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Speech enhancement with an acoustic vector sensor: an effective adaptive beamforming and postfiltering approach
EURASIP Journal on Audio, Speech, and Music Processing volume 2014, Article number: 17 (2014)
Abstract
Speech enhancement has an increasing demand in mobile communications and faces a great challenge in a real ambient noisy environment. This paper develops an effective spatialfrequency domain speech enhancement method with a single acoustic vector sensor (AVS) in conjunction with minimum variance distortionless response (MVDR) spatial filtering and Wiener postfiltering (WPF) techniques. In remote speech applications, the MVDR spatial filtering is effective in suppressing the strong spatial interferences and the Wiener postfiltering is considered as a popular and powerful estimator to further suppress the residual noise if the power spectral density (PSD) of target speech can be estimated properly. With the favorable directional response of the AVS together with the trigonometric relations of the steering vectors, the closedform estimation of the signal PSDs is derived and the frequency response of the optimal Wiener postfilter is determined accordingly. Extensive computer simulations and a real experiment in an anechoic chamber condition have been carried out to evaluate the performance of the proposed algorithm. Simulation results show that the proposed method offers good ability to suppress the spatial interference while maintaining comparable log spectral deviation and perceptual evaluation of speech quality performance compared with the conventional methods with several objective measures. Moreover, a single AVS solution is particularly attractive for handsfree speech applications due to its compact size.
1 Introduction
As the presence of background noise significantly deteriorates the quality and intelligibility of speech, enhancement of speech signals has been an important and challenging problem and various methods have been proposed in the literature to tackle this problem. Spectral subtraction, Wiener filtering, and their variations [1] are commonly used for suppressing additive noise, but they are not able to effectively suppress spatial interference. In order to eliminate spatial interferences, beamforming techniques applied to microphone array recordings can be employed [2–9]. Among these, the minimum variance distortionless response (MVDR) beamformer known as the Capon beamformer and their equivalent generalized sidelobe cancellers (GSC) work successfully in remote speech enhancement applications [2]. However, the performance of MVDRtype methods is proportional to the number of array sensors used, thus limiting their application. Moreover, the MVDR beamformer is not effective at suppressing additive noise, leaving residual noise in its output. As a result, the wellknown Wiener postfiltering solution normally can be employed to further reduce the residual noise from the output of the beamformer [7]. Recently, speech enhancement using the acoustic vector sensor (AVS) array has received research attention due to the merit of spatial colocation of microphones and signal time alignment [5, 10–12]. Compared with the traditional microphone array, the compact structure (occupying a volume of approximately 1 cm^{3}) makes the AVS much more attractive in portable speech enhancement applications. Research showed that the AVS array beamformer with the MVDR method [5, 10] successfully suppresses spatial interferences but fails to effectively suppress background noise. The integrated MVDR and Wiener postfiltering method using AVS array [12] offers good performance in terms of suppression of spatial interferences and background additive noise, but it requires more than two AVS units as well as the good voice activity detection (VAD) technique.
In this paper, we focus on developing a speech enhancement solution capable of effectively suppressing spatial interferences and additive noise at a less computational cost using only one AVS unit. More specifically, by exploring the unique spatial colocation property (the signal arrives at the sensors at the same time) and the trigonometric relations of the steering vectors of the AVS, a single AVSbased speech enhancement system is proposed. The normconstrained MVDR method is employed to form the spatial filter, while the optimal Wiener postfilter is designed by using a novel closedform power spectral density (PSD) estimation method. The proposed solution does not depend on the VAD technique (for noise estimation) and hence has advantages of small size, less computation cost, and the ability to suppress both spatial interferences and background noise.
The paper is organized as follows. The data model of an AVS and the frequency domain MVDR (FMV) with a single AVS are presented in Section 2. The detailed derivation of the closedform estimation of the signal PSDs for an optimal Wiener postfiltering (WPF) using the AVS structure is given in Section 3. The proposed normconstrained FMVeffective Wiener postfiltering (NCFMVEWPF) algorithm for speech enhancement is presented in Section 4. Simulation results are presented in Section 5. Section 6 concludes our work.
