Weighted pressure matching with windowed targets for personal sound zones

Personal sound zones (PSZ) systems use an array of loudspeakers to render different audio signals to different users with minimum leakage between them.^1,2 To achieve this, a set of filters is used to process the audio signals that are fed to the loudspeakers. Different techniques have been proposed to compute the filters, such as beamforming,^3,4 soundfield synthesis,^5–7 energy cancellation approaches,^8,9 or hybrid approaches.^10,11 Among these, acoustic contrast control (ACC)¹² is the algorithm that can achieve highest isolation between the bright and dark zones, where the terms bright and dark zone refer to the regions where we want high and low acoustic energy, respectively.¹² However, ACC cannot synthesize a specific target response in the bright zone. To solve this limitation, the weighted pressure matching (WPM) algorithm has been proposed,¹³ which offers the possibility to render a target response in the bright zone while keeping control over the energy in the dark zone. To do so, the authors in Ref. 13 proposed a novel cost function in which a weighting parameter is used to balance the components of the energy in the dark zone and the error with respect to the desired response in the bright zone. The WPM algorithm can be formulated in time-domain,¹⁴ subband-domain¹⁵ and frequency-domain.¹⁶ In this paper, because of its simplicity, we will use the frequency-domain formulation. Nevertheless, our findings can be generalized to the other formulations of WPM.

The main advantage of WPM over ACC is that it allows us to synthesize a desired target response in the bright zone; however, this is at the cost of higher energy in the dark zone (i.e., higher interference between the users of the system).¹³ The performance of ACC and WPM has been compared under reverberant conditions in Refs. 17 and 18. The target response selected for the bright zone is a key choice for WPM systems, as different targets responses can lead to different levels of energy in the dark zone. However, this is an aspect that, to the best of our knowledge, has not been previously studied in the literature. The most usual approach in reverberant environments is to select the target as the room impulse response (RIR) produced by one of the speakers of the system in the control points of the bright zone.^{14–16,19–21} This approach aims to synthesize all the direct and reverberant components of the RIR in the bright zone while minimizing the energy in the dark zone. The late reverberation components, however, can be assumed to be diffuse for frequencies above the Schroeder frequency. We will show that there is no set of filters that can achieve high energy differences for the diffuse reverberant components in the bright and dark zones. Therefore, trying to synthesize these components in the bright zone while minimizing their energy in the dark zone does not give the best overall performance. We therefore propose a variation of the WPM approach, in which a window function is applied to the target impulse response for the bright zone. By windowing this response, we can control which reverberant components are synthesized and which are minimized in the bright zone. Windowing has been previously used in the context of PSZ systems in Refs. 18, 22, and 23. In these works, the authors propose to window the RIRs used to compute the filters. The effect of windowing in these cases is similar to regularization, as it makes the filters more robust to inaccuracies in the RIRs. The approach proposed in this paper is conceptually different, as the RIRs used to compute the filters are not windowed, but windowing is instead applied to the targets for the bright zone. We present experimental evaluations in two scenarios that show the performance obtained with different window lengths. The results indicate that windowing the target can lead to performance improvements with respect to the case without windowing. In general, it seems that the optimal window length is frequency and scenario dependent, but the improvements that can be obtained are more significant for mid-high frequencies. Moreover, the results show that the higher the room reverberation, the higher the performance improvements obtained by windowing the target. Finally, we present evaluation results that show that the improvements in the performance obtained with the proposed method are robust to perturbations in the environment.

The paper¹ is structured as follows. Section II studies the WPM algorithm. Section III presents the novel target selection for WPM. Section IV presents experimental results to show the performance of the proposed strategy to select the target under different reverberation levels. Finally, Sec. V summarizes the main conclusions.

