New analytical calculation and estimation of TDOA
for underdetermined BSS in noisy environments
Takuro Maruyama∗† , Shoko Araki† , Tomohiro Nakatani† ,Shigeki Miyabe∗ , Takeshi Yamada∗ , Shoji Makino∗ and Atsushi Nakamura†∗ Graduate School of Systems and Information Engineering, University of Tsukuba

1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan† NTT Communication Science Laboratories, NTT Corporation

2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
Email: maruyama@mmlab.cs.tsukuba.ac.jp
Abstract—We have proposed a new algorithm for sparsenessbased underdetermined blind source separation (BSS) that cancope with diffused noise environments. This algorithm includes
a technique for estimating the time-difference-of-arrival (TDOA)
parameter separately in individual frequency bins for eachsource. In this paper, we propose methods that integrate thefrequency-bin-wise TDOA parameter to estimate the TDOA ofeach source. The accuracy of TDOA estimation with the proposed
approach is shown experimentally in comparison with a conventional approach. The separation performance and calculation
time of the proposed approach is also examined.

I. I NTRODUCTION
Blind Source Separation (BSS) has been intensively investigated because the problem setting matches the real environment very well. With overdetermined BSS cases, source
separation can be performed satisfactorily, especially in a clean
environment, for example by using Independent Component
Analysis (ICA). To be able to handle a more realistic situation,
however, we must consider the underdetermined case, where
there are fewer sensors than sources.
Many methods have been proposed for underdetermined
BSS [1], [2], [3], [4], [5]. They employ the sparseness characteristics of the source signals as a clue for BSS, and are
based on the clustering of the source location information,such as time differences of arrival (TDOAs). However, mostsuch methods tend to be weak in the presence of reverberation
and background noise.
Recently, Izumi et al. [6] proposed a BSS method thatcan overcome this problem. The method assumes a diffuse
noise environment to make the estimation robust, and solves
the TDOA clustering based on the Expectation-Maximization(EM) algorithm. However drawback with this method is thatit cannot analytically update the TDOA parameters. As a consequence, this method requires a time-consuming exhaustive
search for the TDOA update.
We have proposed a new efficient BSS algorithm to remove
the need for such an exhaustive search [7] within the samerobust BSS framework proposed in [6]. The new update rule
has eliminated the need for the exhaustive search of TDOA,
and reduced the computational load. The new update ruleprovides the TDOA estimation results for each frequency bin.

Such frequency-wise TDOA estimation is sufficient for
BSS, however, it is insufficient for TDOA estimation of eachsource. In concrete terms, we have to determine one common
TDOA for each source from the frequency-bin-wise TDOA
estimated in individual frequency bins for the source.
So in this paper, we propose methods for estimating the
TDOA of source signals from the estimated TDOA parameters at each frequency bin. We provide experimental results
showing the accuracy of TDOA estimation with the proposed
methods in comparison with that of the conventional method.
We also report experimental results for the separation performance and calculation time of the proposed methods incomparison with those of the conventional method.
II. CONVENTIONAL METHODThis section outlines the method proposed by Izumi et al.
T[6] and the problem with it. Let xf,t = [xf,t,L , xf,t,R ] besignals observed by two microphones represented in the timefrequency domain. Where f , t are the index of the timefrequency slots. If we assume that source signals are sufficiently sparse such that only one source signal is active at eachtime-frequency point, and each source signal is transferred asa plane wave, xf,t can be written as[

xf,t,L
xf,t,R

]

[
=

xf,t

1
ej2πf δk][sf,t,k += bf,k sf,t,k + Nf,t

nf,t,L
nf,t,R](1)(2), where k is the index of the source, sf,t,k is the complex
spectrum of the source signal that is active at a time-frequency
slot, bf,k is the transfer function from the source to themicrophones (δk is the TDOA between two microphones), andnf,t is the observation error, which includes reverberation andbackground noise and is assumed to be uncorrelated with thesource signals.
We assume that Nf,t is time-invariant and follows a Gaussian distribution with a zero mean and a covariance matrix
σf2 Vf where σf2 is the noise power, and V is given as follows
for the diffused noise

