Signal Processing 107 (2015) 185–196

Contents lists available at ScienceDirect

Signal Processing
journal homepage: www.elsevier.com/locate/sigpro

Blind compensation of interchannel sampling frequency
mismatch for ad hoc microphone array based on maximum
likelihood estimation
Shigeki Miyabe a,n, Nobutaka Ono b, Shoji Makino a,1aLife Science Center of Tsukuba Advanced Research Alliance, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
Principles of Informatics Research Division, National Institute of Informatics/Department of Informatics, School of Multidisciplinary
Sciences, The Graduate University for Advanced Studies (SOKENDAI), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japanba r t i c l e i n f oabstract

Article history:
Received 17 February 2014
Received in revised form
10 September 2014
Accepted 11 September 2014
Available online 22 September 2014

In this paper, we propose a novel method for the blind compensation of drift for the
asynchronous recording of an ad hoc microphone array. Digital signals simultaneously
observed by different recording devices have drift of the time differences between theobservation channels because of the sampling frequency mismatch among the devices. On
the basis of a model in which the time difference is constant within each short time frame
but varies in proportion to the central time of the frame, the effect of the sampling
frequency mismatch can be compensated in the short-time Fourier transform (STFT)
domain by a linear phase shift. By assuming that the sources are motionless and havestationary amplitudes, the observation is regarded as being stationary when drift does notoccur. Thus, we formulate a likelihood to evaluate the stationarity in the STFT domain toevaluate the compensation of drift. The maximum likelihood estimation is obtained
effectively by a golden section search. Using the estimated parameters, we compensate
the drift by STFT analysis with a noninteger frame shift. The effectiveness of the proposed
blind drift compensation method is evaluated in an experiment in which artificial drift isgenerated.
& 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Keywords:
Ad hoc microphone array
Drift
Sampling frequency
Maximum likelihood estimation
Blind source separation

1. Introduction
Microphone array signal processing is a framework foranalyzing spatial information of a sound field observed withmultiple microphones to perform speech enhancement,
source separation, source localization, and so forth, which
are difficult by the processing of single-channel observations[1]. Microphone arrays are used in various applications,
including teleconferencing, hands-free speech recognition, hearing aids, acoustic monitoring, spatial audio, andnCorresponding author. Tel.: þ 81 29 853 6566; fax: þ 81 29 853 7387.
E-mail address: miyabe@tara.tsukuba.ac.jp (S. Miyabe).
1
The following co-author is a member of EURASIP: Shoji Makino.

computer games. While conventional microphone array signal
processing assumes that multichannel signals are observed bya unified analog-to-digital converter (ADC), recently increasing
attention has been focused on an extension of the microphonearray framework, the so-called ad hoc microphone array,
where a combination of observations by independent recording devices is treated as multichannel recording [2]. The nonnecessity of wired channels to achieve synchronization
enables the downsizing of the recording devices, which isan important attribute for hearing aids [3]. In addition, thisframework is suitable for recording meetings because of theeasy construction of the microphone array by the combinationof widely available portable recording devices, such as cell
phones, IC recorders, and video cameras, and the freedom of
the microphone arrangement, which enables recording with a

http://dx.doi.org/10.1016/j.sigpro.2014.09.015
0165-1684/& 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/3.0/).

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S. Miyabe et al. / Signal Processing 107 (2015) 185–196

high SNR by setting each device close to each speaker [2,4].
Also, considerable effort has been made to develop wirelessacoustic sensor networks (WASNs), where the recording
devices are connected by wireless networks [5].
However, the increased freedom of ad hoc microphonearrays raises various issues that do not arise in conventional array signal processing. For example, the arraygeometry is unknown [2,6–8], the recording devices havedifferent unknown gains [2], each device starts recording
independently [7,8], and the sampling frequencies are notcommon among the observation channels [9–15]. Also, in
WASNs, the efficiencies of communication and distributed
computation are important issues to achieve array signal
processing with a limited bandwidth and array nodes withlow computational power [5].
Among these issues of ad hoc microphone arrays, oneof the most important is the mismatch of sampling
frequencies. Since each ADC is not synchronized with theothers, the individual variability of clocks results in a slightmismatch of the sampling frequencies, causing a change inthe time difference between channels due to the constantskew, the so-called drift. Array signal processing generally
assumes that the sources do not move and utilizes thephase differences inherent to the positions of the sources.
However, drift causes the phase differences to constantly
change as if the sources are moving, preventing the use ofarray signal processing to analyze phase differences
assuming static sources [10,11]. Also, the asynchronousrecording causes offsets of the recording start time.
Estimation and compensation of the sampling frequency
mismatch and recording start offset are indispensable aspreprocesses in array signal processing.
While there are two types of methods for estimating
the sampling frequency mismatch, i.e., supervised andunsupervised, our research focuses on unsupervised
estimation. The former, supervised estimation, mainly
estimates the relation between the clock time of the ADC
and the absolute time by using the time stamp received
from satellites or wireless networks [9,13–15]. While theadvantage of this approach is that the recording start offset
can be estimated in addition to the sampling frequency
mismatch, the disadvantage is the constraint that therecording devices must have the ability to receive thetime stamp. Another problem is that the accuracy of thetime stamp is generally much lower than that required inarray signal processing, and training involving the longobservation of time stamps is required for accurate estimation. Unsupervised estimation cannot obtain a precise
estimate of the recording start offset without prior information. Inaccurate compensation of the recording start
offset is problematic with particular classes of supervised
array signal processing which receive time differences ofarrival (TDOAs) of positions of specific sources as priorinformation because the offset changes the relation
between the TDOAs and the positions. However, in a blind
scenario of array signal processing such as blind source
separation (BSS) [16], where only the observation is given,a rough compensation of the recording start offset issufficient. Therefore, in this paper we focus on the accuratecompensation of the drift and the rough compensation of
the recording start offset in an unsupervised manner.

