Performance Analysis of ICA in Sensor Array

As the best-known scheme in the field of Blind Source Separation (BSS), Independent Component Analysis (ICA) has been intensively used in various domains, including biomedical and acoustics applications, cooperative or non-cooperative communication, etc. While sensor arrays are involved in most of the applications, the influence on the performance of ICA of practical factors therein has not been sufficiently investigated yet. In this manuscript, the issue is researched by taking the typical antenna array as an illustrative example. Factors taken into consideration include the environment noise level, the properties of the array and that of the radiators. We analyze the analytic relationship between the noise variance, the source variance, the condition number of the mixing matrix and the optimal signal to interference-plus-noise ratio, as well as the relationship between the singularity of the mixing matrix and practical factors concerned. The situations where the mixing process turns (nearly) singular have been paid special attention to, since such circumstances are critical in applications. Results and conclusions obtained should be instructive when applying ICA algorithms on mixtures from sensor arrays. Moreover, an effective countermeasure against the cases of singular mixtures has been proposed, on the basis of previous analysis. Experiments validating the theoretical conclusions as well as the effectiveness of the proposed scheme have been included.


Introduction
The fundamental goal of BSS is to recover the original signals from their mixtures when the mixing process is unknown. Since first considered in the biological problem of motion decoding in vertebrates by Jutten, Ans and Roll [1], the BSS problem has been expanded in many multi-sensor systems: Antenna arrays in electromagnetism or acoustics, chemical sensor arrays, electrode arrays in electroencephalography, etc. This very wide set of possible applications is probably one reason for its success [1,2]. Independent Component Analysis (ICA) is one of the most widely used BSS techniques for revealing hidden factors that underlie sets of random variables, measurements, or signals. The ICA of a random vector consists of searching for a linear transformation that minimizes the statistical dependence between its components [3,4]. The power of ICA resides in the physical assumptions that different physical processes generate unrelated signals. The simple and generic nature of this assumption allows ICA to be successfully applied in a diverse range of research fields [5]. For the last three decades, ICA has received attention from varied domains, including the biomedical applications [6][7][8][9][10][11][12][13] such as the single-channel electromyogram (EMG) classification with ensemble-empirical-mode-decomposition-based ICA for diagnosing neuromuscular disorders in [6], the identification of simple and complex finger flexion movements using surface electromyography (sEMG) and muscle activation strategy with Subband Decomposition ICA (SDICA) receiving array so as to expect better separation performance and more robustness against intricate situations? Why the algorithm works well for some mixtures while being unsatisfactory for the others? Under what kind of scenarios would the ICA algorithms fail? and further, what measures can be taken when faced with such failure? Answers to such questions are interesting and of great importance. However, little attention has been paid to the issue in the current literature, especially analytical form works.
In this manuscript, the typical scenario of antenna array receiving is taken as an illustration for the varied applications involving sensor arrays. The problem of concern is the performance of ICA algorithms under varied situations depicted by the mixing matrix, the sources and the environment noise. The influence of several fundamental factors in practice has been analyzed with quantitative results; factors concerned include the environment noise level, the element spacing of the array, the frequencies of the sources as well as their locations. The analytic connection between the noise variance, the source variance, the condition number of the mixing matrix and the optimal signal to interference-plus-noise ratio is given. Also, the question how the mixing matrix is affected by the element spacing of the array, the frequencies and locations of the radiators is considered, with special concern on the prevention of (nearly) singular mixing matrices. What is more, an effective countermeasure against the situations of singular mixtures has been proposed, on the basis of previous analysis. The manuscript is arranged as follows. Models of the noisy ICA and the sensor array output as well as the definition of the condition number are introduced in Section 2. Analytical works are mainly implemented in Section 3. In Section 4, simulation works are carried out to demonstrate the validity of previous results and conclusions. Summaries and conclusions are stated in Section 5.

