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# Beamforming under Quantization Errors in Wireless Binaural Hearing Aids

*EURASIP Journal on Audio, Speech, and Music Processing*
**volume 2008**, Article number: 824797 (2008)

## Abstract

Improving the intelligibility of speech in different environments is one of the main objectives of hearing aid signal processing algorithms. Hearing aids typically employ beamforming techniques using multiple microphones for this task. In this paper, we discuss a binaural beamforming scheme that uses signals from the hearing aids worn on both the left and right ears. Specifically, we analyze the effect of a low bit rate wireless communication link between the left and right hearing aids on the performance of the beamformer. The scheme is comprised of a generalized sidelobe canceller (GSC) that has two inputs: observations from one ear, and quantized observations from the other ear, and whose output is an estimate of the desired signal. We analyze the performance of this scheme in the presence of a localized interferer as a function of the communication bit rate using the resultant mean-squared error as the signal distortion measure.

## 1. Introduction

Modern digital hearing aids perform a variety of signal processing tasks aimed at improving the quality and intelligibility of the received sound signals. These tasks include frequency-dependent amplification, feedback cancellation, background noise reduction, and environmental sound classification. Among these, improving speech intelligibility in the presence of interfering sound sources remains one of the most sought-after features among hearing aid users [1]. Hearing aids attempt to achieve this goal through beamforming using two or more microphones, and exploit the spatial diversity resulting from the different spatial positions of the desired and interfering sound sources [2].

The distance between the microphones on a single hearing aid is typically less than 1 cm due to the small size of such devices for aesthetic reasons. This small spacing limits the gain that can be obtained from microphone array speech enhancement algorithms. Binaural beamforming, which uses signals from both the left and right hearing aids, offers greater potential due to the larger inter-microphone distances corresponding to the distance between the two ears (16–20 cm). In addition, such a scheme also provides the possibility to exploit the natural attenuation provided by the head. Depending on the location of the interfering source, the signal-to-interference ratio (SIR) can be significantly higher at one ear compared to the other, and a binaural system can exploit this aspect.

A high-speed wireless link between the hearing aids worn on the left and right ears has been recently introduced [3]. This allows binaural beamforming without the necessity of having a wired connection between the hearing aids, which is impractical again due to aesthetic reasons. The two hearing aids form a body area network, and can provide significant performance gains by collaborating with one another. The performance of binaural noise reduction systems has been previously studied in, for example, [4–8]. However these systems implicitly assume the availability of the error-free left and right microphone signals for processing. In practice, the amount of information that can be shared between the left and right hearing aids is limited by constraints on power consumption imposed by the limited capacity of hearing aid batteries. It is known [9] that quantization of a signal with an additional bit causes the power dissipation in an ADC to be increased by 3 dB. Hence to conserve battery in a hearing aid, it is critical to compress with as few bits as possible before wireless transmission occurs. One in five users was reported to be dissatisfied with hearing aid battery life [10], and it is thus an important consideration in hearing aid design. In this paper, we study analytically the trade-off in the performance of a GSC beamformer with respect to quantization bits.

Different configurations are possible for a binaural beamforming system, for instance, both hearing aids could transmit their received microphone signals to a central device where the beamforming is performed, and the result could then be transmitted back to the hearing aids. Alternatively, the hearing aids could exchange their signals and beamforming may be performed on each hearing aid. In this paper, to analyze the effect of quantization errors on beamforming, without loss of generality we assume that each hearing aid has one microphone and that the right hearing aid quantizes and transmits its signal to the left hearing aid, where the two signals are combined using a beamformer. This paper is an extension of our earlier work [11], incorporates the effect of head shadow and presents a more detailed experimental analysis.

If the power spectral density (PSD) of the desired source is known a priori, the two-microphone Wiener filter provides the optimal (in the mean squared error sense) estimate of the desired source. The effect of quantization errors in such a framework has been investigated in [12]. However, in practice the PSD is unknown. In this paper, we consider a particular beamformer, the generalized sidelobe canceller (GSC) [13], which does not require prior knowledge of the source PSD.

The GSC requires knowledge of the location of the desired source, which is available since the desired source is commonly assumed to be located at (in front of the microphone array) in hearing aid applications [2]. The motivation behind this assumption is that in most real-life situations, for instance, a conversation, the user is facing the desired sound source. In a free field, the two-microphone GSC can cancel out an interfering sound source without distorting the desired signal, which is a desirable feature in hearing aids. Thus, the GSC is well suited for hearing aid applications, and we study the impact of quantization errors on the GSC in this paper.