To the best of our knowledge, there have been few
works on the blind estimation of sampling frequency
mismatch. Liu et al. proposed a method of estimation
involving the iteration of independent component analysis(ICA) and evaluation of the correlation between theestimated independent components by utilizing the property that ICA can extract uncorrelated independent sources
only when the drift is well compensated [10]. However, its
applicability is limited to determined systems with equal
numbers of sources and microphones so that the independent components can be extracted. Markovich-Golan et al.
estimated the sampling frequency mismatch as the rate ofchange of the phase in the interchannel correlation of anoise observation [12]. The modeling of the sampling
frequency mismatch using the phase is very similar toour proposed framework, which is discussed later. However, the scenario of this method, which assumes voiceactivity detection (VAD), is different from our fully blindestimation scenario.
In this paper, we propose a novel method for blind driftcompensation by maximum likelihood estimation of thesampling frequency mismatch in the short-time Fouriertransform (STFT) domain. The basic idea of the maximum
likelihood estimation is published in a previous conference
proceedings paper [17]. The optimization algorithm isfollowed by resampling as a modification of the STFT
analysis with a noninteger frame shift, which we proposedin another conference proceedings paper [18]. In addition,
we newly propose an iterative algorithm for estimating therecording start offset and sampling frequency mismatch,which enables the accurate compensation of drift evenwith a long observation. We model the drift in the STFT
domain as a linear phase by ignoring the drift inside each
short time frame. By assuming that the sources areunmoving and have stationary amplitudes, the soundwave to be observed is regarded as stationary regardlessof the number of sources. Since the stationarity collapses
with the pseudo-movement of the sources caused by drift,
the stationarity can be a cue to estimate the drift. Thus, wederive a likelihood function in the STFT domain to measure
the stationarity and evaluate the compensation of thesampling frequency mismatch. The maximum likelihood
estimate is searched for efficiently by performing a golden
section search. We also show that the likelihood function
evaluates the coherence between the channels. To compensate the recording start offset, we shift the observed
signal in the time domain to maximize the interchannel
correlation of the signals with the sampling frequency
mismatch compensated. Since the accuracies of theestimation of the offset and the compensations of the drift
are mutually dependent, particularly when the observation is long, we iterate these procedures. We evaluate theeffectiveness of the proposed method of blind drift compensation in an experiment to emulate the asynchronousrecording of an ad hoc microphone array by giving an
artificial sampling frequency mismatch.
The rest of the paper is organized as follows. In
Section 2, we formulate the asynchronous observation of
the ad hoc microphone array. In Section 3, we describe ourmodeling of the drift in the STFT domain. In Section 4, we
derive the likelihood function used to estimate theS. Miyabe et al. / Signal Processing 107 (2015) 185–196

sampling frequency mismatch, carry out an efficient
search for the maximum likelihood estimate, and analyze
the properties of the likelihood function. In Section 5, we
describe STFT analysis with noninteger frame shift forcomputationally simple resampling. In Section 6, we
describe the whole algorithm used in the proposed
methodofblind drift compensation. In Section 7, we evaluate theeffectiveness of the proposed method. Finally, the paper isconcluded in Section 8.

187

fs

x1 (t )t

x2 (t )tT21(1+ε) fsx1 [n]

2. Effect of sampling frequency mismatch on discrete
signalsx2 [n]

We suppose that sound pressures x1 ðtÞ and x2 ðtÞ on two
microphones are sampled by different ADCs, where tdenotes the continuous time. The sound pressures areobserved as the discretized signals denoted as x1 ½n
for n ¼ 0; …; N1  1 and x2 ½n for n ¼ 0; …; N2  1, where ndenote the discrete time, and Ni for i¼1, 2 is the number ofobserved samples of the ith channel, and N1 and N2 are notnecessarily the same. Throughout this paper, the notations
ðÞ and ½ are used for denoting continuous and discrete
time signals, respectively. Although the signals are available as observation only at the integer-valued discrete
times, we accept the noninteger values of the discrete
times. We also suppose that the sampling frequency ofx1 ½n is f s and that of x2 ½n is ð1 þ ϵÞf s for a dimensionless
number ϵ used to define the sampling frequency mismatch, without loss of generality. In this paper, we assume
that the ADCs have common nominal sampling frequencies and that jϵj 5 1. Note that we focus on the compensation of drift between two channels in this paper, but theextension to an arbitrary number of channels can be
carried out easily. The relations between xi ½n and xi ðtÞ
for i¼1,2 are given by
 
n
;
ð1Þ
x1 ½n ¼ x1
fs
x2 ½n ¼ x2

nþT 21 ;
ð1 þ ϵÞf snð2Þwhere the origin of the continuous time t¼0 is defined as
the time when the sampling of x1 ½n starts, and T21 is thecontinuous time when the sampling of x2 ½n starts. The
discrete times of these two channels have independent
correspondence to the continuous time as described in (1)and (2), and the phase difference between the channels
linearly changes according to the time, hereafter we refer
this behavior as drift. Fig. 1 shows a conceptual diagram of
the asynchronous recording in the case that x1 ðtÞ ¼ x2 ðtÞ.
We denote the pair of discrete times corresponding to theidentical continuous time t as n1 for the first channel andn2 for the second channel. Then the discrete time pair n1
and n2 satisfies the following condition:n2 ¼ ð1 þ ϵÞðn1 D21 Þ;

ð3Þ

D21 ¼ f s T 21 ;ð4ÞnFig. 1. Conceptual diagram of the asynchronous recording in the casewhen the analog waveforms of the two channels are the same, i.e.,x1 ðtÞ ¼ x2 ðtÞ. The sampling frequency mismatch with ϵ4 0 expands thedigital waveform of x2 ½n. (a) Analogue waveform and sampling and (b)Resultant digital waveform with drift.

where D21 denotes the discrete time of the first channel
when the recording of the second channel starts. Notethat hereafter we use the notation of the pair n1 and n2
when we consider the correspondence to the identicalcontinuous time.
According to the above modeling of the sampling of theasynchronous recording, the discretized signal x^ 2 ½n of thesecond channel with precise synchronization to the firstchannel is given as nx^ 2 ½n ¼ x2
fs
ð5Þ
¼ x2 ½ð1 þ ϵÞðn  D21 Þ:Thus, to achieve the synchronization, we have to obtain
the amplitude of the second channel at the nonintegervalued discrete times ð1 þ ϵÞðn D21 Þ, and the problem
reduces to resampling. The precise resampling is givenby the following infinite convolution of the sinc function:x^ 2 ½n ¼

1∑n0 ¼  1sincðð1 þ ϵÞðn  f s T 21 Þ  n0 Þx2 ½n0 :

ð6ÞSince an infinitely long observation is unavailable inpractice, the infinite convolution has to be approximated,for example, by a truncated convolution. Additionally,a simplified model is necessary to formulate an effective
estimate of the unknown mismatch parameters.
3. Modeling of sampling frequency mismatch in STFT
domain3.1. Statement of problem
Here we discuss the modeling and compensation of thesampling frequency mismatch, which is the basis of the
proposed method. Since parameter estimation in arraysignal processing is generally formulated as statisticaloptimization in the STFT domain, it is sufficient to compensate for the effect of drift appearing in the STFT
domain. Thus, the analysis in the STFT domain is appropriate for the estimation and compensation of drift asa preprocessing step in array signal processing.

