Adapting transfer functions to changes in atmospheric conditions for outdoor
sound field control
Franz M. Heuchel(1)∗ , Diego Caviedes-Nozal(1) , Efren Fernandez-Grande(1) , Jonas Brunskog(1) , Finn T. Agerkvist(1)(1) Technical University of Denmark, Acoustic Technology Group

Abstract
The performance of active sound field control solutions is directly dependent on the accuracy of the measured
acoustic transfer functions between the control loudspeakers and control areas. Outdoors, these transfer functions are affected by atmospheric conditions and their variation over time. In this work, we investigate methods
for estimating transfer functions changes due to changing atmospheric conditions with the goal of adapting outdoor sound field control systems to such changes. We train and test the estimation strategies against results of alarge set of outdoor transfer function measurements in different atmospheric conditions.
Keywords: Outdoor sound propagation, Sound Field Control, Machine learning

1INTRODUCTION

Active control of the sound that outdoor concerts radiate into their surroundings might be useful to reduce
the annoyance that such events create. However, the propagation of sound in earth’s atmosphere is complex,because it is influenced by changes in wind, temperature and other conditions [1]. Studies on the influenceof atmospheric variations on loudspeaker and microphone characteristics [2] and on measured outdoor impulse
responses [3] reported that the whole transmission path from signal generation to its recording is sensitive
to atmospheric conditions. Such changes have a crucial effect on non-adaptive sound field control systems[4, 5, 6]. But also active control systems based on adaptive filtering rely on some kind of estimate of thesecondary transmission path. For example, one of the most popular adaptive filtering algorithms, the filteredreference least-mean-square filter [7], needs a secondary path estimate with a maximum phase error of 90◦ forconvergence [8, p. 144]. If the secondary path changes too much, the secondary path model of the FxLMS
filter needs to be updated.
Thus, sound reproduction systems that intend to synthesize a specific sound field accurately, be it for the purposeof accurate reproduction of sound or its control, might need to take changes in transmission paths due toatmospheric changes into account.
In a previous work, Caviedes-Nozal et al. experimentally studied changes in outdoor transmission paths between
two loudspeakers and four microphones with distances of 150 m to 300 m apart under varying temperature
and wind conditions [9]. These atmospheric variations influenced both magnitude and phase of the measured
transfer functions. It was observed that the induced weather dependent change of phase could be reasonably
well modelled by an appropriate delay applied to a single reference impulse response. While this optimal
delay was approximately linearly dependent on temperature and wind variations, it is questionable if a linear
regression estimation predicts the delay accurately enough for use in a secondary path model of a FxLMS
filter. To continue that work, we investigate in the present paper if machine-learning based regression models
can make better estimations of this compensation delay or even the complete transfer functions. Such methods
might be fast and accurate predictors when trained on a meaningful data set compared to estimations based on
physical models [10].
The paper is structured as follows. In section 2 the data set of measured transfer functions is described andthe used machine learning regression methods are explained. Section 3 shows how accurately transfer function∗ fmheu@elektro.dtu.dk1178

changes can be predicted from weather data. These results are discussed in section 4.