Let us consider a PSZ system that uses an array of L loudspeakers, and where the bright and dark zones are spatially sampled using M_b and M_d control points, respectively. Let us denote $Hml,q(f)$ as the room frequency response at frequency f between the l-th loudspeaker and the m-th control point in the q-th zone, where $q∈{b,d}$⁠, and b and d are the indices of the bright and dark zones, respectively. Also, let us define $Gl(f)$ as the frequency response of the filter used to filter the signals that will be fed to the l-th speaker. From now on, we will omit index f for the sake of simplicity. Next, let us define the $Mq×L$ matrix containing the room frequency responses between all loudspeakers and all control points in the q-th zone as

Similarly, let us define the $L×1$ vector containing the frequency responses of the filters for all loudspeakers in the system as

Then, we can write the $Mq×1$ vector containing the combined frequency responses for the control points in the q-th zone as

Moreover, let us define $Dm,b(f)$ as the target frequency response that we want to synthesize at the m-th control point of the bright zone (we assume that a null target response is selected for the dark zone). Then, we can write the $Mb×1$ vector containing the target frequency responses for all the control points in the bright zone as

Once the model is presented, we describe the weighting pressure matching (WPM) algorithm, which was originally proposed in Ref. 13. The algorithm aims to find the filter coefficients g that minimize the following cost function

We can see that Eq. (5) is formed by three terms: (1) the mean energy in the dark zone, (2) the mean square error (MSE) with respect to the target frequency response in the bright zone, and (3) the energy of the filter coefficients. In Eq. (5), λ is a regularization factor that constrains the energy of the filters. Also, κ is the weighting factor satisfying $0≤κ≤1$ that is used to balance the solution, e.g., high values of κ put more effort into minimizing the mean energy in the dark zone whereas low values put more effort into minimizing the MSE in the bright zone. It is straightforward to show that the optimal filter coefficients that minimize Eq. (5) are given by,¹³ 

The target selected for the bright zone heavily influences the performance of WPM; however, this is an aspect that has not been extensively studied in the PSZ related literature. The most common approach in reverberant environments is to select the target impulse response as the delayed response from one of the loudspeakers to all of the control points in the bright zone,^{14–16,19–21} i.e.,

where $lr∈{0,…,L−1}$ is the index of the reference loudspeaker, and τ is a modelling delay that assures the causality of the filters. The target frequency response $Dm,b$ is obtained by computing the DTFT of $dm,b$ in Eq. (7). The previous selection aims to synthesize the direct propagation component and all the reverberant components produced by the reference loudspeaker in the bright zone.

Next, let us define

which is a window function centered on n = 0. From the definition of the window in Eq. (8), it is easy to see that its length is $2Lw−1$⁠; however, from now on, we are going to refer to the length of the window as L_w (i.e., the number of positive time samples in the window). Now, we alternatively propose to define the target impulse response in the control points of the bright zone as

where τ_p is the propagation delay corresponding to the direct component of $hmlr,b$⁠, and in which the window is time-shifted such that its center is located on $n=τp+τ$⁠. Then, we propose to select the target frequency response $Dm,b$ used to form $db$ as the DTFT of $dm,b(Lw)$ in Eq. (9). It is important to highlight that, contrary to what is proposed in Refs. 18, 22, and 23, no windowing is applied to the RIRs used to form matrices $Hb$ and $Hd$⁠. Also, it is relevant to note that the targets in Eqs. (7) and (9) are equivalent when $Lw=∞$⁠. However, Eq. (9) has the advantage that, by selecting the window length, we can choose which reverberant components we want to synthesize and which to minimize in the bright zone. Then, when selecting a window length that removes certain reverberant components, we are aiming to achieve a de-reverberation of the bright zone. We present in Fig. 1 a schematic to illustrate this effect. In the schematic, we show an example of a RIR between the reference loudspeaker and one control point in the dark and bright zones, and also, the target that we aim to achieve at these control points using a window of length L_w. From this schematic, it is easy to see that when $Lw=∞$⁠, we want to synthesize all the reverberant components in the bright zone while minimizing the energy of all the components in the dark zone. However, the effect is different when we select finite values for L_w. For time instants within the window, we seek to synthesize the direct propagation component and some early reflections in the bright zone, while minimizing the energy in the dark zone. For time instants after the end of the window, we want to minimize the energy of the reverberant components both in the bright and dark zones. The proposed target selection can be directly applied for the time-domain formulation of WPM¹⁴ by using Eq. (9) to form the vector with the target impulse responses in all the control points of the bright zone.