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S. Miyabe et al. / Signal Processing 107 (2015) 185–196

correspondence between n1 ¼ m and n2 ¼ ϕ21 ðmÞ, indicating an identical continuous time.
m−

L
2

m+

m

n1

L
−1
2

3.3. Compensation model of sampling frequency mismatchin STFT domain

L/2
L

Fig. 2. Discrete time n1 of the first channel in the frame centered atsample m.

We define the STFT signal X i ðk; mÞ, i ¼ 1; 2; k ¼
 L=2; …; L=2  1 obtained by the frame analysis of xi ½n
centered at the mth sample with a window of length L
samples as




L1
L
2π jkl;ð7Þ
X i ðk; mÞ ¼ ∑ wðlÞxi l þ m  exp 2L
l¼0
where wðlÞ is an appropriate window
pﬃﬃﬃﬃﬃﬃﬃﬃ function, k is thediscrete frequency index, and j ¼  1. We assume that
L is even. Note that the discrete Fourier transform issubstituted by the fast Fourier transform in practicalprocessing.
We discuss the correspondence between n1 and n2
inside the frame centered at sample m in the first channel,i.e., m  L=2 r n1 rm þ L=2  1 as shown in Fig. 2. From (3),the following correspondence is obtained:ðn2 mÞ  ðn1  mÞ ¼ ϵm þ ϵðn1  mÞ  ð1 þ ϵÞD21 :ð8ÞConsidering that ðn1  mÞ is the time inside the frame, theterms on the right side of (8) have the following
properties:

 The first term ϵm increases or decreases in proportion
to the frame central time m.

 The second term ϵðn1  mÞ is proportional to the time
inside the frame.

 The third term ð1 þ ϵÞD21 originates from the differencein the recording start times.

3.2. Constant-drift model within frame
Let us start from the second term on the right of (8).
Suppose that both ϵ and L are small and their product ismuch smaller than one sampling interval,
jϵLj 5 1:ð9ÞThen we obtain the following approximation because
 L=2 r n1 m rL=2 1:ðn2 mÞ  ðn1  mÞ  ϵm ð1 þ ϵÞD21 ;ð10Þwhich indicates that the drift inside each frame is sufficiently small to be ignored. In addition, if the frame of thesecond channel is centered at

ϕ21 ðmÞ ¼ ð1 þ ϵÞðm  D21 Þ;

ð11Þthen the frame is synchronous with the frame of the firstchannel centered at the mth sample because of the

Next, we discuss the third term  ð1 þ ϵÞD21 on the rightof (8), which is caused by the difference in the recording
start times. Although accurate estimation of the differencein the recording start times is difficult, a small error is
acceptable for the class of array signal processing without
the TDOA prior such as BSS, as described in Section 1. Thus,
the satisfactory compensation of the recording start offset
with the error much smaller than the frame size isobtained by the time shift with the maximum correlation.
We now redefine the discrete time of the second channel
so that the recording start offset is compensated asD21  0, with n2 ¼ 0 corresponding to n1  0. Then (10) is
approximated asðn2 mÞ  ðn1  mÞ  ϵm:

ð12Þ

If, for the central samples m of all the frames, theconditionjϵmj5 Lð13Þis satisfied and the shift of the frame is much smaller than
the frame size L, the time difference ϵm in (12) can be
regarded as a small shift of the whole signal inside eachframe. Such a small shift can be compensated by a linear
phase in the STFT domain as
2π jkϵmð14Þ
X^ 2 ðk; m; ϵÞ ¼ X 2 ðk; mÞ exp
L
to obtain the STFT signal X^ 2 ðk; m; ϵÞ with the sampling
frequency mismatch compensated.
3.4. Trade-off in frame size
To obtain the compensation model of (12), we introduced two assumptions in (9) and (13) related to the frame
size L. These assumptions contradict each other when
the sampling frequency mismatch ϵ is large and the signal
lengths N1 and N2 are large, as summarized in thefollowing:
1. If L is large, (9) cannot be satisfied and the drift insideeach frame becomes too large to ignore.
2. If L is small, (13) cannot be satisfied, and the drift
cannot be compensated by a linear phase in the STFT
domain as in (14), particularly when the observation islong and jmj takes a large value.
Thus, the frame size has to be chosen carefully. We show
a conceptual diagram of the trade-off in Fig. 3.
The drift inside each frame is unignorable when thesampling frequency mismatch ϵ and the frame length L arelarge and the condition in (9) cannot be satisfied. As can beseen in Fig. 3, the largest effect of the drift inside the frame
appears in both ends of the frame with the error of thediscrete time of jϵL=2j samples. Such that the time error ofjϵL=2j samples corresponds to the error of the phase jπ jkϵj

S. Miyabe et al. / Signal Processing 107 (2015) 185–196

φ 21(m)− m

ε L /2m

D21

L

φ 21(m)− m

189

these assumptions, we derive a likelihood for the sampling
frequency mismatch using the compensation model givenby (14). Note that in this section we assume that thecondition in (13) is satisfied, and the model mismatch isfixed in the algorithm described in Section 6.
We assume that all the sources are stationary and that
their amplitudes are stationary on a long-term basis. Thenthe compensated observed signal X^ ðk; m; ϵÞ expressed invector notation, given byX^ ðk; m; ϵÞ ¼ ½X 1 ðk; mÞ; X^ 2 ðk; m; ϵÞT ;

ð15Þ

T

Fig. 3. Trade-off in the frame length L.

