1IWAENC 2006 – PARIS – SEPTEMBER 12-14, 2006

UNDERDETERMINED SOURCE SEPARATION BY ICA AND HOMOMORPHIC SIGNAL
PROCESSING
1†

Stefan Winter, 2 Walter Kellermann, 1 Hiroshi Sawada and 1 Shoji Makino
1

wifan@cslab.kecl.ntt.co.jp
NTT Communication Science Laboratories, NTT Corporation
2-4 Hikaridai Seika-cho Soraku-gun Kyoto, 619-0237 Japan2
Department of Multimedia Communication and Signal Processing, University Erlangen-Nuremberg
Cauerstr. 7, 91058 Erlangen
1

ABSTRACT
Nearly all approaches for underdetermined blind source separation (BSS) assume independent and identically distributed (i.i.d.)sources. They completely ignore the temporal structure of colored sources such as speech signals. Instead, we propose a multivariate model based on a multivariate Gaussian distribution thatis then used to determine an unmixing matrix for underdetermined BSS. Based on parameterization by cepstral coefficients
we present a novel ICA-based cost function for estimating the
speech-related parameters of the unmixing matrix. Experimental results support the proposed approach.

1. INTRODUCTION
Blind source separation (BSS) describes techniques that
aim at separating P signals if no information is available
other than Q mixed versions of the original signals. Most
BSS approaches assume that there are at least as many
microphones as source signals (Q ≥ P ), which is called(over-) determined BSS. Since overdetermined BSS canbe reduced to determined BSS [1], we refer to both asdetermined BSS.
Instead, we consider underdetermined BSS, where we havefewer microphones than source signals (Q < P ). In thiscase, (see e.g. [2, 3, 4], the separation quality in terms ofinterference suppression and signal distortion is still not
as good as with determined BSS. This is particularly true
if wideband signals such as speech signals are involved.
The difficulty is that in contrast to determined BSS the solution of underdetermined BSS goes beyond system identification. Even if the mixing system is fully identified,additional effort is required to separate the mixtures.
Nearly all approaches designed to solve the latter problem assume independent and identically distributed (i.i.d.)sources [2, 3, 4].While this assumption may serve well asa first order approximation, it completely ignores the temporal structure of colored sources such as speech signals.
† The author is on leave from the Department of Multimedia Communication and Signal Processing, University Erlangen-Nuremberg

Here we propose a multivariate model that takes the temporal correlation explicitly into account. We consider convolutive mixtures of speech signals in the time domain.
This paper concentrates on the separation and assumes
that the mixing matrix is known.
The contribution of this paper is based on the following
ideas: With the multivariate source model we can derive
an unmixing matrix for underdetermined BSS, which results in high quality source separation if the parameters of
this unmixing matrix are known. We rely on two principles when estimating the source-related parameters of theunmixing matrix. First we reduce the number of parameters by utilizing real cepstral coefficients based on principles of homomorphic signal processing [5]. Then weapply independent component analysis (ICA) to estimate
the reduced set of parameters.
After formulating the problem analytically in Sec. 2, we
provide statistical models for Bayesian inference in Sec. 3
together with the resulting unmixing matrix. Section 4elaborates the concise parameterization of the unmixing
matrix. In Sec. 5 we derive a novel algorithm for estimating the speech parameters based on ICA. Section 6presents experimental results which are discussed in Section 7.
2. PROBLEM FORMULATIONWith sp (t) ∈ R denoting the p-th source signal (1 ≤ p ≤P ) and hqp (τ ) (0 ≤ τ ≤ L − 1) the impulse responseof length L from source p to sensor q, we obtain mixed
signals xq (t) ∈ R (1 ≤ q ≤ Q, Q < P ) byXX
xq (t) =sp (t)h(τ − t) + n(t).
(1)
p

τ

n(t) denotes noise added to the sensors. LetTS p = sp (0) · · · sp (T − 1)
, 1≤p≤P(2)

denote a frame of length T of the p-th original speechTsignal. (·) denotes the transpose. We assume that the2

IWAENC 2006 – PARIS – SEPTEMBER 12-14, 2006

speech signal is stationary within a frame of length T with
autocorrelation
rp (τ ) = E{sp (t)sp (t + τ )}.

(3)

We summarize the P frames of the different source signals
by the vector


S1


S =  ...  ∈ RP ·T .
(4)SPFor each source S p the autocorrelation matrix Rp ∈ RT ×T
is given by a symmetric Toeplitz matrix. Its first row is defined by
r p = rp (0) · · · rp (T − 1) .
(5)
We summarize the P autocorrelation matrices by


R1
0


P ·T ×P ·T
..
R=
.
∈R
.
0

(6)

RP

We define the mixing matrix
H 11 · · · H 1P..
Q·(T −L+1)×P ·T..
H =  ...
, (7)
∈R
.
.
H Q1

···

H QP

where H qp denotes convolution matrices with Toeplitz
structure
hqp (L − 1) . . .
hqp (0)0..
H qp = .
hqp (L − 1) . . .