It is not initially clear whether windowing the target response will lead to performance improvements. However, above the Schroeder frequency, the late reverberation components of the RIR become diffuse.²⁴ These components can then be assumed to be of similar energy but uncorrelated between the different control points. Now, let us define $Hb,dif$ and $Hd,dif$ as the matrices containing the diffuse components of the RIR at single frequency f in the bright and dark zones, respectively. From the ACC algorithm,²⁵ we know that the maximum acoustic contrast (AC) that can be achieved between the diffuse components of the bright and dark zones is given by the highest eigenvalue of matrix $(Hd,difHHd,dif)−1Hb,difHHb,dif$⁠. For the diffuse components, we can assume that $Hb,difHHb,dif≈σI$ and $Hd,difHHd,dif≈σI$ (where σ is the energy of the diffuse field in the room).²⁶ Then, the maximum AC that can be achieved between the diffuse components of the bright and dark zones is approximately 1 (in linear units). This shows that there is no set of filters that is able to provide a significant energy difference between the diffuse components in the bright and dark zones. Thus, selecting a target that tries to synthesize all of the reverberant components, including the diffuse components in the bright zone (i.e., with $Lw=∞$⁠) is not a good choice. In Sec. IV, we will present evaluation results that show that improvements in the performance can be achieved by windowing the target, i.e., aiming the minimization of the energy of the diffuse components in both zones.

In this section, we evaluate the performance of the proposed target selection for different window lengths and for two scenarios with different reverberation times. First, we define the experimental setup and the metrics used for the evaluations. Then, we present evaluation results that show the effect on the performance of windowing the target response for the bright zone. Finally, we evaluate the robustness to perturbations in the filters computed with a windowed target.

The experimental evaluations have been carried out in two scenarios using rooms with different reverberations levels:

• Scenario 1: Office-like room at the Institute of Telecommunications and Multimedia Applications (iTEAM) of size $7.2×11.72×2.63$ m and reverberation time $T60=0.5$ s. The Schroeder frequency for this room is 95 Hz. In this scenario, a linear array of 8 two-way loudspeakers with an inter-element distance of 0.18 m has been used [see Fig. 2(a)]. The setup is formed by one bright and one dark zone [Fig. 3(a)]. In each zone, two different grids of microphones have been used for spatial sampling, such that the RIRs measured with the control grid are used to compute the filters and the RIRs measured with the validation grid are used to evaluate the filters. The highest frequency that can be rendered without spatial aliasing is 1498 Hz.²⁷ 

• Scenario 2: Listening room at iTEAM, which is a rectangular room of size $9.07×4.45×2.65$ m with acoustically treated walls and reverberation time $T60=0.18$ s. The Schroeder frequency for this room is 137 Hz. Similarly to scenario 1, a linear array of 8 two-way loudspeakers with an inter-element distance of 0.18 m [Fig. 2(b)] has been used and two zones are considered, which are spatially sampled using two different grids of microphones [see Fig. 3(b)]. The highest frequency that can be rendered without spatial aliasing is 1401 Hz.²⁷ 

In all cases, the RIRs were measured using the exponentially swept sine technique²⁸ with a sampling frequency of 44 100 Hz. In the following, we use notation $· ¯$ to denote the elements related to the validation grid of microphones. Due to the effects of the spatial aliasing in the arrays of loudspeakers, we limit the study to the frequency range 150–2000 Hz.