at the discrete frequency k, and the effect of the drift inside
the frame is large at high frequencies. The typical quantityof the error is in the order of 10 μs with the sampling
frequency normally in the order of 10  5 and the frame
analysis with the length of the order of 0.1 s. Note that theeffect of the errors is reduced with a typical choice of thewindow function wðlÞ to suppress the amplitude nearboth ends.
In contrast to the condition so that we can ignore thedrift inside frames, the condition in (13) to approximate
the shift of the signal with the linear phase is that theframe length L is much larger than the signal shiftjϕ21 ðmÞ  mj  jϵmj. It is hard to satisfy this condition
when the observation is long and the sampling frequency
mismatch ϵ is large. We show a typical example. Supposing we observe signals of the length 10 s with a nominal
sampling frequency of 16 kHz, and D21  0. With a sampling frequency mismatch of ϵ ¼ 5  10  5 and a window
length of L ¼2048, the maximum shift of the signal in thelast frame is about 8 samples, which is sufficiently small tobe approximated by a linear phase. However, for a longer
observation, the time shift becomes closer to the frame
length L, and exceeds the frame length in the case ofobservation for longer than 2560 s.
To solve the trade-off, we employ the optimization of
the STFT analysis with the noninteger frame shift as
described in Section 5. With the optimized STFT analysis,
we do not need to consider the condition in (13), and theframe length L should be decided considering the condition in (9) and the suitable frame length for the arraysignal processing.

where fg denotes matrix transposition, is regarded asa stationary random variable if the sampling frequency
mismatch ϵ is estimated accurately. We also assume thatthe compensated observation X^ ðk; m; ϵÞ with the stationarity recovered by the precise estimation of ϵ has a zeromean bivariate complex normal distribution for eachfrequency bin k, whose density of each frequency bin isgiven by
 expð  X^ ðk; m; ϵÞH VðkÞ  1 X^ ðk; m; ϵÞÞ;p X^ ðk; m; ϵÞ; VðkÞ ¼
π 2 det VðkÞ
ð16Þ
where VðkÞ denotes the covariance matrix. Thus, accurate
estimation of ϵ recovers the stationarity of X^ ðk; m; ϵÞ andmaximizes the following log likelihood function JðV; ϵÞ,which evaluates the fit with the zero-mean bivariate
normal distribution:
JðV; ϵÞ ¼ ∑ log pðX^ ðk; m; ϵÞ; VðkÞÞk;m¼ ∑ ð  log π 2  log det VðkÞk;m

 X^ ðk; m; ϵÞH VðkÞ  1 X^ ðk; m; ϵÞÞ;

ð17Þ

H

where fg denotes conjugate transposition. V denotes thegroup of all covariance matrices VðkÞ for k ¼  L=2; …;
L=2 1, which can be substituted by the following sample
estimate:
1∑X^ ðk; m; ϵÞX^ ðk; m; ϵÞH ;VðkÞ’
j 8 mj m

ð18Þ

where j 8 mj denotes the number of frames. By substituting(18) and omitting the constants, a simplified version JðϵÞ ofthe log likelihood function JðV; ϵÞ is given byJðϵÞ ¼ ∑ log det ∑X^ ðk; m; ϵÞX^ ðk; m; ϵÞH :
k

ð19Þ

m

4. Maximum likelihood estimation of sampling
frequency mismatch assuming spatial stationarity

Note that the sum of the quadratic forms in the last term of(17) is constant with the sample estimate of the covariancematrix given by (18), as

4.1. Probabilistic model in STFT domain∑X^ ðk; m; ϵÞH VðkÞX^ ðk; m; ϵÞThe drift caused by the sampling frequency mismatch
causes the TDOAs of each sound source to change slowly
with time as if the source is moving. Thus, if the movements of sources are not large, the compensation of thesampling frequency mismatch can be evaluated by considering how static the TDOAs are. Also, by assuming thatthe sources are stationary, spatial stationarity can be used
as a measure of the sampling frequency mismatch. Withm

¼ ∑X^ ðk; m; ϵÞHm¼ 2j 8 mj:∑m0 X^ ðk; m0 ; ϵÞX^ ðk; m0 ; ϵÞHj 8 mj!1X^ ðk; m; ϵÞð20Þ

Unfortunately, an estimate of ϵ that maximizes the likelihood JðϵÞ cannot be obtained analytically. In addition, as
plotted in Fig. 4, JðϵÞ has not only a clear global maximum
but also several local maxima. We discuss an effective

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S. Miyabe et al. / Signal Processing 107 (2015) 185–196-6500

Table 1
Algorithm of the golden section search used to search for the maximumof JðϵÞ.

Log likelihood J(ε)

-7000
-7500
-8000
-8500
-9000

-4

-4.0

-4

-2.0

0

-4

2.0

-4

4.0

Sampling frequency mismatch εFig. 4. Example of the log likelihood function JðϵÞ for the observation ofa two-speech mixture of 10 s with the sampling frequency mismatch
ϵ ¼  1:25  10  4 .

Definition and initialization:
def pﬃﬃﬃ
Set φ ¼ ð 5  1Þ=2, a’ϵdn  1 , b’ϵdn þ 1 .
Step 1:
Set p’b  φðb  aÞ, q’aþ φðb  aÞ.
Calculate JðpÞ and JðqÞ.
Step 2:
If JðpÞr JðqÞSet a’p, p’q, q’a þ φðb  aÞ.
Else
Set b’q, q’p, p’b  φðb  aÞ.
End ifStep 3:
If b  a 4 ρ
Go back to Step 1.
Else
Obtain the result ϵ’ðaþ bÞ=2.
Terminate the algorithm.
End ifsearch method to obtain the global maximum in thefollowing.
continues until the range shrinks to the desired resolution

4.2. Efficient optimization of maximum likelihood estimate
by golden section search
Since the only parameter to be estimated in the maximization of the likelihood is the sampling frequency
mismatch ϵ, the problem reduces to a line search. If we
can assume that the log likelihood function JðϵÞ is unimodal, efficient optimization can be obtained by performinga golden section search. In the example shown in Fig. 4,JðϵÞ given by (19) is usually locally unimodal around theglobal maximum. Thus, after specifying the unimodal
range including the global maximum, we perform a golden
section search.
To specify the unimodal range around the maximum,
we discretize ϵ roughly and select the discretized valuethat maximizes JðϵÞ. We generate D uniform samples in therange ½  E; E (E 40) and define the discretized values ϵd,d ¼ 0; …; D  1 asϵd ¼  E þ

