International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016
BLIND SOURCE SEPARATION USING INDEPENDENT VECTOR ANALYSIS Aditi Singla, Dr. Jyoti Saxena
Electronics and Communication Engineering
ABSTRACT Independent Vector Analysis, an extension of fast independent component analysis from univariate components to multivariate components, is a powerful and interesting Blind Source Separation technique, which is applied to many applications like telecommunication, Radar signal separation, feature extraction and biomedical signal processing. Before this algorithm, many more algorithms were used for blind source separation of equal number of sources and sensors. In this paper, over determined case in frequency domain is considered that is number of sensors is greater than number of sources. The purposed algorithm is implemented in four steps: Centering, Whitening, joint Diagonalization and source separation. The efficacy of the proposed technique is computed and it outperforms in terms of MSE. Keywords- Stationary, noisy mixture, Time frequency domain, random process, uncorrelation 1. INTRODUCTION
Blind source separation (BSS) recovering source signals from their linear or non-linear mixtures, without knowing the mixing process has attracted considerable interest [4, 7, 8]. The term ‘Blind’ indicates that there is no a priori information about the original sources. So original source signals are unknown or latent, they are characterized by random variables or vectors. Independent Component Analysis (ICA) is a method to find statistically independent sources resorting to higher order statistics. It has been extended to the deconvolution of mixtures in both time and frequency domain. Although frequency domain approach is preferred because of the intense computations and slow convergence of ISSN: 2395-1303
the time-domain (TD) approach, the permutation problem must be resolved. Independent vector analysis (IVA) can effectively avoid this problem and improve the separation performance[2]. Earlier methods (e.g. Independent Component Analysis (ICA), Algorithm for Multiple Unknown Signals Extraction (AMUSE), Second Order Blind Identification(SOBI),Principle Component Analysis(PCA),Generalized Morphological Component Analysis(GMCA),Equivalent Adaptive Source Separation Via Independence(EASI) etc.) have recently gained popularity in source separation invoking the assumption of statistical independence sources.Later,research efforts mainly focused on improving the performance of BSS methods
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International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016
for over determined system, highly noisy mixtures. 2. PROPOSED METHOD The IVA is a statistical algorithm that is expressed as a set of multivariate observations. These observations are combinations of unknown variables where the underlying independent observed variables are called sources. This model is expressed as: M=g∗d+N (1) Where M = [m , m … … … … m ] is noisy mixed signal or noisy mixture, g is unknown mixing matrix, d is statistically independent signal represented as a vector and N is assumed to be stationary, spatially and temporally white, zero mean random process and independent of sources. It is essential to discuss preprocessing steps (centering and whitening) that are generally computed before using proposed IVA method in BSS problems [6]. 2.1 Centering To simplify the implementation for IVA, a preprocessing step Centering is normally applied to center the observation vector by subtracting its mean vector v=E{M}, as follow: X = M − E[M] (2) This implies that signal d is zero mean and the mixing matrix g remains the same after centering process. 2.2 Whitening It is usually considered as a necessary condition but not the sufficient condition for independence criteria. After Whitening, the task of Blind Source Separation becomes ISSN: 2395-1303
easier. It is done by transforming the correlated signals to uncorrelated signals [1, 3].
