POLAR COORDINATE BASED NONLINEAR FUNCTION
FOR FREQUENCY-DOMAIN BLIND SOURCE SEPARATION
Hiroshi Sawada

Ryo Mukai

Shoko Araki

Shoji Makino

NTT Communication Science Laboratories
2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japanfsawada,ryo,shoko,makig@cslab.kecl.ntt.co.jp
ABSTRACTThis paper presents a new type of nonlinear function for independent component analysis to process complex-valued
signals, which is used in frequency-domain blind source
separation. The new function is based on the polar coordinates of a complex number, whereas the conventional one
is based on the Cartesian coordinates. The new function isderived from the probability density function of frequencydomain signals that are assumed to be independent of thephase. We show that the difference between the two types
of functions is in the assumed densities of independent components. Experimental results for separating speech signals
show that the new nonlinear function behaves better than theconventional one.

2. FREQUENCY-DOMAIN BSS BASED ON ICA
Suppose that there are N source signals si (t) that are mutually independent, and these signals are observed at M miN
crophones xk (t) = i=1 hki (t)  si (t), where hki (t) represents the impulse response from source i to microphone
k , and  denotes the convolution operator. The goal of BSS
is to separate observed signals xk (t) into N unmixed signals yi (t) that are mutually independent. The separation
has to be done without knowing impulse responses hki (t)nor original source signals si (t).
To cope with convolutive mixtures, time-domain signals xk (t) are converted into frequency-domain time-series
signals Xk (!; m) by a T -point windowed DFT (discrete
Fourier transform):

P

1. INTRODUCTION
Blind source separation (BSS) is a technique to estimate
original source signals using only sensor observations thatare mixtures of the original signals. Independent component
analysis (ICA) [1–3] works well for BSS, if the mixture is
instantaneous (non-convolutive). In a real room environment, however, sounds are mixed in a convolutive mannerwith reverberations, and long reverberations make the problem difficult. One of the major methods to cope with reverberations is frequency-domain BSS [4–7]. In this approach,a convolutive mixture in the time domain is converted into
multiple instantaneous mixtures in the frequency domain,
and ICA is applied to the instantaneous mixture in every
frequency bin.
In frequency-domain BSS based on ICA, we have todeal with complex-valued signals. An extension of an ICA
algorithm to process complex numbers was proposed [4],where the nonlinear function was based on the Cartesian coordinates of a complex number: nonlinearities are applied to
the real and imaginary parts separately. This nonlinear function actually works and widely used by other researchers[5–7]. However, there has been presented no appropriate
interpretation of this function. Moreover, it imposes an additional constraint that prevents a learning algorithm fromconverging unless a non-holonomic algorithm [8] is employed.

0-7803-7402-9/02/$17.00 ©2002 IEEE

In this paper, we propose a new type of nonlinear function for an ICA algorithm to process complex numbers.1 It
is derived from the probability density function of frequencydomain signals that are assumed to be independent of thephase. As a result, the new function turns out to be basedon the polar coordinates of a complex number. We also givean interpretation of the Cartesian coordinate based function.
With experimental results for separating speech signals in areverberant environment, we compare the behaviors of these
two types of nonlinear functions, and discuss the differences
between them.

Xk (!; m) =X x  mS w  e

T 1

k(

+

)

( )

j!

(1)

 =0

where w( ) denotes a window function, S is a shifting interval of the window, and ! = 0; T1 2; : : : ; T T 1 2 . Now,
we have (!; m) = [X1 (!; m); : : : ; XM (!; m)]T for eachfrequency ! . Then, an unmixing N  M matrix (! ) andunmixed signals (!; m) = [Y1 (!; m); : : : ; YN (!; m)]T
are obtained by solving an ICA problem:XY
Y !; m W ! X !; m
(

)=

( )

(

W

)

in each frequency bin.

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1 Our preliminary work on this topic is presented in [9].

Before explaining a complex-valued ICA, let us review
an ordinary real-valued ICA algorithm. Based on the information maximization approach [1, 2] combined with thenatural gradient [3], an unmixing matrix W is gradually improved by the learning rule:W =  [I h'(Y)Y i] W:

3. NEW NONLINEAR FUNCTION
In this section, we propose a new type of nonlinear function
derived from the complex counterpart of equation (2):(Yi ) =T

In this formula,  is a step size parameter that has an effect on the speed of convergence, hi denotes the averaging
operator, and '  is a nonlinear function defined as:()'(Y) = ['(Y ); : : : ; '(YN )]T@
log p(Yi )'(Yi ) =
@Y
1

( )

(2)

i

where p Yi is the probability density function (pdf) of Yi .
2=
Yi , then the function isIf we assume p Yi
Yi , which is widely
hyperbolic tangent ' Yi
used for super-gaussian distributions [1, 2].
In frequency-domain BSS, signals obtained by DFT arecomplex. To deal with complex signals in ICA at each frequency, the calculation of W and the nonlinear function
were extended [4]:( ) = cosh ( )( ) = 2 tanh( )W =  [I h(Y)Y i] W(Yi ) = tanh[re(Yi )] + j  tanh[im(Yi )]
H

( )

( )

[

( ) tanh( )

( )

]



h(Yi )Yi i = 1

( )

( )



W =  [diag(h(Y)YH i) h(Y)YH i] W;

( )
( )

( )=

=

( )

This assumption is natural for a frequency-domain signal,
since the phase of Yi depends on the position of windowsw  of a windowed DFT (1) and the windows can be shifted
arbitrarily.
Then, let us consider the derivative of a real-valued function
p Yi . Generally speaking, a real-valued function
whose argument is a complex is not analytic: the derivative
is not well-defined. Throughout this paper, we use the following definition of the derivative of a real-valued function.