2METHOD

Estimation changes in transfer functions from weather data can be seen as a regression problem. The main
goal in regression analysis is to find an accurate estimation model f with model parameters β that estimates
the dependent variables Y from the known variables X, i.e. f (X, β ) ≈ Y . In the present paper, two regression
problems are studied which intend to estimate the changes with weather in transfer functions (or equivalently inimpulse responses) relative to a reference transfer function Hre f ( f ). In the first problem, the complex transferfunction coefficients (Y ) are estimated using a Feed Forward Neural Network ( f ) from the weather conditions(X), where the network parameters (β ) are computed by Deep Learning and back-propagation. In the secondproblem, the effective delay (Y ) by the weather conditions (X) is estimated using an automated machine-learning
framework that finds both useful regression functions f and their parameters β .
In the following, the data set and the two machine-learning approaches are shortly summarized.
2.1 Data set
In the previous study [9], outdoor transfer-functions were measured between two loudspeakers and eight microphones together with wind speed, wind direction and temperature at different heights over a period of 6 days.
The present study focuses on one of the loudspeaker-microphone pairs (L2, M7), which were separated by adistance of approximately 160 m. The used loudspeaker was a subwoofer with a frequency range of around40-140 Hz.
The measurement series consisted of 77 measurement sets, with each set comprising 10 recordings of 10 second
exponential sweeps, totalling 770 impulse response measurements. The sets were separated by a 30 minutes
break during the day or a whole night between the last and first measurement of a day. Thus, the weather
conditions can be very different between sets while they are rather similar during the 10 sweeps of one set.
The first transfer function measurement of the series is chosen as the reference Hre f ( f ).
For the first regression problem, the independent variables are the wind speed v, wind direction α and temperature T at 5 different heights and the frequency vector f
X = (v8m , v31m , v44m , v57m , v70m , α8m , α31m , α44m , α57m , α70m , T8m , T31m , T44m , T57m , T70m , f).
The complex transfer functions changes are reconstructed using a Feed Forward Neural Network, the dependent
variables Y are the changes in phase and magnitude of the frequency responses H( f ) due to varying weather X
relative to the reference frequency transfer function Hre f ( f )HH,∠.
Y = 10 log10Hre fHre f(1)For the second regression problem, the independent variables are
X = (v8m , v31m , v44m , v57m , v70m , α8m , α31m , α44m , α57m , α70m , T8m , T31m , T44m , T57m , T70m ).
The dependent variable,Z

Y = arg max (h ? hre f )(∆t) = arg max∆th(t)hre f (t − ∆t)dt,(2)∆tis the time delay that maximizes the cross-correlation between the impulse responses h(t) measured during
weather condition X and the first measured impulse response hre f (t).
The measurement series data is divided into 51 sets for model training and 26 sets for testing for both problems.

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Hidden Layers

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tanh(x)

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sigmoid(x)

x1
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𝑓 (L)

Output layer

Input layer

𝑓𝑓 (𝑛)
(1)

1.0
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6 4 2 0 2 4 6x(b) Activation functions from top tobottom: hyperbolic tangent and sigmoid.

(a) Feed forward neural network diagram.

Figure 1. Feed forward neural network representation and activation functions.
2.2 Feed Forward Neural Networks
A feed forward neural network (FFNN), also known as multilayer perceptrons, is an artificial neural networkthat approximates a function Y = f ∗ (X) through the connection of several simple functions [11]f ∗ (X) ' f (X, β ) = f (L) (...( f (n) (... f (1) (X, β1 ), βn ), βL ),(3)where L is the number of hidden layers in the network. Figure 1a shows a classical representation of a FFNN.
The input layer is the vector of known variables X. This vector is connected to the first hidden layer, which
is subsequently connected to the following hidden layers until the output layer (i.e. it is forward-propagated).
The overall length of the chain gives the depth of the network. Each layer consists of smaller units called
perceptrons (represented by circles in Figure 1a) which output is ruled by an activation function g. For example,
the output of the first perceptron in the first hidden layer in the figure is!
Np=g∑ wi xi + b ,(4)i=1where wi are the weights for each of the elements of the previous layer and b is an overall bias. The w’s andb’s for each perceptron in the network that best approximate the desired dependent variables f (x) are foundvia back-propagation, which is the essential algorithm to train a deep neural network. In back-propagation, thetraining data Y is compared to the output of the network at the output layer based on a defined cost function(mean squared error in the present study). The different contributions of each perceptron to the total error are
estimated by calculating the partial derivatives of the error with respect to the different weights and biases using
the chain rule and backpropagating of the error over the layers. For more details on the derivation, refer to [11].
There are multiple types of activation functions that are standard choices in this type of networks. In this work
we use two different activation functions: the hyperbolic tangent in the first hidden layer and the sigmoid in
the second hidden layer, as they were proven to be useful in similar problems [10] (see Fig. 1b).
To avoid overfitting, the model is trained using 10-fold cross validation over the data and dropout of 20% in
both hidden layers, which consists on the random deactivation of 20% of the perceptrons during training [12].

