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Residual feedback suppression with extended modelbased postfilters
EURASIP Journal on Audio, Speech, and Music Processing volume 2021, Article number: 21 (2021)
Abstract
When designing closedloop electroacoustic systems, which can commonly be found in hearing aids or public address systems, the most challenging task is canceling and/or suppressing the feedback caused by the acoustic coupling of the transducers of such systems. In many applications, feedback cancelation based on adaptive filters is used for this purpose. However, due to computational complexity such a feedback canceler is often limited in the length of the filter’s impulse response. Consequently, a residual feedback, which is still audible and may lead to system instability, remains in most cases. In this work, we present enhancements for modelbased postfilters based on a priori knowledge of the feedback path which can be used cooperatively with the adaptive filterbased feedback cancelation system to suppress residual feedback and improve the overall feedback reduction capability. For this, we adapted an existing reverberation model such that our model additionally considers the presence and the performance of the adaptive filter. We tested the effectiveness of our approach by means of both objective and subjective evaluations.
1 Introduction
Signal processing in a closed electroacoustic loop is a challenging task. It occurs in various applications such as hearing aids [1, 2], public address (PA) systems [3, 4] or socalled incar communication (ICC) systems [5, 6]. In all these systems, feedback occurs because the signal that is played back using a loudspeaker is recorded by a microphone, processed, and then played back again using the same loudspeaker. This may lead to an instability of the system, namely when the loop gain for at least one frequency is larger than 0 dB and the phase is a multiple of 2π. Even if the system is in fact stable, the additional reverberation may make the signals sound unnatural or—more generally—degraded with respect to quality. To reduce the described effects, different methods already exist. The state of the art approach is to use an adaptive filter to estimate the acoustic path utilizing methods like the normalized least mean square (NLMS) algorithm or a Kalman filter [7]. However, besides the fact that operating in a closed acoustic loop requires a sophisticated control mechanism for a robust application of adaptive filters, there are some more limitations. One is that the filter converges towards a bias due to the high correlation between the local and the excitation signal. This makes an additional decorrelation stage essential for many approaches. Another limitation is that usually a filter with limited length will be used when implementing the adaptive filter. Consequently, the filter must be designed in such a way that its length covers at least the most important part of the room impulse response (RIR). Sometimes this is not possible, especially in cases of multichannel applications where a multitude of filters have to be implemented or in cases when the reverberation time is long, e.g., for large rooms like concert halls. The other limitation that leads to feedback never being completely removed is that there will always be a residual misalignment in the adaptive filter, which in turn leads to an error in the estimated feedback signal. Figure 1 depicts an example for a true impulse response h_{LM,i} as well as the part \({\hat h}_{\text {LM},i}\) that an adaptive filter has estimated. The bottom plot shows the difference between the impulse response h_{LM,i} and \({\hat h}_{\text {LM},i}\). The estimation error h_{Δ,i} is the impulse response which causes the residual feedback in such a system.
A different method to increase the stability gain in electroacoustic loops is to estimate the shortterm power spectral density (PSD) of the feedback by using the energy envelopes of the room’s subband impulse responses. These envelopes can be obtained by a priori or online measurements as well as simulations. With this information, a model can be derived which is then used for a convolution with the loudspeaker subband power signal. This results in an estimate of the feedback’s shortterm PSD. This estimate is then used within a socalled Wiener filter (or a variant of it) to attenuate the feedback components within the microphone signal. Except from online measurements, the envelope is assumed to be constant. However, it can be shown that the modelbased methods are robust against room changes and that the envelopes vary only slightly over time. The main advantage of this method is that the model can be implemented recursively and, thus, very efficiently in terms of computational complexity. There will not be any length limitations as described when using adaptive finite impulse response (FIR) filters. However, there are disadvantages, too. The main one lies in the derivation of the Wiener filter, which assumes that both the desired and the undesired signals are orthogonal. In the presented application this is not the case, since the feedback signal (undesired) is only a delayed and processed version of the local speech signal (desired). This means that not only feedback will be reduced, the modelbased approach will also affect the desired signal. However, due to the fact that speech is assumed to be shorttime stationary and there is a delay in the processing and also in the path between loudspeaker and microphone, it is usually observed that the attenuation of the desired signal is small compared to the attenuation of the feedback signal. Hence, this method is able to increase the stability in closedloop systems.
In [8], we presented a method that makes use of the advantages of both described systems. Therefore, we introduced three ways to estimate the residual feedback PSD recursively, taking an adaptive filter into account. We also compared this with the modelbased feedback suppression which was presented in [9]. In this work we made some improvements regarding the models. Furthermore, we show more implementation details. The objective evaluation was improved by adjusting the features. Additional simulations were also performed to investigate the performance during room changes and the convergence of the adaptive filter in the presence of a postfilter. In addition, further acoustic paths were simulated.
1.1 Organization of this paper
The paper is organized as follows: after this introduction, previous research work is summarized in Section 2. Afterwards the modelbased feedback suppression approach is explained in Section 3. In Section 4, we show how we adapted the modelbased approach to use it as a postfilter. After that, we present different methods to derive the required model parameters in Section 5. Finally, we show the evaluation procedure in Section 6 before a conclusion is provided in Section 7.
1.2 Notation
Throughout this contribution the notation will follow some basic rules:

Scalar quantities such as timedomain signals are written in lowercase, nonbold letters such as s(n) for a signal at time index n.

Shortterm frequencydomain quantities are described by upper case letters such as X(μ,k), with k being the frame index and μ as frequency index.

Vectors are noted as bold letters, e.g., H(μ,k) represents a vector containing filter coefficients in subband μ at frame index k.

Smoothed signals are noted by overlined letters such as \(\overline {x}(n)\) and estimated signals are written as letters with a hat such as \(\hat {x}(n)\).

All signals are represented in discrete time.
2 Previous and related work
Electroacoustic feedback is a challenge in various technical systems. The most prominent ones are hearing aids, public address systems, and incar communication systems. Therefore, lots of research has been done in those domains in recent years. A comprehensive overview of different approaches regarding feedback suppression can be found in [10]. In this work, we will focus on room modeling methods.
To fully erase the feedback and, therefore, to allow arbitrary gains, the impulse response of the feedback path must be estimated by means of an adaptive filter. Early approaches use a standard echo canceler to fulfill this task [11–13]. Here, the impulse response is estimated e.g. with a normalized least mean square (NLMS) algorithm in the time domain. If the local signal and the excitation signal are correlated, the problem is that adaptive filters converge to a biased solution. This is strongly the case in closedloop electroacoustic systems [14].
One solution to overcome this problem is to decorrelate the signals. This can, for example, be realized by frequency shifting. It is shown in [15–17] that a slight frequency shift within the frequency range of speech is sufficient to improve the convergence of the adaptive filter. The signals can also be decorrelated with linear prediction or prewhitening [18]. In addition to the decorrelation of the signals, a special stepsize control can further improve the convergence of the adaptive filter. In [1, 19], the decorrelation methods frequency shift and prewhitening are compared and combined with a stepsize control, based on a derivation of the socalled pseudooptimal step size. Another stepsize control that is able to improve the convergence of the adaptive filter without the need of any decorrelation method is described in [20, 21]. Here, the reverberation of the system is exploited to adapt the filter, since signals are not correlated during reverberation. With this stepsize control, both stability and speed of convergence can be improved also for high system gains.
One drawback of the feedback cancelation approaches is that the adaptive filter must cover the relevant length of the room’s impulse response. Otherwise, residual feedback is audible and may even cause the system to become instable. Since long filters increase the computational complexity as well as the convergence time, short filter lengths are often preferred.
In the field of acoustic echo cancelation (AEC), postfilters based on frequencydomain Wiener filters are commonly used [22–25]. The idea is that the residual echo is nothing but the undisturbed error signal which is the signal after the subtraction of the AEC took place assuming the absence of any local speech and noise signals. A very similar approach was already used for residual feedback suppression [13]. The downside of this technique is that the PSD estimation should only be done in remote single talk conditions [26]. Such a situation does not exist in case of closedloop systems. There is however one exception which is the end of a speech segment when there is still some power in the loop due to the loop delay.
In [9], the authors present a feedback suppression method based on well known speech dereverberation techniques [27, 28]. Here, the feedback path is modeled with an statistical model. Based on this the feedback’s PSD is estimated.
In [8], the modelbased feedback suppression is tailored in a way that it can be used as a residual feedback suppression in combination with an adaptive feedback canceler. Therefore, three adapted statistical models have been proposed which can be used to model the feedback path taking an adaptive filter into account. Modelbased approaches have already been used in adaptive echo cancelation systems [29, 30]. In [31], the authors also use a modelbased approach as a postfilter for adaptive echo cancelation. The idea is to use adaptive approaches to model the residual echo power spectral density. However, in all of these approaches the adaptive filter is assumed to work perfectly and only the acoustic path, which is not covered by the filter is taken into account. In AEC applications, this might be sufficient as reasonable steadystate performance can be reached. However, this is not the case in adaptive feedback cancelation.
In the presented paper, the adapted models for residual feedback suppression are further investigated and a more detailed insight, as well as more simulations and results are given.
3 Modelbased feedback suppression
