Bluetooth Medical Sensor Node

Beamforming In Wireless Acoustic Sensor Networks

Xinwei Guoa,b, Minmin Yuanc, Chengshi Zhenga,b,∗, Xiaodong Lia,b aKey Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy

Of Sciences, 100190, Beijing, China

cResearch Institute of Highway Ministry of Transport, 100088, Beijing, China

Abstract

This paper derives the analytical solution of a novel distributed node-specific block-diagonal linearly constrained minimum variance beamformer from the centralized linearly constrained minimum variance (LCMV) beamformer when considering that the noise covariance matrix is block-diagonal. To further re- duce the computational complexity of the proposed beamformer, the Sherman- Morrison-Woodbury formula is introduced to compute the inversion of noise sample covariance matrix. By doing so, the exchanged signals can be computed with lower dimensions between nodes, where the optimal LCMV beamformer is still available at each node as if each node is to transmit its all raw sensor signal observations. The proposed beamformer is fully distributable without imposing restrictions on the underlying network topology or scaling computa- tional complexity, i.e., there is no increase in the per-node complexity when new nodes are added to the networks. Compared with state-of-the-art distributed node-specific algorithms that are often time-recursive, the proposed beamformer exactly solves the LCMV beamformer optimally frame by frame, which has much lower computational complexity and is more robust to acoustic transfer func- tion estimation error and voice activity detector error. Numerous experimental results are presented to validate the effectiveness of the proposed beamformer.

Keywords:

distributed beamforming, node-specific, speech enhancement, wireless acoustic sensor networks.

1. Introduction

Wireless acoustic sensor networks (WASNs) generally consist of several nodes, where each node has one or many sensors, a processing unit, and a wireless communication module allowing them to exchange data. Compared with the traditional and single sensor array , WASNs can physically cover a wider area, which have more opportunity to select a subset of nodes close to some target sources, and thus higher signal-to-noise ratio (SNR) and direct-to-reverberant ratio (DRR) can be expected [2, 3]. As the next-generation technology for au- dio acquisition and processing, WASNs have many potential applications, such as binaural hearing aids [4, 5, 6], (hands-free) speech communication systems [7, 8, 9], and acoustic monitoring systems [10, 11, 12, 13].

In principle, all the sensor signal observations from different nodes can be transmitted to a fusion center, and then an optimal beamformer can be com- puted, where this approach is known as the centralized estimation [14, 15, 16].

The centralized estimation requires a large communication bandwidth, a large transmission power consumption at the individual nodes, and a nonnegligible computational complexity at the fusion center. However, both the power and the communication bandwidth resources in WASNs are often limited. Further- more, in many WASNs applications, the fusion center may be undesirable due to privacy considerations . A trivial solution is obtained by only utilizing the local sensor signal observations at a single node without any communication link with other nodes. Whereas, this solution cannot utilize the entire informa- tion from the WASNs and hence is only sub-optimal. A promising solution is to develop a suitable distributed approach, which often has three stages . At the first stage, each node processes its own sensor signal observations to obtain some compressed signals. At the second stage, only these compressed signals are transmitted to reduce the communication bandwidth. At the last stage, a

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target signal is obtained by merging all these compressed signals properly. Distributed speech enhancement algorithms can be roughly divided into two main categories: node-specific and non-node-specific. For the node-specific es- timation algorithms, each node in the WASNs can estimate a different target signal, that is to say, a target source for one node may be an interfering source for another node, and vice versa. The node-specific estimation problem is intrinsic in a blind beamforming framework where the acoustic transfer functions (ATFs) between the target sound sources and the sensors are generally unknown. For these blind beamformers, some subspace estimation algorithms can be used to estimate the subspace of the ATFs [19, 20], and then the target signal can be estimated as it is observed at a reference sensor. If each node in the WASNs chooses its own local sensor as reference, the spatial information of the target source can be preserved. Therefore, the node-specific estimation algorithms are preferred for many practical applications .

Several non-node-specific speech enhancement algorithms have been pre- sented in [17, 18], and , where different nodes shared a common reference sensor. In , each node was assumed to have one sensor and a distributed delay and sum (DDS) beamformer in a randomly connected network was pro- posed.

