Data-driven threshold selection for direct path dominance test
Orhun Olgun and Hüseyin Hacıhabiboğlu
1 Graduate School of Informatics, Middle East Technical University (METU), Ankara, TR-06800, Turkey

Abstract
Direction-of-arrival estimation methods, when used with recordings made in enclosures are negatively affectedby the reflections and reverberation in that enclosure. Direct path dominance (DPD) test was proposed as apre-processing stage which can provide better DOA estimates by selecting only the time-frequency bins witha single dominant sound source component prior to DOA estimation, thereby reducing the total computationalcost. DPD test involves selecting bins for which the ratio of the two largest singular values of the local spatial
correlation matrix is above a threshold. The selection of this threshold is typically carried out in an ad hocmanner, which hinders the generalisation of this approach. This selection method also potentially increases thetotal computational cost or reduces the accuracy of DOA estimation. We propose a DPD test threshold selection
method based on a data-driven statistical model. The model is based on the approximation of the singular value
ratio distribution of the spatial correlation matrices as a generalised Pareto distribution and allows selecting
time-frequency bins based on their probability of occurrence. We demonstrate the application of this threshold
selection method via emulations using acoustic impulse responses measured in a highly reverberant room witha rigid spherical microphone array.
Keywords: direction-of-arrival estimation, spherical microphone arrays, direct path dominance test (DPD)1

INTRODUCTION

Direction of arrival (DOA) estimation is one of the essential stages of many different audio and acoustic signal processing applications. Rigid spherical microphone arrays (RSMA) can be used to obtain the sphericalharmonic decomposition of sound fields which enables the development of 3D DOA estimation methods.
Several DOA estimation methods have been proposed for RSMAs. For example steered response power (SRP)
maps [8] are obtained by steering a maximally directive beam and identifying directions that maximize theoutput power as source DOAs. Other methods such as pseudo-intensity vector (PIV) [7] and its improved
version, augmented intensity vectors (AIV) [5] are based on the energetic analysis of sound fields. A subspacebased extension of PIV method was also proposed [11]. EB-MUSIC [18] and EB-ESPRIT [19] are methods thatare based on the adaptation of high resolution spectrum estimation methods to the spherical harmonic domain.
More recently, hierarchical grid refinement (HiGRID) was proposed as a spatial entropy-based DOA estimation
method [1]. It was shown that HiGRID has a lower computational cost for time-frequency bins that contain asingle sound source.
One of the main problems with EB-MUSIC is the necessity for prior information on the number of sources ina time-frequency bin. Direct path dominance (DPD) test which can identify time-frequency bins with a single
dominant source was proposed to address this problem [12]. DPD test has since been used together with PIVand SSPIV to improve the DOA estimation accuracy of these methods [10].
DPD test is based on the calculation of the ratio of the effective rank of the spatial correlation matrix andselecting the time-frequency bins that have unit effective rank. This is done by calculating the ratio of thelargest two singular values of the spatial correlation matrix. Time-frequency bins for which this ratio is above
a given threshold are selected as bins containing contributions only from a single source and used for DOA
estimation. DPD test is very effective in selecting time-frequency bins with a single sound source when theemployed threshold is selected appropriately. When it is selected to be higher than necessary, it eliminates binsthat contain useful DOA information, potentially resulting in the detection of only the most prominent sourcesin multiple source localisation scenarios. In contrast, carrying out the DPD test with a lower than necessary3313

threshold results in selecting time-frequency bins that are possibly contaminated by other sources or coherent
reflections, degrading the DOA estimation accuracy, and increasing the computational cost of DOA estimation.
We present a data-driven threshold selection method based on the approximation of the long-term statistical
distribution of singular value ratios as a generalised Pareto distribution. The method allows selecting bins whichhave a prescribed probability of occurrence. It also allows an approximate specification of the number of binsthat will be selected, thereby making it possible to adapt the DOA estimation process to the available computational power. We evaluate the capabilities of the method using emulations obtained with acoustic impulse
responses measured using an RSMA and with HiGRID as the DOA estimation method.
This paper is organized as follows. Sec. 2 presents a brief overview of spherical harmonic decomposition. Sec. 3
explains DPD test and DPD test threshold selection using the generalized Pareto distribution (GPD). Sec. 4
presents an evaluation of DPD threshold with the proposed approach for bin selection and DOA estimation
using the hierarhical grid refinement (HIGRID) algorithm for DOA estimation. Sec. 5 concludes the paper.

