Complex FastICA for Convolved Mixtures - (Artificial) Neural Networks Project

Report on the project Complex FastICA for Convolved Mixtures Author: Antonio Cifonelli Andrea Lacagnina Alessandro Spada

Supervisor: Michele Scarpiniti

Report submitted in fulfillment of the requirements for the project of Neural Networks at Sapienza - University of Rome.

Contents 1 Introduction

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2 FastICA

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3 Complex FastICA 3.1 Complex Random Variables . . . 3.2 Indeterminacy of the Independent 3.3 Choice of the contrast function . 3.4 Fourier transform . . . . . . . . . 3.5 Permutation . . . . . . . . . . . .

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4 Implementation

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5 Experimental Results 5.1 Anechoic Room . . 5.2 Acoustic Room . . 5.2.1 t=60ms . . 5.2.2 t=150ms . .

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1

Abstract In this report we examine the problem of blind audio source separation using Independent Component Analysis, in particular the fast fixed point algorithm (FastICA). In order to separate audio sources recorded in a real recording environment, we need to model the mixing process as convolutional. The purpose of this report is to investigate the applicability of FastICA for performing fast ICA algorithm in frequency domain. The goal is to evaluate (provide) an implementation in frequency domain, involving complex valued signals in order to have advantages over time-domain implementations. Keywords: Independent Component Analysis, FastICA, Complex valued signals,

1

Introduction

Blind Source Separation (BSS) is a frequently arising problem in signal processing; a problem studied in depth during the last decades. Several methods to overcome this problem are presented in literature. The purpose of our study is to investigate the applicability of one of the proposed methods on a BSS problem. Our attention is focused on the Independent Component Analysis (ICA) algorithm for the separation of complex valued signals. The ICA algorithm is a computational method for separating a multivariate signal into additive sub-components. In particular, the observed data is expressed as a linear combination of underlying latent variables. Moreover the latent variables are assumed non-Gaussian and mutually independent. The ICA algorithms are gaining more and more attention in the last years and now several improvements of this algorithm are being discussed and implemented. One of the most efficient and popular solution for ICA is the fast fixed point algorithm (FastICA). The most attractive property of FastICA (as the name suggests) is the faster and more accurate learning obtained by estimation of the parameters related to the probability density function (PDF) of signals. In general, the PDF estimation problem represents one of the most critical issues of common ICA algorithm [3]. In order to improve the PDF matching for the learning algorithm, FastICA seeks an orthogonal rotation of prewhitened data. This rotation is found through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. More particularly this method is used in signal processing in conjunction with a set of complex valued signals, for example in case of signal processing in frequency domain. Indeed implementations in frequency domain, involving complex valued signals, have advantages over time-domain 2

implementations, especially in the separation of convolutive mixtures. Thus it is a common practice to Fourier transform the signals, which results in complex valued signals. In particular we use the Short Time Fourier Transform (STFT) in order to maintain temporal information. In this report we describe how FastICA faces a special case of blind source separation that is the ”Cocktail Party Problem”. This is the phenomenon of being able to focus one’s auditory attention on a particular stimulus while filtering out a range of other stimuli. This is one of the most challenging problem in the field of signal processing due to the noise and reverberation of the environment. Furthermore we consider a simplification of the problem, i.e. the number of independent component variables is at most the same as the number of observed linear mixtures. Moreover, before applying the algorithm on the data, we proceed with two very useful pre-processing steps that are the centering and the whitening. The goal of this work is to provide a method able to extrapolate the information of the original set of signals from a set of mixtures. The signals are emitted by two source positioned at three meters of distance from the other. The mixtures are provided from two pairs of microphones. Each microphone is arranged perpendicularly to each other in different places within a reverberation room.