2dE
:
D1

ð21Þ

Then we compare all the values of Jðϵd Þ and find the
optimal index dn that maximizes Jðϵd Þ as
d ¼ argmaxd Jðϵd Þ; d ¼ 0; …; D  1:n

ð22Þ

The range parameter E can be defined easily by considering the possible range of ϵ. Since the sampling frequency
mismatch normally takes a value on the order of 10  5,
E can be set to approximately 10  4 or larger. The appropriate choice of D depends on the setting of E and thesignal characteristics. At least for the observation of
speech, the shape tends to be similar to the example inFig. 4, and the appropriate value of D should satisfy thecondition 2E=ðD 1Þ o 10  4 .
After the search of the discretized values to obtain thecoarse estimate ϵdn , the golden section search refines theestimation by narrowing the search range in an iterative
manner. We show the algorithm in Table 1. The initial
search range is set to ½ϵdn  1 ; ϵdn þ 1 , and the iteration

ρ( 40). Finally, the estimate ϵ is given as the center of thenarrowed search range.
4.3. Robustness of proposed likelihood evaluation
Here we discuss the robustness of the evaluation of
the log likelihood given by (19) against noise and themismatch of the Gaussian assumption. Before the discussion of robustness, we simplify the log likelihood function
given in (19) by discarding the residual constants. By using
the coherence γ 212 ðk; ϵÞ between channels after compensating the sampling frequency mismatch ϵ, defined asγ^ 212 ðk; ϵÞ ¼

j∑m X 1 ðk; mÞn X^ 2 ðk; m; ϵÞj2;ð∑m jX 1 ðk; mÞj2 Þð∑m jX^ 2 ðk; m; ϵÞj2 Þ

ð23Þ

the log likelihood JðϵÞ is further simplified by separating
the constants as follows:∑jX 1 ðk; mÞj2
∑jjX^ 2 ðk; m; ϵÞj2JðϵÞ ¼  ∑ logkmm ∑X 1 ðk; mÞ X 2 ðk; m; ϵÞ
m




2
¼  ∑ log ð1  γ^ 12 ðk; ϵÞÞ ∑jX 1 ðk; mÞj2
∑jX 2 ðk; mÞj2
n

2

m

k

¼  ∑ logð1  γ^ 12 ðk; ϵÞÞ þ const
2

m

.,

ð24Þ

k

where fgn denotes the complex conjugate. Note that in(24) we used the equality jX^ 2 ðk; m; ϵÞj ¼ jX 2 ðk; mÞj, which istrivial from (14).
First, we discuss the effect of the non-Gaussianity of theobserved signal. The complex amplitudes of the acoustic
signal in the STFT domain can be regarded as having zeromean, but it is well known that many sound sources
including speech tend to have super-Gaussian complex
amplitudes in the STFT domain. As can be seen in (24),only the interchannel coherence is evaluated in the log
likelihood function JðϵÞ, it can be expected that nonGaussianity does not affect the estimation of the samplingS. Miyabe et al. / Signal Processing 107 (2015) 185–196

-8820
-8840
Log likelihood J(ε)

-8860
-8880
-8900
-8920
-8940
-8960
-8980
-9000
-9020

-4

-4.0

-4

-2.0

0

-4

2.0

-4

4.0

Sampling frequency mismatch εFig. 5. Example of the log likelihood function JðϵÞ for the noisy observation of a two-speech mixture with the sampling frequency mismatch
ϵ ¼  1:25  10  4 . The noise is uncorrelated white Gaussian with the SNRof 0 dB.

frequency mismatch ϵ so much. We will investigate this byexperiments using speech.
Next, we discuss the robustness against noise. The
estimation of ϵ is possible as long as the amplitudes at
the two microphones are correlated because the log likelihood function JðϵÞ evaluates only the interchannel coherence γ 212 ðk; ϵÞ by effectively weighting the significant
frequency bins. In the frequency bins where only a few
unmoving sources are observed, high stationarity is maintained only when drift is compensated properly. Particularly in the frequency where only one source is dominant
2
and the coherence γ^ 12 ðk; ϵÞ is close to one, the kth
frequency significantly affects the log likelihood JðϵÞ with2
infinitely high value of  logð1  γ^ 12 ðk; ϵÞÞ. In contrast, JðϵÞ
is not strongly affected by noisy frequency bins with low
2coherence because of the low value of  logð1  γ^ 12ðk; ϵÞÞ  0. Therefore, the log likelihood JðϵÞ automaticallytakes only the significant frequency bins into account, andthe maximum likelihood function estimation of the sampling frequency mismatch is expected to have high robustness against noise. Fig. 5 shows an example of the log
likelihood of an asynchronous observation of asynchronous speech mixture used in Fig. 4 with the uncorrelated
white Gaussian noise imposed. We can see that the log
likelihood function JðϵÞ is slightly flattened by the noise
but the peak is still clear under this severe condition withthe very low SNR of 0 dB.
Note that we have confirmed that the stationarity canbe evaluated satisfactorily using JðϵÞ even when there aremany sources in the observation. However, the proposed
method is weak against the large movement of the sources
because the movement degrades the stationarity andcauses the time variation of the interchannel correlation.
5. Estimation of drift-compensated signal by STFT
analysis with noninteger frame shift
In this section, we describe an efficient resampling
method based on modified STFT analysis to estimate
the STFT signal with the drift compensated under thecondition that the sampling frequency mismatch ϵ and the191

recording start offset D21 are given. Here we assume thatthe drift inside each frame can be ignored, as discussed in
Section 3.2. This signal estimation procedure is used inboth the parameter estimation and the drift compensation
as discussed in Section 6. Although it is possible to use
various resampling methods to compensate for the driftwith the parameters ϵ and D21 given, many of the accurate
resampling methods are computationally complex. Our
proposed method is a simple modification of STFT analysis
and its computational cost is slightly larger than that of theoriginal analysis. Also, the time-domain signal is obtainedby inverse STFT (ISTFT) analysis.
As we showed in Section 3.2, the drift inside each frame
can be ignored under the condition given by (9) that jϵLj issmall, and the drift can be compensated by shifting theframe central time. According to (11), the time of thesecond channel synchronous with the time m at the centerof the frame of the first channel is given by ϕ21 ðmÞ. Thus,
we estimate the STFT signal X 2 ðk; ϕ21 ðmÞÞ whose frame
center is positioned at the discrete time ϕ21 ðmÞ.
In the beginning, we perform the STFT analysis of x1 ½n
as in (7):




