2019 27th European Signal Processing Conference (EUSIPCO)

CNN-based virtual microphone signal estimation for
MPDR beamforming in underdetermined situations
Kouei Yamaoka∗ , Li Li∗ , Nobutaka Ono† , Shoji Makino∗ , and Takeshi Yamada∗∗ University of Tsukuba, 1-1-1 Tennodai, Tsukuba-shi, Ibaraki 305-8577, Japan{yamaoka@mmlab.cs, lili@mmlab.cs, maki@tara, takeshi@cs}.tsukuba.ac.jp† Tokyo Metropolitan University, 6-6 Asahigaoka, Hino-shi, Tokyo 191-0065, Japan, onono@tmu.ac.jp

Abstract—In this paper, we propose a novel approach tovirtually increasing the number of microphone elements between
two real microphones to improve speech enhancement performance in underdetermined situations. The virtual microphone
technique, with which virtual signals in the audio signal domain
are estimated by linearly interpolating the phase and nonlinearlyinterpolating the amplitude independently on the basis of βdivergence, has been recently proposed and experimentally shown
to be effective in improving speech enhancement performance.
Furthermore, it has been reported that the performance tends toimprove as the nonlinearity is improved. However, one drawbackof this method is that the interpolation is employed in eachtime-frequency bin independently, in which the spectral andtemporal structures of speech signals are ignored. To address
this problem and improve the nonlinearity, motivated by thehigh capability of neural networks to model nonlinear functions
and speech spectrograms, in this paper, we propose an alternative
method of amplitude interpolation. In this method, we employ
a convolutional neural network as an amplitude estimator thatminimizes the mean squared error between the outputs of aminimum power distortionless response (MPDR) beamformer
and the target speech signals. The experimental results revealed
that the proposed method showed high potential for improving
speech enhancement performance, which was not only superior
to that of the conventional virtual microphone technique but also
the performance in the corresponding determined situation.

I. I NTRODUCTION
The technique of reducing undesirable noise while enhancing the target speech in recorded mixture signals, whichis referred to as speech enhancement, plays an importantrole in many audio signal processing applications such as
automatic speech recognition [1]. Beamforming and blindsource separation (BSS) [2] are commonly used methods forspeech enhancement and can yield good performance as longas a sufficient number of microphones are available, namely,
the number of microphones M equals or exceeds the numberof sound sources N (M ≥ N ) to suppress N −1 interferers bynull steering. Otherwise, the performance of speech enhancement tends to decrease considerably. However, commonly usedsmall recording devices such as voice recorders often have
only two microphones, which is insufficient to meet the determined condition. To achieve satisfactory speech enhancement
performance with such devices having a small microphonearray, many methods have been proposed to enhance speech inunderdetermined situations (M < N ) such as time-frequency
masking [3]–[5], multichannel Wiener filtering [6], [7], andnonnegative matrix factorization (NMF) [8]–[10]. Althoughthese methods are noteworthy in that they can significantly
improve speech intelligibility in underdetermined situations,
there is a tradeoff between low signal distortion and high noise
reduction performance.

978-9-0827-9703-9/19/$31.00 ©2019 IEEE

Real microphone𝑥1
𝑣𝑖

Virtualmicrophones

𝑣𝑖+1
𝑥2

Real microphone signal
Virtualmicrophonesignalsα𝑖Interpolation

・・・

outputs

Signalprocessing・・・

1 − α𝑖Real microphone signal

Real microphone

Fig. 1. Microphone array signal processing with virtual microphone technique

On the other hand, the virtual microphone technique [11]allows the well-studied methods for determined situations(M = N ) to be applied to signals recorded in underdeterminedsituations by virtually increasing the number of channels.
As shown in Fig. 1, with two real microphone signals x1
and x2 , the virtual microphone technique is used to estimate
the observed signal vi at a position where there is no real
microphone placed by interpolating the phase and amplitude
independently. Note that since the virtual microphone signals
are generated in the audio signal domain, this technique can beapplied to not only speech enhancement but also other typesof signal processing [12], which is different from techniques
in which signals are generated in the power domain [13]–[15]or a higher-order statistical domain [16].
In the virtual microphone technique, on the basis of the Wdisjoint orthogonality (W-DO) [3], [17] assumption, phasesof virtual signals can be obtained using linear interpolation by approximately modeling propagating waves as planewaves. For amplitude estimation, since modeling the amplitudes of propagating waves is difficult owing to the complicated acoustic environment, complex logarithmic interpolation
and a generalized version, where the interpolation rules arederived as closed-form solutions of an optimization problem formulated using β-divergence, have been proposed andexperimentally shown to be effective in improving speechenhancement performance using a maximum signal-to-noise
ratio (MaxSNR) beamformer. Furthermore, the results reported
in [11] indicated that speech enhancement performance tends
to increase when the improved nonlinearity is applied to amplitude interpolation. However, one drawback of this method
is that the interpolation is applied in each time-frequency bin
independently, in which the spectral and temporal structuresof speech signals are ignored.