2

FastICA

In the preceding section we introduced the problem of Blind Source Separation, whereas in this section we examine the FastICA algorithm; a very efficient, and robust method for the source separation. The task of this algorithm is to transform the observed data x, using a linear static transformation W as s = W x, into maximally independent components s measured by some function F (s1 , . . . , sn ) of independence. The data are represented by the random vector x = (x1 , . . . , xm )T and the components as the random vector s = (s1 , . . . , sn )T . The W matrix represents the estimation of the A matrix that is the mixing matrix. So W is an approximation of A−1 if x = As. Before proceeding with the explanation of the algorithm, we will introduce two very useful pre-processing tools: • Centering: the most basic and necessary pre-processing is to center x, which denotes the input vector data matrix. The result is the subtraction of its mean vector m = E(x) so as to make x a zero-mean 3

variable. This implies that the signal s is zero-mean. The centering is made solely to simplify the ICA algorithms; • Whitening: another useful pre-processing strategy in FastICA is to first whiten the observed variables. The idea is to apply the whitening process after the centering. This means that before the application of the FastICA algorithm, we transform the observed vector x linearly so that we obtain a new vector x0 which is white. A whitening transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables. The covariance of this new variables is the identity matrix. This means that the variables are uncorrelated and have variance 1, thus the covariance matrix of x0 is equal to the identity matrix: E(x0 x0T ) = I The whitening transformation is always possible [2]. To achieve a whitened signal we use the eigen-value decomposition (EVD) of the covariance matrix E(x0 x0T ) = EDET . The E matrix is the orthogonal matrix of eigenvectors of E(x0 x0T ) and D is the diagonal matrix of its eigenvalues. Whitening can now be done by: x0 = ED1/2 ET x Once a signal is centered and whitened, we can proceed with the FastICA algorithm. In case of Single Deflationary approach, where each column is evaluated singularly, this algorithm iteratively finds the direction for the weight vector w ∈ RM , where M represent the number of signals. This vector maximizes a measure of non-Gaussianity of the projection wT x0 where, as described above, x0 represent the whitened input vector data matrix. In the following of this report we will omit the apex on x because we will use only whitened signal. To measure non-Gaussianity, FastICA relies on a non-quadratic and non-linear function f (u), its first derivative g(u), and its second derivative g(u)0 . This set of function is called Contrast Functions or Activation Function and gives us information about the negentropy of the projection. In probability theory and statistics the negentropy is a measure of the signal skewness and a descriptor of the shape of a probability distribution. The goal of a FastICA algorithm for a single component extraction is to define a direction vector w that gives us information about the signal. The steps for extracting the weight vector w for single component in FastICA are the following: 4

• Randomize the initial weight vector w; • Compute w+ n ← E(xg(wT x)T )−E(g 0 (wt x)), where E means averaging over all column-vectors of matrix X; • Compute w ←

w+ . ||w+ ||

The single unit iterative algorithm estimates only one weight vector. The aim of this work is to estimate a set of components, e.g. a set of separate speakers from a cocktail party scenario. Estimating additional components requires repeating the algorithm. The components must be mutually independent in order to obtain linearly independent projection vectors. In FastICA we can define as mutual independent components the set of components that maximizes non-Gaussianity. In [1] and [2] are provided several ways of extracting multiple components. In the following algorithm we show the simplest. Algorithm 1: FastICA for Multiple component extraction Input : C number of desired components, X ∈ RN ×M Output: W ∈ RC×N , S ∈ RC×M 1 Function FastICA(X, C) 2 for ∀p ∈ X to C do 3 Wp ← Random vector of length N ; 4 while Wp changes do 1 1 5 wp ← Xg(wpT X)T − g 0 (wpT X)1wp ; M P M 6 wp ← wp − pj=0 −1wj wpT wj ; wp 7 wp ← ||wp ||T 8

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Output: W = [w1 , ..., wc ] , S = W T X

Complex FastICA

In this section we describe the FastICA algorithm for the separation of complex convolutive mixtures. This is a frequently arising problem in signal processing and in particular it is possible to address it in frequency domain. This is due to the common practice to Fourier transform the signals, because the frequency domain offers several computational advantages. The transformed signal is composed of complex-values whose absolute value represents the amount of that frequency present in the original function, and 5

whose complex argument is the phase offset of the basic sinusoid in that frequency. This kind of approach is called bin-wise FastICA separation. So it is very useful to provide a way to use the FastICA algorithm in the frequency domain.

