TDOA Estimation by Mapped Steered Response
Power Analysis Utilizing Higher-Order Moments
Xiao-Dong Zhai, Yuya Sugimoto, Shigeki Miyabe, Takeshi Yamada, and Shoji Makino
Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, Japan.
E-mail: {zhaixiaodong, sugimoto}@mmlab.cs.tsukuba.ac.jp, {miyabe, maki}@tara.tsukuba.ac.jp, takeshi@cs.tsukuba.ac.jp
Abstract—In this paper, we propose a new estimation method
for the time difference of arrival (TDOA) between two microphones with improved accuracy by exploiting higher-ordermoments. In the proposed method analyzes the steered response
power (SRP) of the observed signals after nonlinearly mapped
onto a higher-dimensional space. Since the mapping operation
enhances the linear independence between different vectors
by increasing the dimensionality of the observed signals, the
TDOA analysis achieves higher resolution. The results of anexperiment comparing the TDOA estimation performance ofthe proposed method with that of the conventional methods
reveal the robustness of the proposed method against noise andreverberation.

I. I NTRODUCTION
In array signal processing, the time delay of arrival (TDOA)
is used in various applications as prior information [1]. Also,
TDOA estimation is used in direction of arrival (DOA) estimation, which is an important issue in audio scene analysis. In
Particular, in the simultaneous localization problem of soundsources and microphones [2], accuracy of TDOA estimationis critical.
The general approach to estimating the TDOA is to analyze
the peak of the interchannel correlation. The most representative method is the generalized cross-correlation (GCC) method[3], which sharpens the peak of the correlation function usinga whitening filter. While many methods have been proposed
for DOA estimation for a microphone array with three ormore elements [4][5][6][7][8], the conventional GCC method
is still the most reliable method for TDOA analysis for two
microphones. Although the DOA estimation problem withtwo microphones is equivalent to TDOA estimation when thevelocity and the distance between the microphones are given,high-resolution DOA estimation methods such as multiple
signal classification (MUSIC) [5] and its alternatives [7][8]cannot achieve high accuracy because they assume that the
number of microphones is much larger than the number ofsources.
In this paper, we propose a new TDOA estimation method,referred to as mapped steered response power (SRP), which
improves the estimation accuracy. First, we map the observed
signal onto a higher dimensional space using the nonlinear
function we used in our previous work for the extensionof MUSIC based on high-order moment analysis [8]. Next,
in the mapped higher-dimensional space, we conduct SRP
analysis [6], which is a multichannel extension of the GCC
method. Owing to the improved linear independence resulting
from the increase in the dimensionality, the SRP analysis
after the nonlinear mapping achieves higher-resolution TDOA
estimation. Through a simulation experiment to verify theeffectiveness of our proposed method, we show that mapped
SRP achieves higher TDOA estimation accuracy than the

978-616-361-823-8 © 2014 APSIPA

conventional methods of GCC, MUSIC, and 2q-MUSIC [7]in noisy and reverberant environments.
II. S TATEMENT OF P ROBLEM
Here we formulate the signal model and the TDOA estimation problem. Suppose we have a target source whose
amplitude is given by s (t), where t denotes the continuous
time. Our goal is to estimate the TDOA ∆τ of the target sources (t) between two microphones in a room where reverberationand noise exist. The observed signals are formulated as{}
x1 (t) = α1 sinc (t − τ1 ) + h1 (t) ⊛ s (t) + n1 (t) , (1){}x2 (t) = α2 sinc (t − τ1 − ∆τ ) + h2 (t) ⊛ s (t) + n2 (t) ,(2)∆τ ≜ τ2 − τ1 ,sin πtsinc (t) =
,
πt

(3)
(4)

where i denotes the index of the channel, xi (t) denotes
the observed signal at the ith channel, αi and τi are theattenuation coefficient and the time of arrival of the direct
wave propagating from the target source to the ith channel,
respectively, hi (t) is the impulse response of the indirect
propagation path from the target source to the ith microphone,ni (t) is the observation noise at the ith microphone, and ⊛is the convolution operator between two functions f (t) andg (t) given by∫ ∞f (t) ⊛ g (t) =f (t′ ) g (t − t′ ) dt′ .
(5)−∞

