast affine projection (FAP) and fast Euclidean direction search (FEDS) algorithms to adaptive noise cancellation for speech enhancement.
Source: §I · p. 1, B. Algorithm development · p. 3Results
22.2623 and 24.9078 SNRI for FEDS and FAP, respectively; both converge faster than LMS, NLMS, and AP and comparably to RLS.
Source: §V · p. 6, §VI · p. 6Takeaways & Limitations
The number of iterations per new sample is user-selected, providing an explicit tradeoff between convergence or tracking properties and computational complexity.
Source: §VI · p. 6Adaptive noise cancellation requires high coherence between the noise in the corrupted signal and the reference channel, which limits effectiveness when microphone separation reduces coherence.
Source: §IV · p. 5
Abstract
from arXiv · showhide
In many application of noise cancellation, the changes in signal characteristics could be quite fast. This requires the utilization of adaptive algorithms, which converge rapidly. Least Mean Squares (LMS) and Normalized Least Mean Squares (NLMS) adaptive filters have been used in a wide range of signal processing application because of its simplicity in computation and implementation. The Recursive Least Squares (RLS) algorithm has established itself as the "ultimate" adaptive filtering algorithm in the sense that it is the adaptive filter exhibiting the best convergence behavior. Unfortunately, practical implementations of the algorithm are often associated with high computational complexity and/or poor numerical properties. Recently adaptive filtering was presented, have a nice tradeoff between complexity and the convergence speed. This paper describes a new approach for noise cancellation in speech enhancement using the two new adaptive filtering algorithms named fast affine projection algorithm and fast Euclidean direction search algorithms for attenuating noise in speech signals. The simulation results demonstrate the good performance of the two new algorithms in attenuating the noise.
I. INTRODUCTION
The paper proposes applying FAP and FEDS to adaptive noise cancellation for speech enhancement, comparing them with classical adaptive filters. The motivation is to balance convergence speed against computational complexity.
I. INTRODUCTION
- The paper applies FAP and FEDS to speech-noise cancellation and compares them with LMS, NLMS, AP, and RLS.It also reports experimentally selected parameters for the two proposed algorithms.Source: §I · p. 1
- The proposed approach updates one or more filter coefficients per time instant to trade off convergence rate and computational complexity.Source: §I · p. 1
- FAP and FEDS convergence is described as superior to LMS, NLMS, and AP, and comparable to RLS [11]-.The introduction frames algorithm choice as a tradeoff: NLMS has low computational complexity, while RLS has high complexity but fast adaptation.Source: §I · p. 1
II. BACKGROUND ON LMS, NLMS, APA AND RLS ALGORITHMS
This section introduces the adaptive-filter setup and reviews classical LMS and RLS update rules. It also identifies step size and forgetting factor as algorithm parameters.
II. BACKGROUND ON LMS, NLMS, APA AND RLS ALGORITHMS
- The adaptive-filter setup uses input x(n), desired signal d(n), and output error e(n), with filter coefficients chosen so the output estimates the desired signal.Source: §II · p. 2
- LMS updates the filter vector using the input and error, while its step size μ determines convergence speed and steady-state mean-square error.Source: §II · p. 2
- The section also gives sample-window vectors for the input and desired signal used in the adaptive-filter formulation.Source: §II · p. 2
- RLS updates the filter vector using an autocorrelation-matrix estimate, with λ serving as the forgetting factor.Source: §II · p. 2
III. FAPA AND FEDS ALGORITHMS
The FAPA and FEDS approach frames adaptive filtering as recursively approximating desired-signal samples with linear combinations of input-data columns. It updates the filter to reduce the resulting error.
III. FAPA AND FEDS ALGORITHMS
- The adaptive-filter problem is to update h(n) at each time instant so that the error between the desired signal and filter output is as small as possible.Source: §III · pp. 2–3
- For the described formulation, the data matrix and desired-signal vector collect recent input and desired-signal samples, respectively.Source: §III · p. 2
- The method builds an approximation to d(n) from linear combinations of columns of X(n), using filter coefficients as weights.The procedure is inspired by the greedy BMP algorithm for successively improving vector approximations.Source: §III · p. 3
B. Algorithm development
The algorithm recursively improves the filter approximation by updating one coefficient at a time. FEDS selects coefficients sequentially, whereas FAP selects the update that maximally reduces the residual.