3.1

Complex Random Variables

In our model all convolved signals, i.e. the mixtures, are complex-valued. A complex random variable may be represented as y = u+iv where u and v are real-valued random variables. The associated density of y is f (y) = f (u; v) ∈ R2 , whereas the expectation of y is E(y) = E(u) + iE(v). Two complex random variables y1 and y2 are uncorrelated if E(y1 , y2∗ ) = E(y1 )E(y2∗ ), where y ∗ designates the complex conjugate of y. The covariance matrix of a zeromean complex random vector y = (y1 , . . . , yn ) is: C11 . . . C1n .. . .. E(yy H ) = ... . . Cn1 . . . Cnn where Cjk = E(yj yk∗ ), and y H stands for the Hermitian of y. As said above in a non-complex case, after the centering and whitening steps, all source signals are zero-mean. Moreover the signals have unit variances and uncorrelated real and imaginary parts of equal variances. In short, these requirements are equivalent to E(ssH ) = I and E(ssT ) = O. These assumptions imply that signals must be strictly complex, that is the imaginary part of a signal may not in general vanish.

3.2

Indeterminacy of the Independent Components

The independent components s in our model are found by searching for a matrix W such that s = W x up to some indeterminacies. The reference model is x = As where A is the mixing matrix, the signal is complex-valued. In the real case a scalar factor αj ∈ R, αj 6= 0 can be exchanged between sj and a column αj of A without changing the distribution of x = αj sj = (αj aj )(αj−1 sj ). In other words the order, the signs and the scaling of the independent components cannot be determined. Anyhow, the order of sj may be chosen arbitrarily and it is a common practice to set E(s2j ) = 1; thus only the signs of the independent components are uncertainties. Similarly in the complex case there is an unknown phase vj for each sj . In this case:   sj , |vj | = 1vj ∈ C aj sj = (vj aj ) vj 6

The distribution depends on the modulus of sj only and the multiplication by a variable vj does not change the distribution of sj . Thus the distribution of x remains unchanged as well. Summarizing it is impossible to retain the phases of sj and the W A is a matrix where for each row and column there is a non zero elements vj ∈ C that is unit modulus. This matrix structure is due to the indeterminacy of the components.

3.3

Choice of the contrast function

As shown in section 2, this algorithm iteratively finds the direction for the weight vector w ∈ RM in order to maximize the non-Gaussianity of the mixture signal. Our approach relies on a non-quadric and non-linearity set of functions. Hyv¨arinen in [2] proposes two way to define these functions. 1. The first is defined ”top-down”. In this approach the non-gaussianity or more in general the independence is measured often by using comlulants. However in general it is not robust; 2. The second approach is called ”bottom-up”, where the higher order statistics are implicitly embedded into the algorithm by arbitrary nonlinearities. The presented FastICA algorithm starts with an arbitrary non-linear contrast function using random vector w of values. Each element of w is a columnvector with number of rows equal to the number of components. These set of elements represent the columns of the matrix W . The contrast functions used during the test phase are f (u), its first derivative g(u), and its second derivative g(u)0 : f (u) = log(a + u) g(u) = 1/(a + |u|)2 g(u)0 = (−1/(a + |u|)2 )2 where a1 and a2 are some arbitrary constants for which values a1 ≈ 0.1 and a2 ≈ 0.1. In general each non-linear learning function G divides the space of probability distributions into two half-spaces. Independent components can be estimated by either maximizing or minimizing a function similar to : 2

JG(w) = E{G(|wH x| )} 7

In this case the goal is to maximize the negentropy. The learning function used during this work is: 2

2

2

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w+ = E{x(wH x) ∗ g(|wH x| )} − E{g(|wH x| ) + |wH x| g 0 (|wH x| }w

3.4

Fourier transform

Before the ”contrast” step, we prefer to Fourier transform the mixtures in order to obtain computational advantages. Moreover, in order to maintain information about the time domain, we use the Short-time Fourier transform (STFT): this type of transformation is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. Therefore the Short-time Fourier transform returns a set of equal length segments that represents the entire mixture broken down into frequency sub-band. So the STFT is the way to overcome the first time ambiguity [1].