2019 27th European Signal Processing Conference (EUSIPCO)

To overcome this drawback and improve the nonlinearityof amplitude generation, motivated by the high capability ofneural networks to model nonlinear functions and speech spectrograms, in this paper, we propose the use of convolutional
neural networks (CNNs) to learn the rules for estimating amplitudes of virtual signals. Specifically, the proposed method
trains a CNN to estimate a virtual signal that can minimize
the mean squared error of the outputs of a minimum power
distortionless response (MPDR) beamformer and the target
speech signals. Note that this is another merit of replacing
the ad hoc interpolation rules with rules learned in a datadriven manner, namely, it allows a complicated model to belearned with a loss function that is easy to formulate or istask-dependent.
II. RULE - BASED VIRTUAL MICROPHONE TECHNIQUE
We model the microphone signals in the short-time Fouriertransform (STFT) domain. Here, let xm (ω, t) be the mthreal microphone signal (m = 1, 2) at angular frequency
ω in the tth time frame. The amplitudes of these signals
are denoted as Am (ω, t) = |xm (ω, t)| and the phases aredenoted as ϕm (ω, t) = ∠xm (ω, t). A virtual microphonesignal v(ω, t, α, β) is defined as the observation estimated at
the point obtained by internally dividing the line joining two
real microphones in the ratio α : (1 − α) (see Fig. 1). Here,β is the hyperparameter of β-divergence, which controls thenonlinearity of amplitude interpolation. Hereafter, when there
is no need to distinguish ω, t, α, or β, the signal is simply
denoted as v.
To interpolate a virtual microphone signal, we need to define
the distance between the real and virtual microphones on thebasis of an appropriate model that is extremely complex. To
simplify the model, we here consider the models for phase andamplitude interpolation separately. Additionally, interpolating
the phase and amplitude separately introduces nonlinearity into
virtual signal generation, which is necessary to increase thenumber of channels.
A. Phase interpolation based on plane wave model
We assume W-DO [3], [17] for mixed signals, that is,each time-frequency bin is dominated by at most one soundsource. Then, the observed signal in each time-frequency
bin can be regarded as a single wave. On the basis of thisassumption, the physical model of propagating waves can thenbe approximated as that of a plane wave. The phase ϕv of avirtual microphone signal v can then be interpolated linearlyon the basis of the model as
ϕv = (1 − α) ϕ1 + αϕ2 .

(1)Since the observed phase has an aliasing ambiguity given by
ϕi ±2ni π with integer ni , this interpolation requires no spatial
aliasing, that is,|ϕ1 − ϕ2 | ≤ π.

(2)

B. Amplitude interpolation based on β-divergence
Since there are many acoustic conditions such as the distance between the sound sources and microphones and the
directions of arrival (DOAs), it is difficult to faithfully model
the amplitude of a propagating wave. Thus, instead of somephysical assumptions, we utilize β-divergence as an adjustablemeasure of distance to quantify the distance between the realand virtual microphones.
The β-divergence between the amplitude of the virtual
microphone Av and that of the ith real microphone Ai is
defined as
Dβ (Av , Ai ) =
Av (log Av − log Ai ) + (Ai − Av )



 Av
Av

− log
−1
Ai
Ai

 Aβ

Aβ Av Aβ−1
v
i


+ i −
β(β − 1)ββ−1(β = 1),(β = 0),(3)(otherwise),

where Dβ (Av , Ai ) is continuous at β = 0 and β = 1.
Then, we derive the interpolation rule of the amplitude Av
that minimizes σDβ , the sum of Dβ (Av , Ai ) weighted by
the hyperparameter of the virtual microphone interpolation α,
which indicates the position of the virtual microphone,σDβ = (1 − α)Dβ (Av , A1 ) + αDβ (Av , A2 ),Avβ = argminAv σDβ .