[
Vf =

1
sinc(2πf D/c)sinc(2πf D/c)
1

]
(3)

. Here c is the velocity of sound, and D represents the distance
between the two microphones. The purpose of the conventional
method [6] and this paper is to estimate the source signals
sf,t,k solely from the mixed observation xf,t .
The likelihood function for the observation xf,t is1p(xf,t |k, θ) = 2 4exp
π σf |Vf |(1− 2 NHV−1 Nf,t,k
σf f,t,k f

)

σf2 =

sf,t,k =Let θ = {σf2 , δk , sf,t,k } be the parameter set. The log
likelihood function is
∑∑
t

log

f

∑

p(xf,t |k, θ)p(k|θ)−1bHf,k Vf bf,k(5),

and the mixing weight p(k|θ) (where
calculated by(10)∑

k p(k|θ)= 1) is1 ∑∑
mf,t,k
TF t

(11)

f

, where T and F are the numbers of time frames and frequency
bins, respectively.
In [6], as δk cannot be solved analytically, the update is
performed by calculating Q(θ|θ′ ) for all the discretized δkand selecting δk that gives the maximum of Qk∑
, where p(k|θ) is the mixing weight ( k p(k|θ) = 1), and k is
an index of a source that is assumed to be dominant at a time
frequency point according to the sparseness characteristics
assumption.
In [6], this log likelihood is maximized with the EM algorithm. The parameters to be estimated are θ = {σf2 , δk , sf,t,k }where k is the hidden variable and the auxiliary function inthis problem is
Q(θ|θ′ )−1bHf,k Vf xf,tp(k|θ) =

.

(9)

k

(4)

L=

1 ∑∑
−1
mf,t,k NH
f,t,k Vf Nf,t,k
T t

E [log p(xf,t ; θ)|θ′ ]∑∑∑=
mk,f,t log p(xf,t |k, θ)p(k|θ). (6)δk = argmaxδk Q(θ|θ′ )(12).
This update rule has two problems. One is that we cannot
obtain an exact, optimal update of δk but only its discretizedapproximation in each iteration. The other problem is that
this exhaustive search requires a large computational cost. To
overcome these problems, we derive an analytical update rulefor estimating the TDOA parameter δk .

=

t

f

k

. Here, time-frequency mask mk,f,t is the posterior probability
that source k is active at a time-frequency slot, and θ′ is theparameter set obtained by the previous iteration.
Moreover, mk,f,t functions as a soft mask for separating thek-th source in the EM algorithm, and after the convergence of
the EM algorithm we can estimate the separated source signal
yf,t,k as the expectation using the estimated parameters andmf,t,k given by
yf,t,k = mf,t,k−1
bH
f,k Vf xf,t−1
bH
f,k Vf bf,kIII. PROPOSED METHOD
In this section, we provide an analytical update rule for the
TDOA parameter δk .
A. Calculation of TDOA parameter
The right-hand side of the likelihood function (4) is denotedby using the components of the vectors x, b and the matrix
V−1
NH
f,t,k Vf Nf,t,k
[
]
[
1
1
= n∗f,t,L n∗f,t,R 2
σf (1 − ϕ2f ) −ϕf{1=|nf,t,L |2 + |nf,t,R |222
σf (1 − ϕf )(7).
In this paper, the E step updates mk,f,t and the auxiliary
function, and the M step updates the other parameters.
The time-frequency mask mk,f,t is updated byp(k|θ′ )p(xf,t |k, θ′ )mf,t,k = p(k|xf,t,k , θ′ ) = ∑′′ ′′ ′k p(k |θ )p(xf,t |k , θ )][nf,t,L
nf,t,R}∗−ϕf n∗f,t,R nf,t,L − ϕf Nf,t,R Nf,t,L,

where ϕf denotes ϕf = sinc(2πf D/c).
From (1)
[
]
[
] [
]
nf,t,L
xf,t,L1=−
sf,t,knf,t,R
xf,t,Rβf,k(8).
The parameters σf2 and sf,t,k are estimated by differentiating the auxiliary function with respect to each parameter, andsetting them at zero,