L1
L
2π jkl;ð25Þ
X 1 ðk; mÞ ¼ ∑ wðlÞx1 l þ m  exp 2L
l¼0with a uniform integer frame shift of R samples, e.g.,
m ¼ 0; R; 2R; …. With such integer values of m, the corresponding frame central times ϕ21 ðmÞ of the second channel are generally noninteger-valued. Since the precise
estimation of the amplitude at a noninteger discrete timeis too complicated as shown in (6), we approximate theframe analysis with the frame center rounded to thenearest integer in the time domain, and compensate theround-off error of the frame center by a linear phase in the
STFT domain. The rounded central time is written as⌊ϕ21 ðmÞ⌉, where ⌊  ⌉ denotes the rounding operation to
the nearest integer, i.e.,⌊ϕ21 ðmÞ⌉ ¼ arg minn jϕ21 ðmÞ  nj;

n A Z:

ð26Þ

Since the error appears as the delay of ðϕ21 ðmÞ ⌊ϕ21 ðmÞ⌉Þ
samples, the linear phase compensation gives X 2 ðk; ϕ21ðmÞÞ asL1LX 2 k; ϕ21 ðmÞ ¼ ∑ wðlÞx2 l þ ⌊ϕ21 ðmÞ⌉ 
2
l¼0


2ϕjkðϕ21 ðmÞ  ⌊ϕ21 ðmÞ⌉Þ
exp
:
ð27Þ
L
Now the obtained STFT signal X 2 ðk; ϕ21 ðmÞÞ is synchronous with X 1 ðk; mÞ, which is analyzed using the frame shift
R. Thus, when we need to obtain the time domain signalx^ 2 ½n with the drift compensation, a good approximation is
obtained by the ISTFT analysis of X 2 ðk; ϕ21 ðmÞÞ using theoriginal integer frame shift R.
6. Algorithm of blind drift compensation
Here we describe an algorithm for blind compensationof the drift combining the maximum likelihood estimationof ϵ and the optimized STFT analysis with the noninteger
frame shift. The algorithm iteratively estimates the offset
D21 in the time domain and the sampling frequency

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S. Miyabe et al. / Signal Processing 107 (2015) 185–196

mismatch ϵ in the STFT domain. Although an error of theestimation of D21 much smaller than the frame length in
STFT analysis is accepted in array signal processing
without the use of TDOA priors, as described in Section
1, satisfactory estimation cannot be guaranteed under the
existence of drift. Also, the unignorable error of D21
degrades the estimation accuracy of ϵ owing to theinsufficient STFT analysis. Thus, the estimations of D21
and ϵ are mutually dependent, and the estimation procedures are iterated to resolve the dependence.
6.1. Initial frame analysis
First we obtain the initial STFT signal with the appropriately compensated offset to evaluate the sampling
frequency mismatch in the STFT domain.
In this initial frame analysis, we assume that sampling
frequency mismatch does not occur, i.e.,

ϵ’0:

ð28Þ

Under this assumption, an integer value of D21 with anacceptable error much smaller than the frame length isobtained to maximize the correlation between channels:
D21 ’arg maxδminðN1 ;N 2 þ δÞ  1∑

n ¼ maxð0;δÞx1 ½nx2 ½n  δ; N2 o δ oN 1 :ð29ÞNote that the error of this estimation of D21 cannot bemade sufficiently small when the assumption ϵ  0 in (28)
has a large error and the observation length is large. Thus,
the estimation of D21 is renewed after the estimation of ϵin the STFT domain. Using this estimate of D21, we obtain
the STFT signal x0i ðnÞ, i¼1,2 with the redefined discrete
time that cancels the offset asx01 ½n’x1 ½n;ð30Þx02 ½n’x2 ½n D21 :ð31ÞFinally, we perform the initial STFT analysis to obtain
X 0i ðk; mÞ, i¼1,2 using the uniform frame shift R, e.g.,
m ¼ 0; R; 2R; …, for both channels, as




L1
L
2π jkl:ð32Þ
X 0i ðk; mÞ ¼ ∑ wðlÞx0i l þ m  exp 
2
L
l¼0

M’

minðN 1  D21 ; N 2 Þ  maxð0; D21 Þ 1
:
2

ð34Þ
0

Thus, the drift-compensated STFT signal X^ ðk; m; ϵÞ withthe minimal effect of the third term  ð1 þ ϵÞD21 in (8) is
obtained by a shift of the ϵðm  MÞ samples to X 02 ðk; mÞ as
0
2π jkϵðm  MÞ:
ð35ÞX^ 2 ðk; m; ϵÞ ¼ X 02 ðk; mÞ exp
L
According to (19), the log likelihood function J 0 ðϵ0 Þ used0
to evaluate the compensated STFT signal X^ 2 ðk; m; ϵ0 Þ isgiven as00
J 0 ðϵ0 Þ ¼  ∑ log det ∑X^ ðk; m; ϵ0 ÞX^ ðk; m; ϵ0 ÞH ;

ð36Þ

0
0
X^ ðk; m; ϵ0 Þ ¼ ½X 01 ðk; mÞ; X^ 2 ðk; m; ϵ0 ÞT :

ð37Þ

m

k

The value of ϵ that maximizes J ðϵ Þ is estimated bya simple modification of the procedure in Section 4.2,
where ϵ is substituted with ϵ0 and JðϵÞ is substituted withJ 0 ðϵ0 Þ in which the current estimate ϵ is treated as aconstant.
Finally after the estimation of ϵ0 by the golden section
search, we update ϵ using ϵ0 as000

ϵ’ð1 þ ϵÞð1 þ ϵ0 Þ  1;

ð38Þ

which gives the update of the estimated sampling frequency ð1 þ ϵÞf s in (33).
6.3. Estimation of offset
Now we describe the update of the estimation of theoffset D21. First, we obtain an estimate of the driftcompensated time-domain signal x^ 2 ½n by the procedurein Section 5 with the substitution of the estimated ϵ andthe assumption D21 ¼ 0. Then the offset D21 is estimated by
maximizing the correlation between x1 ½n and x^ 2 ½n:D21 ’arg maxδminðN1 ;N 2 þ δÞ  1∑

n ¼ maxð0;δÞx1 ½nx^ 2 ½n  δ; N2 o δ oN 1 :ð39ÞFinally, we estimate the drift-compensated STFT signal
X 2 ðm; ϕ21 ðmÞÞ by substituting the estimated parameters ϵand D21 again into the procedure in Section 5.