2 Problem formulation
2.1 Data model for an AVS unit
An AVS unit generally consists of four colocated constituent sensors, including one omnidirectional sensor (denoted as the osensor) and three orthogonally oriented directional sensors depicted as the usensor, vsensor, and wsensor, respectively. As an example, Figure 1 shows a data capture system with an AVS unit. In this paper, focusing on deriving the algorithm and making the derivation clear, let us assume that there is one target speech s(t) at (θ_{ s }, ϕ_{ s }) = (90°, ϕ_{ s }) and one interference signal s_{ i }(t) at (θ_{ i }, ϕ_{ i }) = (90°, ϕ_{ i }) impinging on this AVS unit, where ϕ_{ s }, ϕ_{ i } ∈ [0°, 360°) are the azimuth angles. Since s(t) and s_{ i }(t) arrive in the horizontal plane, we only need the usensor, vsensor, and osensor to capture signals from the AVS unit. The angle difference between s(t) and s_{ i }(t) is defined as
The corresponding steering vectors are given by
where [.]^{T} denotes the vector/matrix transposition.
In the cases that room reverberation does not exist, the received data of the AVS can be modeled as [13]
where n_{avs}(t) is assumed as the additive white Gaussian noise at the AVS unit. Specifically, we have the following definitions:
where x_{ u }(t), x_{ v }(t), and x_{ o }(t) are the received data of the u, v, and osensor, respectively, and n_{ u }(t), n_{ v }(t), and n_{ o }(t) are the captured noise at the u, v, and osensor, respectively. The task of speech enhancement with an AVS is to estimate s(t) from x_{ avs }(t).
In this study, without loss of generality, we follow the commonly used assumptions [4]: (1) s(t) and s_{ i }(t) are mutually uncorrelated; (2) n_{ u }(t), n_{ v }(t), and n_{ o }(t) are mutually uncorrelated.
2.2 FMV with a single AVS
The MVDR beamformer is considered as one of the most popular beamforming methods for suppressing spatial interferences in remote speech applications. In this subsection, we present the formulation of the frequency domain MVDR beamformer (FMV) with two sensors (usensor and vsensor) of the AVS unit. From (2) to (4), the data received by the usensor and the vsensor can be modeled as [14]
where
and
The frequency domain formulation of the data model of (7) is given by
where X(f) = [X_{ u }(f), X_{ v }(f)]^{T} and N(f) = [N_{ u }(f), N_{ v }(f)]^{T}. The beamforming is then performed by applying a complex weight to the captured signals, and the output of the FMV can be denoted as
where (.)^{H} denotes the Hermitian transposition. w^{H}(f) = [w_{ u }(f), w_{ v }(f)] is the weight vector of the FMV. Let us define
Obviously, g(ϕ_{ s }, f) and g(ϕ_{ i }, f) can be viewed as the spatial response gains of the FMV to the target spatial signal S(f) and the spatial interference signal S_{ i }(f), respectively. Substituting (12) and (13) into (11), we can get
The basic idea of designing the optimal FMV is to maintain g(ϕ_{ s }, f) = 1 for S(f) while minimizing the output signal power (P_{ YY } = E[Y(f)Y*(f)]) of the FMV to suppress other undesired sources. Hence, the optimal weight vector of the FMV can be obtained by solving the constrained optimization cost function [2]:
where R_{ x }(f) = E[X(f)X^{H}(f)] is the autocorrelation matrix of the received data of the FMV. The optimal solution of (15) is given as [2]
Equation 16 is the standard form of the FMV. It is clear that when a(ϕ_{ s }) is fixed (speech target is static), w_{FMV}(f) depends on the estimate of ${\mathbf{R}}_{\mathbf{x}}^{1}\left(\mathit{f}\right)$. There are several methods that have been proposed to estimate R_{ x }(f) [1], and the diagonal loading technique is one of the robust algorithms aiming at avoiding the nonsingularity in (16), which leads to a normconstrained FMV (NCFMV) as shown in (17) [3]:
where I is an identity matrix, γ is the positive loading factor, and σ is a small positive number to avoid the denominator becoming zero. It is expected that the NCFMV will greatly suppress the spatial unwanted signals. Obviously, the output of the NCFMV can be derived as follows with (17), (12), (13), and some simple manipulations:
2.3 The estimation of the power spectral density
As discussed above, the NCFMV is only effective in suppressing the spatial interferences. In this section, a new solution has been proposed by incorporating the wellknown Wiener postfilter (WPF) to further suppress the residual noise in beamformer output Y(f) in (18).