The performance of the GSC may be affected by other sources of error such as microphone mismatch, errors in the assumed model (the desired source may not be located exactly at , reverberation, and so forth. Variations of the GSC that are robust to such imperfections are discussed in [14–16]. In this paper, we exclude such errors from our analysis to isolate the effect of the errors introduced by quantization on the performance of the GSC.

The remainder of this paper is organized as follows. We introduce the signal model and the head shadow model we use in Section 2. The binaural GSC and its behavior in the presence of quantization errors are discussed in Section 3. The performance of the GSC at different bit rates is analyzed in Section 4. Finally, concluding remarks and suggestions for future work are presented in Section 5.

## 2. Signal Model

Consider a desired source in the presence of an interferer , where represents the time index. A block of samples of the desired and interfering signals can be transformed into the frequency domain using the discrete Fourier transform (DFT) as

where is the frequency index. Let , and , where indicates complex conjugation. We assume that the left and right microphones each have one microphone. The signal observed at the microphone in the left hearing aid can be written as

where and are the transfer functions between the microphone on the left hearing aid and the desired and interfering sources, respectively, and corresponds to uncorrelated (e.g., sensor) noise with . The transfer functions and include the effect of head shadow. For each , we model , , and as memoryless zero mean complex Gaussian sources, with variances , , and , respectively. Their real and imaginary parts are assumed to be independent with variances , , and , respectively.

The signal observed at the right ear can be written as

where the relevant terms are defined analogously to the left ear. We assume that , and that , , and are pairwise independent.

We use the spherical head shadow model described in [17] to obtain the head related transfer functions (HRTFs) , , and . Define the origin to be the center of the sphere. Let be the radius of the sphere, be the distance between the origin and the sound source, and define . Let denote the angle between a ray from the origin to the sound source and a ray from the origin to the point of observation (left or right ear) on the surface of the sphere as shown in Figure 1. The HRTF corresponding to the angle of incidence is then given by [17]

with

where is the Legendre polynomial of degree , is the spherical Hankel function of order , and is the derivative of with respect to its argument.

Let denote the angle between the vertical -axis and a ray from the origin to the desired source. Let be defined similarly for the interfering source. The microphones on the left and right hearing aids are assumed to be located at and respectively, on the surface of the sphere. For example, if in Figure 1, , then the location of the source relative to the left ear is . We have

Similarly, the transfer functions corresponding to the interferer are given by

We consider the case where the quantities , , , and are all unknown. As is typical in hearing aid applications [2], we assume the desired source to be located in front of the user, that is, . Thus, due to symmetry, the HRTFs between the desired source and the left and right microphones are equal (this is valid in anechoic environments, and only approximately satisfied in reverberant rooms). Let . The GSC structure [13] depicted in Figure 2 can then be applied in this situation. The fixed beamformer simply averages its two inputs as the desired source component is identical in the two signals. The blocking matrix subtracts the input signals resulting in a reference signal that is devoid of the desired signal, and forms the input to the adaptive interference canceller.

We assume that the hearing aid at the right ear quantizes and transmits its signal to the hearing aid at the left ear where the two are combined. Let represent the reconstructed signal obtained after encoding and decoding at a rate bits per sample resulting in a distortion , where . The *forward channel* with respect to the squared error criterion can be written as [18, pages 100-101],

where , , and is zero mean complex Gaussian with variance . Recall that we model , , , and as memoryless zero mean complex Gaussian random sources for each , with independent real and imaginary parts. The rate-distortion relation for the complex Gaussian source follows from the rate-distortion function for a real Gaussian source [18, Chapter 4],

so that the distortion is obtained as . The signals and form the two inputs to the GSC.

If the PSDs , and are known, more efficient quantization schemes may be designed, for example, one could first estimate the desired signal (using a Wiener filter) from the noisy observation at the right ear, and then quantize the estimate as in [12]. However, as the PSDs are unknown in our model, we quantize the noisy observation itself.

## 3. The Binaural GSC

We first look at the case when there is no quantization and the left hearing aid receives an error-free description of . This corresponds to an upper bound in our performance analysis. We then consider the case when is quantized at a rate bits per sample.