2

SPHERICAL HARMONIC DECOMPOSITION

It is possible to approximate a spatially band-limited sound field using a linear combination of a finite numberof spherical harmonic functions. For a band-limited pressure distribution p(k, r, Ω) defined on a sphere:Np(k, r, Ω) = ∑n∑ pnm (k, r)Ynm (Ω).

(1)

n=0 m=−nwhere pnm (k, r) are the SHD coefficients that depend on the wave number k and the radius r of sphere, Ynm (Ω)
is the spherical harmonic function of order n ∈ N and degree m ∈ Z evaluated at the direction Ω = (θ , φ ), N is
the maximum order of the decomposition. SHD coefficients are related to the sound field, such that:ZZ

pnm (k, r) =S

p(k, r, Ω) [Ynm (Ω)]∗ dΩ(2)where S is a spherical surface.
In practice, sound fields can be recorded via spherical microphone arrays. Such arrays sample the sound field
at a finite number of Q points, {Ωq }q=1···Q on a spherical surface. If the sampling scheme satisfies the orthogonality conditions of the spherical harmonic functions [9], the following cubature can be used to obtain the SHD
coefficients:Q

pnm (k, r) = ∑ wq p(k, r, Ωq ) [Ynm (Ω)]∗

(3)

q=1

where {wq }q=1···Q are the cubature coefficients. The maximum SHD order and the number of microphones on
the array are related by Q ≥ (N + 1)2 .
SHD coefficients of the sound field around a rigid sphere due to a single monochromatic plane wave withfrequency f , wavenumber k = 2π f , arrival direction Ωs , and amplitude al (k) ∈ C are given as [16]:(l)pnm (k, r) = al (k)e jk·r = 4πin al (k)bn (kr) [Ynm (Ωs )]∗(4)

where bn (kr) is a frequency dependent weight that is due to a combination of the incoming and scatteredsound. Notice that we omitted the time dependence of the plane wave for simplicity. SHD coefficients canbe normalized to obtain the spherical harmonic functions. For a monochromatic plane wave with the complex
amplitude αl (k) ∈ C, the normalized SHD coefficients are:(l)p̃nm (k) =3314

pnm (k, r)= αl (k) [Ynm (Ωs )]∗4πin bn (kr)(5)

Sound fields can be represented as linear combination of multiple plane waves. Since SHD is a linear operation,
the SHD coefficients of a sound field comprising multiple plane waves will consist of a linear combination of
the SHD coefficients of its constituents. For a sound field that consist of L plane waves:Lp̃nm (k) = ∑ αl (k) [Ynm (θ , φ )]∗ ,

(6)

l=1
T
which can be expressed in matrix notation as pnm (k) = YH
s a(k) where a(k) = [a1 (k), a2 (k), ..., aL (k)] is the L ×1
2
complex amplitude vector, Ys is L × (N + 1) beamspace manifold matrix with the l-th row given by:y(Ωl ) = [Y00 (Ωl ),Y1−1 (Ωl ),Y10 (Ωl ), ...,YNN (Ωl )](7)

In general terms, DOA estimation with RSMAs corresponds to the identification of Ωl given pnm (k) subject tocertain practical constraints such as sensor noise, and by using certain assumptions appropriate for the task suchas a priori knowledge on the number of sound sources.

