Spatial soundfield reproduction using deep neural networks
Hanchi Chen(1) , Thushara D. Abhayapala(1)(1) Audio & Acoustic Signal Processing Group, College of Engineering and Computer Science, Australian National University

Abstract
Sparsity-based sound field reproduction algorithms often result in improved localization and larger reproduction
region, but also lead to high computational cost. In this work, we present a novel approach for sparse reproduction, where a deep neural network (DNN) is trained to determine the optimal driving signals for a loudspeakerarray, given the desired sound field coefficients as the input. We show that when trained using the proposed
method, the DNN-based algorithm can outperform existing Lasso-based algorithms in terms of noise sensitivity
and computation speed.
Keywords: Soundfield reproduction, deep neural networks

1

INTRODUCTION

Sound field reproduction systems aim to reproduce a desired sound scene so that listeners in a certain reproduction region can perceive the reproduced sound as if they were located in the original sound field where therecording was made. A loudspeaker array is often placed close to the reproduction region, and various approaches have been developed to determine the placement of these loudspeakers as well as the driving signals
for each loudspeaker. One main approach for sound field reproduction is Wave-Field Synthesis [1, 2], which
is based on the Kirchhoff-Helmholtz integral, this method delivers high reproduction accuracy over a reasonably large region, however the number of loudspeakers increases significantly for high frequencies. Anotherapproach is Ambisonics / Higher Order Ambisonics [3, 4, 5, 6, 7], where the sound field is represented using
first order or higher order spherical harmonics. This approach typically produces a “sweet spot” or a sphericalreproduction region which scales with the frequency as well as the number of loudspeakers available, for higher
frequencies and large reproduction region, it is necessary to use an unrealistic number of loudspeakers.
Sound field reproduction techniques which exploit the sparsity of the sound field present a new approach tosolving the problem. The 1-norm regularized LMS algorithm, or Lasso, can be utilized to effectively reduce
the number of active loudspeakers [8, 9, 10, 11, 12]. In [8], a Lasso algorithm is used first to determine asubset of loudspeakers for producing a certain sound field, then the driving signal for each active loudspeaker iscalculated using a pressure matching approach. This method is shown to use a smaller number of loudspeakers
to produce satisfactory reproduction quality. In [13], a Lasso-based algorithm is developed to optimize the soundfield reproduction for under-sampled sound fields and judicious loudspeaker placements. A latent group Lasso
algorithm is proposed in [14] to improve the accuracy of Lasso-based reproduction algorithms.
The solution of Lasso problems are not linear, and they need to be calculated in an iterative manner, thus
resulting in high computational cost. On the other hand, Deep Neuron Networks (DNN) have predictable computational cost, and dedicated hardware such as GPUs can further accelerate the computation of DNN outputs.
More importantly, DNNs are well know for their ability in predicting non-linear transfer functions. Neural Networks have already seen wide use in the audio signal processing area, including speech recognition [15], source
separation [16], beamforming [17] and DOA estimation [18]. However, to the best of our knowledge, its application in sound field reproduction has yet to be developed. In this work, we present a DNN-based sound field
reproduction algorithm, including its training method. We show that the DNN can be trained to generate sparse
loudspeaker driving signals to reproduce plane wave sound fields. The solution is compared with Lasso-basedalgorithm, and we show that the DNN algorithm is able to generate more stable driving signals in the presenceof noise and reverberation.

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2

PROBLEM FORMULATION

It has been shown that Least-Mean-Square based Pressure-Matching methods for spatial sound reproduction canlead to degraded localization performance, due to the fact that all the loudspeakers in the reproduction array
been activated [8]. Therefore, in this work, we aim to derive a sparse reproduction algorithm which minimizes
the number of active loudspeakers for a given sound field. For this purpose, it is necessary that the desired
sound field exhibit some form of sparsity in nature. Thus, we make the following assumptions regarding thesound field model:
1: At each frequency k, there exists one dominant acoustic source, the sound field due to which is to be
reproduced by a loudspeaker array.
2: The dominant source is sufficiently far, such that its sound field at the reproduction region can be seen as aplane wave.
3: The reverberations, noise and other interferences are weaker than the direct-path signal from the dominantsource, and are uncorrelated with the direct-path signal.
We further assume that a higher order microphone array is placed at the observation point to record the original sound field, which is capable of decomposing the captured sound field into Nth order spherical harmoniccoefficients [19, 20]NP(r, θ , φ , k) ≈ ∑n∑ Cnm (k) jn (kr)Ynm (θ , φ ),(1)