()log ( )= +@f (Y ) @f (Y )=+ j @f (Y ) :
@Y

def

@YR

@YI

The relevance of this definition is in the fact that the result
points to a direction in which f Y increases. Using thisdefinition, we derive( )@ jY j
@
@ q 2
=
(
+
j
) YR + YI2 = ej(Y )
@Y
@YR
@YI

(7)

which is used in the following theorem.

(Yi ) in (6) is(Yi ) = '(jYi j) e@
log p(jYi j)
where '(jYi j) =@ jYi jTheorem 1 Taking Assumption 1,

j  (Y i )Proof: By transforming equation (6) into

(Yi ) ==@(jY i j )log  p(jYi j) = p(j1Y j) @p@Y@Yiii
1 @p(jYi j) @ jYi jp(jYi j) @ jYi j @Yi

and taking equation (7) into account, we prove the theorem.
Here, we have a nonlinear function based on the polar
coordinates of a complex number. By using this type offunction, constraint (5) does not appear. Since Yi is a complex conjugate of Yi ,(Yi )Yi = '(jYi j) ej Y jYi j e j Y = '(jYi j) jYi j:( i)

we can avoid constraint (5). Researchers [5, 6] used this algorithm combined with a Cartesian coordinate based function. However, there is still another convergence problem,
which is also shown in Sec. 5.

( )Definition 1 Let Y
YR jYI be a complex and f Y bea real-valued function: C ! R. We define the derivative as:

htanh[im(Yi )]re(Yi ) tanh[re(Yi )]im(Yi )i = 0: (5)Equation (5) imposes the additional constraint that re Yi
and im Yi should be mutually independent. This constraint
is too strong, and there are cases where W does not converge well because of this. We show such a case in Sec. 5.
If we use non-holonomic algorithm [8]:( )jYi j ej(Yi ) be a complex-valued
Assumption 1 Let Yi
signal. The pdf p Yi of Yi is independent of the phase:p Yi
 p jYi j , where p jYi j is the pdf of jYi j andis a constant.

(4)

where Yi is the complex conjugate of Yi . This makes theaverage amplitude of Yi converge to some value. Extracting
the imaginary part of this equation, we have(6)

At first, we make an assumption on the density p Yi ofa complex-valued signal Yi in the frequency domain.

(3)

where YH represents the conjugate transpose of Y, andre Yi and im Yi are the real and imaginary parts of Yi ,respectively. In nonlinear function
Yi , is applied separately in the real and imaginary parts. We call thistype of function a Cartesian coordinate based function.
Although function (3) actually works, no appropriate interpretation of this function has been presented yet. Moreover, it has a convergence problem. Looking into the diagonal elements of I h Y YH i , we see that W converges to a point that satisfies@
log p(Yi ):
@Yi

( i)

Hence, the imaginary part of (4) becomes 0.
If we assume a super-gaussian distribution p jYi j
jYi j ej(Yi ) .
=jYi j , Yi

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cosh( ) ( ) = tanh( )

( )=

17.5

Table 1. Experimental conditions

30

40

I
Diag

17

Æ

and
(two sources)
4 cm
3 seconds
TR = 300 ms
8kHz
w  : Hanning
T = 2048 points (256 ms)
S = 512 points (64 ms)
 = 0.2

16.5
16
15.5

()

Polar

direction of sources
distance of two microphones
length of source signal
reverberation timesampling rate
window function
window length
shifting interval
step size
gain parameter
number of iterations

Æ

15
14.5

 = 100

Average SNR (dB)

14

100

13.5
13
12.5

4. INTERPRETATION OF THE CARTESIANCOORDINATE BASED FUNCTION

13

14Polar−I

15.45

Cartesian−I

14.78

Polar−Diag

15.38

Cartesian−Diag

14.89

15
Cartesian16

17

In this section, we give an interpretation of the conventional
Cartesian coordinate based function (3). We assume the definitions in Sec. 3 in the following theorem.