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8∠Y

2404
2
3440(a)Amplitude difference [dB]10log10|Y|

Phase difference [rad]34

Output layer

2560 perceptrons𝑣𝛼𝑇f640 perceptrons

Input layer

Hidden Layers

6080

100 120 140

Frequency [Hz]6420246406080

100 120 140

Frequency [Hz](b)

Figure 2. (a) Designed FFNN to reconstruct the normalized transfer functions Y = HHre f . (b) Resulting error in
the test set. A phase error below 90◦ suggests that the estimator might be suitable for adapting the secondary
path model of an FxLMS algorithm.
2.3 Automated machine learning with auto-sklearn
Selecting the right regression method f and its hyperparameters can be a tedious process as no single algorithm is known to work best on all data sets. For the second regression problem where the effective delay of
the impulse response due to weather changes is estimated from weather data, the automatic machine learning
framework auto-sklearn [13] is used. It builds a regression estimator from an ensemble of regression algorithmsof the well known scikit-learn Python library [14] and optimizes their hyperparameters automatically based on
the training data.

3RESULTS

In this section, the error between the predicted and measured transfer functions changes is shown for bothregression problems. The phase change estimation errors are discussed in terms of its suitability for updating
the secondary path model of an FxLMS algorithm. A delay error below 1.8 ms or phase error of 90◦ suggeststhat the FxLMS algorithm converges. Below 0.9 ms or 45◦ convergence might be fast.
3.1 Transfer-function estimation using Feed Forward Neural Networks
The changes in magnitude and phase of the measured transfer functions with respect to the reference (Eq. (1))
were estimated with a simple FFNN consisting of two hidden layers of 640 and 2560 perceptrons correspondingly (see Fig. 2a).
Figure 2b shows the error between the measured and the predicted changes in the transfer functions both inphase and magnitude. The amplitude error is between [-2, 2] dB for most of the frequency range. The phaseerror is below 90◦ up to 110 Hz.
3.2 Delay estimation
The auto-sklearn estimator was trained with default settings on the training set and predict the compensation
delays on the test set. For comparison a linear regression estimator was used to find the same compensation
delays. Figure 3 a) displays the delay prediction error of both estimators together with the the delay error
without compensation (y pred = 0). Compared to the reference impulse response, the atmospheric variations leadto effective delays in the transmission path of up to 7 ms when no compensation is applied. This might becritical for an FxLMS adaptive filter, because for convergence, a pure delay error in the secondary path model1181

64202
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200250Phase difference: measured - compensated [rad]Prediction error ytest ypred [ms]no compensation
Linear regression
auto-sklearn

Optimal delay

Linear regression

auto-sklearn

34240423450
100
Frequency [Hz]150 50100
Frequency [Hz]150 50100
Frequency [Hz]150

Figure 3. Left: Difference of delay estimations to optimal delay. Right: Corresponding phase errors. A delay
error below 1.8 ms or 90◦ suggests that the estimator might be suitable for adapting the secondary path modelof an FxLMS algorithm. Below 0.9 ms or 45◦ the convergence might be fast.
must be less than 1.8 ms for a minimum phase error of 90◦ at 140 Hz. The auto-sklearn estimation error isless then 0.9 ms in most of the responses in the test set, which translates to a phase error below 45◦ , if thevariations would only induce a pure delay. Such a phase error has only a small effect on the convergence of aFxLMS algorithm [8].
As described in [9], the atmospheric variations are only approximately modelled as a delay of the impulse
responses. The actual phase difference of the impulse responses in the test set and the delay compensatedreference impulse response is shown on the right hand side of figure 3 (optimal delay). The phase error using
the optimal delay is small, which shows that modelling the influence of atmospheric variations as a pure delay
is a reasonable choice. Estimating the delay with the auto-sklearn estimator leads to phase errors that are below
90◦ in the frequency range of the subwoofer (30-140 Hz) and thus in the region of convergence.

4DISCUSSION

Results show that machine learning regression techniques are useful tools to model the relation between thetransfer functions and weather conditions. It seems more difficult to predict changes at higher frequencies for
all the suggested solutions.
The results suggest that a single channel outdoor active noise control system based on FxLMS might adopt its
secondary path model by estimating a compensation delay from the current weather conditions.

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