(4)
(5)

By differentiating σDβ with respect to Av and setting it to 0,
the interpolated amplitude is obtained as
Avβ =exp ((1 − α) log A1 + α log A2 ) (β = 1),() 1β−1 β−1 (1 − α) Aβ−1+αA(otherwise).
1
2

(6)

Note that Avβ is continuous at β = 1 and this interpolation
is equivalent to complex logarithmic interpolation [18] with
β = 1.
From the above, the virtual microphone signal v is represented as
v = Avβ exp (jϕv ).

(7)

Note that the phase can be interpolated with arbitrary real
numbers α, whereas the amplitude interpolation is defined onlyin the domain of 0 ≤ α ≤ 1 when β ̸= 1. The extrapolationof a virtual microphone in the domain α < 0, 1 < α wasconsidered in [12].
III. P ROPOSED METHOD : CNN- BASED AMPLITUDE
ESTIMATION FOR MPDR BEAMFORMER
The effectiveness of the virtual microphone technique in
improving speech enhancement performance using a MaxSNR
beamformer has been confirmed in [11]. According to theexperimental results in [11], the performance improvement
tends to increase when a larger β is used for amplitudeinterpolation, which indicated the importance of the nonlinearity in generating a virtual signal. However, with the virtual
microphone technique, the amplitude is estimated in each timefrequency bin independently, in which the structure of speech
signals is ignored. To further improve the nonlinearity for
amplitude interpolation and estimate the amplitude by taking
account of all the observed time-frequency bins, in this paper,
we introduce the use of a CNN as an alternative means ofamplitude estimation (see Fig. 2). Furthermore, learning therules in a data-driven manner facilitates the modeling process
and allows the use of task-dependent loss functions.

STFT

CNN
Virtual phase
spectrogram

Position of VM
Observed phase
spectrograms

Linearinterpolation

Concatenate

Observed signals

Virtual amplitude
spectrogram

iSTFT

Observed amplitude
spectrograms

Speech enhancement with MPDR

2019 27th European Signal Processing Conference (EUSIPCO)

Enhanced signal

Fig. 2. Flowchart of proposed method

A. Loss function with MPDR beamformer
In this paper, we introduce a task-dependent loss functionthat forces the generated amplitude to become optimal for
constructing an MPDR beamformer [19] and minimizes themean squared error between the output of the beamformer andthe target signal s(ω, t). For the observed signal consistingof two real and I virtual microphone signals x(ω, t) =T
[x1 (ω, t), v1 (ω, t), · · · , vI (ω, t), x2 (ω, t)] , an MPDR beamformer enhances the source of interest byy(ω, t) = wH (ω)x(ω, t),(8)Tw(ω) = [w1 (ω) · · · wM (ω)] ,(9)

where y(ω, t) denotes the output signal of the beamformer,
w(ω) the spatial filter vector, (·)T the transpose, and (·)H the
Hermitian transpose. The spatial filter w(ω) is given asw(ω) =

Φ(ω)−1 a(ω)
aH (ω)Φ(ω)−1 a(ω)Φ(ω) = E[x(ω, t)x(ω, t)H ],

,

(10)
(11)

which is a well-known closed-form solution of the optimization problem
J =
∑{ [
]
}
E |wH (ω)x(ω, t)|2 + 2Re[λ∗ (wH (ω)a(ω) − 1)] ,ω(12)which is derived using the method of Lagrange multipliers.
Here, E[·] is the expectation operator, Re[·] takes the real partof the input argument, λ∗ is the complex-valued Lagrange
multiplier, and a(ω) is the relative transfer function (RTF) of
the target, which is defined as the ratio of the acoustic transfer
Tfunctions h(ω) = [h1 (ω) · · · hM (ω)] from the target source[]ThM (ω).
to the microphone array, i.e., a(ω) = 1 hh21 (ω)(ω) · · · h1 (ω)
The training loss function of the CNN can then be written as
∑ [
]
Jc =
E |wH (ω)x(ω, t) − s(ω, t)|2 .
(13)
ω