The optimal filters have been computed using Eq. (6), where the weighting parameter κ is set to 0.5, if not indicated otherwise. Moreover, the regularization parameter λ is selected such that the array effort [which will be defined next in Eq. (12)] is upper bounded as $AE≤AEmax$⁠. The value of AE_max will be indicated in each case. The target for the bright zone has been selected using Eq. (9), where we select $wLw$ as a Tukey window with cosine fraction $α=0.3$²⁹ and the reference loudspeaker is $lr=3$ [as indicated in Figs. 2(a) and 2(b)]. We select a Tukey window because, when applied to the target response, it does not modify the amplitude of the first early reflections of the response and it leads to a smooth transition between the reverberant and null components of the target. Other types of windows have also been investigated but gave similar results. Moreover, when $Lw<∞$⁠, we apply a $13$ octave band equalizer to the target to obtain the same energy in the bright zone as for the case with $Lw=∞$⁠. In Fig. 4(a), we show an example of the causal part of a Tukey window with $Lw=529$ (12 ms), and Figs. 4(b) and 4(c) show examples of a target impulse response and a frequency response in the bright zone for scenario 1 with L_w = 529 (12 ms) and $Lw=∞$⁠.

Next, we present the metrics used for the evaluations. First, we define the MSE in the bright zone as

where the target is selected using the same criterion used to compute the filters, meaning that the same window $wLw$ is applied to obtain the targets $db$ and $d¯b$ that are used to compute and to evaluate the filters, respectively. Next, we define the mean energy in the dark zone as

and the array effort as

where E_ref is the energy required by the reference loudspeaker to provide the same energy in the bright zone as the array of loudspeakers using the set of filters g.

So far, we have defined metrics that are directly related to the three components of the cost function in Eq. (5), and whose influence can be adjusted with λ and κ. Now, we define the acoustic contrast (AC) as

which is a metric related to the level of acoustic isolation between the bright and dark zones. The AC is not directly present in cost function [Eq. (5)] but is commonly used as a performance indicator for PSZ systems. Finally, we define the mean Kurtosis of the measured RIRs at sample time n as

where $μml,q(n)$ and $σml,q(n)$ are the mean value and the standard deviation of $h¯ml,q$⁠, respectively, over the interval $n,…,n+Ls−1$⁠, and the expected value for time n is computed over the interval $n,…,n+Ls−1$⁠. In Eq. (14), N_h denotes the total number of RIRs for which the mean Kurtosis is computed. The Kurtosis is a measure of the tailedness of a sample distribution. For a Gaussian probability density function (PDF), the Kurtosis value is 0. The authors in Ref. 30 show that the Kurtosis is closely related to the diffuseness of late reverberations. In particular, they show that the early part of the RIR, containing the direct propagation component and strong deterministic reflections, is unlikely to have a Gaussian distribution, so it presents high Kurtosis levels. However, the late diffuse components of the RIR present Kurtosis values close to 0. The motivation is that the reflection density is high for the diffuse part of the RIR, which makes it more likely that the RIR has a Gaussian distribution. For the computation of the Kurtosis, we used a length segment of L_s = 882 (20 ms) and all the RIRs have been aligned, such that their direct propagation component is located at sample index n = 0. From now on, we use a third-octave averaging³¹ for all frequency-domain plots to improve the readability of the results.

In this subsection, we present experimental results to evaluate the performance of the proposed target selection for scenario 1.

First, we compare the proposed target selection with the approach employed in Refs. 18, 22, and 23 in which not only the target used to form $db$ is windowed but also the RIRs used to form $Hq$⁠. We show in Fig. 5 the performance of the proposed approach as a function of L_w in terms of: mean energy in the dark zone [in Figs. 5(a) and 5(d)], MSE in the bright zone [in Figs. 5(b) and 5(e)], and array effort [in Figs. 5(c) and 5(f)]. Similarly, the performance of the approach used in Refs. 18, 22, and 23 is shown in Fig. 6. As expected, the performance in both cases is identical for very long windows, e.g., 200 ms. We can see in Fig. 6 that windows shorter than 100 ms highly degrade the performance for the case with windowed RIRs and targets. This is because relevant reverberation components of the RIRs are not taken into account in the optimization. On the contrary, the proposed approach takes into account all the reverberant components of the RIR in the optimization but tries to minimize the energy of some of these components. The results in Fig. 5 show that short windows can offer better performance in this case. These results indicate that the performance of the system can be improved with the proposed approach by selecting short windows, while this is not possible with the approach used in Refs. 18, 22, and 23. From now on, we focus on further evaluating the performance of the proposed method.

