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IEEE SIGNAL PROCESSING MAGAZINE [26] MARCH 2015

formulate the demixing problem as a scalar source separation

problem for each frequency independently (instantaneous ICA),

and implement scalar ICA algorithms for each STFT bin [33],

[36], [37].

Similar to adaptive filtering, where time-domain approaches

imply a significantly higher computational complexity for

obtaining a similar performance as frequency-domain

approaches, frequency-domain implementations of ICA (FD-

ICA) are computationally more attractive. On the other hand, if

these are straightforwardly formulated as independent ICA

problems in each STFT bin, the resulting demixing system does

not perform a linear but a circular convolution, which is inade-

quate for demixing a linear mixing system [35]. As an immedi-

ate consequence, the so-called internal permutation and scaling

problems result: as the outputs of any unconstrained ICA sys-

tem are only determined up to an unknown scaling factor and a

permutation of their order, for FD-ICA the order and the scaling

of the outputs may be different for each STFT bin. Therefore the

outputs of the scalar ICA units have to be realigned so that for a

given output channel

y all frequency bins belong to the same

source [37] and are properly scaled, e.g., by minimizing the

average power difference of the outputs

y relative to the inputs

(minimum distortion principle) [38].

In the acoustic signal extraction context, the mechanism of

BSS based on ICA has been shown to be equivalent to a set of

P adaptive beamformers, each of which aims to extract one

source by suppressing all other sources, thereby exploiting the

spatial diversity of the microphone signals [39]. Note that for

adaptive beamforming, the DOA or the RTFs of the target

source should be known, and that it can adapt the required

statistics only with given source activity information, while

ICA does not need such information.

APPLICATION IN ALDs

To illustrate the spatial filtering capacity of ICA, Figure 6 depicts

the overall transfer function

from a given source position in

a reverberant environment for a null-steering (delay-and-subtract)

beamformer and one output channel of an ICA system, thereby

demonstrating the actual interference suppression performance in

a reverberant environment [40]. The improved spatial null

achieved by ICA confirms the hypothesis that, due to capturing all

correlated components belonging to the same source in the same

output, ICA does not only suppress the direct path but also reflec-

tions of an interfering source, e.g., [40]. Nevertheless, one has to

bear in mind that the suppression of reflections results from a

compromise in the spatial directivity, which a null-steering beam-

former cannot offer. Obviously, using the same number of micro-

phones, ICA cannot use more spatial degrees of freedom than a

supervised beamformer, and therefore the spatial selectivity of ICA

remains limited to what an optimum and ideally controlled beam-

former can achieve, as long as it uses the same statistics for deter-

mining its parameters [39].

The fact that ICA does not require prior knowledge about

source positions, microphone topology, and source activity, and

can adapt well during the activity of multiple sources, renders it

a highly attractive method for ALDs in complex acoustic envi-

ronments with unpredictable interference and noise, and usu-

ally unknown source and microphone topologies.

Unfortunately, however, ICA systems that can robustly and

quickly separate more than three sources in real-world environ-

ments have not been presented yet, so that scenarios with an

unknown number of interferers cannot be handled by such a

generic ICA system.

ESTIMATION OF INTERFERENCE AND NOISE STATISTICS

The performance of the signal extraction algorithms discussed in

the sections “MVDR Beamformer” and “Multichannel Wiener Fil-

ter” critically depends on the estimates of the statistics of the

desired and the undesired signal components, respectively. When

implementing these algorithms, it is typically assumed that there

is a domain where either the desired or the undesired components

can be observed alone. While in selected cases, stationarity

assumptions may hold reliably to justify a predetermined estimate

[41], it must usually be assumed that the statistics of both the

Frequency (kHz)

φ (°)

Magnitude

Response for Delay and Subtract BF

0 1 2 3 4 5 6 7 8

Frequency (kHz)

(a) (b)

012345678

−80

−60

−40

−20

φ (°)

−80

−60

−40

−20

−35

−30

−25

−20

−15

−10

−5

Magnitude

Response for Directional BSS

−35

−30

−25

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−5

−

[FIG6] Interference cancelation in a reverberant environment obtained by (a) null-steering beamformer and (b) ICA ,(T 300 ms

interfering point source at 0c at a distance of .1 1m of a two-microphone array with spacing d 15 cm) .=

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