Approximated affine projection algorithm for feedback cancellation in hearing aids
Introduction
The LMS adaptive filter has been widely used for feedback cancellation in hearing aids thanks to its simplicity and efficiency [1], [2], [3]. However, the convergence performance of the normalized least mean square (NLMS) algorithm is often deteriorated by coloured input signals. To overcome this problem, the affine projection (AP) algorithm that updates the weight vector based on a number of recent input vectors can be used [4]. This allows a higher convergence speed than the LMS algorithm, especially for coloured input signals, but it is computationally complex. Many fast versions of the AP algorithm have been suggested to provide significant simplifications [5], but they require a process of matrix inversion, which is not only computationally expensive but also a source of numerical instability. Recently, an algorithm approximating the process of matrix inversion using Gauss–Seidel (GS) iteration has been suggested [6]. GS iteration has stable convergence behaviour, especially when the input autocorrelation matrix is diagonally dominant. However, the algorithm in Ref. [6] suffers from a convergence problem when the algorithm is associated with small step sizes.
In this paper, we present a new approximated AP algorithm based on GS iteration. This new algorithm shows stable convergence even with small step sizes and is applied here to the problem of feedback cancellation in hearing aids.
A long-standing issue regarding feedback cancellation in hearing aids is the correlation between the input and output signals of the hearing aid, which leads the adaptive feedback cancellation system to create a bias in the estimate of the feedback path [7], [8]. This issue is often resolved by applying delays in the forward or the control paths. In [9], it was shown that, if a delay is inserted in the forward path, identification of the feedback path and the desired signal model is possible. For a wideband input, such delays enable an adaptive system to converge to an accurate estimate of the feedback path. However, for a narrowband input, even with delays, the feedback cancellation system tends to minimize the error signal by cancelling the input instead of modelling the feedback path. Adaptation with a sinusoidal input in general causes a large mismatch between the estimated and actual feedback paths. This mismatch results in signal cancellation, system instability and ringing, or colouration, of the output signal [8]. One approach to maintaining system stability is to use constrained adaptation [10]. However, this does not distinguish between a deviation caused by error in the noise model and one caused by a change in the external feedback path.
In this paper, we propose a method of controlling the learning rate of the adaptive feedback cancellation filter to minimize the system's instability and signal cancellation caused by narrowband inputs. The proposed method is combined with the approximated AP algorithm and varies the step size in relation to the prediction factor of the input signal. It provides fast convergence to changes in the feedback path and can prevent signal cancellation and colouration artefacts for narrowband inputs.
In Section 2, we provide an AP algorithm with orthogonalized input vectors, which is approximated in Section 3. In Section 4, an algorithm for controlling the step size of the adaptive feedback cancellation is presented. We present simulation results in Section 5. Section 6 concludes this paper.
Section snippets
Affine projection algorithm with orthogonalized input vectors
The affine projection algorithm updates the weight vector based on M most recent input vectors. Similar to the well-known NLMS algorithm, a given step size is used to control the rate of convergence and the steady-state excess mean square error. Let w(n) be an estimate of an unknown weight vector at the time index n; the affine projection algorithm computes w(n) aswhere x(n) and d(n) denote (N × 1)
Approximated AP algorithm
The AP algorithm shown in Eq. (8) requires a stack of M linear prediction filters of orders from 0 through M − 1. Although this approach can avoid the process of matrix inversion, this remains computationally expensive. In [6], the Gauss–Seidel (GS) iteration was used to solve the (M − 1)th-order LS linear prediction problem. To simplify the process, a single GS iteration per sample was performed to estimate the solution vector of the LS problem. This is equivalent to solving the system R(n) αM = bM
Learning rate control of the approximated AP algorithm for the feedback cancellation in hearing aids
In the adaptive feedback cancellation system, increased step size will give faster adaptation, and reduced step size can improve the sound quality of the system with slow adaptation speed. When the spectrum of the input signal varies with time, the desired adaptation speed is a compromise between the rapid adaptation required to track changes in the feedback path and the slow adaptation required to avoid bias caused by temporary pure tones [10]. In this study, we propose a learning rate control
Computer simulations
For the simulations, a 20 dB hearing aid gain (G = 10) was assumed and an 80 sample decorrelation delay (Δ = 80) was inserted in the forward path together with a probe noise, which provides conditions for identifiable feedback path with a insignificant bias [7]. The feedback path was modelled using a 128-tap FIR filter at a sampling rate of 16 kHz. The feedback path models used in the simulations are shown in Fig. 1. Adaptive filters of N = 30 and η = 2 were used for the simulations presented in this
Conclusions
In this paper, we propose a new approximated AP algorithm and a learning rate control method for feedback cancellation in hearing aids. The proposed algorithm showed stable convergence behaviour even with small step sizes. In addition, by controlling the learning rate in relation to the prediction factor of the input, system instability and colouration artefacts caused by narrowband inputs could be prevented. Simulation results verified the efficiency of the proposed algorithm.Acknowledgements
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