Next, we discuss the effect of the window length in the performance of the proposed target selection. In Fig. 5, the metrics in the top and bottom rows are computed with AE_max = 0 dB and AE_max = 15 dB, respectively, but the performance for both effort constraints is equal for frequencies above 800 Hz. The motivation is that the matrix that must be inverted in Eq. (6) is well conditioned above 800 Hz, and then, even with a regularization parameter of λ = 0 does not lead to a high array effort. For frequencies below 800 Hz, the performance is different for the two effort constraints, however, the effect of the window length in the performance is similar in both cases. Then, the following analysis is valid for the two studied constraints. We can see in Figs. 5(b) and 5(e) that very short windows (of 1 ms or less) lead to an MSE that is at least 10 dB worse than for longer windows, while the energy generated in the dark zone is similar [in Figs. 5(a) and 5(d)]. This degradation appears because very short windows aim to suppress the first early reflections, and also, to equalize the response of the reference loudspeaker in the bright zone, which is too restrictive and leads to important errors in the bright zone. Also, we can see that, in general, short windows (of about 12 ms) present a significantly lower energy in the dark zone than longer windows. The motivation is that these windows remove the diffuse components from the target in the bright zone, and then, the designed filters are capable of minimizing the diffuse field, both in the bright and dark zones. Then, the improvements obtained by windowing the target are not obtained because we force the cancellation of some high energy early reflections, but because we target the cancellation of the diffuse reverberant tail. In particular, for frequencies in the range of 150–200 Hz, 400–500 Hz, and above 700 Hz, the improvements in the energy in the dark zone are not at the cost of substantially higher MSE. For frequencies 200–400 Hz and 500–700 Hz, short windows lead to lower energy in the dark zone at the cost of worse MSE. For these frequency bands, lower energy in the dark zone may be achieved for long windows by tuning the weighting parameter κ; therefore, at this point, it is not yet clear if real improvements in the performance are achieved with short windows in all the studied frequency range.

Now, we present additional results where we compare the following configurations:

• Win. Target (WT): L_w = 529 (12 ms) and $κ=0.5$⁠.

• Full Target (FT): $Lw=∞$ and $κ=0.5$⁠.

• Full Target with Frequency Dependent κ (FT-FD): $Lw=∞$ and κ selected as shown in Fig. 7 to achieve the same MSE as WT.

It is important to note that in order to fairly determine which window length is able to provide lower energy in the dark zone, we must compare their performance for the case in which their MSE is equal. This means that reductions in the energy in the dark zone for one of the configurations compared to another are then not at the cost of higher MSE in the bright zone. This is the motivation to include FT-FD in the comparison. The performance of the three configurations is shown in Fig. 8 in terms of: mean energy in the dark zone [in Fig. 8(a)], MSE in the bright zone [in Fig. 8(b)], array effort [in Fig. 8(c)], and acoustic contrast [in Fig. 8(d)]. We only present results for AE_max = 15 dB, as we can see in Fig. 5 that the effect of the window length on the performance is similar for different effort constraints. We see that WT can offer lower energy in the dark zone than FT and also lower or equal MSE in certain frequency bands, e.g., 150–200 Hz, 400–500 Hz, 700–900 Hz, and 1.1–1.5 kHz. This means that a longer window cannot offer better performance than Lw = 529 for these frequencies, neither in terms of energy in the dark zone nor in terms of MSE (even if we tune κ). Next, if we compare WT and FT-FD, we can see that WT leads to lower energy in the dark zone for all the studied frequencies except for the band 200–360 Hz, where FT-FD presents slightly lower energy. For example, WT leads to almost 6 dB lower energy in the dark zone than FT-FD around 1 kHz, and 5 dB lower energy around 180 Hz. Obtaining lower energy in the dark zone does not necessarily mean that the acoustic contrast is higher, since the energy reduction could be produced because the energy in the bright zone is also lower. This aspect is especially important for the comparison of targets windowed with different window lengths, as these targets present different energy levels for the bright zone at individual frequencies (although their mean energy is equal across $13$ octave bands). Nonetheless, the results in Fig. 8(d) indicate that, in general, the acoustic contrast follows an inverse trend compared with those for the energy in the dark zone, since for the same MSE level, WT can offer higher contrast than FT-FD across almost all of the studied frequencies (with improvements of more than 5 dB). Moreover, the array effort required by WT is lower than for FT and FT-FD for frequencies above 700 Hz. The MSE is almost the same for WT and FT-FD, as expected, and is broadly similar for FT, although slightly higher or lower for different frequencies. From these results, we can conclude that, for scenario 1, windowing the target response with a short window of length L_w = 529 (12 ms) leads to significantly better performance than the case without windowing for most of the studied frequencies. This indicates that targeting the minimization of the energy of the diffuse reverberant components in the bright and dark zones can lead to great improvements in the performance for this scenario.