In [9] it was shown that room dereverberation techniques as they were introduced in e.g. [27, 28] can be used to increase the stability of closed electroacoustic loops as we face them in ICC systems. In this section, the modelbased feedback suppression will be described before adapting it for a system with feedback cancelation. We will start with linear, timeinvariant systems with coefficient index i. Of course, we can assume here only shortterm stationarity. Therefore, we will introduce timevariance (by adding also a frame index k) after this generic view on the entire system.
A simple example of a timedomain system operating in a closed electroacoustic loop can be seen in Fig. 2.
The signal y(n) is the microphone signal at time index n and g is a Wiener filter with coefficients based on the estimated feedback which is used to suppress the recorded feedback that is present in y(n). h_{SE} is the impulse response that belongs to the system of the individual application, where SE stands for signal enhancement. It differs with the individual application and may include noise suppression in case of an ICC system or an equalization filter in case of a public address (PA) system. After the signal enhancement stage, x(n) is played back using a loudspeaker resulting in a feedback r(n). The latter is obtained by a convolution of x(n) with the room impulse response h_{LM}. As mentioned above, the room impulse response is assumed to be constant for now. Thus, the time index n can be dropped and we obtain the feedback signal as
r(n) is recorded again by the microphone together with additional background noise b(n) and local speech s(n), yielding the microphone signal
The transfer function of the entire electroacoustic loop can be described as follows:
Here, Ω=2πf/f_{S}∈[0,2π) is the frequency f normalized with respect to the sampling frequency f_{S}. Furthermore,
is the socalled openloop gain. For stability reasons
and
should hold. G(e^{jΩ}) can be used to fulfill this condition. One possibility would be to choose the transfer function based on the estimated feedback
Therefore, an estimate of the feedback PSD \(\hat {S}_{{rr}}(e^{j\Omega })\) is required, which can be derived from Eq. (1).
To make this method more robust against small variation in h_{LM} and to get the ability to save computational complexity we define a model of h_{LM} based on its socalled reverberation time T_{60} and some other parameters, which will be explained next.
This finally results in an exponentially decaying model of the power envelope of the subband version of H_{LM}(μ)
where μ is the discrete subband index, k is the frame index and A(μ) are coupling factors that describe the coupling properties of the acoustic path. P(μ) is the delay of the acoustic path in frames and
describes the decay behavior, where L denotes the frameshift in samples. Using Eq. (8) the estimated shorttime PSD of the current feedback \(\hat {S}_{rr,\mathrm {A}}(\mu,k)\) can be calculated as the convolution of the shorttime PSD of the loudspeaker signal \(\hat {S}_{{xx}}(\mu,k)\) with the magnitude square of the modeled subband impulse response
On the top left of Fig. 3, the energy envelope of the modeled subband impulse response for a single subband is depicted. This can now be used for a subband version
of Eq. (7), with X indicating the individual model type, e.g., X=A.
4 Modelbased feedback suppression as postfilter
Due to stability reasons, FIR filters are commonly used in adaptive filter applications like echo or feedback cancelation. If this kind of method is used in a closed electroacoustic loop system, it is capable of subtracting parts of the feedback signal r(n) from the microphone signal y(n) depending on how good it is adapted to the true room impulse response. However, there are some limitations in the steadystate performance. One of them is that there will always be a residual system mismatch which is caused by nonoptimal control or estimation errors, even if robust adaptive control schemes are used. If an efficient implementation in the subband domain is chosen, the performance is also limited due to aliasing effects caused by the filter banks. The other limitation is due to the part of the true room impulse response which cannot be covered by the adaptive filter. This happens because the FIR filter needs to be implemented with a fixed length, which is often restricted by computational complexity.
Here, a postfilter is usually used to suppress the parts of the feedback which remain after a feedback cancelation approach as it is depicted in Fig. 4.
The idea is to use the method proposed in the previous section and adapt it, so it can be used as a postfilter. Because of its recursive nature, the model will also cover rooms with a long reverberation time without significant impact on the complexity. However, it has to be adapted with respect to the presence of the adaptive filter. Therefore, the effective impulse response, which is a combination of the true impulse response and the one estimated by the adaptive filter, needs to be computed to derive a new model. The effective impulse response can be derived using the signal e(n) from Fig. 4. For this signal holds:
Here,
is a vector containing the adaptive filter coefficients and
contains the m latest samples of x(n), where m is the number of filter coefficients in the adaptive filter. With this the effective impulse response can be derived as:
As one may expect, knowledge about the actual system mismatch for each subband is necessary as it has an influence on the amount of residual feedback. Another difference compared to model A is that there is a direct connection from the loudspeaker signal to the error signal now which is caused by the misalignment in the adaptive filter.
However, this is unknown and has to be estimated. Furthermore, assumptions about the shape of the system misalignment over filter taps has to be made.
During the time when adaptive filters were being studied very extensively—decades ago—two different ideas of the progress of the filter coefficients during an adaptation period coexisted—and still do so today.
4.1 Model B
One idea is that adaptive algorithms spread the error more or less equally over all coefficients. As a consequence, the system mismatch vector can be modeled as a white process with zero mean and a timevariant, but lagindependent variance. Our investigations showed that this seems to be correct if the filter is well converged. So the first approach is to model a constant system mismatch for all filter taps, yielding a power envelope of the residual subband impulse response:
In this work, this approach is named model B. M is the filter length in frames. In case of subband processing, H_{Δ}(k) is a vector containing the system mismatch vector in every subband.
is a vector containing the norms of the system mismatch vectors. This is often estimated within adaptive control schemes. An overview about several estimation procedures can be found in [32]. The estimated residual feedback \(\hat {S}_{rr,\mathrm {B}}(\mu,k)\) can be obtained by convolving this modeled subband impulse response with the PSD of the loudspeaker signal yielding a solution consisting of two parts:
Each of these two parts can be calculated recursively, leading to a very low computational complexity:
In the top right of Fig. 3, the energy envelope of the subbandimpulse response for a single subband of model B compared to model A (introduced in the previous section) can be seen.
4.2 Model C
The second idea is that the system mismatch of the individual coefficients is more or less proportional to the magnitude of the room impulse response of the system that should be identified. This behavior is also observable, but mainly at the beginning of adaptation processes or—in general—whenever the filter is not well adapted. Since feedback cancelation approaches for ICC systems face generally hard conditions e.g. permanent doubletalk and high background noise levels, this model would be an option here.
To model this we assume the first interval for the direct part of the residual impulse response to be zero. This is usually the case only when the adaptive filter is initialized. After some iterations the coefficients will differ from zero. However, the system mismatch in this interval will always stay small compared to the interval between the largest coupling and the rest of the adaptive filter. Here, the system mismatch is modeled as exponentially decaying with the same T_{60} as it is used in all other approaches. Furthermore, we introduce Q(μ) which is the power of the maximum value in the system mismatch vector. This leads to the following power envelope of the model
Using this model, the PSD of the residual feedback can be calculated as follows:
4.3 Model D
The drawback of the two proposed models is that the performance will depend on how accurate the estimation of the system distance is. An easier method compared to models B and C is to assume that the adaptive filter operates perfectly well and there is consequently only the length limitation which has to be covered by the postfilter. This would be advantageous, but it is not very realistic in practical approaches. Models B and C are better in this regard. In this case, ∥H_{Δ}(μ,k)∥^{2} can be set to zero and the estimated PSD of the feedback signal simplifies to
which corresponds to \(\hat {S}_{{mm}}(\mu,k)\) in Eq. (22).
5 Model parameters
To use the proposed modelbased approach, a priori knowledge about the room is needed. In this work, the room is assumed to be power stationary, meaning that the power envelope of the room impulse response does not vary much over time. Depending on the particular control mechanism used for the adaptive filter, a correction of the model parameters based on the adaptive filter is also possible. The parameters can be extracted using a measured impulse response like it was proposed in [33].
To follow this approach, the timedomain impulse response has to be transformed into the subband domain
where N is the window length and, thus, the length of the DFT, and h_{ana} is the window function used in the filter bank. The absolute value of the subband impulse response is smoothed along the frequency axis in both positive (Eq. (28)) and negative (Eq. (29)) direction for every frame with the smoothing constant ζ:
This way, a zerophase lowpass filter is realized to reduce the variance along the frequency axis. As a first step, the delay can be determined for each subband by finding the index of the first maximum value of the smoothed magnitude subband impulse response in each subband
with L_{LM} representing the considered length of the impulse response in frames and λ^{k} representing an exponentially decaying series with λ∈(0,1), which can be used to avoid choosing late constructive interferences as maxima.