Fig1. Block diagram of the proposed IVA BSS
It is well known that convergence speed of source separation algorithm mainly depends on the step size parameters. If the step size is large, the convergence speed is high because of lesser number of iterations and vice versa. After applying the centering process, whitening is generally used for: 1. Data suitable for the ICA based separation algorithms. 2. Speed up the ICA convergence. 3. Better stability properties for the IVA separation. In this study, whitening is used to remove noise from the signal mixture by subtracting the covariance matrix of the noise ḉ from covariance matrix of noisy mixture T, that is: Ȃ = (T − ḉ) / x (3) In other words, the modified covariance matrix: Ĉ=T−ḉ (4) is used instead of the original covariance matrix C.Hence, whitening is capable of
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International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016
removing noise from the noisy mixtures. The source signal vector with zero mean is transformed into a new vector possibly of lower dimension by a linear transformation, whose elements are independent or uncorrelated with each other. The linear transformation is found by calculating the Eigen value decomposition of the covariance matrix. 2.3 Source Separation Despite utilizing information in the entire time domain, the purposed technique can selectively utilize the ‘effective’ information in local regions in time-frequency plane, where only one mode has energy [5]. So, spatial time frequency distribution (STFD) matrices (obtained from local auto and cross covariance functions) which can be derived based on the short time Fourier transform (STFT).First, define the STFT of a signal s(t) as: R(t, f) =
∞ s(w) ∞
d(t − w)e
π
dw
(5)
Where d(t) is a window function [10]. Then auto and cross terms of STFD matrices are defined as: P
,
w ′ )e
(t, f) = ∬ s (w)s ∗ (w ′ )d(t − w)d(t − Π
dwdw ′
(6)
However there might be some local frequency overlap, there may also exit some local regions in the time-frequency plane where the frequency contents are well separated. P , (t, f) at these local regions is diagonal. Assume that we can find the timefrequency points in these local regions, these are referred as ‘effective’ time-frequency ISSN: 2395-1303
points and a whitening matrix L which can transform the mixing matrix M to a unitary matrix V.So, mixing matrix can be computed as: M=L+V (7) Because P , (t, f) is diagonal and V becomes unitary matrix that diagonalizes L. For minimizing the error and speeding up the IVA convergence, an adaptive step size is implemented. The idea of the adaptive step size is to make the step size dependent on gradient norm in order to get a fast evaluation. Summary of all steps: 1. Represent source signals as vectors by using ‘New Variable’ in MTALAB. 2. Centering and Whitening of the data 3. Computing STFD of the whitened observation signals 4. Joint-Diagonalization 5. Source Separation 3. SIMULATION AND RESULT The IVA proposed method is evaluated on three statistically independent signals with noise [9]. These signals are of 256 lengths. These are listened through four sensors. To estimate the original source signals from their mixtures, the mixed signals must pass through several prewhitennig stages: Centering and Whitening. In fig. 2, 3 and 4, X-axes represent the number of samples and Y axes represent the amplitude of the signal in volts. Results by this proposed method are shown below:
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International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016 Fig.5 Three estimated sources
Fig.2 Original Three signals
Fig3. Time Frequency distributions of three signals
Fig.4 Four observations after randomly mixing and adding noise (SNR=20)
In Fig.5 and 6 the original and estimated signals are shown.
Fig.6 Three original sources
Fig.7 Comparison of estimated and original sources
In Fig.7, original and estimated signals are compared. From Fig.7, it is clear that original and estimated signals are approximately same. From Fig.5 and Fig.6, amplitude value (in volts) corresponding to different time instants (in seconds) are computed and tabulated in Table 1(for source 1), Table 2(for source 2) and Table 3(for source 3).Fig.7 shows the comparison between original and estimated signals. Table 1. Amplitude values of original and estimated source 1
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International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016