Fig. 1. SNRs compared between Polar and Cartesian( ) = p(YR )  p(YI ), then(Y ) = '(YR ) + j  '(YI )@log p(YI ).
where '(YR )= @Y@ log p(YR ), '(YI )= @Y@@
Proof: (Y ) = @Y log p(Y ) j  @Y log p(Y )= @Y@ log p(YR) j  @Y@ log p(YI ).

dB. Each point represents the results of Polar and Cartesianwith the same gradient and with the same combination of
speech signals.
In order to compare the results as correctly as possible,
we avoided the influence of the permutation problem [5–7]of frequency-domain BSS. We selected the best permutation by actually calculating the SNR in each frequency bin.
Therefore, the results in Fig. 1 are ideal ones under the condition that the permutation problem is perfectly solved. Before applying ICA, we whitened observed signals X to be
uncorrelated and to have unit variances. This pre-process
was very important to make the ICA algorithm stable especially for the Diag cases, where equation (4) is not concerned. Without this process, the Diag cases could haveexhibited irregular convergence speeds among the different
frequency bins.
We can see that the result of Polar is better than that ofCartesian in most cases. At first, we discuss the additional
constraint (5) imposed in the Cartesian–I case. Figure 2
shows the values of I h Y YH i at some frequency
bin. The horizontal axis corresponds to the number of iterations. The first graph shows the absolute values of eachelement for the Cartesian–I case. We see oscillations thathinder convergence. They come from the imaginary partsof the diagonals as shown in the second graph. If we use apolar coordinate based function, we can eliminate such oscillations as discussed in Sec. 3. The third graph shows the
Polar–I case. We can see a smooth convergence. Clearly,
the mutual information among Y is well minimized in thiscase unlike in the Cartesian–I case.
If we use Diag as the calculation of W, we can eliminate the additional constraint (5) even in the Cartesian case.
Accordingly, by investigating the results of Diag, we cansee the differences that purely come from the difference

Theorem 2 If p Y

R

I

R

I

R

I

This shows that a Cartesian coordinate based function assumes that the pdf p Y can be factorized into the productof p YR and p YI , and therefore YR and YI are mutuallyindependent. We can notice that the additional constraint(5) is satisfied if this assumption is met. By this theorem,
the assumed density in the nonlinearity (3) turns out to bepY
YR  I =
YI .
R=

( )

( )

( )

( )=

cosh( )

cosh( )

5. EXPERIMENTS AND DISCUSSIONS
To compare the two types of nonlinear functions, we conducted experiments to separate speech signals. Actually, weused the following two nonlinear functions:

Polar
Cartesian(Y ) = tanh(jY j) ej  Y(Y ) = tanh(YR ) + j  tanh(YI )

()

and the following two gradients of W

I
Diag

W =  [I h(Y)YHHi] W
W =  [diag(h(Y)Y i) h(Y)YH i] W:

The other conditions are summarized in Table 1.
The experiments were performed for 48 combinations (2nonlinear functions, 2 gradients, and 12 pairs of speech signals). Figure 1 shows the overall results. The measurement
is the average of two output SNRs (signal-to-noise ratios) in

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[

( )

]



Polar−Diag
Cartesian−Diag

0.1

0.2

0
0

10

0.2

20

30

40

50

0.08
0.06
0.04
0.02

Cartesian−I

initial

0

0

2.7

−0.2
0
0.4

10

20

30

40
[1,1]
[1,2]
[2,1]
[2,2]

Polar−I
0.2

0
0

10

20

30

40

2.9
Real part50h(Y)YH i]

1

1

0.5

0.5

0

0

−0.5

−0.5

−1
−1

of the nonlinearities between Polar and Cartesian. In fact,
we found another convergence problem in a Cartesian–Diagcase. Figure 3 shows the trajectory of element W11 of W atsome frequency bin. We see that the direction of the movement changes gradually in Polar–Diag, whereas it changes
sharply and frequently in Cartesian–Diag. The difference
comes from the assumed densities, as discussed in Sec. 3
and Sec. 4. Figure 4 shows the contour and gradient oflog p(Y ), being p(Y ) = = cosh( jY j) in Polar, andp(Y ) = =[cosh( YR )  cosh( YI )] in Cartesian. The gradient corresponds to (Y ). We see that the direction of
the gradient smoothly changes around the neighborhood in
the Polar case, whereas it changes steeply near the verticaland horizontal axes in the Cartesian case. By increasing thenumber of samples, this steepness may be smoothed out by
the averaging operator h(Y )Y  i. However, we conjecturethat the jag in Fig. 3 comes form this steepness.
6. CONCLUSIONSWe proposed a polar coordinate based nonlinear function toprocess complex-valued signals in ICA. Compared to the
Cartesian coordinate based function, the main difference isin the assumed densities of independent signals. In frequencydomain BSS, the assumption that the density is phase independent is more natural than the assumption that the real andimaginary parts are mutually independent, since signals areproduced by a windowed DFT. This consideration is supported by the experimental results: the Polar results were
better than the Cartesian ones in most cases, and there weresome convergence problems in the Cartesian case.

3

3.1

Cartesian

Polar

Iteration

Fig. 2. Values of [I2.8

Fig. 3. Trajectory of W11 on the complex plane

50

Imaginary part

Absolute value

Imaginary part

0.12

Cartesian−I
Imaginary part

Absolute value

0.4

0Real part

1

−1
−1

Fig. 4. Contour and gradient of0
Real part1

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