−ϕf
1

]

(13)

(14)

,
where βf,k denotes βf,k = ej2πf δk . By substituting (14)into (13), we obtain (15)−1NH
f,t,k Vf Nf,t,k{1|xf,t,L − sf,t,k |2 + |xf,t,R − βf,k sf,t,k |222
σf (1 − ϕf )−ϕf (xf,t,R − βf,k sf,t,k )∗ (xf,t,L − sf,t,k )}−ϕf (xf,t,R − βf,k sf,t,k )(xf,t,L − sf,t,k )∗{ ∗1sβ ∗ βf,k sf,t,k2
2
σf (1 − ϕf ) f,t,k f,k∗−βf,ks∗f,t,k [xf,t,R − ϕf (xf,t,L − sf,t,k )]==−βf,k sf,t,k [xf,t,R − ϕf (xf,t,L − sf,t,k )]∗+|xf,t,R |2 − ϕf x∗f,t,R (xf,t,L − sf,t,k )}−ϕf xf,t,R (xf,t,L − sf,t,k )∗ + |xf,t,L − sf,t,k |2
(
)∗ (
)
1
{ ξf,t,k − βf,k sf,t,k
ξf,t,k − βf,k sf,t,k2
2
σf (1 − ϕf )=+(1 − ϕ2f )|xf,t,L − sf,t,k |2 },(16)

Furthermore, the clause containing the above-mentioned βf,kcan be rewritten as()∗ ()ξ − βf,k sf,t,k
ξ − βf,k sf,t,k= |ξf,t,k |2 + |sf,t,k |2∗
sf,t,k ej2πf δk − ξf,t,k s∗f,t,k e−j2πf δk−ξf,t,kwhere ψSk and ψξk represent the phases of sf,t,k and ξf,t,k ,respectively (i.e, sf,t,k = |sf,t,k |ejψSk , ξf,t,k = |ξf,t,k |ejψξk )If ϕf = 0, ψSk − ψξk is the phase difference between two
microphones.
By using (15) and (17), the likelihood function (4) can be
rewritten as√2π1σf2 |Vf |

( )
exp C

( |ξ
)
f,t,k ||sf,t,k |
· exp
cos(ψ
−
ψ
−
2πf
δ
)
s
ξ
f,k
σf2 (1 − ϕ2 ), where C is independent of δf,k .
The last term,( |ξ
)f,t,k ||sf,t,k |
exp
cos(ψ
−
ψ
−
2πf
δ
)
s
ξ
f,k
σf2 (1 − ϕ2 )

(18)

(19)

has the shape of the von Mises distribution [8]g(x|κ, µ) =1eκ cos (x−µ)
2πI0 (κ)

It should be noted that the function arctan(x) is unique only
if −π/2 < x < π/2. However, 2πf δf,k can fall in the −π
to π range. Therefore, when |x| ≥ π/2, we have to modify
the estimated value by checking the inflection point of theauxiliary function. To accomplish this, we calculate the second
order differential of the auxiliary function, and modify thevalues as follows∂2Q• If δf,k < 0,
≥ 0, then 2πf δf,k ← 2πf δf,k + π∂(δ 2 )

(20)

, where −π < x ≤ π, µ is the mean of the distribution(−π < µ ≤ π), κ > 0 is a concentration parameter, andI0 (x) is a modified Bessel function of the first kind and order

f,k
2

∂ Q
If δf,k > 0, ∂(δ≥ 0, then 2πf δf,k ← 2πf δf,k − π2f,k )• Otherwise, we do not modify 2πf δf,kIn summary, the proposed method estimates the parameters
σf2 , sf,t,k and mf,t,k in the same ways as described in
Section 2, and the frequency-dependent TDOA parameter δf,kis calculated with the update rule (21).
•= |ξf,t,k |2 + |sf,t,k |2−2|ξf,t,k ||sf,t,k | cos(ψSk − ψξk − 2πf δk ), (17)p(xf,t |k, θ) =∑mf,t,k |ξf,t,k ||sf,t,k | sin(ψξk − ψSk )∑2πf δf,k = arctan tmf,t,k |ξf,t,k ||sf,t,k | cos(ψξk − ψSk )t(21)(15) .