n=0 m=−nwhere P(r, θ , φ , k) denotes the sound pressure at position (r, θ , φ ), k is the wave number, jn (kr) is the sphericalBessel function;, Ynm (θ , φ ) is the spherical harmonic coefficient and Cnm are the spherical harmonic coefficients.
Using Assumptions 1 and 3, the spherical harmonic coefficients can be further decomposed as
Cnm (k) = Dnm (k) + Rnm (k) +Unm (k),(2)

where Dnm (k), Rnm (k) and Unm (k) are the spherical harmonic coefficients of the direct-path signal, reverberation/interference, and measurement noise, respectively.
Assuming that a loudspeaker array is used to reproduce the sound field due to the dominant acoustic source,` , H ` ...H ` ]T ,with the spherical harmonic based channel vector of the lth loudspeaker expressed as h ` = [H00NN
11the goal of this paper is to find a set of loudspeaker driving signals Wl , such that the resulting sound field
best approximates the desired direct-path signal, while minimizing the total number of active loudspeakers. The
corresponding cost function can be written as
W H − D k22 + λ kW
W k1 ,
arg minkW(3)

W

where k · k2 and k · k1 denote vector 2-norm and 1-norm, respectively, W = [W1 ,W2 , ...WL ] is the vector of driving
signals for all the L speakers in the array, H = [hh1 , h 2 , ...hhL ] is the channel matrix between the loudspeaker arrayand the reproduction region, and D = [D00 , D11 , ...DNN ] is the vector of desired spherical harmonic coefficients.
We note that instead of attempting to reproduce the complete sound field represented by Cnm , the cost function(3) aims to minimize the reproduction error for the direct-path signal only, and ignore the noise and interferencecomponents that may be present in the recorded sound field, which differs from existing sparse reproduction
algorithms in the literature. The motivation is that, in a non-anechoic reproduction environment, the recorded
reverberation will add to the reverberation of the reproduction room, hence leading to excessive reverberationenergy which degrades the clarity of the desired signal; furthermore, the reverberations may activate additional
loudspeakers, which further degrades the localization of the system.

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3

DNN-based sound field reproduction

3.1 Sparse driving signal calculation using complex Lasso
The cost function (3) has the standard Lasso problem formulation, where the penalty parameter λ can be adjusted to change the balance between the reproduction accuracy and the sparsity of the loudspeaker driving
signals. Various algorithms have been proposed for solving the Lasso optimization problems, including Coordinate Descent and Alternating Direction Method of Multipliers (ADMM). In general, these algorithms operate on
real-valued inputs. However, the problem (3) is complex-valued. It is thus necessary to transform the problem
into a real-valued equivalent problem so that it can be solved using existing methods.
W ` = R(W` ) I (W` ) , D̂
Dnm = R(Dnm ) I (Dnm )To perform the transform, we first define Ŵ` ) −I (H ` )
`
R(Hnm
nm
H nm =
Ĥ
(4)
` )
` ) .
I (Hnm
R(Hnm




W = Ŵ
W 1 Ŵ
W 2 . . . ŴW L , D̄
D = D̂
D00 D̂
D11 . . . D̂
DNNThen, we construct the real-valued matrices W̄
 1
1
1 
H 00 Ĥ
H 11 . . . Ĥ
H NN
Ĥ
 2
2
2 
Ĥ
H 00 Ĥ
H 11 . . . Ĥ
H NN 

H = 
H̄
(5)
..
..  .
 ..
..
.
 .
.
. 
L
L
L
H 00 Ĥ
H 11 . . . Ĥ
H NN
ĤThen the real-valued equivalent of (3) can be written as [21]LW H̄H − D̄
Dk22 + λ ∑ kWˆ ` k2 .
arg minkW̄
W
W̄

(6)