B. Network architecture design
The network architecture employed in the proposed method
is designed by considering the following two aspects: 1) amplitude spectrograms of speech signals show region dependence,i.e., they have different frequency structures in voiced andFig. 3. Network architectures used for amplitude estimation. The inputs
and outputs are 1D data, where the frequency dimension of spectrograms isregarded as the channel dimension. “w”, “c”, “k” and “d” denote the width,channel number, kernel size, and dilation factor, respectively. “Conv”, “BN”
and “GLU” denote 1D convolution, batch normalization, and gated linear unit,
respectively.

unvoiced segments; 2) the entire amplitude spectrogram shouldbe estimated by taking all the observed time-frequency bins
as a cue, which means a large receptive field is required.
We use gated CNNs [20], which were originally introduced to model word sequences for language modeling andwere shown to outperform long short-term memory (LSTM)based language models trained in a similar setting. Similar
to LSTMs, the gating mechanism of gated CNNs allows
a network to learn what information should be propagatedthrough the hierarchy of layers. There are some attempts havebeen made to adopt CNNs into beamforming techniques. In[21], time-frequency masks for the target and noise signals
are estimated with a CNN so that the power spectral density
matrices for conducting beamforming can be obtained using
the masked signals.
In the gated CNN, by using Hl−1 to denote the output ofthe (l − 1)th layer, the output of the lth layer Hl of a gated
CNN can be written as
Hl = (Hl−1 ∗ Wlf + bfl ) ⊗ σ(Hl−1 ∗ Wlg + bgl ),(14)

where Wlf and Wlg are the weight parameters and bfl and bgl
are the bias parameters of the lth layer, respectively, ⊗ denotes
element-wise multiplication and σ is the sigmoid function.
The main difference between a gated CNN and a regular
CNN layer is that a gated linear unit (GLU), namely, thesecond term of (14), is used as a nonlinear activation function
instead of tanh activation or regular rectified linear units2019 27th European Signal Processing Conference (EUSIPCO)(ReLUs) [22]. Similar to LSTMs, GLUs are data-driven gates,
which control the information passed on in the hierarchy. Thisparticular mechanism allows us to capture long-range context
dependences efficiently without encountering the vanishinggradient problem. We also employ a one-dimensional (1D)
CNN and dilated convolution to capture long-term contextual
dependences. With 1D convolution models, the frequency
dimension is regarded as the channel dimension and an input
spectrogram is convolved with a (1, kT ) filter so that thefeatures extracted in the low-level layers can take into accountall the frequency bins, where kT is the filter width in thetime dimension. Dilated convolution [23] is another effective
approach whereby CNNs can capture wider receptive fields
with fewer layers by convolving a larger filter derived fromthe original filter with dilating zeros, namely, the original filter
is applied by skipping certain elements in the input. Note thatthe network is designed to be fully convolutional so that inputs
having arbitrary lengths can be handled. The details of the
network used in experiments are shown in Fig. 3.

TABLE IE XPERIMENTAL CONDITIONS
Number of real microphones M
2 or 3
Number of sound sources N
3
Distance between microphones
4 cm (M = 2), 2 cm (M = 3)
Reverberation time
120 msSampling rate
8 kHz
Input SNR
0 dB
Window length / shift
1024 / 512 samples
TABLE IIE XPERIMENTAL RESULTS
Closed Test
conditions
SDR
SIR
2 real mic
1.79602.3456
2 real mic + 1 vir mic (conv.)
8.44325.9321
2 real mic + 1 vir mic (prop.)
11.6489 19.6029
3 real mic
8.1513 12.9107
Open Test
conditions
SDR
SIR
2 real mic
2.19392.86245.96982 real mic + 1 vir mic (conv.)
8.5820
2 real mic + 1 vir mic (prop.)
5.3579 15.5531
16.1139 19.5216
3 real mic