Basically, according to the formulation of the Wiener filter in the frequency domain, to estimate S(f) from Y(f), the frequency response of the Wiener filter is given by [6, 8]
where ψ_{ YS }(f) is the crosspower spectrum density (CSD) of S(f) and Y(f) and ψ_{ YY }(f) is the power spectral density (PSD) of Y(f). Generally, Y(f) are considered uncorrelated to interferences, and we can approximately get the second equation in (19) via (18). From (19), it is clear that a good estimate of ψ_{ SS }(f) and ψ_{ YY }(f) from X(f) and Y(f) are very crucial to the performance of the WPF. There are some PSD estimation algorithms that have been proposed under different spatialfrequency joint estimation schemes. For singlechannel application as an example, the voice activity detection (VAD) method is usually applied to get the noise and speech segments, and then the spectrum subtraction algorithm can be used to remove noise components before estimating ψ_{ SS }(f). Moreover, for microphone array postfiltering schemes, ψ_{ SS }(f) can be estimated from the available multichannel signals, which are assumed to be within an incoherent noise environment [6].
Motivated by the unique properties of the AVS, where multichannel signals are available (u, v, and osensor signals) and there exists a trigonometric relationship between the steering vectors a(ϕ_{ s }) and a(ϕ_{ i }) of the AVS, in this paper, we will derive a closedform solution to estimate ψ_{ SS }(f) and ψ_{ YY }(f) to form an optimal WPF. The system diagram proposed is shown in Figure 2.
3 The formulation of the Wiener postfilter
3.1 Derivation of the estimate of CSD and PSD
For presentation clarity, let us define the notation of the crosspower spectral density (CSD) between α(f) and β(f) as
From (10), we have
For presentation simplicity, the frequency index f will be dropped in the following derivation. Ideally, the additive noises of u, v, and osensors have the same power, and then we have
It is noted that the assumption of equal power for all channels used in (23) is not true for the real signals recorded by the AVS unit, but these can be calibrated in practice [15]. With (18), (21), (22), and (23), the CSD and PSD of signals can be derived following the definition given in (20):
From (24) to (30), it is clear that there are seven equations with four unknown variables ψ_{ NN }, g(ϕ_{ i }), ${\mathit{\psi}}_{{\mathit{S}}_{\mathit{i}}{\mathit{S}}_{\mathit{i}}}$, and ${\mathit{\psi}}_{{\mathit{S}}_{\mathit{i}}{\mathit{S}}_{\mathit{i}}}$. Hence, using (28) and (26), the PSD of noise can be derived as
Similarly, the gain response of the NCFMV on the interference S_{ i } can be given by
Moreover, the PSD of the interference S_{ i } and the target speech S can be derived, respectively, as follows:
3.2 The proposed EWPF method and some discussions
Till now, we have mathematically derived the closedform expressions of the ψ_{ SS } in (34), ψ_{ YY } in (27), and W_{ pf } in (19). Since Y, X_{ u }, X_{ v }, and X_{ o } can be measured, the estimates of ψ_{ SS } and ψ_{ YY } can be determined accordingly. Hence, (33), (34), (27), and (19) describe the basic form of our proposed effective Wiener postfiltering algorithm for further enhancing the speech with an AVS (here, we term it as EWPF for short). In the following context, we will have some discussions on our proposed EWPF method.
Firstly, to implement the EWPF, the crosspower spectral density ψ_{ αβ }(f) needs to be estimated. It is well known that the recursive update formula is a popular approach:
where l is the frame index and λ ∈ (0, 1] is the forgetting factor.
