Switched Combination of Simplified Kalman Filter and Affine Projection Algorithm for Acoustic Feedback Cancellation

Acoustic coupling between the microphone and the loudspeaker is a major issue in open-fit digital hearing aids. When compared to a close-fit hearing aid, an open-fit dramatically reduces signal quality and limits the potential maximum stable gain. Adaptive feedback cancellation (AFC) is a practical method for reducing the influence of acoustic coupling. However, because to the high correlation between the loudspeaker signal and the incoming signal, it might induce bias in calculating the feedback path if not carefully considered, especially when the incoming signal is spectrally coloured, as in speech and music. For decreasing this bias, the prediction error method (PEM) is well recognised. In this paper we proposed a switched PEM based hybrid combination of simpliedkalman filter and affine projection algorithms (H-SKF-APA), with soft-clipping that allows for further increase in convergence/tracking rates, resulting in a better ability to recover from an unstable or howling state. Simulation results showed that the proposed algorithm performed better.


Introduction
Acoustic feedback is a serious problem in digital hearing aids [1] [2]. It is the primary source of howling, whistling, or screeching in hearing aids. Acoustic feedback is mostly caused by the acoustic coupling of the desired input signal with the loudspeaker signal at the microphone's input [3]. An acute degradation of the desired signal and troublesome howling increase as the gain increases in acoustic coupling.
There are numerous strategies for acoustic feedback cancellation (AFC); among them, adaptive filtering is effective, as seen in Fig. 1, which comprises the continuous estimation of the feedback signal and its subtraction from the microphone input signal [4]. In fig. 1, W represents the actual acoustic path between loudspeaker and microphone, w represents the estimated acoustic path, which is continuously estimated by an adaptive filter, and G represents the gain of the hearing aid.The primary shortcoming of this continuous adaptation strategy is that the desired signal is a combination of both feedback and input signals [5]. As a result of the correlation between the desired source signal and the loudspeaker signal, the feedback signal is in correlation with the source signal. Hence, the adaptive filter may suffer from a bias estimation problem [6]. • Email: editor@ijfmr.com

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Volume 5, Issue 4, July -August 2023 2 Figure.1 PEM-AFC System The PEM-based technique is an effective one in obtaining an unbiased model of the feedback path by decorrelating the signals in the feedback loop, i.e., by pre-whitening the loudspeaker and the microphone signals with the inverse of the near-end source signal model and then these prefiltered signals are used to estimate the feedback path using the adaptive algorithm [6][7] [8]. The design of an effective adaptive feedback controller demands a trade-off between improving steady-state performance with an unbiased estimate of the feedback path and the adaptive algorithm's convergence rate. The least mean square (LMS) algorithm has a low convergence rate for coloured signals and a high computational complexity when modelling a acoustic path with a large number of filter coefficients [8] [9]. The step-size used to control the adoption of the adaptive filter coefficients is an important parameter in AFC, since it provides better trade-off between convergence speed and steady state misalignment. Different algorithms such as variable step-size(VSS) schemes [10] [11], subband NLMS with VSS ref, affine projection algorithm with VSS [12] have been developed for better trade-off. The Kalman filter in timedomain and frequency-domain has been used in [13] [14][15] to implement PEM-AFC system. Due to limited power handling capability of hearing aids, simplified versions of time-domain Kalman filter has been developed for PEM-AFC [13]. In [16], multiband structure Kalman filter is proposed for system identification problem, the same can be used for PEM-AFC system as it is less complex and having low processing delay due to parallel processing with subbands. In [14], [17], hybrid combination of SKF and NLMS algorithms has been proposed to overcome reconvergence inability when the howling occurs due to sudden change in feedback path.
In this article contents are organised as follows: section 2 discusses the signal model of PEM-AFC system; section 3 discuss about simplified Kalman filter for PEM-AFC and APA algorithm for AFC; section 4 discuss about the switched combination of SKF and APA algorithms; section 5 contains the simulation results and the final section provides the conclusion.