3.5

Permutation

By means of the contrast function, the FastICA algorithm models the W matrix in order to separate the original signals in the mixtures. This matrix is then used to detect the components of the mixtures, however the resultant set of separated components of the signal is not labeled. Indeed after the separation step we lose information about the original signal affinity, thus we need a process to reconstruct the original set of signals: this step is called permutation process. In general this notion relates to the act of arranging all the members of a set into some sequence or order, in this case rearrange the segments of separated signals. To achieve this result the algorithm uses two approaches: the first based on the direction of arrival and the second one based on correlation. • First we discuss the direction of arrival approach (DOA) where the directions of source signals are estimated and permutations are aligned based on them. This approach finds the direction of arrival among two consecutive column w of W (for each bin of frequency) in order to detect the correct sequence of bin of frequency of the reconstructed signals. More in general, in signal processing direction of arrival denotes the direction from which usually a propagating wave arrives at a 8

point, where a sensor array is located. Thus we can use this method to detect the direction of the propagating wave and provide a well made separation of the signals. In other words the goal is to estimate the directions θ(f ) = [ϑ1 (f ), .., ϑN (f )]T for each bin of frequency extracted by every row of W (f ). Then we may obtain a permutation matrix P (f ) by sorting θ(f ). We use the closed-form formula for DOAs estimation proposed in [5]. For each bin of frequency it is computed a set of direction of arrival θ(f ) where each ϑk (f ) is computed using the general formula shown below: ϑk (f ) = arccos(

arg([W −1 ]j /[W −1 ]j 0 ) ) 2πf c( − 1)(dj − d0j )

where k represents the k-source signal, j and j 0 represent the combination of the elements of the columns of W (f ), f represents the value of frequency and c the air temperature. Once the DOAs estimation is done we can fix the permutations at some frequencies where the confidence of the DOA approach is sufficiently high. Our criteria for the decision are: – the number of estimated directions is the same as the number of sources; – the directions θ(f ) do not differ greatly from the averaged directions θbs , i.e.|θs (f ) − θbs | is smaller than a threshold thθ Unfortunately there are three problems with this method: – the directions of arrival cannot be well estimated at some frequencies, especially at low frequencies where the phase difference caused by the sensor spacing is very small, and also at high frequencies where spatial aliasing might occur; – the calculation of null directions by plotting directivity patterns is time consuming; – estimating DOAs from null directions is difficult when there are more than two sources. We can improve the permutation of the components of the separated signals introducing a second method described above. 9

• This approach is based on inter-frequency correlations of signals, called the correlation approach and is used where the previous one is not quite accurate enough. We define the correlation of two signals x(t) and y(t) as: cor(x, y) = (µx∗y − µx ∗ µy )/(σx ∗ σy ) where µx is the mean and σx is the standard deviation of x. Based on this definition, cor(x, x) = 1, and cor(x, y) = 0 if x and y are uncorrelated. Therefore there is high correlations among neighbors bin frequency (and in general between two signal) if separated signals corresponds to the same sources signal. So a simple criterion for deciding if apply the permutation process (Πf ) at frequancy f in this context is to maximixe the sum of the correlation between neighboring bin frequencies within distance δ : Πf = argmaxΠ

N XX

f cor(vΠ(i) )

|g−f | i=1

In other words we can define the correlations between bin neighbors parts of separated signals and decide to swap the components in order to maximixe the correlation and output a well reconstruction set of separated signals. In conclusion, as described in [5], we can review the characteristics of the two used approaches: The DOA approach is robust since a misalignment at a frequency does not affect other frequencies. On the contrary it is not precise enough since the evaluation is based on an approximation of a mixing system. The correlation approach is precise as long as signals are well separated by ICA since the measurement is based on separated signals. On the contrary it is not precise since the evaluation is based on an approximation of a mixing system.