Secondly, it is noted that when Δϕ defined in (1) is close to or equal to 0, the denominator in (32) goes to 0. To avoid this situation, one small positive factor σ_{ r } should be added to the denominator of (32) and we get
Thirdly, analyzing the properties of g(ϕ_{ i }), we observe the following: (1) If the target source s(t) is considered as shorttime spatially stationary (approximately true for speech applications), w_{ NC } in (17) can be updated every L_{ u } frames for reducing computational complexity. Therefore, from the definition of (13), the gain g(ϕ_{ i }) will remain unchanged within L_{ u } frames. However, ${\widehat{\mathit{\psi}}}_{\mathit{\alpha \beta}}\left(\mathit{f},\mathit{l}\right)$ is estimated frame by frame via (35); therefore, a more accurate estimation of g(ϕ_{ i }) can be achieved by averaging over L_{ u } frames. (2) From (36), it is clear that the small denominator will lead to a large variation of g(ϕ_{ i }), reflecting incorrect estimates since the NCFMV is designed to suppress rather than to amplify the interference. Hence, it is reasonable to apply a clipping function f_{ c }(x, b) (see (43)) to remove the outliers in the estimate of $\widehat{\mathit{g}}\left({\mathit{\varphi}}_{\mathit{i}}\right)$.
4 The proposed NCFMVEWPF algorithm
Similar to the existing remote speech enhancement applications, our proposed algorithm is implemented in the frequency domain by segmenting the received signal into frames and then the shorttime Fourier transforms (STFTs) are applied. Specifically, to determine w_{ NC } in (17), the estimate of the R_{ x } is given by [10]
where k is the frequency bin index and k = 1,2,…, K. C_{ d } is a constant slightly greater than the one that helps avoid matrix singularity. F is the frame number used for estimating R_{ x }(k), and in our study it is set as F = 2L_{ u }. Let us define X_{ u }(k,l) and X_{ v }(k,l) as the k th component of the spectrum of the l th frame of x_{ u }(n) and x_{ v }(n), respectively, and we have
From (37) to (41), we can see the autocorrelation matrix R_{ x }(k) is estimated by using the F most recent fast Fourier transforms (FFTs). Therefore, the robust estimation of W_{ pf }(k) in (19) asks for the robust estimation of g(ϕ_{ i },k). According to the discussions in Section 3.2, we adopt the following estimation:
where L_{1} = fix((l − 1)/L_{ u })L_{ u } + 1, L_{2} = fix((l − 1)/L_{ u })L_{ u } + L_{ u }, fix(.) is the floor operation, b is a predefined threshold, and f_{ c }(x,b) is the clipping function and defined as
For presentation completeness, the proposed NCFMVEWPF algorithm is summarized in Algorithm 1.
5 Simulation study
The performance evaluation of our proposed NCFMVEWPF algorithm has been carried out in this section. The commonly used performance measurement metrics have been adopted, which include the following:

1.
Output signal to interference plus noise ratio (SINR) defined as [7]
$$\mathrm{SINR}=10log\left({\u2225{\mathit{z}}_{\mathit{s}}\left(\mathit{t}\right)\u2225}^{2}/{\u2225{\mathit{x}}_{\mathit{o}}\left(\mathit{t}\right){\mathit{z}}_{\mathit{s}}\left(\mathit{t}\right)\u2225}^{2}\right)$$(44)
where z_{ s }(t) is the enhanced speech of the system and x_{ o }(t) is the received signal of the osensor. Moreover, a segmental output SINR is calculated on a framebyframe basis and then averaged over the total number frames to get more accurate prediction of perceptual speech quality [7].

2.
Log spectral deviation (LSD), which is used to measure the speech distortion and defined as [16]
$$\mathrm{LSD}=\u2225ln\left({\mathit{\psi}}_{\mathit{ss}}\left(\mathit{f}\right)/{\mathit{\psi}}_{\mathit{zz}}\left(\mathit{f}\right)\right)\u2225$$(45)
where ψ_{ ss }(f) is the PSD of the target speech and ψ_{ zz }(f) is the PSD of the enhanced speech. It is clear that the smaller LSD indicates the less speech distortion. Similar to the calculation of SINR, the segmental LSD is computed.

3.
Perceptual evaluation of speech quality (PESQ) [17]: To evaluate the performance of the speech enhancement algorithms, ITUPESQ software [17] is utilized.