3

THRESHOLD SELECTION FOR DIRECT PATH DOMINANCE TEST

Direct path dominance (DPD) test is a time-frequency bin selection method that was originally proposed foruse with eigenbeam multiple signal classification (EB-MUSIC) algorithm [12]. Apart from its advantages whenused with EB-MUSIC, DPD test can be used as a preprocessing stage for any DOA estimation method. DPD
test operates on the windowed Fourier transform of the SHD coefficients of the sound field. Therefore, thediscussion that follows assumes the SHD coefficients pnm (τ, κ) are available up to a given order, N.
DPD test [12] is based on the observation that, the spatial correlation matrix has unit rank for time-frequency
bins where only a single direct component exists. While the real spatial correlation matrix cannot be exactly
known, its expectation in a given time-frequency bin can be used as an approximation. This can be calculated
by averaging the observed spatial correlation matrices over time and frequency, such that:R̃p (τ, κ) =JτJκ1∑ ∑ Rp (τ + jτ , κ + jκ )Jtot jτ =−Jτ jκ =−Jκ(8)where Jtot = (2Jτ + 1)(2Jκ + 1) and the spatial correlation matrix Rp (τ, κ) = pnm (τ, κ)pnm H (τ, κ) is evaluatedat the time-frequency bin (τ, κ). Time-frequency averaging across 2Jτ + 1 time and 2Jκ + 1 bins acts as the
expectation. Notice that Rp (τ + jτ , κ + jκ ) is by itself unit rank, but the time-frequency averaged spatial correlation matrix need not be, unless the SHD coefficient vectors, {pnm H (τ, κ)} in the time-frequency averaging
window are scaled copies of each other. The latter case would only occur if there is a single source in the timefrequency averaging window. However, there will always be multiple components (e.g. sources and coherent
reflections) in a real sound field, and the spatial correlation matrix will be full rank.
In order to differentiate the case where the direct sound is dominant for cases where the spatial correlation
matrix is numerically calculated to be full rank, the DPD test involves the identification of time-frequency bins
where the effective rank of the time-frequency averaged spatial correlation matrix is unity. This is achieved via
obtaining the singular value decomposition (SVD) of the averaged spatial correlation matrix and calculating theratio of the two largest singular values. In other words time-frequency bins for which the effective rank is unity
are selected such that:no
PDPDtest = (τ, κ) : erank R̃p (τ, κ) = 1(9)where,

E = erank R̃p (τ, κ) = 1

if R =σ1 (τ, κ)> TDPD ,σ2 (τ, κ)(10)Here, σ1 (τ, κ) and σ2 (τ, κ) are the largest and second-largest singular values of R̃p (τ, κ) and TDPD  1 is a
threshold. Notice that the ratio of the singular values given in (10) can never be less than unity.

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0.2

0.175

0.175

0.15

0.15

Normalized Histogram

Normalized Histogram

0.2

0.125
0.1
0.075
0.050
0.025

0

0.125
0.1
0.075
0.050
0.025

10

20

30

40

Singular Value Ratio

(a)

50

60

0

10

20

30

40

50

60

Singular Value Ratio(b)

Figure 1. Normalised histograms of R for sources at a distance of (a) 0.5 m and (b) 2.6 m. The fitted
generalized Pareto distributions are also shown with red curves.
The selection of the DPD threshold is typically made in an ad hoc manner by selecting a satisfactory value
which which will eliminate a majority of time-frequency bins and at the same time provide good DOA estimates
subject to the capabilities of the selected DOA estimation method [10, 17]. Therefore, DPD threshold selection
has a profound effect on the trade-off between the accuracy and computational cost of DOA estimation.
The data-driven threshold selection approach we propose in this paper is based on the observation that the
singular value ratios of spatial correlation matrices come from heavy right-tailed probability distributions. These
distributions were observed to change with the reverberation time, direct-to-reverberant (D/R) ratio, as well as
the number and types of sources to be localized.
A flexible probability distribution that can be used to approximate such distibutions is the generalised Pareto
distribution (GPD). GPD is a three-parameter probability distribution that is effective when used to model thetails of other probability distributions [15]. Cumulative distribution function (CDF) of GPD [3] is given as:
Fζ ,σ ,µ (x) = 1 − [1 − ζ (x − µ)/σ ]1/ζ

(11)

Note that Fζ ,σ ,1 (1) = 0. Probability density function (PDF) of GPD is given as:1ζ (x − µ) (1−ζ )/ζ
fζ ,σ ,µ (x) =
1−
σ
σ

(12)

where ζ ≤ 0 is the shape parameter, σ > 0 is the scale parameter, and µ > 0 is the location parameter.
In reverberant environments, the probability of observing a high ratio of singular values is low. Similarly, formost realistic cases involving speech or music signals, the time-frequency representation is sparse, resulting ina majority of time-frequency bins which have a low singular value ratio. If we assume that effective ranks
calculated for each time-frequency bin as in (10) come from a GPD with a location parameter, µ = 1, theselection of the DPD threshold can be made based on the probability of observing a ratio greater than a desired
ratio threshold, TDPD such that:
P[R > TDPD ] = 1 −Z TDPD1fζ ,σ ,1 (x)dx = 1 − Fζ ,σ ,1 (TDPD ) + Fζ ,σ ,1 (1)(13)

where P[·] represents the probability. In other words, if the scale and shape parameters can be estimated fromthe singular value ratios of time-frequency bins, the threshold for the DPD test can be selected as the minimum
value which satisfies:
P[R > TDPD ] = 1 − Fζ ,σ ,1 (TDPD ) < Pb
(14)