0∈ RT −L+1×T .

hqp (0)(8)xq (0) · · ·xq (T − L)T

as illustrated in Fig. 1. N ∈ RQ·(T −L+1) is derived
from n(t) in a similar way to that used to derive X from
x(t). The final goal in BSS is the estimation of signals Y
that resemble the original signals S as closely as possible.
Since only the mixed signals are available, this is at best
possible up to arbitrary permutation and scaling. With determined BSS it is sufficient to estimate the mixing matrix
H or its inverse. Since the mixing matrix is square andtherefore invertible (assuming that H is non-singular), theinverse can be used to separate the signals. In contrast
to determined BSS, in the underdetermined case estimation of the mixing matrix H is insufficient. Even if H
is available, estimating the original signals from the mixtures poses its own problem, since H cannot be simply
inverted. In the following we concentrate on estimating
the original signals and assume that the mixing matrix is
known or else can be estimated [2, 6, 7].
3. MODELS
We make the common assumption that the speech signals
are mutually independentYp(S) =p(S p ).
(12)p

Similar to the source signals we define
Xq =Figure 1: Mixing process for underdetermined BSS with
Q = 2, P = 3, T = 5 and L = 3, 1 ≤ q ≤ Q (9)and summarize X q by


X1


X =  ...  ∈ RQ·(T −L+1) .

(10)

We further assume that the speech signal is approximately
stationary within a frame of appropriately chosen lengthT.
Nearly all approaches to underdetermined BSS assume independent and identical prior distributions. In contrast, in
this section we propose a multivariate prior based on theanalytically tractable, multivariate Gaussian distribution

XQp(S p ) = N (S p |µp , Rp )This results in a compact description of the mixing process
for one frame given by
X = HS + N

(11)

(13)

with zero mean vector µp and covariance matrix Rp .
We also define the likelihood derived from the commonly
assumed additive white Gaussian sensor noise model with

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IWAENC 2006 – PARIS – SEPTEMBER 12-14, 2006

identical variance σ 2 across all sensors. Based on the mixing process (11), this assumption results in the Gaussian
likelihoodp(X|H, S, σ 2 I) = N (X|HS, σ 2 I).

(14)

From the Bayes’ rule [8] we then obtain the posterior of
the source signal given the mixed signals asp(S|X) ∝ p(S) · p(X|H, S, σ 2 I)
∝ N (S|µS , ΣS )with the correlation matrix
−1

1 T
−1
ΣS = R + 2 H H
σ

µS

=

σ 2 R−1 + H T H

−1

∂r p
=
∂ SbpdThen, the derivative of the correlation matrix Rp is given∂rby a symmetric Toeplitz matrix with ∂ Sbpd as its first colp
umn.
(16)

H T X.

(20)




n o
bp −  2πd k
2 · IDFT exp 2 · DFT S
.
T

(15)

and the meanof the autocorrelation sequence r p with respect to the cepstral coefficient Sbpd , 1 ≤ d ≤ D, which denotes the d-th
bp . The derivative yields
element of S

(17)

Since the posterior distribution is again Gaussian, the MMSE
estimate is given by its mean, resulting in the estimatedsource signals

−1
Y =
σ 2 R−1 + H T H
H T X.
(18)|{z}

5. PARAMETER ESTIMATION
To estimate the cepstral coefficients of each source in agiven frame, we apply the principles of ICA. Based onminimum mutual information (MMI) [10], which is themost general approach to ICA [11], we define the costfunction
JICA= −E {log |det R| + log | det C|+)Xlog pY p (Y p )

(21)
(22)

p

W

As shown in Sec. 6, the unmixing matrix W is sufficientfor high quality source separation. The problem is now toestimate the parameters of the unmixing matrix.
4. PARAMETERIZATION
The fewer the parameters that need to be estimated in relation to the available data, the better the estimate will be. In
order to reduce the number of parameters in the unmixing
matrix, we describe the autocorrelation by cepstral coefficients. We exploit the fact that speech signals can generally be concisely described by a few cepstral coefficients[5]. In addition, the autocorrelation is related to the real
cepstrum by a one-to-one mapping based on the WienerKhintchine theorem [9].
We summarize the Dp real cepstral coefficients used forbp . By zero-padding S
bp to
the p-th source by the vector S
length T , the autocorrelation sequence r p can be shown
to be given by
n