B. Algorithm development
- Each iteration projects the current approximation error onto a selected data-matrix column and uses that projection to update one filter coefficient.The step size is introduced to control convergence speed and stability.Source: B. Algorithm development · p. 3
- FEDS selects the coefficient index sequentially modulo M, while FAP chooses the index that maximally reduces the residual.The selection is related to choosing a column similar to the a-priori approximation error.Source: B. Algorithm development · p. 3
- Repeating the update P times at a sample can update the same coefficient more than once, and the resulting vector is the filter estimate for that time.Source: B. Algorithm development · pp. 3–4
- For exponentially weighted or sliding-window versions, implementations can achieve multiplicative complexity of M(P+5), while a block exponentially weighted version can achieve M(P+3),.Source: B. Algorithm development · p. 4
IV. ADAPTIVE NOISE CANCELLATION
Adaptive noise cancellation estimates noise in the primary signal from a correlated reference-noise input, then subtracts that estimate to produce a denoised output. The method assumes both noise signals are uncorrelated with speech, and the residual error is minimized rather than eliminated.
IV. ADAPTIVE NOISE CANCELLATION
- The reference-noise input is filtered to estimate the noise component in the primary input, and that estimate is subtracted to produce the denoised signal.The reference input should contain noise correlated with the primary-input noise but no speech-related content.Source: §IV · p. 5
- Minimizing the error energy also minimizes the difference between the system output and primary-input noise, but the residual error cannot become zero.The error is intended to converge toward the speech signal, not equal it exactly.Source: §IV · p. 5
- The adaptive filter is trained to match the unknown noise path from the noise source to the primary sensor, enabling noise cancellation when its estimate closely matches that path.The path is modeled as an unknown FIR channel.Source: §IV · p. 5
- The noise signals must be uncorrelated with speech and correlated with each other, preferably equal, for the setup to obtain its best results.Source: §IV · p. 5
V. EXPERIMENTAL RESULTS
In the office-noise speech experiment, FEDS and FAP converged faster than LMS, NLMS, and AP, with convergence comparable to RLS. Their SNRI values were higher than those reported for LMS, NLMS, and AP, while RLS had the highest reported SNRI.
V. EXPERIMENTAL RESULTS
- 13.5905 (LMS), 16.8679 (NLMS), and 20.0307 (AP) SNRI; NLMS and AP converged faster than LMS in the M=8 experiment.The original speech was corrupted with office noise, and the primary signal SNR was -10.2180 dB.Source: §V · p. 5
- 22.2623 (FEDS) and 24.9078 (FAP) SNRI, versus 29.7355 (RLS); FEDS and FAP converged faster than LMS, NLMS, and AP, with convergence comparable to RLS.Source: §V · p. 6, Table II · p. 6
- Filter-coefficient evolution comparisons also show FEDS and FAP performing better than LMS, NLMS, and AP, and comparably with RLS.Source: §V · p. 6
- The FEDS and FAP filter-order sweep from 1 to 300 found maximum SNRI at M=8, and their maximum SNRI over L occurred at L=25.Source: §V · p. 6
VI. CONCLUSION
The paper applies FEDS and FAP to adaptive noise cancellation and reports convergence rates comparable to RLS. It also describes user-selected iteration counts as a tradeoff between convergence/tracking and computational complexity.
VI. CONCLUSION
- FEDS and FAP achieve convergence rates comparable to RLS in the noise-cancellation simulations.Source: §VI · p. 6
- Experimentally calculated optimum values are reported for FEDS and FAP.Source: §VI · p. 6
- The user-selected number of iterations per sample trades convergence and tracking properties against computational complexity.Source: §VI · p. 6