3316

The threshold to be used in the DPD test in order to guarantee that the ratio for the selected bins have theprobability of occurrence, Pb should then be selected to satisfy:
TDPD > σ (1 − Pbζ )/ζ

(15)

The scale and shape parameters of the underlying distribution can be trivially obtained via maximum likelihood
estimation [6, 2]. This way it becomes possible to select the DPD threshold in a deterministic way. The numberof selected bins can also be approximately controlled, i.e. for K time-frequency bins that are analysed, the theexpected number of bins selected this way will be approximately K P̂.
Fig. 1 shows the normalised histograms and the fitted generalized Pareto distributions for two sources at thedifferent distances of 0.5 m and 2.6 m. It may be observed that moving the source away from the RSMA
results in a higher number of bins with lower R since the D/R ratio decreases. The parameters, (ζ , σ , µ) of
the fitted GPD are (−1.7241, 10.1377, 1) and (−0.53987, 6.14617, 1).

4

EVALUATION

We evaluated the DPD threshold selection method using emulated sound recordings. The details of this evaluation is given in this section.
4.1 Sound scene emulations
We used two sets of anechoic audio signals for the evaluations: music and speech. Music signals consisted of
4 seconds of anechoic recordings of Mahler’s Symphony Nr. 1, Mvt. 4 [14] at the sampling rate of fS = 48
kHz. The four tracks where violins are playing in unison are selected. Speech signals consisted of the first 4
seconds of energy normalized dry speech signals from two male and two female speakers recorded at METUSPARG audio lab. The sampling rate was fS = 48 kHz.
We used the acoustic impulse responses (AIRs) recorded with Eigenmike em32 microphone array in a classroom
with a reverberation time of T60 ≈ 1.12 s [13]. Signals used in the evaluations were obtained by convolving thesource signals with measured AIRs.
Three simple scenarios were simulated. In all three scenarios, the four sources were positioned at the drections(π/2, 0), (π/2, π/2), (π/2, π), and (π/2, 3π/2). In the first, second and third scenarios, the sources were positioned at a the distances of 0.5 m, 1m and 1.5 m from the RSMA, respectively. These distances correspond to
average direct-to-reverberant (D/R) ratios of 10.72, 5.69, and 2.12 dB, for the scenarios 1, 2, and 3, respectively.
4.2 DPD test parameters
Windowed Fourier transform with a 1024-point FFT and a Hamming window with 75% overlap are used in theevaluations. The frequency range of analysis was between 2608 and 5216 Hz which allowed a decompositionof order N = 4. Smoothing parameters for the DPD test were Jτ = 4 and Jν = 15. Only the SHD coefficients
up to an order of N = 3 were used for calculating the spatial correlation matrices.
Selection of bins using the DPD test was made for five different probability levels, Pb = 0.00625, 0.0125, 0.025,
0.05, and 0.1 after fitting a GPD to the observed data via maximum likelihood estimation as described in [2].
4.3 DOA Estimation Method
We used hierarchical grid refinement (HiGRID) [1] as the DOA estimation method. HiGRID is based on thecalculation of a sector averaged power of a plane-wave decomposition steered response and identifies the DOAs
of sources by seeking a data-adaptive multresolution steered response representation that minimises spatial entropy. The algorithm provides very accurate DOA estimates on par with other state-of-the-art DOA estimation
methods such as EB-MUSIC at a lower computational cost. It is also robust to noise and reverberation, and canlocalise multiple coherent sources. HiGRID is an output sensitive algorithm which has a lower computational
cost for a low source count. Therefore, using the DPD test as a pre-processing stage to identify bins with a