The DDS with the randomized gossip algorithm is an iterative algorithm for solving averaging consensus problems in a distributed way, where all nodes’ outputs are expected to converge to the same optimal average value.

The DDS typically needs many iterations to converge to the optimal solution, as well as multiple (re)-broadcasts of the intermediary variables.

The Dds

is more suitable for estimating the fixed or slowly varying parameters . In , a time-recursive distributed generalized sidelobe canceler (DGSC) was pro- posed for a fully connected network. The DGSC has two components including constraints subspace and its corresponding null-space, and updates the filter coefficients during speech-absent segments. The DGSC needs to transmit S- dimensional raw signal observations, with S the number of target sources, in addition to the compressed signals to construct the constraints subspace com- ponent.

The DGSC requires a larger communication bandwidth than those

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distributed algorithms transmitting only the compressed signals when one aims to get the estimation of the S target source signals separately. In , the pro- posed block-diagonal LCMV (BD-LCMV) beamformer utilizes a set of linearly equality constraints to reduce the full-element noise sample covariance matrix to a block-diagonal form, and the imposed block-diagonal structure of the esti- mated sample covariance matrix results in a naturally separable objective func- tion. Then the distributed optimal problem can be solved by the primal-dual method of multipliers (PDMM) . The BD-LCMV requires lots of iterations to achieve high performance, and therefore we need to make a trade-offbetween per-frame optimality and communication overhead in practice.

Several node-specific estimation algorithms have been proposed in [4, 5, 14, 24, 26, 27], and , where two main criteria including the minimum mean square error (MMSE) and the minimum variance distortionless response (MVDR) are used. The mean square error (MSE) between the output signal and the de- sired signal comprises two components, namely the desired signal distortion and the residual noise . The MVDR, first proposed by Capon , minimizes the noise power at the output signal while maintaining a distortionless response towards the desired direction. Er and Cantoni generalized the single dis- tortionless response to a set of linear constraints, and denoted this beamformer as LCMV.

In , a distributed node-specific speech enhancement algorithm was pro- posed using the MMSE criterion in a 2-node network for binaural hearing aids applications. The node-specific estimation is required to preserve the auditory cues at the two ears. This method relies on the speech-distortion-weighted mul- tichannel Wiener filter (SDW-MWF), and was referred to as the distributed MWF (DB-MWF). In , an iterative distributed MVDR (DB-MVDR) beam- former was introduced for a similar binaural hearing aids setting. Both meth- ods assume a single target source to obtain convergence and optimality, and are equivalent when the trade-offfactor between noise reduction and distortion is zero in the SDW-MWF. A more general case was presented in , , and , where multiple target sources and J(J ≥2) nodes are considered in a

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so-called distributed adaptive node-specific signal estimation (DANSE) scheme. The scheme considers each node in the WASNs as a data sink, gathering the compressed signals from other nodes, and then estimates the optimal filter co- efficients in an iterative fashion. In and , the algorithms were proposed for a fully connected network and a network with a tree topology (T-DANSE), respectively. In , the algorithm is topology-independent (TI-DANSE). The TI-DANSE algorithm has a slower convergence rate compared to and , and requires a larger number of frames to obtain near optimal performance . In , a distributed LCMV beamformer referred to as LC-DANSE was proposed by combining the DANSE scheme with the LCMV beamformer. For the DANSE algorithms in , , and , they attempt to align the signal components from the same source in different microphone signals. However, the alignment of the signal components is only possible when the filter length is at least twice the maximum time difference of arrival (TDOA) between all the sensors. This means that in general, the noise reduction performance degrades with increasing TDOA and a fixed filter length . For the LC-DANSE algo- rithm, the raw signal observations at one node and the compressed signals from other nodes are concatenated into a new vector. In fact, the new vector still has very high dimensions, especially when the number of target sources S is large, and requires a high complexity to compute the inverse of its covariance matrix.