Finally, Fig. 9 shows the mean Kurtosis of the RIRs for scenario 1. We can observe that the Kurtosis has high values for the early part of the RIR, where the direct component and the early reflections are located, and drops to values below 3 after about 20 ms, indicating that the reverberant components of the RIRs can be assumed diffuse, with a Gaussian PDF, after this time.³⁰ It is for window lengths greater than about 20 ms that the performance starts to deteriorate in Fig. 5. Then, the Kurtosis provides a useful metric for the selection of the window length.

Next, we study the performance of the proposed target selection for scenario 2, with the lower reverberation time. Figure 10 shows the performance as a function of L_w in terms of: mean energy in the dark zone [in Figs. 10(a) and 10(d)], MSE in the bright zone [in Figs. 10(b) and 10(e)], and array effort [in Figs. 10(c) and 10(f)]. In Fig. 10, the metrics in the top row are again computed with AE_max = 0 dB, and AE_max = 15 dB for the bottom row. As for scenario 1, the performance is similar for both constraints at frequencies above 800 Hz, as are the effects of the window length for frequencies below 800 Hz in both cases. We can see in Figs. 10(b) and 10(e) that very short windows (of 1 ms or less) lead to a much higher MSE than longer windows, while the energy generated in the dark zone is similar [in Figs. 10(a) and 10(d)]. Similarly to scenario 1, this is produced because very short windows aim to suppress the first early reflections and to equalize the response of the reference loudspeaker in the bright zone, which leads to high MSE. Then, selecting a window that is too short can significantly degrade the performance. The mean energy in the dark zone is lower for shorter windows (of about 8 ms) than longer ones, particularly for frequencies above 1 kHz. However, we can also see in Figs. 10(b) and 10(e) that, in general, the MSE is higher for short windows. In this case, the lower energy generated in the dark zone when using short windows comes at the cost of higher MSE in the bright zone.

To further study if windowing the target offers performance improvements in this scenario, we present additional results in Fig. 11 where we compare three configurations which are similar to those used above:

• Win. Target (WT): L_w = 353 (8 ms) and $κ=0.5$⁠.

• Full Target (FT): $Lw=∞$ and $κ=0.5$⁠.

• Full Target with Frequency Dependent κ (FT-FD): $Lw=∞$ and κ selected as shown in Fig. 7 to achieve the same MSE as WT.

We again include FT-FD to fairly determine which window length is able to provide lower energy in the dark zone, while giving the same MSE. The performance of these configurations is shown in Fig. 11 in terms of: mean energy in the dark zone [in Fig. 11(a)], MSE in the bright zone [in Fig. 11(b)], array effort [in Fig. 11(c)], and acoustic contrast [in Fig. 11(d)]. As for scenario 1, we only present results for $AEmax=15$ dB to avoid redundancy. Above about 700 Hz, the energy in the dark zone is lower for WT than for FT, but even lower than for FT-FD above 1 kHz. In this case, however, FT has a significantly lower MSE than WT or FT-FD. From these results, we can conclude that windowing the target response for the bright zone with a short window of 8 ms can lead to better performance in terms of acoustic contrast than the case without windowing for frequencies above 1 kHz in this scenario. These results and the results in Sec. IV C show that the optimal window length is frequency and scenario dependent. Comparing the results for the two scenarios, we can see that the higher the reverberation in the room, the greater the improvement obtained by windowing the target response in the bright zone.