We assume that the impulse responses are time-invariant andthat the sound source s (t) the observation noise ni (t) fori = 1, 2 are stationary with zero mean. Also, we assume thats (t) and ni (t) for i = 1, 2 are statistically independent.
III. C ONVENTIONAL GCC- BASED TDOA ESTIMATION
A. TDOA Estimation Model in Time Domain
In this section, we describe the principle of TDOA estimation based on GCC analysis. For the signal model in Sect. II,
the cross-correlation function Cx1 x2 (τ ) between the channels
can be formulated as follows:
Cx1 x2 (τ ) = Et [x1 (t) x2 (t + τ )] ,∫1 T
x (t) dt,
Et [x (t)] = lim
T →∞ T 0(6)(7)

where Et [·] denotes the expectation of the argument in thetime domain and T represents the length of the observation

APSIPA 2014

interval 0 ≤ t < T . Furthermore, we assume that the source
signal s(t) and noise ni (t) are uncorrelated and that
Csni (τ ) ≈ 0.

(8)

According to Eqs. (7) and (8), the cross-correlation function
Cx1 x2 (τ ) can be expanded as
Cx1 x2 (τ ) ≈ α1 α2 Css (τ − ∆τ ){+ α1 h2 (−τ ) + α2 h1 (τ )}+ h1 (τ ) ⊛ h2 (−τ ) ⊛ Css (τ ) .

∫ ∞−∞h21 (t) dt ≈∫ ∞−∞h22 (t) dt.

(9)

(10)

(11)

Moreover, if the source signal s(t) is white, the crosscorrelation function Cx1 x2 (τ ) can be rewritten as:[]
Cx1 x2 (τ ) ≈ α1 α2 E s2 (t) sinc (τ − ∆τ ) .

(12)Under the assumptions given by Eqs. (10) and (12), ∆τ
can be estimated by finding the maximum of Cx1 x2 (τ ).
However, when the above assumptions have an error owing to a reverberant and noisy recording environment anda colored signal, the peak of Cx1 x2 (ω) becomes unclear.
To ensure robustness against noise and reflected waves, thecross-correlation function Cx1 x2 is substituted for its whitened
version ψ12 (τ ) ⊛ Cx1 x2 (τ ) in the GCC method by applying
the whitening filter ψ12 (τ ) to clarify the peak:∆τ ← arg max ψ12 (τ ) ⊛ Cx1 x2 (τ ) .

−π

Gx1 x2 (ω) = En [x∗1 (ω, n) x2 (ω, n)] ,1 ∑x (ω, n) ,
N n=1

(15)
(16)

N

According to the assumptions of Eq. (10), the whole crosscorrelation function Cx1 x2 (τ ) can be approximated as
Cx1 x2 (τ ) ≈ α1 α2 Css (τ − ∆τ ) .

∆τ ← arg max JGCC (τ ) ,
(14)
∫ πτ
JGCC (τ ) =Ψ12 (ω) Gx1 x2 (ω) exp (−ȷωτ ) dω,En [x (ω, n)] =Assuming that the direct waves have sufficiently larger gain
than the interference waves, we introduce the following relationship among αi and hi :α12 ≈ α22 ≫transform (DFT) of the cross-correlation function Cx1 x2 (t):

(13)

τ

where [·]∗ denotes the complex conjugate, En [·] denotes the
expectation of the argument in frequency domain, and Ψ12 (ω)
is the frequency-domain whitening filter corresponding to the
DFT of ψ12 (t). The design of Ψ12 (ω) will be described in
Sect. III-D.
C. Multichannel extension by SRP
SRP analysis is an extension of the GCC method for
estimating multiple TDOAs of all the pairs of three channels
or more simultaneously. Although this paper focuses on the
TDOA estimation of a two-channel pair, we review the SRP
process because the methodology is utilized in the formulationof the proposed method.
Without loss of generality, the estimation of the TDOAs of
M channels can be formulated as the estimation of the TDOAs∆τ1j (j = 1, · · · , M ) between the jth channel and the firstchannel regarded as the reference channel as follows:∆τ1j = τj − τ1 .