In addition, we also compared the performance of the Zelinski postfilter (ZPF) [4], NCFMV [5], and NCFMVZPF [6] algorithms under the same conditions to our proposed algorithm. The setup of the single AVS unit is shown in Figure 1.
In computer simulation studies, for each trial, a male speech lasting about 5 s acts as the target speech s(t) and babble speech taken from the Noisex92 database [18] acts as the interference speech s_{ i }(t). One set of the typical waveforms used in our simulation studies is shown in Figure 3.
5.1 Experiments on simulated data
5.1.1 Experiment 1: the SINR performance under different noise conditions
In this experiment, we have carried out nine trials (numbered as trial 1 to trial 9) to evaluate the performance of the algorithms under different spatial and additive noise conditions [9]. The experimental settings are as follows: The sampling rate is set to be 16 kHz and a 512point FFT is used. The target speaker is located at (90°, 45°) and the interference speaker is set at (90°, 0°). For the proposed NCFMVEWPF algorithm, parameters are set as λ = 0.6, σ_{ r } = 10^{−3}, L_{ u } = 4, γ = σ = 10^{−5}, C_{ d } = 1.1, and b = 6, which produced the best experimental results under this specific setup. For comparison algorithms, the parameter settings are set as the same as those in the relevant papers. The experimental results are listed in Table 1.
As shown in Table 1, the best performance for different conditions is addressed in italics. The proposed NCFMVEWPF algorithm outperforms other algorithms in terms of SINRout in trials 1 to 4 and trials 7 to 9, and gives comparable performance in trial 5 and inferior performance in trial 6. It is noted that, in trials 4 to 6, there is no spatial interference considered (i.e., s_{ i }(t) = 0). The performance for trial 5 indicates that the proposed NCFMVEWPF is not as effective as the ZPF in suppressing the additive noise with higher SNR (SNR > 10 dB) when spatial interference is not present. Therefore, these experimental results demonstrate the superior capability of the proposed NCFMVEWPF in suppressing the spatial and adverse additive interferences. For visualization purposes, the results in Table 1 have also been plotted in Figure 4, where the xaxis represents the SINR of the signal captured by the AVS (termed as SINRinput) and the yaxis represents the SINR of the enhanced speech (termed as SINRout).
5.1.2 Experiment 2: the impact of the angle between the target and interference speakers
This experiment evaluates the impact of the angle between the target and interference speakers (Δϕ = ϕ_{ s } − ϕ_{ i }) on the performance of the NCFMVEWPF algorithm. The results of the SINRout versus Δϕ are shown in Figure 5, where the same experimental settings as those used for trial 7 in experiment 1 were adopted except the target speech location ϕ_{ s } varied from (90°,0°) to (90°,360°) with 45° increments. From Figure 5, it is clear to see that when Δϕ → 0° (the target speaker moves closer to the interference speaker), for both algorithms, the SINRout drops significantly and almost goes to 0. This means the speech enhancement is very much limited under this condition. However, when Δϕ > 0°, the SINRout gradually increases. It is quite encouraging to see that the SINRout of our proposed NCFMVEWPF algorithm is superior to that of the NCFMV algorithm for all angles. Moreover, the SINRout of our proposed NCFMVEWPF algorithm maintains about 15 dB when Δϕ ≥ 45°.
5.1.3 Experiment 3: SINR, LSD, and PESQ performance
In this experiment, we adopted three performance metrics (SINR, LSD, and PESQ) to evaluate the performance of the algorithms. The same experimental settings of those used in experiment 1 were employed, where the SINRinput is set as 0 dB, the target speaker is located at (90°, 45°), and the interference speaker is at (90°,0°) (Δϕ = 45°). The experimental results are given in Table 2. It can be seen that the overall performance of our proposed NCFMVEWPF algorithm is superior to that of other comparison algorithms. The LSD and PESQ performance of the NCFMVEWPF algorithm is comparable to that of the NCFMVZPF [6] algorithm. It is encouraging to see that the proposed NCFMVEWPF algorithm is able to effectively suppress the interference and additive noise while maintaining good speech quality and less distortion.