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b
Table 1. DPD test thresholds for different probabilities, P.
bP
0.00625 0.0125 0.025 0.05
0.1
Scenario 1 (Violins)
79.6
53.7
35.6 23.0 14.2
Scenario 2 (Violins)
46.7
34.9
25.5 18.0 12.2
Scenario 3 (Violins)
34.7
26.6
20.0 14.5 10.0
Scenario 1 (Speech)
459.5
226.5 110.9 53.6 25.2
Scenario 2 (Speech)
94.0
58.7
36.2 21.8 12.6
Scenario 3 (Speech)
31.6
22.7
16.0 11.0 7.2
bTable 2. Number of bins selected for different probabilities, P.
Pb
0.00625 0.0125 0.025 0.05
0.1
Scenario 1 (Violins)
379
675
1414 2536 4426
Scenario 2 (Violins)
386
727
1344 2433 4733
Scenario 3 (Violins)
502
828
1328 2335 4570
Scenario 1 (Speech)
55
191
797
2389 5102
Scenario 2 (Speech)
672871013 2638 5317
Scenario 3 (Speech)
1645191364 2825 5494

single dominant component would reduce its computational cost even further.
HiGRID was used on the time-frequency bins selected via the DPD test. The maximum SHD order was N = 4,
and the maximum resolution level used was 3 which which corresponds to a spherical grid with 768 elements
and provides angular resolution of 7.33◦ . Implementations of HiGRID and DPD in Python 3.6 were used [4].
4.4 Results
Table 1 shows the thresholds calculated for different scenarios at different probability levels. Values shown withbold typeface indicate cases where less than four sources were localised. It may be observed that decreasing
Pb corresponds to a higher DPD test threshold and it is possible to use high DPD thresholds without degrading
DOA estimation performance.
Table 2 shows the number of bins selected from a total of 42552 time-frequency bins in each tested case. It
may also be observed that the thresholds and the number of selected bins are different for speech and music aswell as for different scenarios. The number of selected bins approximately coincide with P̂ × 100% of the total
number of tested bins.
The time-frequency bins that pass the DPD test are processed with HiGRID to determine the source DOAs.
Fig. 2 shows the spherical DOA histogram for Scenario 2 with violins using bins selected by setting P̂ = 0.025.
All sources are clearly represented by a distinct peak in the histogram.
Table 3 shows the DOA estimation errors using data-driven DPD test threshold selection and HiGRID for theemulated cases. Except for very low values of P̂, all four sources were correctly localised with a very low
average DOA estimation error below half of the maximum resolution used in nearly all cases. Another observation is that higher thresholds can be used for closer sources and vice versa. Overall, P̂ = 0.0125 and P̂ = 0.025
are good selections for music and speech, respectively, providing low DOA estimation errors at a low computational cost for all the different cases tested. The effect of reverberation time and other acoustical features of
the recording venue on threshold selection remains to be investigated.

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Table 3. DOA estimation errors in degrees.
0.00625 0.0125 0.025 0.05
Pb
Scenario 1 (Violins)
3.27◦
2.77◦
2.24◦ 1.91◦◦◦
Scenario 2 (Violins)
4.36
2.92
2.06◦ 1.10◦◦◦
Scenario 3 (Violins)
4.01
3.34
1.56◦ 0.99◦
Scenario 1 (Speech)
8.31◦
3.31◦
3.53◦ 3.14◦◦◦
Scenario 2 (Speech)
0.02
2.41
2.61◦ 2.74◦◦◦
Scenario 3 (Speech)
1.12
3.32
2.87◦ 1.56◦

0.1
1.52◦
1.38◦
1.21◦
2.30◦
1.92◦
1.05◦
400

200

Inclination histogram

200
100

Azimuth Histogram

Figure 2. Mollweide projection of DPD-HiGRID DOA histogram showing the peaks of source locations on aspherical grid consisting of 768 pixels.

5

CONCLUSIONSWe proposed a data-driven threshold selection method for the direct path dominance test in this paper. The
method is based on the observations that the singular value ratios used in the DPD test come from a heavy
right-tailed distribution which can be approximated well by using the generalised Pareto distribution, and alsothat time-frequency bins that contain a single source only are rare events. These observations allow the selectionof a DPD test threshold using the probability of occurence, effectively allowing the control of approximate
number bins to be processed. We presented an evaluation of the proposed method for speech signals and noise.
The evaluations revealed that DPD test thresholds that are much higher than those reported in literature can beused without degrading the performance of DOA estimation.

ACKNOWLEDGEMENTS
The work reported in this paper is supported by the Turkish Scientific and Technological Research Council(TÜBİTAK) Research Grant 113E513 “Spatial Audio Reproduction Using Analysis-based Synthesis Methods”(2014-2018).

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