whereξf,t,k = [xf,t,R − ϕf (xf,t,L − sf,t,k )].

zero. Thus, (19) indicates that the phase difference ψs − ψξ ≈arg (xf,t,R ) − arg (xf,t,L ) follows a von Mises distribution
whose mean value corresponds to the frequency-dependent
TDOA µ = 2πf δf,k , and the concentration parameter is the|ξ||sf,t,k |signal to noise ratio (SNR) related 1 value κ = f,t,k.
σf2 (1−ϕ2f )Therefore, we can derive the update rule for δk using amethod similar to a maximum likelihood estimation of the
mixture model of the von Mises distribution. However, because
the cosine part of (19) depends on the frequency f , we haveto derive the update rule for δf,k at each frequency, which isdifferent from the previous frequency independent update rule(12). By substituting (19) into (6) and setting ∂δ∂Q
= 0, thef,kupdate rule becomes

B. Estimation of TDOA parameters
We described how to compute the TDOA parameters using the method proposed in [6]. However, the TDOAs are
estimated independently in individual frequency bins by thismethod. We propose the following three methods for estimating the frequency-independent TDOAs.
• PROPOSED METHOD A
A presumed δf,k is used as an independent TDOA for
every frequency bin.
• PROPOSED METHOD B
An average value of δf,k over all the frequency bins isused as a TDOA estimate
for the source indexed by k.
∑(This means δk = F1 f δf,k )Let this be PROPOSED METHOD B.
• PROPOSED METHOD C
Figure 1 shows an example set of estimated TDOAs
obtained by the method proposed in [6].
The solid lines show the value of the estimated TDOA,
and the dashed lines show the values of the correct
1 When we assume that ϕ = 0, |ξf,t,k ||sf,t,k | = |sf,t,k |f
σ 2 (1−ϕ2 )
σ2

SNR.

f

f

2

, which is the

−4

x 10

15
0o

6

Reverberation time:
130 ms4

Sampling rate:
8000 HzTDOA [sec]

2

4 cm
0

70 o
−2

o

30

Room size:
4.45 m × 3.55 m × 2.50 m

−4

Door
−6

0

500

1000

1500

2000
frequency [Hz]

2500

3000

3500

4000

Fig. 2. Experimental setup

Fig. 1. Sample of calculated δf,k

B. DOA estimation accuracy
TDOAs. Figure 1 shows that the estimation error is verylarge in low frequency bins. Moreover, many of theestimated TDOAs in high frequency bins also include
many errors. So, in PROPOSED METHOD C, the rangeof f to be used for the estimation is restricted. An averagevalue of δf,k (200 Hz < f < 3000 Hz) is adopted and it
is used as a TDOA estimateof each source signal. (This∑1means δk = 3000−200
200<f <3000 δf,k )Let this be PROPOSED METHOD C.
C. Separate source signals
The proposed separation method employs the proceduredescribed in section II, except that (12) is replaced by (21).
IV. EXPERIMENTS
A. Experimental conditions
We performed experiments with measured impulse responses in a room with a reverberation time of 130 ms.
The experimental setup is shown in Fig. 2. We used two
microphones, whose spacing was 4 cm. The number of sources
K was K = 2 (70◦ and 150◦ ), or K = 3 (30◦ , 70◦ and150◦ ). Observed signals are made by convolving the measured
room impulse responses and 5-second English speech signals
sampled at 8 kHz. The frame size and frame shift for STFT
were 64 ms and 16 ms, respectively.
In our experiments we used a Gaussian noise with zeromean and a covariance matrix of σf2 Vf , where Vf was givenby (3), and σf was determined so that the SNR with respect
to source 1 had a preset value.
We compared the direction of arrival (DOA) estimation
accuracy and computational times of the conventional method
and PROPOSED METHODS A, B and C. For a discrete searchof the DOA parameters δk with the conventional method,
we compared the Q functions of the 0◦ to 180◦ range inincrements of 1◦ to find the optimum according to Eq, (12).