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4

Implementation

Figure 1: A Fast Fixed Point scheme for Frequency-Domain Blind source separation The purpose of this section is to describe shortly our complex FastIca proposed approach for convolutive mixtures. In order to achieve such result we can divide the problem in a set of main stages, every one of them with its own input and output and organized in a pipeline as depicted in Figure 1. At the beginning the algorithm receives a set of mixed signals perceived by a set of input devices. As said before, we focus on a particular case study, where the mixtures numbers is equal to the number of source presents into a room. Each mixture must include information of the signals propagated from the sources of sounds and the reverberations of the environment. More in detail, the blind source separation is a technique for estimating original source signals from their mixtures at sensors. If signals are mixed instantaneously, we can directly employ an instantaneous ICA algorithm to separate the mixed signals. However in a real room environment, signals are mixed in a convolutive manner with reverberations. Consequently the observed signal from each microphone is the result of a convolution between the original sources and the impulsive response of the room. The observed signals are captured by a set of microphones and are the inputs of the algorithm. In particular we use complex microphones as input devices. Each input device is defined as the composition of two cardioid microphone positioned at 90◦ to each other (figure 2 and 3). Each pair of microphones provide a single output i.e. a mixture that is composed as: s = s1 + is2 11

Figure 2: A complex microphone composed by two cardoid microphones positioned at 90◦ to each other.

Figure 3: This image gives us an idea on how sensitive it is to sounds arriving at different angles about its central axis.

where s is the output of complex microphone, s1 and s2 represent the output provided by the two cardioid microphone. It is clear from the formula that the output of a complex microphone is the composition of the real part of s1 and the imaginary part of s2 . Once the mixed signals are obtained we can proceed with the separation. Two approaches have been proposed in literature. The first approach is time-domain, where ICA is applied directly to the convolutive mixture model. This approach achieves good separation, but it is not as simple as ICA for instantaneous mixtures, and computationally expensive. The other approach is frequency-domain BSS, where complex-valued ICA for instantaneous mixtures is applied in each frequency bin. The merit of this approach is that the ICA algorithm becomes simple and can be performed separately at each frequency. We use this approach, so we need to transform the perceived signals using the STFT as shown in section 3.4. We can proceed with the fast fixed-point phase, which consists of a set of iterative steps. The algorithm iteratively finds for each bin of frequency the direction of weight vector w that maximizes the negentropy for each source component, n components for n mixed signal. In order to obtain n w-vectors 12

we need to estimate elements that are mutually ”independent”. So we apply the deflationary approach iteratively in order to obtain a set of linearly independent projection vectors. As described in section 2, before the iterative process it is important to model the mixture signals with a centering and pre-whitening step. The output of the FastIca step is the unmixing matrix W that represents an approximation of the inverse of the mixing matrix A. At this point any complex-valued ICA algorithm can be employed with a frequency-domain approach. However the permutation ambiguity of the ICA solution becomes a serious problem, i.e. we need to align the permutation in each frequency bin so that a separated signal in the time domain contains frequency components from the same source signal. In order to obtain an aligned result we use the method presented in [5] for solving the permutation problem robustly and precisely by integrating the Doa approach, and interfrequency correlations approach, presented in section 3.5. After a fast scaling step, the algorithm proceeds with the inverse of the STFT to recover the time information of the detected signals and outputs the separated signals.

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5

Experimental Results

The purpose of this section is to present results from a set of experiments that were designed to assess the validity of the proposed methods. In particular we show six experiments based on three different rooms with different reverberation. Moreover in this section we introduce some tools in order to evaluate the outputs. The accuracy of the proposed results can be estimated in many ways and a first estimation of the goodness may be made by naked eye from the graphs. However, to provide detailed information about the quality of the separation, we provide two numerical measures. • The first numerical measure proposed is the SIR, acronym of signal to interference ratio (SIR). SIR is computed in order to check the accuracy of the separation of the latent sources by an algorithm performing ICA. It is defined as N max( Qi )2 1 X 10 log10 T SIR = N i=1 Qi Qi − max( Qi )2

where Q is the overall transforming matrix of the latent source components, Qi is the i-th column of Q and N is the number of the source signals. More in detail Q = Wi ∗ Hi with Wi and Hi components of the estimation of the mixing matrix W and the mixing matrix A for the i-th bin of frequency. • The second numerical measure proposed is the performance index PI that we use in order to measure the performance of the algorithms. Its defining equation is

N N N X X X |gki | |gik | 1 − 1) + ( − 1) PI = ( N (N − 1) i=1 maxj |gij | maxj |gji | i=1 i=1

!

where gij is the (i, j)-element of the matrix G = W H. The term maxj gij represents the maximum value among the elements in the i-th row vector of G. Similarly the term maxj gji represents the maximum value among the elements in the i-th column vector of G. When the perfect separation is achieved, the performance index is zero.