Besides, both the DANSE and the LC-DANSE algorithms are time-recursive and require multiple frames to reach optimality, which incurs a slow tracking performance . In , each node was assumed to have more than one sen- sor. The recursive estimation of the inverse noise or noisy sample covariance matrix is structured as a consensus problem and is realized in a distributed manner via the randomized gossip algorithm for arbitrary topologies, similar to . In each iteration, each node needs to transmit M-dimensional, with M the total number of sensors in the WASNs, signals to obtain the product of the M × M inverse sample covariance matrix and the M-dimensional sensor signal observations. The communication cost may be higher than that of the centralized algorithm, and the convergence error accumulates across time when

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the aforementioned product is not accurately estimated. In this paper, we propose a distributed node-specific block-diagonal lin- early constrained minimum variance (DNBD-LCMV) beamformer. The DNBD- LCMV utilizes a set of linear equality constraints to reduce the full-element noise sample covariance matrix to a block-diagonal form and then its analyt- ical solution can be derived from the centralized LCMV beamformer directly.

The inverse noise sample covariance matrix at each node is proposed to update by the Sherman-Morrison-Woodbury formula and is used to compute the exchanged signals. The proposed DNBD-LCMV can significantly reduce the number of signals exchanged between nodes, yet obtains the optimal LCMV beamformer at each node as if each node can transmit its all raw sensor signal observations. The DNBD-LCMV is fully distributable for any network topology and is completely scalable, i.e., there is no increase in the per-node computa- tional complexity when new nodes are added to the networks. Compared with the state-of-the-art distributed node-specific algorithms, the DNBD-LCMV ex- actly solves the LCMV beamformer optimally in each frame, which has much lower computational complexity and is more robust to ATF estimation error and voice activity detector (VAD) error.

The remainder of this paper is organized as follows. In Section 2, the signal model is introduced, and the centralized LCMV is also presented in this section. In Section 3, the DNBD-LCMV and extension of DNBD-LCMV are shown, and its computational complexity and communication bandwidth are analyzed in Section 4. The experimental results are presented in Section 5 and some conclusions are given in Section 6.

2.1. Signal Model

We consider the WASNs with J sensor nodes, where the set of nodes is de- noted as J = {1, · · · , j, · · · , J}. Each node j is equipped with Mj microphones and thus the total number of microphones is M = PJ

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speech enhancement problem is often formulated in the short-time Fourier trans- form (STFT) domain and the vector y (f, l) is given by

(1)

where f denotes the frequency index, and l denotes the time-frame index. yj (f, l) is an Mj × 1 vector consisting of locally received microphone signals at the jth node, and the superscript T denotes the transpose operator. y (f, l)

(2)

where s (f, l) is a signal vector containing S speech sources, n (f, l) is a noise

(3)

is a full-rank M × S ATF matrix. In particularly, Hj (f, l) is the ATF matrix between the S speech sources and the Mj microphones at the jth node. In the following, H (f, l) can be approximated as time-invariant in each frame and all derivations refer to a single frequency bin. The frame index l and the frequency index f will be omitted when no confusion arises.

2.2. Centralized Lcmv Beamforming

For the centralized LCMV beamformer, node j applies a M-dimensional estimator wj to the M-dimensional microphone signals y to obtain the node-

Specific Output Dj = Wh

j y, where the superscript H denotes the conjugate transpose operator. wj can be obtained from the following general optimization

Nnh

is the noise covariance matrix and E{·} denotes the expected value operator. gj is an S ×1 desired response vector for the S speech

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sources. Its entries usually consist of ones and zeros to preserve the target sources and eliminate other interfering sources simultaneously. The solution of

Nnh

−1 gj.

(6)

where gj (k) and s (k) are the kth entries of gj and s, respectively. The super- script “∗” marker denotes the conjugate operator. Equations (5) and (6) require each node to have access to all microphone signals y to estimate Rnn and then obtain dj. Therefore, all the locally re- ceived signals yj at the jth node need to be transmitted, which results in a large communication bandwidth and a large transmission power in the WASNs.

Besides, the computational power grows dramatically with the increase of M when computing the inversion of Rnn.

3. Method

In the previous section, it is assumed that each node transmits all its mi- crophone signals to every other node in the WASNs such that each node can compute (5) and (6). We now look, instead, to the case where each node only transmits a linearly compressed version of its microphone signals by means of the distributed node-specific block-diagonal LCMV (DNBD-LCMV) beamformer.