Next, a vector of length M_{LM} is defined for every subband. It contains the logarithmic impulse responses, starting at the delay which was found before:
This can be modeled linearly for each subband
Written in matrix notation, the logarithmic subband impulse responses can be combined to
where Γ is the observation matrix and Θ represents the parameter vector
To find the unknown parameters in Θ, Eq. (33) can be modified according to
with [Γ^{T}Γ]^{−1}Γ^{T}=Γ^{†} being the pseudo inverse of Γ. The last step is to convert the logarithmic values to linear values
A simplification, which we also used for our simulations, would be to assume the model parameters except the coupling factors to be identical for all frequencies. In this case, the delay and the decay constants can be computed directly from the energy decay curve (EDC) defined as
The EDC describes the remaining energy in the system at time instance i normalized to 0 dB. An example of an EDC measured in a car cabin can be seen in Fig. 5. The delay T_{D} is the time instant where the EDC begins to drop. Using this value, the delay in frames can be derived as
with ⌊...⌋ denoting rounding towards the next smaller integer. The reverberation time T_{60} is the time instant when the energy that remains in the tail of the room impulse response reaches − 60 dB. Often, this cannot be found in the EDC, because there is measurement noise dominating when using a measured impulse response. In this case, the EDC has to be extrapolated linearly, as it was done in the example shown in Fig. 5. Using this value the decay instant can be derived:
The remaining parameters which need to be estimated are the coupling factors. These are the absolute squared values of the maximum of the subband impulse response, which can be found at frame index k=P according to Eq.(30). Therefore, the smoothed version of Eq. (29) should be used to reduce the variance along the frequency axis to yield
In Fig. 6, a timefrequency analysis of the measured impulse response and the corresponding model can be seen.
6 Results and discussion
The proposed schemes were tested in an ICC application. For this, an impulse response measured in a van was used. It is the same one as shown in Fig. 5. The complete setup is shown in Fig. 7.
For our simulation we used clean speech signals from different male and female speakers sampled at 44.1 kHz. The DFT order was set to N=512 and we used a frameshift of L=256. The NLMSbased adaptive filter
with a fixed stepsize α and E(μ,k) and X(μ,k) being the error signal or the excitation signal vector, respectively, was adapted until a specific system distance for all subbands was obtained. Afterwards the adaption was stopped by setting α to zero. As the system distance for this particular simulation was known, we used this value also as a parameter in model B and C to avoid the influence of estimation errors. M was set to four frames, corresponding to a filter length of 46.4 ms. Afterwards the stepsize was set to zero for the simulation. This was done since the models would affect the convergence behavior of the adaptive filter, so the results would also be affected. The aim of the postfilter is to reduce the residual feedback in the microphone signal as much as possible without affecting the desired speech signal. To prove this, the ratio of the mean logarithmic speech power and the mean weighted logarithmic speech power was calculated as
with
The same can be done for the reverberation parts of the signal
with
where N_{Frames} is the total number of processed frames in this simulation and μ_{z}∈[μ_{Start},μ_{Start+1},⋯,μ_{Stop}] are the investigated subbands representing frequencies between about 90 Hz and 8000 Hz, where most of the speech power is located. S_{ss}(μ,k) is the shortterm power spectrum of the input frame of the clean speech signal. Φ_{s}(μ,k) is a binary mask based on a subband voice activity detection which is one for subbands where voice is detected and zero for those without voice. Φ_{r}(μ,k) is the equivalent for the reverberation signal with the additional condition that all timefrequency bins where Φ_{s}(μ,k)=1 are set to zero. An example of the described masks can be seen in Fig. 8, where the speech signal of the investigated subbands is at the top, the mask for the clean speech is in the middle, and the mask for the reverberation is at the bottom. Using these masks, the impairment is only evaluated where the respective signals are present. The reason for using this method is that we wanted to observe the ratio between speech impairment and feedback suppression as well as both quantities alone. This is important because a large speech impairment could lead to audible artifacts. More established methods like segmental SNR would only show the feedback reduction, which is not sufficient for our purpose.
Furthermore, we used a shorttime objective intelligibility (STOI) measure, which was proposed in [34]. This measurement shows good performance in evaluating degradation caused by timefrequency based algorithms e.g. noise reduction. As a reference signal we used the clean speech s(n). The loudspeaker signal x(n) was the signal to be evaluated.