B. GCC Algorithm in Frequency Domain
Although we described the model of GCC-based TDOA
estimation in the continuous time domain, the practical processing is usually conducted in the discrete time-frequency
domain. First, we obtain the short-time Fourier transform(STFT) x(ω, n) of the time domain observation x (t), where−π ≤ ω < π denotes the angular frequency and n = 1, . . . , N
is the time index of the N STFT frames. Since a delay of thelength τ is expressed by the phase rotation exp(−ȷωτ ) in thetime-frequency domain, the estimation of the TDOA ∆τ isequivalent to finding the filter exp(−ȷω∆τ ) that maximizes
the cross-spectrum Gx1 x2 (ω), which is the discrete Fourier(18)

Note that the TDOA ∆τ11 is obviously zero. We define thefollowing vector τ consisting of all TDOAs ∆τ11 , · · · , ∆τ1M :T

τ = [∆τ11 , · · · , ∆τ1M ] ,(19)where [·]T denotes the transpose. Then, the correct estimationof the vector τ ought to maximize the cross-correlation functions of all channel pairs. Hence, the SRP method estimates
the vector τ by maximizing the summation of the GCC criteriaof all channel pairs.
τ ← arg max JSRP (τ ′ ) dω subject to τ1′ = 0, (20)
JSRP (τ ′ ) =

τ′
M
M ∫ π
∑∑
i=1 j=1

By designing an appropriate filter for the conditions, the
TDOA estimation accuracy is greatly improved even in a noisy
or reverberant environment.

(17)

−π

( (
))
Ψij (ω) Gxi xj (ω) exp ȷω τi′ − τj′ ,(21)′
τ ′ = [τ1′ , · · · , τM
] .
T

(22)

Also, the TDOA estimation that maximizes an SRP JSRP (τ )can be expressed by a quadratic form of the generalized
correlation matrix Ψ(ω) ⊕ R(ω) and the steering vectorb(ω; τ ′ ) as follows:
JSRP (τ ′ ) = b (ω; τ ′ ) (Ψ (ω) ⊕ R (ω)) b (ω; τ ′ ) ,
(23)
1
T
′
b (ω; τ ′ ) = √ [exp (−ȷωτ1′ ) , · · · , exp (−ȷωτM)] , (24)MΨ = [Ψij (ω)]ij ,
(25)
[
]
H
R (ω) = En x (ω, n) x (ω, n)[](26)= Gxi xj (ω) ij ,HT

x (ω, n) = [x1 (ω, n) , · · · , xM (ω, n)] ,

(27)

H

where, [·] denotes the complex conjugate transpose, ⊕ denotes the Hadamard product, Ψij (ω) is the GCC filter for theijth channel pair, and [·]ij denotes a matrix consisting of theargument as its (i, j) entry. Note that TDOA estimation using
the SRP method when M = 2 is equivalent to that using the
GCC method because()
T
JSRP [τ1′ , τ2′ ] = 2JGCC (τ2′ − τ1′ )
2 ∫ π
∑
+
Ψii (ω) Gxi xi (ω)dωi=1−π∝ JGCC (τ2′ − τ1′ ) .