For the sake of an easy exposition, we first assume that the ATF matrix H is known, and describe the DNBD-LCMV for a fully connected network, where each node is able to directly communicate with every other node in the WASNs.

Then, all derivations are extended to a blind beamforming framework and any topology, similar to .

3.1. Dnbd-Lcmv With Known Atf Matrix

If the reverberation time of a room is moderate or large enough and/or the noise is far away from the microphones, i.e., the distance between the noise and the microphones is larger than the reverberation radius, its reverberant sound field is diffuse, homogenous, and isotropic and . In this case, the normalized correlation Ωm,m1 (f) between two microphones m and m1 with

(7)

where c = 343 m/s is the sound speed. The correlation can be roughly divided into two frequency regions: one highly correlated at low frequencies and the other much less correlated at high frequencies. The boundary between the two regions occurs at the first zero-crossing frequency fc = c/ (2λm,m1). When the distance λm,m1 is large, the frequency fc is small. For example, fc equals 171.5 Hz for λm,m1 = 1 m.

For the WASNs, the microphones within a node are often nearby, whereas the microphones from different nodes are further away. The noise can be assumed to be uncorrelated across the different nodes , and then Rnn (l) has the

Following Block-Diagonal Form Approximately

Rnn (l) = Blockdiag (∆nn,1 (l) , · · · , ∆nn,j (l) , · · · , ∆nn,J (l))

0M1×Mj

... ... ...

... ...

0Mj×Mj

... ... ...

... ...

(8)

where ∆nn,j (l) is the noise covariance matrix at the jth node, and 0Mj1×Mj2 is an Mj1 × Mj2 null matrix. Note that the noise covariance matrix used in the existing node-specific algorithms, such as [14, 24, 26, 27], and , is full-element. In the non-node- specific algorithm BD-LCMV , the block-diagonal Rnn (l) is adopted and

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the weight vector associated with the jth node is given by

(9)

where µ (l) is a Lagrange multiplier shared by all nodes. Then the dual opti- mization problem is introduced to compute the optimal µ (l) and is solved by

J (L)Yj(L) Is Exchanged Between Nodes And

the output dj(l) is obtained by summing all ej(l). In this paper, the proposed beamformer has a completely new scheme. From (5) and (8), the weight vector corresponding to the lth-frame microphone signals

Nn,1 (L) H1

...

Nn,J (L) Hj

...

−1

gj.

(10)

The S-dimensional compressed signals zj (l) and the S × S matrix Dj (l)

J ∆−1

nn,j (l) Hj.

(11)

From (6) and (10), the node-specific output can be rewritten as

Where ˜Z (L) Is The Product Of D−1 (L) And Z (L),

˜z (l) = D−1 (l) z (l) .

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In particularly, D (l) and z (l) are obtained by summing all zj (l) and all Dj (l)

J=1

zj2 (l) .

(14)

The above derivations assume that the noise covariance matrix ∆nn,j (l) at the jth node is perfectly known. However, for practical applications, ∆nn,j (l) is unknown and needs to be estimated from the noisy observations. This often requires a hard or soft VAD to determine whether the speakers are present or not. It is noted that the design of a VAD mechanism is a hot research topic on its own, and is out of the scope of this paper [18, 24, 35]. In the following, two methods including non-recursive (moving average) smoothing and first-order recursive smoothing are considered to estimate ∆nn,j (l).

3.1.1. Non-Recursive Smoothing Method

For the non-recursive smoothing method, the set of microphone signals frames and the set of noise-only frames for the current block of microphone signals are denoted by Ly and Ln (Ln ⊆Ly), respectively. The block has |Ly| frames microphone signals and |·| denotes the cardinality of a set. The noise covariance matrix at the jth node can be estimated using the set of noise-only

Yj (Ln) Yh

j (ln) .

(15)

Rigorously, all the estimated values need to use “b” to distinguish them from their true values. Analogously to ∆nn,j (l) in (15), we omit “b” in the following for the sake of brevity when no confusion arises. For the current block of mi- crophone signals, ∆nn,j (l) only needs to be estimated once. From (11), Dj (l) also needs to be estimated once.