We compared DOA estimation errors. For each K, 10
speaker combinations were tested and the results were averaged. Table I shows the errors of the angular estimation. We
can see from table I that PROPOSED METHOD C was thebest error of the proposed methods. However, the conventional
method was comparable to or even better than all the proposed
methods.
To investigate the influence of noise power and reverberationon the TDOA estimation we also compared the estimation
errors of the conventional and proposed methods by changing
the SNR and the reverberation time. Table II shows theinfluence of the noise on the TDOA estimation performance.
The number of sources K was 3 (30◦ , 70◦ and 150◦ ), and theother conditions were the same as those in Section IV-A. By
adjusting the power of the noise σf2 , the SNR was set at 27.5
dB (= Table IV. (b)), 20 dB (Table II. (a)), and 10 dB (Table
II. (b)).
Table III shows the influence of the reverberation time on
the TDOA estimation performance. The number of sources K
was 3 (30◦ , 100◦ and 135◦ ), and the other conditions were thesame as those in Section 4.1. The reverberation time was set
at 300 ms.
We can see from Tables I, II and III that PROPOSEDMETHOD C was the best of the three proposed methods.
However, the conventional method was comparable to or evenbetter than all the proposed methods.
C. Separation performance and computational time
We compared the separation performance and computationaltime of the conventional method and the proposed methods A,
B and C. The separation performance was evaluated in termsof the signal to interference-plus-noise ratio (SINR) and thesignal-to-distortion ratio (SDR) [2]. For each K, 10 speakercombinations were tested and the results were averaged.
Figure 3 shows example spectrograms of the signals separated by the different methods. The figure shows that thesource signal was successfully separated from the observed

Fig. 3. Signal spectrogram. Top left: observed signal, top center: one example of source image, top right: separated signal with conventional method, bottom
left: separated signal with proposed method A, bottom center: separated signal with proposed method B, bottom right: separated signal with proposed method
C.
TABLE IA NGULAR ESTIMATION ERROR(a) K = 2 (70◦ and 150◦ )MethodSource1
Conventional 1.1[deg]Proposed A
2.2[deg]
Proposed B
3.9[deg]
Proposed C
1.3[deg]

Source2
3.0[deg]
6.1[deg]
3.0[deg]
0.6[deg](b) K = 3 (30◦ , 70◦ and 150◦ )MethodSource1
Source2
Conventional
2.8[deg]
1.5[deg]
Proposed A
12.1[deg] 6.3[deg]Proposed B
4.1[deg]
1.9[deg]
Proposed C
5.6[deg]
2.4[deg]

Source3
3.5[deg]
7.7[deg]
6.4[deg]
1.6[deg]

TABLE IIDOA ESTIMATION ACCURACY UNDER SEVERAL SNR CONDITIONS(a) SN R = 20 [dB]
MethodSource1
Conventional
1.7[deg]
Proposed A
12.2[deg]
Proposed B
5.0[deg]
Proposed C
4.9[deg]

Source2
0.3[deg]
6.2[deg]
0.7[deg]
2.2[deg]

Source3
4.2[deg]
7.2[deg]
6.4[deg]
3.2[deg](b) SN R = 10 [dB]
MethodSource1
Conventional
0.1[deg]
Proposed A
13.0[deg]
Proposed B
5.3[deg]
Proposed C
4.6[deg]

Source2
1.7[deg]
8.9[deg]
9.5[deg]
0.4[deg]

Source3
5.0[deg]
7.4[deg]
6.4[deg]
3.4[deg]