6.2. Estimation of sampling frequency mismatch
Here we describe the estimation of the sampling
frequency mismatch ϵ. The estimation is given as the
update for the estimate of ϵ from the current one. We
formulate this problem to obtain a parameter ϵ0 which
updates the estimation of the sampling frequency ð1 þ ϵÞf s
as follows:ð1 þ ϵÞf s ’ð1 þ ϵ0 Þð1 þ ϵÞf s :maximize the correlation, the gap between the corresponding discrete times jn2  n1 j is expected to be minimum around the central sample M of the samples
overlapping between the channels, where M is given by

ð33Þ

Although in Section 4 we assumed that the third term ð1 þ ϵÞD21 on the right of (8) is zero, this term should beconsidered in practical processing even though we do nothave accurate estimates of ϵ and D21 here. Since the timedomain signal x0i ½n; i ¼ 1; 2 has been given a time shift to

6.4. Iterative update
The estimation procedure for ϵ in Section 6.2 and thatfor D21 in Section 6.3 are conducted separately. However,
the estimates of these parameters are mutually dependentand the error of one parameter propagates to the other. In
particular, when the observation is long and ϵ is large, theinitial STFT analysis with the assumption ϵ ¼ 0 is strongly
affected by the mismatch of the assumption in (12) anddegrades the estimation accuracy of ϵ in Section 6.2; thus,
the estimation accuracy of D21 in Section 6.3 is also
subsequently degraded. To achieve accurate parameterS. Miyabe et al. / Signal Processing 107 (2015) 185–196

193

Table 2Iterative algorithm of blind synchronization.
Initialization:
Set the iteration number j’1.
Set ϵ’0.
Initialize X 0i ðk; mÞ; i ¼ 1; 2 according to Section 6.1.
Step 1:
Update ϵ according to Section 6.2.
Step 2:Update D21 and obtain X 1 ðk; mÞ and X 2 ðk; ϕ21 ðmÞÞaccording to Section 6.3.
Step 3:
Set j’j þ 1.
If j reaches the maximum iteration number
Terminate the algorithm.
Else
Update X 02 ðk; mÞ as in (40).
Go back to Step 1.
End if

−10°
−50°
−60°

2.15 cm

1.5 m

Fig. 6. Placement of microphones and sources. We evaluated two
combinations of source positions with the horizontal angles of [  501,
301] and [  601,  101].

estimation in such a long observation, we propose theiteration of these procedures to reduce the propagation of
the error in each stage to the next stage.
Table 2 summarizes the iterative optimization algorithm. After the processing described in Sections 6.1–6.3,
we return to the procedure in Section 6.2 after thefollowing update of the STFT signal in the second channel
X 02 ðk; mÞ for the update of ϵ:
X 02 ðk; mÞ’X 2 ðk; ϕ21 ðmÞÞ:

30°

ð40Þ

The procedures in Sections 6.2 and 6.3 after the update
given by (40) are repeated until the iteration reaches the
maximum number of iterations. The iterative algorithm issummarized in Table 2. Note that such iteration is required
only for particularly long observations such as those for10 min.
7. Experimental evaluation
In this section we evaluate the effectiveness of the
proposed method of blind drift compensation. First,
we evaluate the estimation accuracy of the sampling
frequency mismatch by the proposed method. Second,
we evaluate the performance of BSS with the drift compensation to assess the suitability of the proposed driftcompensation as a preprocessing step in array signal
processing.

10  6), respectively, which are realistic values for thepractical bias of clock generators in real ADCs. To generate
an artificial sampling frequency mismatch, we performed
resampling with a polyphase filter, which is implemented
as a standard command in MATLAB. We used auxiliaryfunction-based independent vector analysis [20] to conduct two-channel BSS of the two speakers. The placementsof microphones and speakers are shown in Fig. 6. The
sources are placed 1.5 m apart from the middle of the two
microphones. We evaluated two different source positions,and the combinations of the horizontal angles of the twosources are [ 501, 301] and [  601, 101]. Other conditions are listed in Table 3.
7.2. Accuracy of sampling frequency mismatch estimation
Fig. 7 shows the root mean squared errors (RMSEs) of
the estimates of the sampling frequency mismatches fordifferent signal lengths. We plotted the results of one andtwo iterations of the iterative estimation algorithm
described in Section 6.4. We can see that the proposed
estimation algorithm works appropriately even with shortobserved signals, and the accuracy improves with increasing length of the observed signals. Also we can see that theiterative procedure performs better when the observation
is long, and the improved performance becomes clear for
observations longer than 120 s. Thus, we confirmed thatthe estimation algorithm without iteration is satisfactoryfor short observations, but the iterative estimation
algorithm is effective in the case of long observations.
7.3. Robustness

7.1. Experimental setup
The observed signals are two-channel recordings of
two-speaker mixtures made by the convolution of measured impulse responses and speech signals. The speech
signals are made by the concatenation of word utteranceschosen from the utterances in the ATR Japanese speech
database [19]. We evaluated all 12 combinations of two
speakers selected from two male and two female speakers.
The original sampling frequency of the observation was
16,000 kHz, and we modified the sampling frequency of
one channel by 70.5, 71, and 71.5 Hz. These modifications correspond to sampling frequency mismatchesof 731.25, 762.5, and 793.75 ppm (parts per minute,

We analyzed the behavior of the proposed method ofsampling frequency mismatch estimation to evaluate
robustness. First, to evaluate the effect of the nonGaussianity of the observed signal, we analyzed the
kurtosis, which is a popular measure of Gaussianity. The
kurtosis of complex variables [21] was evaluated in eachfrequency bin for each observation of 30 min, and theresults were averaged over all the observations. Note thatfor the stability of estimation, the fourth-order moments
were not obtained from the sample average but from theparameters of the gamma distributions fitted to theabsolute squared amplitudes by using maximum likelihood estimators. The estimated kurtosis is shown in Fig. 8.