In Table 1, the simulation results for different system distances are shown. P_{s} is the unwanted impairment of the clean speech signal, P_{r} is the equivalent for the reverberation signal. P_{r}−P_{s} is the distance between both. Since one impairment is wanted and the other one is not, it describes the attenuation of the unwanted signal. Consequently P_{s} has to be treated as an offset and needs to be compensated.
When the system distance is at − 40 dB, the results in terms of the different approaches (B–D) for P_{r}−P_{s} as well as STOI are similar. However, it can also be seen that the impairment of the clean speech is significantly higher in case of model A, which is the original unadapted model. Model B, which is assumed to be correct for this simulation because the filter is well adapted and has a constant system mismatch, is very similar to models C and D with respect to P_{s} and P_{r}. This happens due to the fact that in case of a small system distance models (B–D) are nearly the same. When the system distance is increased, the impairment of the clean speech caused by models A and D are nearly constant whereas it increases in case of models B and C. This is exactly what one would expect because of the shorttime stationary nature of speech leading to a large correlation when reducing the lag between an input frame and the models response. The system distance is used as a weight. By increasing it while decreasing the response time the correlation between the wanted and the unwanted signal increases.
In case of model B, there is an immediate response to an input signal. Model C also produces a response in the region of early feedback, resulting in a higher value of P_{s} when increasing the system distance. However, P_{r}−P_{s} is slightly better than in case of all other models. The best compromise regarding the impairment of the clean speech is reached with models C and D. Evaluating with STOI shows similar results. It can be seen that the results improve when the system distance decreases. For large system distances the scores are low even if there is no postfilter applied (). This is due to the existing feedback in the processed signal. For a system distance of −10 dB it can also be seen that the score of model D is slightly higher, although P_{r}−P_{s} of model C is slightly larger. This is due to the fact that model D causes less speech impairment in this particular setup. Furthermore, a relation between STOI and P_{s} can be observed. A large value for P_{s} leads to a low STOI rating, which is due to the fact that STOI evaluates the degradation of the speech signal, which is mainly influenced by P_{s}. The best results in terms of STOI are reached when there is no postfilter at all. However, this does not mean that it makes no sense to use a postfilter at all, because one of its main purpose is to increase stability while saving computing power.
The impulse responses used for the modelbased approach were measured under certain conditions. For example, this could be an empty vehicle at a certain temperature. In reality, however, these are subject to permanent fluctuations due to room changes. For example, a car could be fully loaded and fully occupied or empty. In addition, objects directly in front of sound sources or microphones could cause large attenuation.
Even changes in the distance between loudspeaker and microphone are conceivable. In the following, we will investigate such a situation where the model parameters are determined based on an impulse response of an empty van, but in fact there is a fully occupied interior. This variation of the acoustic path results in a reduced T_{60}=80.3 ms compared to the original 119.9 ms as well as different coupling factors, which can be seen in Fig. 9.
The results (see Table 2) show that both STOI and (P_{r}−P_{s}) are only slightly worse than before, but still very good.
In order to have a more robust evaluation, we simulated a second acoustic path. This time it is one that was recorded in a lecture hall and has a significantly longer decay time T_{60} of 777.8 ms. The delay T_{D} is 18.7 ms. The results can be seen in Table 3. The higher reverberation time results in a slightly higher influence on the desired signal than in the simulation before. However, a clear attenuation of the feedback can still be seen. The previously discussed effects of the different models apply here without restriction.
In order to evaluate the subjective impairment of the desired signal we conducted a listening test with 26 untrained participants aged between 21 and 46 years. We used the same setup as shown before with a system distance of −30 dB.
Overall the procedure was a degradation category rating (DCR) according to ITUT Rec. P.800 [35], which was modified for our purpose. We always played the unprocessed clean speech signal as reference and then the simulated versions with either one of the models (A–D) or the cancelation only (–) in random order. We used three female and two male speakers saying German sentences according to ITUT Rec. P.501 [36]. In sum, every participant had to rate 25 signals. One of the female speakers was used for a trial run which we did not take into account to give the participants the opportunity to get used to the test procedure. The signals are provided on a web page [37].
The rating was defined as following:

5. Excellent – Speech sounds like the unprocessed signal

4. Good – Speech is slightly impaired, but sounds natural

3. Fair – Speech is impaired, but the artifacts are not disturbing

2. Poor – Speech quality degrades, interfering artifacts are clearly audible

1. Bad – Speech is heavily impaired
The results in terms of a mean opinion score (MOS) can be seen in Fig. 10.
The unadapted model and the version without any postfilter were rated with a mean opinion score below 2.5. This shows that artifacts caused by the postfilter are as bad as the residual feedback when there is no postfilter at all. All of the adapted models are rated with a MOS between 3.2 and 3.6 which means that the impairment is less compared to the unadapted model. Here model C shows the best results compared to models B and D. However, a Tukey honest significant difference (HSD) test with α=0.05 as suggested in [35] shows that there is no significant difference between models B, C, and D as well as between model A and no postfilter at all. However, the approaches can be grouped in model (B, C, and D) and (A and −).
The results in Table 4 are consistent with the objective results, with the exception of STOI. In the hearing test, the subjects rated a larger feedback as disturbing as a strong degradation of the speech signal. In contrast, the results according to STOI must be interpreted in such a way that a stronger feedback has less influence than the degradation of speech.
In a next step, we want to evaluate the model influence on the convergence behavior of an adaptive filter. For this, we used an NLMSbased adaptive filter based on pseudooptimal stepsize
according to [26], where the expected value E{·} was approximated by first order IIR smoothing. The socalled undisturbed error E_{u}(μ,k), which is the error signal E(μ,k) without local signals, must be estimated as well. For this, it is replaced by
with H_{Δ}(μ,k) being still not known. First, we replace it with an estimate yielding
In acoustic echo cancelation, \(\beta _{x}^{2}(\mu,k)\) could be estimated by minimum tracking the power of the noisereduced error signal, which is then divided by the smoothed power of X(μ,k). However, this is not possible in feedback cancelation due to the permanent presence of local speech. Here, an approach [38] is to split \(\beta _{x}^{2}(\mu,k)\) into
with \(\beta _{\text {\tiny LEM}}^{2}(\mu,k)\) being a premeasured quantity based on A(μ) and \(\beta _{y}^{2}(\mu,k)\) being the smoothed power ratio of the microphone and the error signal.
For the simulation, we used speech signals recorded in a car at 100 km/h. Now we replaced the fixed values of H_{Δ}(μ,k)^{2} with its estimate which we get from the stepsize control \(\beta _{x}^{2}(\mu,k)\). Q(μ,k) in model C was replaced with \(M A(\mu)\beta _{x}^{2}(\mu,k)\). The loop gain initially was at 5 dB and was increasd with 0.8 dB/second until it reached the final value of 28 dB. The results in terms of the system distance over time with the same adaptive filter and the different models for the postfilter are shown in Fig. 11.