We assume that each block has B frames microphone signals, i.e., |Ly| = B. The scheme of the DNBD-LCMV with the non-recursive smoothing method is shown in Fig. 1 and it consists of the following steps: 1) Initialize: i ←0.

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Figure 1: The scheme of the DNBD-LCMV with the non-recursive smoothing method for the current block of microphone signals (J = 2). Each node j computes its node-specific output dj (l) using its own S-dimensional compressed signals zj (l) and S-dimensional compressed signals transmitted by the other node (Dj (l) only needs to be transmitted once for the current block of microphone signals).

2) Each node j ∈J performs the following operation cycle: • Collect the current block of microphone signals yj (l) , l ∈Ly = {iB, iB + 1, · · · , (i + 1) B −1}.

• For the current block of microphone signals, ∆nn,j (l) is estimated with (15).

J ∆−1

nn,j (l) is applied to Mj-dimensional microphone signals yj (l) to obtain the S-dimensional compressed signals zj (l) with (11). Besides, Dj (l) can also be obtained.

• zj(l) and Dj (l) are transmitted. In particularly, Dj (l) only needs to be transmitted once for the current block of microphone signals. • Each node can have access to all zj(l) and Dj (l), and computes z(l) and D (l) with (14). Then, the inverse matrix D−1 (l) is multiplied with z(l) to obtain ˜z(l). Finally, gj is applied to ˜z(l) to obtain dj(l).

3) i ←i + 1. 4) Return to step 2). From Fig. 1, for each block of microphone signals, Dj (l) only needs to be

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transmitted once. Since Dj (l) is a Hermitian matrix and its main diagonal entries are real numbers, the transmission of Dj (l) results in a total of S2 transmitted real numbers for each frequency bin per block. The transmission cost is slight and can be neglected.

3.1.2. First-Order Recursive Smoothing Method

For the first-order recursive smoothing method, the noise sample covariance

(16)

where Pj (l) = 1 when speech component is detected in the lth-frame micro- phone signals yj (l); and Pj (l) = 0, otherwise. When the binary VAD decision is replaced by a soft speech presence probability (SPP) , Pj (l) can vary from 0 to 1, which will not be further considered here. α is a forgetting factor ranging from 0 to 1.

When Pj (l) = 1, ∆nn,j (l) is not updated, i.e., ∆nn,j (l) = ∆nn,j (l −1). Similar to Section 3.1.1, only the transmission of zj(l) is required. While, for Pj (l) = 0, i.e., yj(l) = nj(l), the current noise-only frame is included to update

∆Nn,J (L) = Α∆Nn,J (L −1) + (1 −Α) Yj (L) Yh

j (l) .

(17)

A naive solution is obtained by transmitting Dj(l) and zj(l) for each noise-only frame, where a total of (S2 + 2S) transmitted real numbers and the inversion operation of ∆nn,j (l) are required. The communication cost and computational load are huge and need to be further reduced.

With the help of the Sherman-Morrison-Woodbury formula , the inversion

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We further define ¯cj (l) , cj (l), and ˜cj (l) as follows:

˜Cj (L) = ∆−1

nn,j (l −1) yj (l) .



.

(21)

By substituting (19) into (18) , (18) can be further written as

!

.

(22)

For the noise-only frame, i.e., Pj (l) = 0, from (20), (21), and (22), the scheme of the DNBD-LCMV with the first-order recursive smoothing method is shown in Fig. 2, and it consists of the following steps: 1) Each node j ∈J performs the following operation cycle:

• The Inverse Matrix ∆−1

nn,j (l −1) is applied to Mj-dimensional microphone signals yj (l) to obtain the Mj-dimensional vector ˜cj (l) and the real number cj (l) with (19), where the S-dimensional vector ¯cj (l) can also

Be Obtained By Using Hj. Then, ∆−1

nn,j (l) is estimated with (22). • ¯cj (l) and cj (l) are transmitted. • Each node can have access to ¯cj (l) and cj (l), and reconstructs Dj (l) and zj (l) with (20) and (21), respectively.

Then, D (L) And Z (L) Are

computed with (14). The inverse matrix D−1 (l) is multiplied with z (l) to obtain ˜z (l). Finally, gj is applied to ˜z (l) to obtain dj (l).