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5.1

Anechoic Room

The first experiment is done using an anechoic room, i.e. a non-reflective, non-echoing room designed to completely absorb reflections of either sound or electromagnetic waves. In particular we tested it in two different situations: with omnidirectional microphones and with cardioid ones, i.e. with a ”heartshaped” sensitivity pattern. The original sounds are shown in fig.4.

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Figure 4: Original Sounds Results from the former are shown in fig.5, from the latter in fig.6.

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Figure 5: Anechoic Room with omnidirectional microphones

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Figure 6: Anechoic Room with cardioid microphones In the first case the SIR got value equal to 8.9062 dB, in the second to 8.7127 dB. The same test has been done with a complex microphones, in order to show the effectiveness of the approach. Indeed we noticed a clear improvement of the separation results. See fig.7. In this case SIR = 9.33104 dB. 16

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Figure 7: Anechoic Room with complex cardioid microphones

5.2

Acoustic Room

In order to get in a more real situation we decided also to test our algorithm in a acoustic room, i.e. a situation where the walls do not have the same absorption factor. Besided we tested this situation with two different reverberation time and also with and without the presence of a complex microphones. 5.2.1

t=60ms

The results with non-complex microphones are in fig.8, instead the complex one in fig.9. The non-complex case returned SIR = 7.4395 dB, and the opposite one equal to 7.34717 dB.

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Figure 8: Acoustic Room with non-complex cardioid microphones, t=60ms

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Figure 9: Acoustic Room with complex cardioid microphones, t=60ms 5.2.2

t=150ms

The results with non-complex microphones are in fig.10, instead the complex one in fig.11. In thi case we observed SIR = 7.65121 dB for the non-complex and SIR = 7.69116 dB for its counterpart. 18

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Figure 10: Acoustic Room with non-complex cardioid microphones, t=150ms

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Figure 11: Acoustic Room with complex cardioid microphones, t=150ms All the experiments were conducted on a Alienware 15, with 8Gb of RAM R CoreTM i7-6700HQ on Windows 10 64-bit. The algoand a processor Intel rithm was developed in MATLAB and tested with MATLAB R2015b. The room’s impulsive response for the mixed signals are obtained with RoomSim 3.4 [6]. 19

6

Conclusions

In this work we presented a project in the field of Blind Source Separation, applying Fast Iterative Component Analysis. In particular we tested the algorithm in case of two sources and two microphones, but adding more elements may be a source of future work. Besides we introduced also the complex domain and we dealt with some difficulties introduced with it, i.e. the Permutation problem. We tested different scenarios, such as anechoic rooms and acoustic ones, varying from time to time the reverberation time. What concerns about future work, one may also consider to test other room configurations with different reverberation. Besides one may also investigate further the benefits of the introduction of the complex domain, eventually consider other complex microphone configurations.

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References [1] Hyv¨arinen, Aapo, Juha Karhunen, and Erkki Oja. ”Independent component analysis”. Vol. 46. John Wiley & Sons. (2004) [2] Bingham, Ella, and Aapo Hyv¨arinen. ”A fast fixed-point algorithm for independent component analysis of complex valued signals.” International journal of neural systems 10.01 page(s): 1-8. (2000) [3] Shi, Xizhi. ”Blind signal processing.” Springer. (2012) [4] Geravanchizadeh, Masoud, and Masoumeh Hesam. ”Convolutive ICA for Audio Signals.” Edited by Ganesh R. Naik (2012): 137. [5] Sawada, Hiroshi, et al. ”A robust and precise method for solving the permutation problem of frequency-domain blind source separation.” Speech and Audio Processing, IEEE Transactions on 12.5 (2004): 530-538. [6] http://media.uws.ac.uk/ campbell/Roomsim/ [Accessed 05 May 2016].

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