`=1

Eq. (6) represents a group Lasso problem with a constant group size of 2, which can be solved using theADMM method [22].
3.2 Training dataset for the DNN
The Lasso solution in Section 3.1 cannot be used directly to compute the desired driving signal, because Dnm
is combined with the unknown interfere and noise components in the received coefficients Cnm . In this section,
we propose a method to train a deep neural network, such that the DNN learns to estimate the desired plane
wave component, and produce the desired sparse driving signals in the presence of noise and interference.
In the ideal case where Rnm and Nnm equal to zero, for a given loudspeaker channel matrix H and a given
impinging direction (ϑ j , ϕ j ), we can use the complex Lasso algorithm to solve for the optimal driving signal
vector W ` where based on Assumption 2 in Section 2, the spherical harmonic coefficients due to the dominant
plane wave can be expressed as [3]j∗(ϑ j , ϕ j ).
Dnm
= in ·Ynm(7)

Considering a total of J impinging directions which are uniformly sampled over the sphere, we can then solvejjjD j } = {[D00W j } form
for the corresponding J number of driving vectors. The total set of {D, D11, ...DNN]} and {W
the basis of the training dataset for the DNN.
j
In reality, the phase of Dnm
will vary, and thus phase of the optimal driving signal will change accordingly.
j
To model this behavior, we augment the training dataset by adding a random phase shift to both Dnm
and W j ,such that
D j,z = D j · ρ j,z ,W j,z = W j · ρ j,z ,

1126

(8)
(9)

where ρ j,z is a complex random variable with unity amplitude and a uniform phase distribution over [0, 2π).
Our goal is to train the neural network such that its output is not affected by the presence of the reverberations
and noises that may be present in the recorded coefficients Cnm . To model this behavior, we further augment
the training data by adding a complex gaussian noise σ to each plane wave coefficients, such that
D j,z,v = D j,z + σ j,z,v ,

(10)

j,z,v
j,z,v
j,z,v
where σ j,z,v = [σ00
, σ11
, ...σNN] is a vector of instances of the random variable σ , corresponding to eachspherical harmonic coefficient. Since the DNN is expected to produce the same output regardless of the valueof σ j,z,v , the desired output signal W j,z is not further augmented, thus all of D j,z,v are paired with the samedesired output vector W j,z . This augmentation procedure results in a total of J × Z ×V training samples.
To improve the training speed as well as outcome, the training data for the neural network should be normalized.
norm
norm
D j,z,v and W̃
W j,z , where
We define the normalized training input and output data as D̃norm

D j,z,v =D̃D j,z,k
W j,z
D̃
W̃
norm
W j,z =, W̃.
D j,z,k k2
D j,z,k k2
kD̃
kD̃

(11)

Thus, it is necessary to also normalize the input data when using the trained DNN to calculate driving signal,and consequently the DNN output needs to be multiplied by the corresponding input coefficient norm to obtainthe correct loudspeaker driving signals.
3.3 Structure of the DNN
In general, DNNs operate on real-valued data. Thus it is necessary to convert the complex valued input signals
into real-valued vectors for the DNN. A natural choice is to use the real and imaginary parts of D j,z,k and W j,zdirectly, however, it was found that the complex and non-linear relationship between the real and imaginary part
results in poor learning outcome for the neural network.
Therefore, we propose to instead compute the absolute value and phase of the training data, so that
 norm
D j,z,k |, ∠D
Dnorm
D j,z,k = |DD̃
(12)
j,z,k ,
 norm
norm
W j,z |, ∠W
W j,z ] .
W j,z = |WW̃(13)
We propose to use a neural network with two fully connected hidden layers, each consisting of 1,000 neurons
and using the rectified linear unit (ReLU) activation function. The input layer is of size 2(N +1)2 , corresponding
D j,z,k ; while the output layer consists of (N + 1)2 neurons, corresponding to the size of W̃
W j,z . The
to the size of D̃linear activation function is used for the output layer.