n oo
bp
r p = IDFT exp 2 · DFT S, (19)
where IDFT {·} and DFT {·} denote the (inverse) Discrete Fourier Transform and exp(·) is applied elementwise.
To estimate optimal real cepstral coefficients based on thecost function presented in Sec. 5, we need the derivativewith E{·} denoting the expectation and C being definedas
−1
C := σ 2 I + HRH T.
(23)
For a gradient-based optimization of the cost function wedetermined the derivative of the cost function by using themultivariate source model (13). The result is given in (24),where tr (·) denotes the matrix trace.
6. EXPERIMENTAL RESULTS
We performed experiments with three speech signals (twomale, one female) of 1.62 seconds (8 kHz sampling rate).
We generated two convolutive mixtures with artificial impulse responses of length L = 11 and assumed that the
mixing matrix was available. The signals were processedin frames of T = 128 samples and shifted by 64 samples. No noise was added to the mixtures. Therefore, thevariance σ 2 served as a regularization parameter and wasset at σ 2 = 10−4 . The sources had the same variance of
0.0037. We compared the following approaches:
E1
E2

This constitutes the reference of the unprocessed
mixed signals
The autocorrelation was directly estimated fromthe original signals to provide an upper limit forthe separation performance of W (18).

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IWAENC 2006 – PARIS – SEPTEMBER 12-14, 2006

(

∂JICA
∂ Sbpd= E

Method
E1
E2
E3

SDR
−20.80
18.39
6.64

tr





1
T
H CH − R−1 − I − 2 (W H) IH T CXX T C T H
2
T

SIR
−6.01
33.13
8.88

SAR
−10.4718.60
11.60

Table 1: Experimental results
The autocorrelation was parameterized by D =30 cepstral coefficients. These, in turn, were estimated by the ICA-based approach as described inSec. 5. To obtain values for the real cepstral coefficients that led to invertible correlation matrices
we provided lower and upper bounds. They weretrained based on clean speech signals.
To evaluate of the separation results we used the signal-todistortion ratio (SDR), signal-to-interference ratio (SIR)
and signal-to-artifact ratio (SAR) as defined in [12]. The
averaged results for the different approaches are shown in
Table 1.
E3

7. CONCLUSION
The results in Table 1 together with a subjective evaluationsuggest that knowing the autocorrelation together with themixing matrix is sufficient to perform high quality underdetermined source separation in noiseless environments(E1 compared to E2). In other words, the problem of underdetermined source separation can be reduced to the estimation of the underlying autocorrelations once the mixing matrix is available. We proposed a novel ICA-based
cost function for estimating the autocorrelations based ona parameterization by cepstral coefficients. An algorithm
based on the gradient of the cost function provides a good
estimate of the source signals in underdetermined BSS(E3).
In the future we plan to estimate both the speech parameters of the unmixing matrix, and the mixing parameters
based on ICA. The estimation of both kinds of parameters would result in a fully blind algorithm for underdetermined source separation.
8. REFERENCES[1] S. Winter, H. Sawada, and S. Makino, “Geometrical interpretation of the PCA subspace approach for
overdetermined blind source separation,” EURASIP
Journal on Applied Signal Processing, vol. 2006,
2006, Article ID 71632, 11 pages.

∂R∂ Sbpd!)(24)[2] A. Blin, S. Araki, and S. Makino, “Underdeterminedblind separation of convolutive mixtures of speechusing time-frequency mask and mixing matrix estimation,” IEICE Trans. Fundamentals, vol. E88-A,
no. 7, pp. 1693–1700, 2005.
[3] C. Févotte and S. J. Godsill, “A Bayesian approachfor blind separation of sparse sources,” Tech. Rep.,
Cambridge University Engineering Dept., January
2005.
[4] S. Winter, H. Sawada, and S. Makino, “On real andcomplex valued L1-norm minimization for overcomplete blind source separation,” in 2005 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), New Paltz, NY, USA,2005, pp. 86–89.
[5] L.R. Rabiner and R.W. Schafer, Digital processingof speech signals, Prentice Hall, 1978.
[6] S. Winter, H. Sawada, S. Araki, and S. Makino,“Overcomplete BSS for convolutive mixtures basedon hierarchical clustering,” in Proc. ICA 2004, Sep
2004, pp. 652–660.
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[8] A. Papoulis and S.U. Pillai, Probability, randomvariables, and stochastic processes, McGraw-Hill,
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[9] J.G. Proakis and D.G. Manolakis, Digital signal processing, Prentice Hall, 3rd edition, 1996.
[10] D.J.C. MacKay, Information theory, inference, andlearning algorithms, Cambridge University Press, 7
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[11] H. Buchner, R. Aichner, and W. Kellermann, Audio signal processing for next-generation multimedia communication systems, chapter Blind Source
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[12] C. Févotte, R. Gribonval, and E. Vincent,“BSS EVAL toolbox user guide – Revision2.0,” Tech. Rep. 1706, IRISA, April 2005.