Signal Processing - July 2016 - 20

signal with the calibration sound. The corresponding objective function, which jointly estimates all source and sensor
positions, is
M

t ,E
t ) = argmin /
(M
M, E

K

/ ^ m m - e k - du k,mh2 .

(7)

m =1 k =1

Note the similarity with (1). Thus, a direct solution of
(7) is possible by employing a variant of the MDS algorithm called base point MDS (BMDS). It infers the microphone positions from PDs between the microphones and the
source positions [4]. First, it computes the coordinates of a
P-dimensional basis from the PDs of P + 1 nodes. Second,
the distance measurements to the base points are used to infer
the microphone positions relative to the base points. Third,
a full distance matrix is constructed to run the conventional
MDS algorithm.
Sachar et al. [38] used a fixed loudspeaker construction
composed of four speakers on the edges of a pyramid that
emitted calibration pulses. They calibrated a small 16-element
microphone array with an accuracy of 0.8 cm and a large aperture array consisting of 448 microphones with an accuracy of
3 cm. Contini et al. [6] used a synchronized loudspeaker, which
was moved to 25 # 5 positions of a rectangular grid, and white
noise as a calibration signal. They were able to calibrate a linear array with an accuracy of 1 cm in both an anechoic and a
strongly reverberant room.
Crocco et al. [8] used chirp signals with known emission
times to obtain precise ToA estimates. An elaborate formulation of the objective function allows exploiting the constraints
that the sensor and event locations are rank P matrices. By
using SVD and a rank approximation technique, they were able
to derive a closed-form solution in the affine space. The transformation of this solution into the Euclidean space requires an
estimation of a matrix with P2 unknown parameters. Thus, the
number of unknowns became independent of the number of
sensors and events. The estimation of this matrix still leads to
a nonlinear optimization problem. However, this is much simpler to solve than an optimization of (7). In an experiment with

eight microphones and 21 source positions, they reported an
accuracy of 1 cm [7].

Active devices
Several methods use active devices such as smartphones or
laptops. Here, microphones and loudspeakers are colocated.
PDs can be obtained from the correlation of a calibration
pulse played back by the loudspeaker at one node with the
signal received by the microphone at other nodes. No time
synchronization is necessary, since the emission offset is
canceled out by using a pair of devices as both sender and
receiver. Typically, an initial estimate is computed by MDS.
Thereafter, a maximum likelihood (ML) estimation is performed, incorporating the known distance between loudspeaker and microphone in an active device [13], [35].
Raykar et al. [36] derived a joint clustering-based ML estimation. In an experiment with laptops on a table, they achieved
an accuracy of around 7 cm without time synchronization and
3 cm with it. Hennecke et al. [13] used smartphones lying
close to one another on a table as shown in Figure 3. Using the
known speaker-microphone distance and the known smartphone orientation in the second step improved the estimate
and allowed resolution of the invariance to mirrored solutions.
They achieved an accuracy of 7 cm for four smartphones,
which were approximately 40 cm apart from each other.

Joint localization and synchronization
If the signals' onset times and recording delays of the ToA
measurements are unknown, the localization problem is considerably more difficult. Gaubitch et al. [11] suggested a twostep approach, where first the timing information and then the
locations were estimated. For both estimation procedures, the
low-rank structure of the sensor and source location matrices
was exploited.
First (5) is rewritten as
m Tm m m + e Tk e k - 2m Tm e k
=
c2
t 2k, m + t k2 + d 2m - 2 (t k, m t k - t k, m d m + t k d m) .

(8)

Next, the equations for m = 1 and for k = 1 are subtracted
from (8). If this is done for k = 2, f, K and m = 2, f, M, and
the ToA measurements utk, m are used, the resulting system of
equations can be expressed in matrix form as
r T Er
- 2M
u + C(i),
=T
c2

FIGURE 3. An ad hoc array composed of smartphones. (Photo used
courtesy of TU Dortmund.)

20

(9)

r is the P # (M - 1) dimensional matrix of the microwhere M
phone locations relative to the first microphone, with entry
(m m - m 1) in the (m - 1) st column, and where Er is the
P # (K - 1) -dimensional location matrix of the acoustic events
relative to the first event, with entry (e k - e 1) in the (k - 1) st
u contains the squares of the measured
column. Furthermore, T
ToA values, and the matrix C(i) gathers the terms that depend
on the unknown timing parameters i = [t 2, f, t k, d 1, f, d M] .

IEEE Signal Processing Magazine

|

July 2016

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Table of Contents for the Digital Edition of Signal Processing - July 2016

Signal Processing - July 2016 - Cover1
Signal Processing - July 2016 - Cover2
Signal Processing - July 2016 - 1
Signal Processing - July 2016 - 2
Signal Processing - July 2016 - 3
Signal Processing - July 2016 - 4
Signal Processing - July 2016 - 5
Signal Processing - July 2016 - 6
Signal Processing - July 2016 - 7
Signal Processing - July 2016 - 8
Signal Processing - July 2016 - 9
Signal Processing - July 2016 - 10
Signal Processing - July 2016 - 11
Signal Processing - July 2016 - 12
Signal Processing - July 2016 - 13
Signal Processing - July 2016 - 14
Signal Processing - July 2016 - 15
Signal Processing - July 2016 - 16
Signal Processing - July 2016 - 17
Signal Processing - July 2016 - 18
Signal Processing - July 2016 - 19
Signal Processing - July 2016 - 20
Signal Processing - July 2016 - 21
Signal Processing - July 2016 - 22
Signal Processing - July 2016 - 23
Signal Processing - July 2016 - 24
Signal Processing - July 2016 - 25
Signal Processing - July 2016 - 26
Signal Processing - July 2016 - 27
Signal Processing - July 2016 - 28
Signal Processing - July 2016 - 29
Signal Processing - July 2016 - 30
Signal Processing - July 2016 - 31
Signal Processing - July 2016 - 32
Signal Processing - July 2016 - 33
Signal Processing - July 2016 - 34
Signal Processing - July 2016 - 35
Signal Processing - July 2016 - 36
Signal Processing - July 2016 - 37
Signal Processing - July 2016 - 38
Signal Processing - July 2016 - 39
Signal Processing - July 2016 - 40
Signal Processing - July 2016 - 41
Signal Processing - July 2016 - 42
Signal Processing - July 2016 - 43
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Signal Processing - July 2016 - 45
Signal Processing - July 2016 - 46
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Signal Processing - July 2016 - 50
Signal Processing - July 2016 - 51
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Signal Processing - July 2016 - 58
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Signal Processing - July 2016 - 60
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Signal Processing - July 2016 - 73
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Signal Processing - July 2016 - 80
Signal Processing - July 2016 - 81
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Signal Processing - July 2016 - 84
Signal Processing - July 2016 - 85
Signal Processing - July 2016 - 86
Signal Processing - July 2016 - 87
Signal Processing - July 2016 - 88
Signal Processing - July 2016 - 89
Signal Processing - July 2016 - 90
Signal Processing - July 2016 - 91
Signal Processing - July 2016 - 92
Signal Processing - July 2016 - 93
Signal Processing - July 2016 - 94
Signal Processing - July 2016 - 95
Signal Processing - July 2016 - 96
Signal Processing - July 2016 - 97
Signal Processing - July 2016 - 98
Signal Processing - July 2016 - 99
Signal Processing - July 2016 - 100
Signal Processing - July 2016 - 101
Signal Processing - July 2016 - 102
Signal Processing - July 2016 - 103
Signal Processing - July 2016 - 104
Signal Processing - July 2016 - Cover3
Signal Processing - July 2016 - Cover4
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