4

PERFORMANCE ANALYSIS

4.1 System setup
To validate the performance of the proposed DNN-based sound field reproduction system, we train the proposed
DNN using a realistic system setup, consisting of 30 loudspeakers surrounding the reproduction region. The
loudspeakers are arranged in a dodecahedron geometry, with each loudspeaker placed on the middle point of
every edge of the dodecahedron. This placement allows sound field reproduction up to 3rd order via nearuniform sampling of the sphere[23].
For simplicity, we model the loudspeakers as plane wave sources, hence each element in the channel matrix
can be written as`∗
Hnm= in ·Ynm(θ` , φ` ),(14)
where (θ` , φ` ) represent the impinging direction of the `th speaker in the array. We uniformly sample J = 1348
directions over the sphere, and generate the corresponding plane wave coefficients up to order N = 1 using1127

group Lasso with λ = 0.02. The plane wave coefficient vectors from each impinging direction is then augmented
with Z = 50 random phase angles. Finally, we add V = 10 complex random noise vectors to each augmented
coefficient vector at SNR = 10, resulting in a total of 674, 000 training samples. The DNN is then trained using
the MatLab Deep Learning Toolbox.
4.2 Behavior analysis
4.2.1 Driving signal sparsity
To analyze the behavior of the trained neural network, we use both the Lasso algorithm and the DNN tocalculate the loudspeaker driving signals for plane waves of various directions. For comparison, we also theresults using the Least-Mean-Square (LMS) Mode Matching method, which is calculated by [6]H H H )−1 H H ,W = D (H(15)

where (·)H denotes Hermitian transpose. The loudspeaker driving signals for impinging directions (ϑ = 36◦ , ϕ =−31◦ ) and (ϑ = 104◦ , ϕ = 140◦ ) are shown in Fig. 1. We note that neither of these two impinging directions
are included in the training dataset.
0.5

0.5
Lasso
DNN
LMS

Lasso
DNN
LMS

0.4

Amplitude

Amplitude

0.4
0.3
0.2
0.1

0.3
0.2
0.1

0

0
1

5

10

15

20

25

30

1

5

Loudspeaker Position

10

15

20

25

30

Loudspeaker Position

(a)

(b)

Figure 1. Comparison of loudspeaker driving signal amplitudes calculated using Lasso, DNN and LMS mode
matching, for impinging direction (a) ϑ = 36◦ , ϕ = −31◦ , and (b) ϑ = 104◦ , ϕ = 140◦ .
From Fig. 1, we can observe that the complex Lasso algorithm is able to generate very sparse solutions to
reproduce the plane waves, with only 3 loudspeakers active in each case, and the activated loudspeakers are thenearest ones to the desired impinging direction. The DNN is able to produce driving signals that are nearly
identical to the Lasso solution, indicating that the network has successfully learned the behavior of the Lasso
algorithm. On the other hand, the LMS Mode Matching algorithm results in solutions where nearly all the
loudspeakers are activated with similar driving amplitude. It can be expected that both the Lasso and DNN
algorithm can produce better localization performance than the LMS algorithm.
We also notice in Fig. 1 that the DNN tends to produce very low amplitude driving signals for many unnecessary speakers. The amplitude of these normalized outputs are typically less than 0.015, which is more than 10times lower than the dominant driving signals. This artefact can be rectified by applying a simple threshold tothe normalized DNN output which forces small driving signals to 0.
4.2.2 Noise resistanceTo verify the DNN-based algorithm’s ability to reject noise / interference, we compare the variation of its output
under noisy input conditions. First we define the performance parameterη,∑L`=1 E{|W`true −Wlnoise |}
,
L

(16)

where E{·} denotes expectation, W`true and W`noise denote the calculated driving signal due to noise-free input
coefficients and noisy input coefficients, respectively. Essentially (16) calculates the average variation of driving

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0.02
Lasso
DNN

0.015

0.01

0.005

0
5

10

15

20

25

30

SNR

Figure 2. Mean variation of driving signal amplitude at various SNR.
Table 1. Computation time for rendering an audio stream using ADMM Lasso, and DNN using both CPU andGPU.
Audio Length
0.1 second
1 second
10 second