TABLE IIIDOA ESTIMATION UNDER A REVERBERATION CONDITION

speech mixture. Table IV shows the SINR, SDR and computational time obtained in the experiments. We can see fromtable IV that our proposed method achieved a comparable
performance to the conventional approach and greatly reduced
the computational time by 1/10 to 1/20. This result shows thatour proposed method can successfully separate signals using
the estimated TDOA parameter δk without using an exhaustive
search that is essential with the conventional method.
To investigate the influence of noise power and reverberation
on separation performance, we also compared the SINR, SDR,
and computational time of the conventional and the proposed
methods by changing the SNR and reverberation time.
Table V shows the influence of noise on the separation

reverberation time = 300 [ms]MethodSource1
Source2
Conventional 25.3[deg] 26.0[deg]Proposed A
11.2[deg] 26.5[deg]Proposed B
26.3[deg] 39.0[deg]Proposed C
30.0[deg] 41.4[deg]Source3
6.8[deg]
5.6[deg]
2.8[deg]
7.6[deg]

performance. The number of sources K was 3 (30◦ , 70◦ and150◦ ), and the other conditions were the same as those in
Section 4.1. By adjusting the power of the noise σf2 , the SNR
was set at 27.5 dB (= Table IV (b)), 20 dB (Table V (a)), and10 dB (Table V (b)).

TABLE IV
S OURCE SEPARATION RESULTS(a) K = 2 (70◦ and 150◦ )MethodSINR
Conventional 15.3[dB]
Proposed A
14.8[dB]
Proposed B
15.1[dB]
Proposed C
15.3[dB]

SDR
6.7[dB]
7.7[dB]
6.6[dB]
6.8[dB]

Calculation Time
398.6[s]25.7[s]16.5[s]16.4[s](b) K = 3 (30◦ , 70◦ and 150◦ )MethodSINR
SDR
Conventional 7.6[dB]
5.4[dB]
Proposed A
6.7[dB]
5.9[dB]
Proposed B
7.6[dB]
5.2[dB]
Proposed C
7.7[dB]
5.4[dB]

Calculation Time
451.5[s]39.1[s]15.0[s]14.8[s]TABLE VP ERFORMANCE UNDER SEVERAL SNR CONDITIONS(a) SN R = 20 [dB]
MethodSINR
Conventional 7.6[dB]
Proposed A
6.7[dB]
Proposed B
7.5[dB]
Proposed C
7.6[dB]SDR
5.4[dB]
5.9[dB]
5.3[dB]
5.4[dB]

Calculation Time
470.1[s]41.0[s]17.8[s]14.5[s](b) SN R = 10 [dB]
MethodSINR
Conventional 5.3[dB]
Proposed A
6.1[dB]
Proposed B
4.9[dB]
Proposed C
5.3[dB]SDR
5.5[dB]
5.9[dB]
5.3[dB]
5.4[dB]

Calculation Time
512.2[s]47.1[s]31.6[s]15.4[s]Table VI shows the influence of the reverberation on theseparation performance. The number of sources K was 3(30◦ , 100◦ and 135◦ ), and the other conditions were the sameas those in Section 4.1. The reverberation time was set at 300
ms.
We can see from Tables V and VI that the performanceof our proposed method was comparable to that of the conventional method in that if reduced the computational time,
regardless of noise and reverberation.
V. CONCLUSION
This paper proposed new methods for estimating the
TDOAs of multiple sources for sparseness-based blind source
separation in noisy and reverberant environments. The proposed methods determine a unique TDOA for each soundsource by taking the average of the TDOAs over different
frequency bins, which can be estimated in a computationallyvery efficient manner by using an analytical update equation.
We confirmed that our proposed methods can greatly reducethe computation time without degrading the quality of theseparated signals by comparison with the conventional method.
However, we also confirmed that the TDOA estimation accuracy of the proposed methods was slightly degraded comparedwith that of the conventional method. We plan to improve ourmethods as regards accuracy of TDOA estimation, to evaluate
them in real environments, and to introduce a noise model thatis more appropriate for reverberation.

TABLE VI
P ERFORMANCE UNDER A REVERBERATION CONDITION
reverberation time = 300 [ms]MethodSINR
SDR
Conventional 7.0[dB] 2.9[dB]
Proposed A
6.3[dB] 2.7[dB]
Proposed B
5.1[dB] 3.0[dB]
Proposed C
5.9[dB] 2.9[dB]

Calculation Time
962.5[s]65.9[s]24.7[s]24.9[s]R EFERENCES[1] O. Yilmaz and S. Rickard, “Blind separation of speech mixtures via timefrequency masking,” IEEE Transactions on Signal Processing, Vol. 52,
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