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S. Miyabe et al. / Signal Processing 107 (2015) 185–196

Table 3
Experimental conditions.
Length of observation
Reverberation time
Frame length L
Frame shift R
Microphone spacing
Discretization search range E

3, 5, 10, 20, 30, 60, 180, 300 and 1800 s
T60 of 130 ms
4096 samples
2048 samples
2.15 cm
5  10  4
20
10  13Discretization division D
Resolution ρ of golden section search

-5

-5

10

10

One iteration
Two iterations

-6

Clean
SNR of 20 dB
SNR of 10 dB
SNR of 0 dB

-6

10

-7

10

RMSE

RMSE of ε

10

-8

10

-7

10

-8

10

-9

10

-9

-10

10

10

10

100
Observation length [s]1000

Fig. 7. Root mean squared errors (RMSEs) of estimates of sampling
frequency mismatch.

6
5

Kurtosis

4
3
2
1
0

0

1000 2000 3000 4000 5000 6000 7000 8000
Frequency [Hz]Fig. 8. Plot of kurtosis in each frequency bin.

The kurtosis of complex variables with a complex normal
distribution is zero, and the kurtosis is larger than one
when the variables are super-Gaussian. Fig. 8 shows that inmost of the frequency bins the kurtosis is highly superGaussian, which is a characteristic of speech. Since accurate estimation was obtained for the super-Gaussian
observation, it is confirmed that the non-Gaussianity ofthe observation does not affect the estimation.
To evaluate the robustness against noise, we superimposed uncorrelated white Gaussian noise on the observation and evaluated the RMSEs of the sampling frequency
mismatch estimation. We compared the accuracy with aclean observation and noisy observations with SNRs of 20,

10100
Observation length [s]Fig. 9. Root mean squared errors (RMSEs) of estimated sampling frequency mismatch under the existence of uncorrelated white noise.

10, and 0 dB. The result is shown in Fig. 9. The RMSE
increases with increasing noise, but the RMSE is only
several times larger under a severe condition with an
SNR of 0 dB than that for the clean observation. Therefore,
it is confirmed that the proposed maximum likelihood
estimation method is robust against noise as discussed in
Section 4.3.
Since the proposed maximum likelihood estimation
method assumes that the sources do not move so thatthe stationarity of the observation is maintained, themovement of sources degrades the estimation performance. To evaluate the effect of the movement of thesources, we changed the positions of the speakers from thedirections [  501, 301] to [  601, 101] in the middle of theobservation. Fig. 10 shows the RMSEs of the sampling
frequency mismatch estimation when the speakers changed the positions. Although the estimation does not failunder this condition, the RMSEs increased approximately
ten-fold. Thus, the proposed maximum likelihood estimation is weak against position shifts of the sources becauseof the degradation of stationarity.
7.4. Contribution to BSS
To assess the contribution of the proposed method ofdrift compensation to BSS, we evaluated the source
separation performance in terms of the signal-todistortion ratio (SDR) [22] at the first microphone asshown in Fig. 11. The SDRs were averaged for the twosources. We evaluated the following conditions:

S. Miyabe et al. / Signal Processing 107 (2015) 185–196-510

Without position shift
With position shift
-6

RMSE

10

-7

10

-8

10

-9

10

10

100

Observation length [s]Fig. 10. Root mean squared errors (RMSEs) of estimated sampling
frequency mismatch when the positions of the speakers changed in the
middle of the observation.

Synchronous

Unprocessed

STFT

ML

WO

Proposed20

195

Fig. 11 shows a comparison of the BSS performances.
Since the SDRs of the unprocessed signal are very low, we
can see that BSS is difficult without drift compensation
under these conditions. The SDRs in the case of resamplingwith the true parameters are only 1 dB lower than those in
the case of synchronous recording. Thus, we can conclude
that the error of the signal expression in the STFT-based
resampling is insignificant in array signal processing. The
degradation in the SDRs of the maximum likelihood linear
phase compensation is considerable for observations
longer than 20 s compared with the other methods. Thus,
resampling is essential and the proposed STFT-based
resampling is satisfactory. The degradation in the SDRs of
the BSS without offset reestimation is also considerable forobservations longer than 30 s, but the proposed method
does not exhibit such degradation. Thus, the effectivenessof the reestimation procedure for the offset described in
Section 6.3 is confirmed. The SDRs of the proposed method
exhibit insignificant degradation compared with STFTbased resampling with the true parameters, and thedegradation from the synchronous recording was only
about 1 dB. Thus, it is confirmed that the parameter
estimation is sufficiently accurate. Therefore, the validityof the proposed compensation algorithm is verified, andits suitability for array signal processing as a process is also
confirmed.

SDR [dB]

15

8. Conclusion

10
5
0
-5

10

100

1000

Signal length [s]Fig. 11. Signal-to-distortion ratios (SDRs) of BSS performances for different conditions of drift compensation.

 Synchronous: BSS without artificial sampling frequency
mismatch to obtain an upper limit for this task.

 Unprocessed: BSS without compensation of the asynchronous recording.

 STFT-based resampling with true parameters (STFT):
BSS of the STFT-based resampling with the true valueof ϵ given and D21 estimated by the procedure in
Section 6.3.
 Maximum likelihood linear phase compensation (ML):BSS of the STFT signal obtained by the procedure in
Section 4.2 without STFT-based resampling.
 Without offset reestimation (WO): BSS of the STFT-based
resampling without the reestimation of D21 obtained by
the procedure in Section 6.3.
 Proposed: BSS of STFT-based resampling with driftcompensation by the procedures in Section 6 without
iteration of the procedures. The number of iterations in
Section 6.4 was two.

In this paper, we proposed a novel method for the blindcompensation of drift arising in the asynchronous recording of an ad hoc microphone array. We modeled the drift
as a shift of short time frames while ignoring the drift
inside each frame. Thus, the compensation of drift wasgiven by a linear phase in the STFT domain. By assumingthat the sources are unmoving and have stationary amplitudes, we used stationarity as the cue to evaluate the driftcompensation. A likelihood function to measure stationarity was formulated, and an efficient maximum likelihood
search of the sampling frequency mismatch by a golden
section search was described. We also proposed an efficient resampling method involving the modification of
STFT analysis with a noninteger frame shift, and described
the algorithm of the introduced blind drift compensation.
In an experiment on the compensation of drift introduced
artificially, we confirmed the accurate estimation andsufficient compensation of drift by the proposed method.
The future work includes the online adaptation of the driftcompensation.
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