5.2 Experiments on recorded data in an anechoic chamber
5.2.1 Experiment 4: the SINRout performance with different speakers
In this experiment, we conducted the speech enhancement by using the recorded data from Ritz's lab [19]. The experimental setup is shown in Figure 6. The AVS has been built by two Knowles NR3158 pressure gradient sensors (usensor and vsensor) and one Knowles EK3132 sensor (osensor) (Knowles Electronics Inc., Itasca, IL, USA). Recordings were made of 10 different speech sentences from the IEEE speech corpus [20] in an anechoic chamber and background noise only from computer servers and air conditioning. The anechoic chamber is similar to the noise field: n_{avs}(t) = 0 and s_{ i }(t) ≠ 0. The sampling rate was 48 kHz and then downsampled to 16 kHz for speech enhancement. The speakers were placed in front of the AVS at a distance of 1 m. Target speech was located at a fixed position (90°, 45°), while interference speech was located at (90°, 90°). Ten trials were carried out using the 10 different target speeches.
The experimental results are shown in Figure 7. The xaxis represents the number of trials, and the yaxis represents the SINR of the enhanced speech (in dB). It is clear to see that the proposed NCFMVEWPF algorithm provides superior SINRout performance for all trails when the SINRinput of the recorded data is at about −5 dB. The experimental results with the real recorded data further validate the effectiveness of the proposed NCFMVEWPF in suppressing the strong competing speech.
5.2.2 Experiment 5: the impact of the angle between the target and interference speakers
Similar to experiment 2, this experiment evaluates the impact of the angle between the target and interference speakers (Δϕ = ϕ_{ s } − ϕ_{ i }) on the performance of the NCFMVEWPF algorithm. The results of the SINRout versus Δϕ are shown in Figure 8, where the experimental setup is the same as that of experiment 4 except that the angle of the target speaker (ϕ_{ s }) varies from (90°,90°) to (90°,0°) with 15° decrement.
From Figure 8, it is clear to see that the performance of the proposed NCFMVEWPF algorithm is superior to that of the NCFMV algorithm for all Δϕ values. Compared to the results shown in Figure 5 using the simulated data, similar conclusions can be drawn for the proposed NCFMVEWPF algorithm. More specifically, with the recorded data, when Δϕ > 15°, the proposed NCFMVEWPF algorithm can effectively enhance the target speech.
5.2.3 Experiment 6: PESQ performance versus Δϕ
In this experiment, we only adopted one performance metrics (PESQ) to evaluate the performance of the algorithms. The same experimental settings as those used in experiment 5 were employed, where the angle of the interference speaker (ϕ_{ i }) was fixed at (90°,90°) and the angle of the target speaker (ϕ_{ s }) varied from (90°,90°) to (90°,0°) with 15° decrement. The experimental results are given in Figure 9. It can be seen that the overall performance of PESQ for our proposed NCFMVEWPF algorithm is superior to that of the comparison algorithm for all angle differences. This experiment also demonstrates the ability of the proposed NCFMVEWPF algorithm in effectively suppressing the interference and additive noise while maintaining good speech quality and less distortion when Δϕ > 15°.
6 Conclusions
In this paper, a novel speech enhancement algorithm named as NCFMVEWPF has been derived with a single AVS unit by an efficient closedform estimation of the power spectral densities of signals. The results of computer simulation show that the proposed NCFMVEWPF algorithm outperforms the existing ZPF, NCFMV, and NCFMVZPF algorithms, in terms of suppressing the competing speaker and noise field. The results of real experiments show that compared with the NCFMV algorithms, the proposed NCFMVEWPF algorithm can effectively suppress the competing speech and additive noise while maintaining good speech quality and less distortion. In addition, it is noted that the NCFMVEWPF algorithm does not require the VAD technique, which not only reduces the computational complexity but also provides more robust performance in a noisy environment, such as the higher output SINR, less speech distortion, and better speech intelligibility. It is expected that this novel approach developed in this paper is a suitable solution for implementation within handsfree speech recording systems.
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Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (No. 61271309) and the Shenzhen Science & Technology Fundamental Research Program (No. JCY201110006). It was also partially supported by the Australian Research Council Grant DP1094053.
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Keywords
 Speech enhancement
 Acoustic vector sensor
 Beamforming
 Postfiltering
 Power spectral density estimation