SAR
13.4448
10.2255
12.4629
10.1648
SAR
12.6560
10.05935.935618.8855

IV. E VALUATION EXPERIMENTS
To evaluate the effect of the proposed method, we conducted
experiments designed to compare the speech enhancement
performance with the MPDR beamformer among the casesof using two real microphones, two real microphones and one
virtual microphone signal estimated by the proposed methodor the conventional method, and three real microphones.
A. Experimental conditions
We prepared two datasets for the closed test (dataset 1) andopen test (dataset 2). Dataset 1 consisted of three speakers andthe audio files for each speaker were one minute long, whichwere used to generate mixture signals. Dataset 2 comprised 10
speakers excerpted from the Wall Street Journal (WSJ0) corpus
and the audio files for each speaker were about 18 minutes
long, where randomly selected half of them (9 minutes) wasused as the training set and the others (9 minutes) served as thetest set. The mixture signals, consisting of a target speech andtwo interferers, were generated by adding each audio signal of
the target speaker to all the paired combinations from the othernine speakers. Therefore, the training dataset included fourhours of data in total. For both datasets, the observed signals
were convolutive mixtures of impulse responses simulated bya room impulse response generator [24]. The experimental
conditions are listed in Table I. The DOA of the target speech
was set to 90◦ , and those of the interferers were set to 50◦
and 150◦ with a reverberation time of 120 ms.
We used the exact RTF for all the methods. The RTF at
the position of a virtual microphone is estimated using theconventional virtual microphone technique. For the proposed
method, the RTF can also be estimated using a neural network,
which is one direction of future work. We used signal-todistortion ratio (SDR), signal-to-interference ratio (SIR), andsignal-to-artifacts ratio (SAR) as the objective criteria forquantifying the speech enhancement performance. A concise
representation of the results was obtained by averaging thesecriteria over the test dataset. Here, the reference signal was thesource image, i.e., the noise-free reverberant speech signal.

B. Results and discussion
The SDR, SIR, and SAR are shown in Table II for eachdataset. Since the MPDR beamformer with two real microphones can suppress only one interferer in each frequency
bin, it is natural that the beamformer failed to enhancespeech in underdetermined situations. By using an additional
virtual microphone signal estimated by the conventional virtual
microphone technique, an improvement of about 4 dB was
obtained in terms of SDR, which showed the effectiveness ofvirtual signals in speech enhancement.
In the closed test, the proposed method achieved a major improvement in all the criteria and even outperformed the methodthat used three observed signals (i.e., determined situations).
The improvements were about 10 dB and 17 dB in terms of
SDR and SIR, respectively. These results are interesting in thatthey reveal not only the high potential of the proposed method
to improve the performance of the conventional method, butalso the fact that the optimal amplitude for constructing an
MPDR beamformer is not equivalent to the observed one. Fig.
4 shows an example of an amplitude spectrogram estimatedby the proposed method and Figs. 5 and 6 show the enhanced
speech signals. However, the performance of the proposed
method decreased significantly in the open test, in which SAR
was lower than that in the underdetermined situations (two real
microphones). Since the estimated virtual signals have nonlinear noise, the output signal computed by linear transformationof a concatenated input signal also contains some artifacts. One
possible reason is that the model trained with dataset 2 was
not generalized well enough to generate amplitudes for all theunseen data, which generated virtual signals including highlevel artifacts. To improve the generalization of the model,a larger dataset and an appropriate network architecture areneeded, which is another direction of future work.
From the above, we concluded that the proposed method has
the potential to improve the conventional virtual microphone
technique and even outperform the MPDR beamformer for
determined situations in terms of speech enhancement perfor-

2019 27th European Signal Processing Conference (EUSIPCO)R EFERENCES

Fig. 4. Amplitude spectrogram of one of the observed two-channel mixture
signals (left) and one estimated using proposed method in closed test (right).

Fig. 5. Target speech signal (left) and signal enhanced using two real
microphones (right).

Fig. 6. Speech signals enhanced using two real microphones and one virtual
microphone signal estimated by conventional (left) and proposed (right)
methods.

mance when the model is trained to be well generalized.
V. C ONCLUSIONSIn this paper, we proposed an alternative method in whicha CNN is used as an estimator of the amplitude of a virtualsignal, which aims to improve speech enhancement performance in underdetermined situations. The CNN is trainedwith a task-dependent loss function that minimizes the meansquared error between the output of an MPDR beamformer
and the target speech signal so that the estimated amplitudes
could be optimal for constructing an MPDR beamformer. An
experiment conducted that involved a closed test showed thatthe proposed method can potentially improve the conventional
method of speech enhancement in underdetermined situations
and even outperform other methods for determined situations
by using the generated amplitudes specialized for the MPDR
beamformer instead of the observed one.
ACKNOWLEDGMENTS
This work was supported by the Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant Numbers
JP16H01735, JPH04131, and JP19J20420.

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