Figure 9 shows the mean Kurtosis of the RIRs for scenario 2, which has high values for times smaller than 10 ms, while it drops below 3 for times greater than about 10 ms. This indicates that the reverberant components of the RIRs are diffuse after about 10 ms, which is the window length after which the performance starts to degrade in Fig. 10, again illustrating the use of the Kurtosis in estimating the window length, although only at higher frequencies, in this case, with a short reverberation time.

We now evaluate whether the improvements obtained by windowing the target are robust to perturbations in the environment. To this end, we present in Fig. 12 evaluations results for scenario 1, where the filters are computed with the RIRs measured at the control points without any perturbation [as in Fig. 13(a)] and evaluated using the RIRs measured at the control points when perturbations in the RIR, due to two people located within the zones, are present [as in Fig. 13(b)]. The control filters are those calculated without any perturbations, as in Sec. IV C, but now the performance, as shown in Fig. 12, has been calculated after these perturbations in the RIR. For comparison purposes, we also include in Fig. 12 the performance of the filters when evaluated without perturbations (as in Sec. IV C). We can see that the perturbations have generally increased the energy in the dark zone and the MSE in the bright zone with respect to the case without perturbations. The mean energy in the dark zone is still significantly smaller for the windowed target in WT than it is without the window, and the MSE is also again broadly similar in the two cases. We can thus conclude that the performance improvements obtained by selecting a short window for scenario 1 are robust to perturbations in the environment.

In this paper, we proposed a novel approach to select the target response in the bright zone for the weighting pressure matching (WPM) algorithm in personal sound zones systems. In previous works, the target for the bright zone has generally been selected to be the room impulse response (RIR) from one loudspeaker to all the control points in the bright zone. The aim is thus to synthesize the direct propagation component and all the reverberant components in the bright zone, while minimizing the energy of all components in the dark zone. The late reverberant components, however, are diffuse above the Schroeder frequency and it is shown that there is no set of filters that can lead to high energy differences between the diffuse reverberant components in the bright and dark zones, so trying to synthesize all of the reverberation components in the bright zone while minimizing their energy in the dark zone is not a good strategy. Alternatively, we proposed to window the RIRs from one loudspeaker to all the control points in the bright zone and use these responses as target for the bright zone. This approach allows us to select which reverberant components we want to synthesize in the bright zone by choosing the window length. Experimental evaluation results in two rooms with different levels of reverberation show the effect of windowing the target response. The results showed that windowing the target response can lead to lower energy in the dark zone and higher acoustic contrast than the case without windowing, with similar mean square error (MSE) in the bright zone in the case of the more reverberant room and with similar array effort. Especifically, improvements of up to 6 dB in the acoustic contrast are observed for a room with $T60=500$ ms when a window length of 12 ms is used in the target impulse responses. The window length that offers best performance is, in general, both frequency and scenario dependent. The Kurtosis of the RIRs, which is related to their level of diffuseness, is also shown to give a good indication of the best window length. Also, we observed that greater improvements with respect to the case without windowing are obtained for mid-high frequencies. The improvements obtained by windowing the target are greater for scenarios with high reverberation level and are also robust to perturbations in the room impulse responses.

Vicent Molés-Cases was supported by the Spanish Ministry of Education through Grant No. FPU17/01288. Stephen J. Elliott was supported by the EPSRC DigiTwin project (EP/R006768/1). Jordan Cheer was supported by the Intelligent Structures for Low Noise Environments (ISLNE) EPSRC Prosperity Partnership (EP/S03661X/1). This research was supported by the Spanish Ministry of Science, Innovation and Universities through Grant No. RTI2018-098085-B-C41 (MCIU/AEI/FEDER, UE). Open access was funded by Universitat Politècnica de València through Grant No. PAID-12-21.