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Figure 2: The scheme of the DNBD-LCMV with the first-order recursive smoothing method (J = 2, Pj (l) = 0). For each node j, the S-dimensional vector ¯cj (l) and the real number cj (l) are transmitted to reconstruct Dj (l) and zj (l) with (20) and (21), respectively, at the other

Node. The Inverse Matrix ∆−1

nn,j (l) is updated with (22). Each node j computes its node- specific output dj (l) use its own S-dimensional compressed signal zj (l) and the reconstructed signal zq (l) , q ∈J \ {j}, where Dj (l) and the reconstructed matrix Dq (l) are also used.

2) l ←l + 1. 3) Return to step 1). From Fig. 2, for each noise-only frame, the S-dimensional vector ¯cj (l) and the real number cj (l) are transmitted to reconstruct the S × S matrix Dj (l) and the S-dimensional vector zj (l) at other nodes instead of the transmission of Dj (l) and zj (l). Besides, the inversion operation of ∆nn,j (l) is only required at the beginning. The communication cost and computational load are greatly reduced compared to the naive solution mentioned above.

3.2. Dnbd-Lcmv With Unknown Atf Matrix

In general, the ATF matrix H is unknown and needs to be estimated online. Some subspace estimation algorithms can be used to estimate the column space

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of H [19, 20]. Define Q = [q1, · · · , qS] to be any basis that spans the column

(23)

where Θ is an S × S matrix comprised of the projection coefficients of the original ATFs on the basis vectors. When the speakers only change their positions slowly with respect to their initial positions, such as teleconferencing, we can estimate the column space Q at the initial stage (e.g., in a centralized way) . This may cause some estimation error of Q if the speakers have some slight movements and therefore robust beamformer is preferred.

The goal for node j is to estimate the target signal from the signal vector s as observed by one of node j’s microphones, referred to as the reference micro- phone. Without loss of generality, the first microphone of each node is chosen as the reference microphone. For node j, the index of its reference microphone

(24)

where Q (m, k) is the entry in the mth row and kth column of Q. ¯gj (k) and gj (k) are the kth entry of ¯gj and that of gj, respectively. The node-specific output in (6) is modified by the following equation

(25)

where H (m, k) is the entry in the mth row and kth column of H. It is obvious that different nodes have different desired response vectors due to different reference microphones, i.e., ¯gj̸ = ¯gq with j̸ = q. Therefore, each node can extract a node-specific target signal based on node-specific reference

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microphone to preserve the spatial information of the target source, such as time difference cues, which are very important in target source localization.

3.3. Extension Of Dnbd-Lcmv

When the noise covariance matrix ∆nn,j in (8) is replaced by the noisy co-

Variance Matrix ∆Yy,J, That Is To Say,

Ryy = Blockdiag (∆yy,1, · · · , ∆yy,j, · · · , ∆yy,J) ,

(26)

one can obtain the distributed node-specific block-diagonal linearly constrained minimum power (DNBD-LCMP) beamformer, which can be given by

Yy Q

−1 ¯gj.

(27)

where this beamformer does not require an estimate of the noise covariance matrix. Particularly, when ∆nn,j is an identity matrix Ij, the DNBD-LCMV becomes the distributed node-specific delay and sum (DNDS) beamformer

Qhq

−1 ¯gj.

(28)

where this beamformer only needs to be calculated once at the beginning. How- ever, it cannot control the sound sources that are not included in ¯gj. Similar to , the proposed beamformers including DNBD-LCMV/DNBD- LCMP, and DNDS can be implemented in the WASNs with arbitrary topologies.

This can be achieved by a slight modification to the data transmission process of each node, where each node aggregates its transmitted data including zj (l) and Dj (l) with the transmitted data from its neighbors. This will allow for the transmitted data to disperse through the WASNs by means of an in-network summation, and is demonstrated for the tree topology, as shown in Fig. 3.

We denote Nj as the set of neighbors of node j in the topology with node j excluded. After the tree is formed with any tree formation algorithm [38, 39], one arbitrary node is assigned as the root node and the nodes only communicate with their neighbors (for example, N6 = {5, 7, 8, 9} and node 1 is the root node

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