Time(sec) 2 4 8 12 16 20 25 50 62.9 75 100 125 150 175 200 225 250
Original source –S1 -1.12 1.93 1.73 0.16 1.32 0.81 1.00 0.97 2.25 2.00 1.93 1.00 0.88 -1.7 0.74 0.5 -1.16
Estimated sources-S1 0.12 0.56 0.98 0.19 0.78 0.81 -0.43 0.55 1.12 1.78 1.98 0.75 1.23 -0.65 -0.78 0.12 -0.98
Table 2. Amplitude values of original and estimated source 2
Time(sec) 2 4 8 12 16 20 25 50 62.9 75 100 125 150 175 200 225
Original source –S2 0.12 -0.23 1.87 0.23 0.25 0.69 1.12 0.54 0.56 -0.13 1.43 0.23 0.42 1.23 0.45 -1.43
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Estimated sources-S2 0.18 -0.35 -0.45 0.13 0.23 0.45 0.98 0.21 0.41 0.45 1.67 0.45 0.98 -0.43 0.57 -1.11
250
-1.89
-0.67
Table 3. Amplitude values of original and estimated source 3
Time(sec) 2 4 8 12 16 20 25 50 62.9 75 100 125 150 175 200 225 250
Original source –S3 -0.11 -0.43 -0.13 -0.23 0.34 0.11 1.46 0.34 -0.61 -0.48 1.65 -2.12 0.12 0.46 0.75 -1.43 0.45
Estimated sources-S3 -0.21 -0.13 0.11 -0.25 0.41 -0.43 -1.51 0.61 -0.42 -0.31 1.89 -2.21 -0.21 0.51 0.83 -0.45 0.61
Another similar experiment is conducted to evaluate the performance of IVA when separating noisy signals of different number of source signals. The same preprocessing steps centering and whitening were conducted. The mean square error for corresponding analysis is computed. As Mean Square Error measures the average of the square of the error, it is given by Error = absolute value of desire signal − absolute value of original signal (8) MSE =
∑
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(9)
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International Journal of Engineering and Techniques - Volume 2 Issue3, May - June 2016
The lower the Mean Square Error, the lower the estimated error on the estimated signal. The performance of the proposed IVA by changing the dimension of sources and observations is shown in Table 4. Table 4. Performance evaluation of Proposed IVA
Number of Sources 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 8 8 9
Number of observations 4 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 6 7 8 9 7 8 9 8 9 9
MSE
0.0013 0.0013 0.0013 0.0013 0.0013 0.0013 0.0223 0.0223 0.0223 0.0223 0.0223 0.0798 0.0798 0.0798 0.0798 0.0798 0.1159 0.1159 0.1159 0.1159 0.1811 0.1811 0.1811 0.2236 0.2236 0.2464
From the above results it can be discussed that as the number of sources increases mean square error increases that is separation performance degrades. ISSN: 2395-1303
4. CONCLUSION This paper studied the problem of separating mixed noisy signals by a proposed Independent Vector Analysis method. The centering preprocessing is used to make source signal zero mean whereas whitening step is adapted to eliminate the noise. This speeded up the convergence of the proposed IVA method. Mean square error shows the better performance of separation of highly noisy mixture. REFERENCES [1]A. Belouchrani and A. Cichocki, “A Robust Whitening Procedure in Blind Source Separation Context 2 Data Model,” pp. 1–7, 2000. [2]H.-M. P. Myungwoo Oh, “Blind source sepaeartion based on independent vector analysis using feed-forward network,” Neurocomputing, vol. 74, no. 17, pp. 3713– 3715, 2011. [3]S. Nakhate, R. P. Singh, and A. Somkuwar, “Robust Preprocessing : Whitening in the Context of Blind Source Separation of Instantaneous Mixture of,” vol. 1, no. 3, pp. 223–232, 2009. [4]M. Ungureanu, C. Bigan, R. Strungaru, and V. Laarescu, “Independent Component Analysis Applied in Biomedical Signal Processing Politehnica,” J. Inst. Meas. Sci. SAS, vol. 4, no. 2, pp. 1–8, 2004. [5]Y. Zhou, “Blind source separation in frequency domain,” Signal Processing, vol. 83, no. 9, pp. 2037–2046, 2003. [6] A. Al-Qaisi, “Blind Source Separation of Mixed Noisy Audio Signals Using an Improved FastICA,” J. Appl. Sci., vol. 15, no. 9, pp. 1158–1166, 2015.
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[7] Z. Yang and C. Li, “Research of City Engineering Speech Blind Separation Algorithms Based on Independent Vector Analysis,” pp. 1581–1584, 2012. [8]J. Escudero, “Tesis Doctoral : Applications of Blind Source Separation to the Magneto encephalogram Background Activity in Alzheimer’s disease.” [9] T. W. Lee, a J. Bell, and R. Orglmeister, “Blind source separation of real world signals,” Proc. Int. Conf. Neural Networks ICNN97, vol. 4, pp. 2129–2134, 1997. [10] Y. Guo and A. Kareem, “System identi fication through nonstationary data using Time – Frequency Blind Source Separation,” J. Sound Vib., vol. 371, pp. 110–131, 2016.
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