### 3.1. No Quantization

The GSC has three basic building blocks. The first is a fixed beamformer that is steered towards the direction of the desired source. The second is a blocking matrix that produces a so-called noise reference signal that is devoid of the desired source signal. Finally, the third is an adaptive interference canceller that uses the reference signal generated by the blocking matrix to cancel out the interference present in the beamformer output.

The output of the fixed delay-and-sum beamformer is given by

where , . We can rewrite as

The blocking matrix is given by , so that the input to the adaptive interference canceller is obtained as

The adaptive filter is updated such that the expected energy of the residual given by is minimized, for example, using the normalized least mean square algorithm [19, Chapter 9]. Since does not contain the desired signal, minimizing corresponds to minimizing the energy of the interferer in the residual. Note that none of the above steps require knowledge of the PSD of the desired or interfering sources.

For our analysis, we require the optimal steady state (Wiener) solution for , which is given by

where

The GSC output can be written as

and the resulting estimation error is

where

### 3.2. Quantization at a Rate R

The beamformer output in this case is given as

Comparing (18) with (11), since , it can be seen that while the fixed beamformer preserves the desired source in the unquantized case, there is attenuation of the desired source in the quantized case. The blocking matrix produces

It is evident from (19) that due to the quantization, the reference signal is not completely free of the desired signal , which will result in some cancellation of the desired source in the interference cancellation stage. The adaptive interference canceller is given by

where

where . The GSC output in this case is

The corresponding estimation error is

where

## 4. GSC Performance at Different Bit Rates

Using (23)-(24), the behavior of the GSC can be studied at different bit rates, and for different locations of the interferer. The solid curves in Figure 3 plot the output signal-to-interference-plus-noise ratio (SINR) obtained from the binaural GSC at different bit rates for an interferer located at . The output SINR per frequency bin is obtained as

For comparisons, we also plot the output SINR obtained using a monaural two-microphone GSC (dotted line). This would be the result obtained if there was only a single hearing aid on the left ear with the two microphones separated by 8 mm in an end-fire configuration. In the monaural case, we consider a rate as both microphone signals are available at the same hearing aid. To obtain Figure 3, the relevant parameter settings were , m, m, m, and m/s. The mean input SIR and signal-to-noise ratio (SNR) were set to 0 dB and 30 dB, respectively, where

It can be seen from Figure 3 that at a rate of 5 bits per sample, the binaural system outperforms the monaural system. Note that by bits per sample we mean bits allocated to each sample per frequency bin. Figure 4 shows the performance of the binaural GSC without considering the effect of head shadow, that is, assuming that the microphones are mounted in free space. In this case, the transfer functions , , and correspond to the appropriate relative delays. The sharp nulls in Figure 4 correspond to those frequencies where it is impossible to distinguish between the locations of the desired and interfering sources due to spatial aliasing, and thus the GSC does not provide any SINR improvement. It is interesting to note that the differences introduced by head shadow helps in this respect, as indicated by the better performance at these frequencies in Figure 3.

The performance of the monaural system varies significantly based on the interferer location. When the desired source and interferer are located close together as in the case of Figure 3, the small end fire microphone array cannot perform well due to the broad main lobe of the beamformer. When the interferer is located in the rear half plane, the monaural system offers good performance, especially at high frequencies. Figure 5 plots the output SINR under the same conditions as in Figure 3 except that the interferer is now located at and thus there is a larger separation between the desired (located at and interfering sources. The monaural system (dotted line) performs better than when the interferer was located at In this case, the binaural system needs to operate at a significantly higher bit rate to outperform the monaural system, and the benefits are mainly in the low-frequency range up to 4 kHz.

For an interferer located at Figure 6 depicts the improvement in SINR averaged over all frequencies after processing by the GSC, for different values of the SIR and SNR. The improvement was calculated as

The largest improvements are obtained at low SIRs and high SNRs, where the adaptive interference canceller is able to perform well as the level of the interferer is high compared to the uncorrelated noise in the reference signal . At high SIR and low SNR values, the improvement reduces to the 3 dB gain resulting from the reduction of the uncorrelated noise due to the doubling of microphones. For low SNR values, the improvement due to the interference canceller is limited across the entire range of SIR values. However, as the SNR increases, the interference canceller provides a significant improvement in performance as can be seen in the right rear part of Figures 6 and 7. At high SNR and SIR values, a low bit rate (e.g., 4 bits per sample) results in degradation of performance as the loss due to quantization more than offsets the gain due to beamforming. At low bit rates, the reference signal , which forms the input to the adaptive interference canceller, is no longer devoid of the desired signal. This is one of the reasons for the poor performance of the binaural GSC at low bit rates as the adaptive filter cancels some of the desired signal. In fact, as observed in [20], in the absence of uncorrelated noise, the SIR at the output of the adaptive interference canceller is the negative (on a log scale) of the SIR in . At high input SIRs and SNRs, even a small amount of desired signal leakage results in a high SIR in , which in turn results in a low SIR at the output as seen in Figure 6. One approach to avoid cancellation of the desired signal is to adapt the filter only when the desired signal is not active [21]. The detections may be performed, for example, using the method of [22].