(28)

D. Design of GCC Filter
The GCC and SRP methods sharpen the peaks of the crosscorrelation function through the whitening of the observation
using whitening filters Ψij (ω) appropriately designed forthe surrounding environment. While several filters have been
proposed on the basis of different criteria, we introduce the
GCC-SCOT and GCC-PHAT filters as examples.
• GCC-SCOT filter1Ψij (ω) = √.
(29)
Gxi xi (ω) Gxj xj (ω)
This filter is known to be robust against noise.
• GCC-PHAT filter1Ψij (ω) =.
Gxi xj (ω)(30)

This filter has good robustness against reflections.
IV. P ROPOSED M ETHOD
In this section, we propose a new extension of GCC-based
TDOA estimation, mapped SRP, by using higher-dimensional
mapping to improve the resolution of TDOA analysis. The
TDOA analysis of mapped SRP is based on a search fora steering vector, similarly in the SRP method. First, boththe two-dimensional steering and observed signal vectors are
mapped nonlinearly onto a higher-dimensional space. Next,
SRP analysis is conducted in the higher-dimensional space tofind the peak of the cross-correlation.
A. TDOA Estimation Algorithm of Mapped SRP
Here we consider a function ϕ : C2 → CM that maps theelements in the two-dimensional space of the observed signals
onto another space with dimensionality M > 2. We assume
that the map ϕ satisfies the following three conditions.
1) The magnitude relation of the norm is retained:∥x∥ ≥ ∥y∥ → ∥ϕ (x) ∥≥ ∥ϕ (y)∥ ,(31)

where ∥ · ∥ denotes the norm.
2) The origin remains intact:x = 0 → ϕ (x) = 0.

(32)3) The orthogonality between vectors is preserved:
xH y = 0 → ϕH (x) ϕ (y) = 0.
ϕ(33)

The correlation matrix R (ω) of the mapped observed
signal ϕ(x(ω, n)) is obtained as[]
Rϕ (ω) = En ϕ (x (ω, n)) ϕH (x (ω, n)) .
(34)

We also map the steering vector b(ω; τ ) given byb (ω; τ ) = [1, exp (−ȷωτ )] ,(35)

and the TDOA ∆τ is estimated as the delay τ that maximizes
the SRP score J ϕ (τ ) in the mapped M -dimensional space:∆τ ← arg max J ϕ (τ ) ,
(36)
∫ πτ
J ϕ (τ ) =J ϕ (ω; τ ) dω,
(37)
−π
(
)
H
J ϕ (ω; τ ) = ϕ (b (ω; τ )) Ψϕ (ω) ⊕ Rϕ (ω) ϕ (b (ω, τ )) ,
(38)
[
]
ϕ
ϕ
Ψ (ω) = Ψij (ω) ,(39)
ijwhere Ψϕij is the whitening filter of the ith and jth dimensionsof the mapped observed signal vector ϕ (x (ω, n)), whose
design is described later.
The TDOA estimations using SRP and mapped SRP, given
by Eqs. (20) and (36), respectively, are conducted by searchingfor a steering vector having higher linear dependence withthe true steering vector. The linear independence between two
different vectors is increased by the map ϕ owing to theincrease in the dimensionality. Therefore, the peak of the SRP
becomes clearer and the resolution of TDOA estimation is
improved.
B. Mapping for Analysis of 2dth-Order Moments
As the function ϕ satisfying the conditions given bydEqs. (31)–(33), we employ the mapping ϕd : C2 → C2 usedin the MUSIC-based DOA analysis in [8]:
]T
[ d
d
∏
∏
,
(40)
xl⊗
xl⊗
ϕd (x(ω, n)) =c1l , · · · ,
c2d l
l=1

l⊗

x

{
x
=
x∗

l=1

(if l is odd)
(otherwise),(41)

where d is the degree of the mapping, ckl is an index specifying the entry number of vector x, which is the (k, l) entryof the 2d × d index matrix C with d repeated permutations of
1 and 2 as its row vectors:
1 1 ··· 1
1 1 · · · 2C ≜ [ckl ]kl = (42) ... ... . . . ...  .
2 2 ··· 2
As discussed in [8], the entries of the correlation matrix
Rϕ (ω) in Eq. (34) correspond to 2dth-order moments. This
characteristic reveals a trade-off between the resolution and therobustness when choosing the degree d. While the mapping
ϕd with a large value of d improves the resolution of TDOA
estimation owing to the increase in the dimensionality, givenby the order of 2d , it loses the robustness against outliers
because of the large variance of the high-order moment
estimation. Thus, the degree d has to be chosen carefully.
In the following experiments, we fixed d to 2 so that robust
TDOA estimation could be obtained with a short observation.
Note that the computational complexity of the proposed
method increases on the order of 22d -fold to obtain the mappedcorrelation matrix compared with the conventional GCC.