ADMM Lasso
1.3s
19s
161s

DNN-CPU
0.61s
4.3s
48s

DNN-GPU
0.67s
1.0s
5.4s

signal magnitude due to noise; a low value of η indicates that the output is able to remain relatively constantin spite of noise and interference.
The value of η at SNR = 5 − 30 is calculated for both the Lasso algorithm and the DNN algorithm. For each
SNR, η is estimated by averaging over 1, 000 iterations. The results are shown in Fig. 2.
It can be seen from Fig. 2 that the DNN algorithm consistently outperforms the Lasso solution in terms ofnoise resistance, this is particularly significant at low SNR. Unsurprisingly, the value of η gradually decreases
with increased SNR, for both the Lasso and DNN algorithm. We can thus conclude that although the DNN
algorithm cannot completely remove the effect of noise and interference, it is able to generate more consistentand accurate driving signals than the Lasso algorithm, especially at lower SNR.
4.3 Computational Complexity
The computation times for rendering an audio stream using ADMM Lasso and the DNN algorithm are shownin Table. 1. All algorithms are implemented in MatLab, running on a PC with 2.6GHz Core-i7, 16GB RAM
and GTX 1060 GPU. We can see that for rendering a very short audio frame (0.1s), the time cost is very
similar for all three algorithms; however, as the audio length increase, the time cost of ADMM Lasso increases
almost linearly, while the DNN algorithms see a much smaller increase, especially when GPU is utilized. Thisphenomenon is mainly due to overheads of the MatLab’s in-built Deep Learning Toolbox. Overall, it can beseen that the DNN algorithm has a much higher computation speed compared to ADMM Lasso, especiallywhen GPU is utilized, hence the proposed method has great potential for real-time rendering applications.

5

EXPERIMENTAL RESULTS

To validate the performance of the proposed DNN algorithm in real-life scenarios, we use the Eigenmike anda spherical 30-loudspeaker array to record and reproduce sound fields in lab environment. The loudspeakerarray is arranged in the same geometry as described in Sec. 4.1, as can be seen in Fig. 3(a). We note that the
acoustic lining on the walls only partially reduce reflection.
In the experiment, we first use loudspeaker No.23 (labeled in Fig. 3(b)) to play a wide band music content. The
audio is recorded by the Eigenmike placed at the origin (center). The Eigenmike is placed with its coordinate
system rotated 45◦ clockwise from the coordinate system of the loudspeaker array, which is equivalent to shifting loudspeaker No.23 counter-clockwise, to the virtual source location indicated in Fig. 3(b), when reproducing
the sound field. This makes the problem of reproducing the recorded sound field no longer trivial, and require1129

more than one loudspeaker.

(a)(b)
Figure 3. (a) The dodecahedron loudspeaker array used to reproduce sound field. (b) The position of the virtual
source direction and the loudspeakers in its proximity.
The recorded music is processed in an overlap-add manner with a frame size of 1024 samples and 75% window
overlap, at 48kHz sample rate. For each frequency bin, we calculate the spherical harmonics up to order N = 1,
and use the trained DNN to generate the driving signal for each loudspeaker. The mean driving signal energyof each loudspeaker is shown in Fig. 4, which is calculated by averaging the frequency-domain signal energy
over 200-4000 Hz.

Driving Energy

0.06

0.04

0.02

0
0

5

10

15

20

25

30

Loudspeaker Position

Figure 4. Driving signal energy of the loudspeaker array to produce the sound field recorded by the Eigenmike.
We can see from Fig. 4 that the output energy shows a very sparse pattern, with loudspeaker 22, 23 and 27
having the highest output energy, followed by loudspeaker 13 and 14. Comparing with Fig. 3(b), we see thatthese are the nearest speakers to the source location, and their driving energy is proportional to the distance
between the speaker and the source location. Subjective listening also confirms that the sound imaging iscorrect over a large area around the origin, further validating that the proposed algorithm is able to reproduce
the desired sound field using sparse driving signals.

6

CONCLUSION

We present a novel DNN-based sound field reproduction algorithm, where a DNN is trained to generate a sparseset of loudspeaker driving signals to reproduce a desired plane wave, while suppressing noise and interference
added to the plane wave. We show that the proposed algorithm is more noise-invariant than the Lasso solution,with lower computation cost. The algorithm is validated in the lab using wide band recording and a 30-channel
loudspeaker array.

ACKNOWLEDGEMENTS
This work is supported by Australian Research Council Grant DP180102375.

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