So far, we have looked at the effect of quantization at a bit-rate independently with respect to each frequency bin. In practice, the available bits need to be optimally allocated to each frequency band . The rate allocation problem can be formulated as

A uniform rate allocation across the different frequency bins cannot exploit the dependence of the output SINR on frequency as seen in Figures 3 and 5, and thus a nonuniform scheme is necessary. The distortion function does not lend itself to a closed-form solution for the rate allocation, and suboptimal approaches such as a greedy allocation algorithm need to be employed. In a greedy rate allocation scheme, at each iteration, one bit is allocated to the band where the additional bit results in the largest decrease in distortion. The iterations terminate when all the available bits are exhausted. Figure 8 shows the output SINR (averaged across all frequencies) at different bit rates for both uniform and greedy rate allocation. Here, the desired and interfering signals were assumed to be speech. The signals, sampled at 16 kHz, were processed in blocks of samples, and the results were averaged over all blocks. Figure 9 shows the PSD of a segment of the signal. It can be seen from Figure 8 that the greedy allocation (dotted) scheme results in better performance compared to the uniform rate allocation (solid) scheme. However, we note that the greedy algorithm requires knowledge of the PSDs and , and the location of the interferer.

## 5. Conclusions

A wireless data link between the left and right hearing aids enables binaural beamforming. Such a binaural system with one microphone on each hearing aid offers improved noise reduction compared to a two-microphone monaural hearing aid system. The performance gain arises from the larger microphone spacing and the ability to exploit the head shadow effect. The binaural benefit (improvement compared to the monaural solution) is largest when an interfering source is located close to the desired source, for instance, in the front half plane. For interferers located in the rear half plane, the binaural benefit is restricted to the low-frequency region where the monaural system has poor spatial resolution. Unlike the monaural solution, the binaural GSC is able to provide a uniform performance improvement regardless of whether the interferer is in the front or rear half plane.

Wireless transmission is power intensive and battery life is an important factor in hearing aids. Exchange of microphone signals at low bit rates is thus of interest to conserve battery. In this paper, the performance of the binaural system has been studied as a function of the communication bit rate. The generalized sidelobe canceller (GSC) has been considered in this paper as it requires neither knowledge of the source PSDs nor of the location of the interfering sources. Both the monaural and binaural systems perform best when the level of uncorrelated noise is low, that is, at high SNRs, when the adaptive interference canceller is able to fully exploit the availability of the second signal. At an SNR of 30 dB and an SIR of 0 dB, the binaural system offers significant gains (15 dB SINR improvement for interferer at even at a low bit rate of 4 bits per sample. At higher input SIRs, a higher bit-rate is required to achieve a similar gain.

In practice, the total number of available bits needs to be optimally allocated to different frequency bands. An optimal allocation would be nonuniform across the different bands. Such an allocation however requires knowledge of the source PSD and the location of the interferer. Alternatively, a suboptimal but practically realizable uniform rate allocation may be employed. It has been seen that such a uniform rate allocation results in a performance degradation of around 5 dB in terms of SINR compared to a nonuniform allocation obtained using a greedy optimization approach.

The main goal of this paper has been to investigate the effect of quantization errors on the binaural GSC. Several extensions to the basic theme can be followed. Topics for future work include studying the effect of reverberation and ambient diffuse noise on the performance of the beamformer. Binaural localization cues such as interaural time and level differences have been shown to contribute towards speech intelligibility. Future work could analyze the effect of quantization errors on these binaural cues.

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Srinivasan, S., Pandharipande, A. & Janse, K. Beamforming under Quantization Errors in Wireless Binaural Hearing Aids.
*J AUDIO SPEECH MUSIC PROC.* **2008, **824797 (2008). https://doi.org/10.1155/2008/824797

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### Keywords

- Sound Source
- Quantization Error
- Rate Allocation
- Microphone Array
- Microphone Signal