Signal Model of PEM-AFC System
The microphone signal is provided by eq (1) polynomial transfer function in q. The loud-speaker signal is given by eq (2), G represents forward gain of the hearing aid, The Prediction-error method (PEM) is widely used to overcome biasing issue while evaluating the filter weights. The prefiltered signals of the microphone () p sk and loud-speaker () p uk are given by, The Prewhitened error signal of PEM-AFC is,

Simplified Kalman filter
The standard or generalized Kalman filtering (GKF) algorithm requires the O(M3) multiplications per sample, where M is length of the filter, which is too expensive in practice. Thus a simplified structure of Kalman filter is used in ref to reduce computational complexity. The simplified Kalman filter (SKF) was proposed in [13]. It was demonstrated that when the appropriate settings are used, it functions as a variable step-size filter. Its quick convergence and low misadjustmenthave been demonstrated for use in PEM-AFC. By using the notations in figure 1, if ( ) represents the correlation matrix of priori state estimation error signal, ( ) is the apriori error signal vector between microphone signal and estimate of Prewhitened feedback signal, the simplified kalman filter algorithm equations [13] for PEM-AFC are given below: When the SKF technique begins to converge or there is a feedback change the possibility of howling is high. As a result, permitting certain NLMS repetitions during these observed times may be advantageous. It is proposed in [14] to use a switching combination of the NLMS and SKF algorithms, which is controlled by a stability detector. When instability is detected, the NLMS iterations are taken for weight estimation, otherwise, the SKF algorithm is used. We propose a novel rule to update the feedback route estimate to increase the convergence/tracking rates and steady-state error while maintaining high output sound quality. Standard adaptive algorithms like LMS, NLMS, and APA exhibit a trade-off between quick convergence/tracking rates and low steadystate error. When the step-size is big, such algorithms provide quick convergence/tracking rates but significant steady-state error, and vice versa. Figure 5 depicts the trade-off for AFC utilizing PEMSC-NLMS with various step-size values. The update rule for APA algorithm is written as,

APA
Furthermore, the affine projection algorithm (APA) is subject to this trade-off in relation to its projection order (P), i.e., the APA delivers quick convergence/tracking but significant steady-state error when P is big and vice versa [17]. When the projection order grows to a specific level, for example, from P=2 to P=6 in the experiment, the PEMSC-APA converges quicker but produces a bigger steady-state error. P=8 results in nearly no improvement in PEMSC-APA convergence, but the worst steady-state error when compared to the identical experiment with P=6.

Proposed algorithm: Switched combination of PEM-SKF and APA algorithms(SW-SKF-APA)
In this paper, we have proposed a hybrid algorithm with the combination of PEM-SKF and APA algorithms. As it is discussed, the PEM-SKF algorithm fails to track the sudden feedback changes and which will result in howling effect. To overcome this we are proposing the switched combination of PEM-SKF and APA. In the normal scenario i.e., when there is no change in feedback path, PEM-SKF runs and provides the good tracking. When the howling is observed with the help of stability detector, the APA algorithm is used in evaluating the feedback path [17].

Figure.2HSKF-APA algorithm for PEM-AFC System
The update equation for the proposed hybrid algorithm is given as,

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Volume 5, Issue 4, July -August 2023 5 The recommended method applies soft-clipping (SC) to the error signal, resulting in the soft-clipping error signal, where  is a scaling parameter. We choose such that the most likely range of the incoming signal is within the linear range of the tanh-function, i.e., where  is a decision threshold that determines the detector's sensitivity and is a binary function that returns 1 if the inequality holds and 0 if it does not.

Simulation Results
The performance of the H-NLMS, H-SKF, and SW-SKF-APA algorithms was investigated using similar feedback path parameters measured in both normal and nearest feedback pathway circumstances. The arriving signal was a concatenation of male and female NOIZEUS database voice. The amplitude responses of the observed auditory feedback channels are presented in Fig. 3, where the first (H1(f)) and second (H2(f)) were measured in free-field and with a telephone receiver situated close to the ear, respectively . Two metrics are used to compare the algorithms: estimation error and achievable stable amplification. The first one, i.e., the estimation erroris measured in terms of misalignment (Mis), which is defined as the normalized difference between the true (acoustic impulse response) and estimated feedback paths, which is often expressed in decibels (dB).    When the incoming music input is set to 0 dB, the MIS and ASG performance of the analysed algorithms is shown in Fig. 6 and 7. The performance of proposed algorithm shown better performance compared to the existing algorithms.

Conclusion:
In this paper, we proposed a switched combination of simplifiedKalman algorithm and affine projection algorithm i.e., HSKF-APA to implement PEM-AFC system. This algorithm helps to overcome the reconvergence inability when the feedback path changes or howling occurs. Computer simulation indicated that the proposed algorithms performance is better in terms of both faster