It can be seen that the best performance is reached when there is no postfilter at all. This is due to the fact that the achievable system distance at a fixed stepsize depends only on the power ratio between feedback signal and local signal [26]. Even with an adaptive stepsize control, as in this case, it does not always work well enough to compensate for this. As mentioned before, this is mainly due to P_{s}. This value attenuates the desired signal, which reduces the power of the loudspeaker signal by the same amount. To adjust the achievable filter performance, the filtered signal must be amplified by the value P_{s}, which is shown in Fig. 12.
Here, we adjusted the gain by an offset of 5 dB in case of models C and D and 10 dB in case of model A. For these three models this nearly matches the individual values of P_{s}. However, this did not work for model B, where we had to add 17 dB to the loop gain. The difference is that P_{s} of models A,C, and D has only a small or no dependency on the system distance, whereas it increases with decreasing system difference in case of model B.
7 Conclusion
In this work, we investigated existing and proposed slightly extended postfilter schemes which are capable of suppressing residual feedback in closedloop systems where adaptive feedback cancelers are used. We showed that there are different ways to adapt the reverberation model with respect to the feedback canceler. We were able to show by means of subjective and objective evaluation that all of our adapted models provide a better performance then using the standard reverberation model (model A) as a postfilter in a system with acoustic feedback canceler. However, there is a drawback. In the model, it was assumed that knowledge about the current system distance is available. This is, however, a quantity which is not available in real systems. But there are several stepsize control methods available where a robust estimation of this quantity is included. In this case, it can also be used for the modelbased postfilter. In all other cases, when no estimation of the system distance exists we propose to use the other adapted model, which assumes the system distance to be zero (model D). Beside models A to D several other models could be thought of and some of them were also tested during this research work, but at the end, we decided to continue only with these four approaches to keep this publication at a reasonable length.
Availability of data and materials
The sample files which we used for the subjective evaluation can be found online [37].
Declarations
Abbreviations
 STOI:

Shorttime objective intelligibility
 DCR:

Degradation category rating
 PA:

Public address
 HSD:

Honest significant difference
 ICC:

Incar communication
 MOS:

Meanopinion score
 RIR:

Roomimpulse response
 EDC:

Energydecay curve
 FIR:

Finite impulse response
 NLMS:

Normalized least mean square
 PSD:

Power spectral density
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MG and PB have conducted the research and analyzed the data. MG, PB, and GS authored the paper. All authors read and approved the final manuscript.
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Gimm, M., Bulling, P. & Schmidt, G. Residual feedback suppression with extended modelbased postfilters. J AUDIO SPEECH MUSIC PROC. 2021, 21 (2021). https://doi.org/10.1186/s13636021002058
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DOI: https://doi.org/10.1186/s13636021002058