TABLE I
E XPERIMENTAL CONDITIONS .

Microphone spacing
Reverberation time (T60 )Noise type
SNR
Frame length
Frame shift length
Frequency band
Audio signal
Sound source
Source directions
Observation length

Room 1: 3.4 × 4 × 2.7 m
Room 2: 5.5 × 9 × 2.6 m
0.2 m
Room.1: 0.3 s, Room.2: 0.8 s
Diffuse pink noise0, −5, −10 dB
512 samples
128 samples
0.2–4 kHz18 male and 14 female utterances
Loudspeaker 1 m from array
Seven azimuths of −90◦ , −60◦ , · · · , 90◦
1s

C. Filters used in Mapped SRP
Here we describe the filter used in mapped SRP. Theircharacteristics are also similar to those for GCC.
• Mapped SRP-SCOT filter′1Ψij (ω) = √ [][] . (43)
2
l⊗
l⊗ 2
d
d
En Πl=1 xcil
En Πl=1 xcjl•

Mapped SRP-PHAT filter′Ψij (ω) =

En

[(

1
) ( d l⊗ )∗ ] .
Πl=1 xcjl
Πdl=1 xl⊗
cil

(44)

V. E XPERIMENT
To confirm the effectiveness of mapped SRP, we conducted
an experiment to evaluate the TDOA estimation accuracy.
As conventional methods for comparison, we implemented
GCC-SCOT and GCC-PHAT, which are discussed in Sect. III,MUSIC, and 2q-MUSIC (q = 2). Note that we excluded theresult for GCC-ML, a well-known whitening filter used in the
GCC method because it did not perform well in this experiment owing to its weakness against noise and reverberation as
pointed out in [6].
A. Experimental Conditions
We employed convolutive mixtures of clean speech with
impulse responses recorded in two different rooms with reverberation times of 0.3 s and 0.8 s. In order to simulate noisy
observation, diffused pink noises [9] with three different SNRsof 0, −5, −10 dB were superimposed on the observed signals.
Table I shows the other experimental conditions.
For the evaluation, we employed the root mean squared error(RMSE) between the estimated TDOA and the true source
TDOA.
B. Experimental Results
Figure 1 shows the experimental results under different conditions. In this TDOA estimation scenario of two-microphone
pair, We can see no clear superiority of the MUSIC-based
methods, which usually performs better when more channels
are available. When the noise and reverberation are relatively
weak, all the methods perform very good and TDOA estimation accuracy of the proposed method is similar to thoseof the conventional GCC methods. However, when the noiseor the reverberation becomes stronger, the TDOA estimation

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accuracy of the proposed method is better than those of theconventional methods. As a result, we find that our proposed
mapped SRP performs better than the conventional methods
when the noise and reverberation are heavy.
VI. C ONCLUSIONSIn this paper, we proposed a new TDOA estimation method
called mapped SRP, which conducts SRP analysis of anobservation nonlinearly mapped onto a higher dimensional
space. The increased dimensionality improves the linear independence between different steering vectors, and the TDOA
estimation by SRP analysis achieves improved resolution. The
covariance of the mapped observation analyzed in mapped
SRP corresponds to higher-order moments. We confirmed that
the SRP with fourth order moment analysis achieves higher
TDOA estimation accuracy than the conventional methods
under in the case of large noise or reverberation.
ACKNOWLEDGMENT
This work was supported by JSPS KAKENHI Grant Number 23240023.
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