Noisy ICA Model
Denote x a data matrix. We think of it as a PˆL object collecting L samples of a Pˆ1 vector, the model of independent component analysis postulates a number N of sources so that the observation matrix x can be explained as: x " As`V (1) where A is the unknown PˆN mixing matrix, s is a NˆL matrix in which each row is called a "source signal" or a "component". Matrix V represents an additive noise item or some other form of measurement uncertainty [1]. ICA is distinguished from other approaches to source separation in that it requires relatively few assumptions on the sources and on the mixing process. Basic assumptions of ICA include [5]: (1) The sources being considered are statistically independent.
The first assumption is fundamental to ICA. Statistical independence is the key feature that enables estimation of the independent components from the observations.
(2) The independent components have non-Gaussian distribution.
The second assumption is necessary because of the close link between Gaussianity and independence. It is impossible to separate Gaussian sources using the ICA framework because the sum of Gaussian sources is indistinguishable from a single Gaussian source, and for this reason Gaussian sources are forbidden. This is not an overly restrictive assumption as in practice most sources of interest are non-Gaussian.
The third assumption is straightforward. If the mixing matrix is not invertible then clearly the demixing matrix we seek to estimate does not even exist.
Generally speaking, the ICA scheme aims at estimating a demixing matrix B based on the output independence that satisfies [3][4][5]: where P is a permutation matrix, D is a diagonal matrix, and the superscript '(˝) -1 ' indicates the matrix inversion. The estimation of sources (with intrinsic scale-permutation ambiguity) is obtained with:

Sensor Array Output
In this manuscript, the typical scenario of a receiving antenna array such as in spaceborne non-cooperative communication is taken as an illustration for the varied applications involving sensor arrays, as depicted by Figure 1. The array consists of P narrowband (NB) omnidirectional antennas and we call x(t) the vector of complex amplitude of signals at the output. Each antenna is assumed to receive the contribution of N zero-mean stationary NB sources that are statistically independent. Under these assumptions, we model x(t) as: where "j" is the imaginary unit, T is the observation length, α i (t), υ i and τ i (t) are respectively the attenuation, the frequency offset and the propagation delay of the channel corresponding to source i at time t; f i is the carrier frequency of the NB source i; s i (t) is the complex amplitude of the corresponding source; a i (t) is the equivalent steering vector (or spatial signature) of source i at time t, which is a Pˆ1 vector depending on the array, the direction of arrival (DOA) and is also relevant with carrier frequencies of sources. v(t) is the noise vector, assumed to be zero-mean, stationary, circular, Gaussian and spatially white, with variance σ 2 v in the sampling band of frequency. In addition, the array is set as a uniform linear array (ULA), the element spacing of which is d, and sensors are disposed along the (Ox) axis. Notice that the sources are presumed to be statistically independent, which is generally the case for physically separated sources in reality.
Regarding the first unit of the array as the datum unit and denoting N complex equivalent sources arriving at the datum unit as r s i ptq, we have: Then Equation (4) can be rearranged as: Regarding the steering vector to be constant within the observation window, then a i (t) can be explicitly expressed as: where θ i and φ i are the azimuth and elevation incident angles of the source i, θ i P[´90˝, 90˝]; ϕ i P r´90 o , 90 o s; λ i is the wavelength corresponding to the carrier frequency of the NB source i, λ i = c/f i (c denotes the speed of light); (x,y,z) are the coordinates of sensors in the array, as depicted in Figure 1; the superscript "T" denotes transpose. Without the loss of generality, we consider the fundamental situation where P = N = 2. The coordinate of the second sensor is (d,0,0) and DOAs of sources are (θ 1 ,φ 1 ) and (θ 2 ,φ 2 ) respectively, as depicted by Figure 2. Moreover, the frequency offsets of sources are supposed to be negligible, and the propagation channels are assumed to be time invariant over the observation duration. Hence, the mixing matrix A whose columns are vectors a i (θ i ,φ i, λ i ) will be:

Condition Number of Matrix
Consider a system of linear equations: x " As (9 The system is featured by the coefficient matrix A, x is the vector of data, and s is the solution vector to be estimated.
The condition number measures the singularity of a matrix or equivalently, the sensitivity of the solution of corresponding linear equation system to errors in the data. Hence, it is a valid indicator of the accuracy and numerical stability of results from matrix inversion and linear equation solution. When a minor perturbation in the observation only leads to a small error in the solution vector, the matrix A is considered to be well-conditioned, and inversely, if minor perturbation of x results in large error in the estimation of s, then the matrix is said to be ill-conditioned.
The connotation of the condition number can be understood in a more straightforward way, that is, it measures the amplification between errors in x (denoted as Δx) and that in the estimation of s (denoted as Δs). In fact, it has been proved in [34] that: In this manuscript, we adopt the Frobenius norm for the calculation of the condition number, which is defined as [34]: (12) in which a ik is the element of A in the i-th row and k-th, 1 |¨| 1 indicates the module value. Combining Equations (8), (10) and (12), the condition number in the 2ˆ2 array receiving case is obtained as:

Environment Noise and the Condition Number
Since restitution of sources via typical ICA algorithms equals solving the linear equation set in nature, it has been well-known that the performance of ICA estimators in the noisy case depends on the condition number of the mixing matrix and the environment noise level. More specifically, it should be intuitively understood that the ICA algorithms would be more sensitive to the noise item when the singularity of the mixing matrix is stronger. Here, we attempt to push the work further via figuring out the upper bound of the ICA separation performance, under certain noise level and the condition number.
The concept of the optimal separator that BSS methods aim at implementing asymptotically is introduced in [35], which indicates the upper limit of the separation performance for all ICA algorithms. In [35], the separation performance is measured by the criterion signal to interference-plus-noise ratio (SINR), and the SINR of source k at output i of separator B is defined therein as: ( 1 E t¨u 1 stands for mathematical expectation), r s k ptq is the k-th equivalent source in Equation (5), a k is the steering vector of source k, R k is the total noise correlation matrix temporal mean for the source k, defined by R zk fi R x´σ is the auto-correlation of the observation, and the superscript "H" indicates conjugate transpose. The restitution quality of the source k by the separator B can be evaluated by the quantity SINRM k , which is the maximum value of SINR k (b i ) among all 1 ď i ď N. It has been verified in [35] that under this criterion, B and BΛΠ have the same performance, in which Λ is a diagonal matrix and Π is a permutation matrix. This indicates that the criterion itself is not susceptible to the intrinsic scale-permutation ambiguity of the ICA.
For the case of two sources, the SINRM 1 at the output of the optimal separator is given by: where ε k fi σ 2 r s k a H k R´1 v a k denotes the signal-to-noise ratio (SNR) at the input of the separator for source k, Rv is the auto-correlation of the noise item. α 12 is the spatial correlation coefficient between sources 1 and 2 in the matrix of R´1 v , defined by: One thing to be noticed is that when the noise on the sensors are additive white Gaussian (as presumed in this paper), we have R v " σ 2 v I (where I is the unit matrix) and thus ε k "´σ 2 (8), the expression can be further simplified as: ( 17) and Equation (16) may be transformed as: Combining Equations (8) and (18), we have: With Equations (13), (15) and (19), there is: and when source 2 is strong enough (ε 2 " 1), Equation (20) can be simplified with approximation as: Similar expression can be obtained for SINRM 2 .
Equations (20) and (21) indicate that the superior limit of the separation performance via ICA algorithms is mainly determined by the SNR at the input of the separator and the condition number of the mixing matrix. Generally speaking, the performance of the algorithms would be more susceptible to the noise with larger condition number of the mixing matrix.
According to Equation (13), cond(A) ě 2, and the condition number reaches its minimum, which indicates the weakest singularity of the mixing process, it is straightforward that cosr2πd¨p cosθ 2¨c osϕ 2 λ 2´c osθ 1¨c osϕ 1 λ 1 qs "´1 and |α 12 | " 0. Then according to Equation (15), the SINRM 1 reaches its maximum and equals ε 1 regardless of ε 2 , which indicates that the SINR at the output equals the SNR at the input of the separator. Since this is theoretically the best result possible, cond(A) = 2 should be the ideal case that is desired in practice.
On the contrary, the case that should be avoided is the singular mixtures, meaning the situation where cond(A) = 8. This is because then the multi-channel ICA problem would degenerate into a single-channel one, which the traditional ICA algorithms cannot handle, and thus, the separation may come to a critical failure in applications.

Condition Number vs. Practical Factors
It can be seen from Equation (17) that under the white Gaussian noise assumption, the SNR at the input of the separator is only determined by the variances of the sources and the noise. That is to say, factors including the element spacing of the array, the frequencies and locations of the NB sources would not affect the signal-to-noise ratio, thus these factors may impact the separation performance via changing the condition number only. In this part, we attempt to analyze their relationships with the condition number in the analytical form and aim at reaching instructive conclusions. Moreover, special attention is paid to the case of singular mixtures.
Without the loss of generality, suppose that in Equation (13) d = βλ 1 , λ 1 ď λ 2 and λ 2 = Dλ 1 . The parameter β represents the element spacing in a relative sense, D describes the divergence of wavelengths of the sources and the influence of the locations of sources will be depicted by the DOAs. With notations above, Equation (13) can be rearranged as:

Element Spacing of the Array
From Equation (22), it is obvious that for fixed D and DOAs, the condition number varies with β in a periodical way, with the period being: The periodicity indicates that, increasing the element spacing of the array may have different impacts under varied situations depicted by D and DOAs.
Taking the partial derivative of Equation (23) against D, we have: where signp¨q is the sign function. Hence, if cosθ 2 cosφ 2 > cosθ 1 cosφ 1 , then the period increases along with the value of D within the interval D P r1, cosθ 2 cosϕ 2 cosθ 1 cosϕ 1 q, cosθ 1 cosϕ 1 ‰ 0; and decreases along with it once D is greater than If cosθ 2 cosφ 2 < cosθ 1 cosφ 1 , the period decreases along with the value of D.
The influence of the element spacing on the condition number is depicted by Figure 3.  The periodicity has been clearly demonstrated, with the period determined by varied D and DOAs. In Figure 3A, , the period is shortened with increasing D. In Figure 3B, θ 1 =´29.3˝, " 0.8520. We can see that the value of β keeps decreasing with greater element spacing.
The minimal and maximal values of the condition number also arise periodically, and they appear respectively at β min and β max that equal: According to previous analysis, the maximal values of the condition number which correspond to the singular mixtures should be avoided in applications, in order to prevent the critical failure of separation. In practice, the expected circumstance is that: with well-designed sensor arrays, the separation via ICA algorithms is satisfactory and can handle with mixtures consisted of sources with diverse carriers (within the band that the array is designed for) and from all possible locations. This is meaningful considering the intricate and unforeseen situations in applications. And here, we attempt to achieve this via designing properly the element spacing of the array.
One straightforward idea is that if the actual β is smaller than all possible β max , then the singular case would not occur. While D and DOAs are fixed, this indicates that: Extending the relationship for all D ě 1 and θ i P[´90˝, 90˝]; ϕ i P r´90 o , 90 o s yields: Since |cosθ 2 cosφ 2 -Dcosθ 1 cosφ 1 | P (0,D). This indicates that so long as the element spacing of the array is smaller than wavelengths of the sources, the cases of singular mixtures can be well prevented for sources with diverse carriers and from all possible directions. The only exception happens when cosθ 2 cosφ 2´D cosθ 1 cosφ 1 = 0, then the mixture would be singular regardless of β. The conclusion can act as a meaningful guidance while designing the sensor array for applications adopting blind source separation. More specifically, if the array is designed for the frequency band (f L~f H ), this means that the element spacing should satisfy d < c/f H . For instance, consider the satellite communication on the C band (4~8 GHz), then the element spacing should be designed below 37.5 mm for the sake of avoiding failures of separation. Moreover, if in the specific application, the DOAs of the sources can be further confined, say θ 1 φ 1 P [-Ang˝, Ang˝] (Ang ď 90), then Equation (27) can be relaxed to: Notice that in many of previous works concerning independent component analysis via multi-sensor arrays, the element spacing is usually presumed as half of the wavelength of the sources, in both theoretical analysis and simulations (for instance, in [16,35]). However, the presumption may be invalid for actual scenarios. Because first, the accurate prior information of the wavelength cannot be obtained in most applications requiring ICA; second, since there may be radiators working on different carriers, the equality is impossible to be maintained. Hence, conclusions here provide a more practical instruction for the selection of the element spacing, and the traditional settlement can be regarded as one of its special case with β = 0.5.

Carrier Frequencies of the NB Sources
From Equation (22), it can be deduced that the condition number varies with the reciprocal of D in a periodical way. The period is determined by β and DOAs as: Take the partial derivative of Equation (29) against β, we have: Thus, the period decreases along with the increase of β.
From Equation (31), it could also be found that the minimum of (1/D) max (denoted as min(1/D) max ) is:

where '[x]' indicates the largest integer no greater than x.
This indicates that for certain β and (θ i, φ i ) (i = 1,2), the cases of singular mixtures could be avoided by sufficiently large values of D, the reciprocal of which is smaller than min(1/D) max . Further, for the adaptation of varied (θ 2, φ 2 ), Equation (33) could be transferred to: Equations (33) and (34) could be meaningful in the situation where one of the sources is fixed in both of the frequency and its location, while the other source is a moving one. Then it can be told from the two formulas how separated the other source should be from the fixed one in terms of carrier frequencies, to ensure the success of their separation while it is moving within certain region. Such scenario may exist in the acoustic applications with one of the speaker walking around the microphone.

Locations of the Sources
First, it has been revealed by Equation (22) one of the edges of separation via ICA algorithms over spatial filtering (beamforming), that is: When sources arrive from identical (or very close) directions, the separation based on spatial filtering will become invalid; while it is still possible for ICA algorithms to work, so long as there is difference in carriers of the NB sources (D ‰ 1).
Second, it can be seen that sources from varied locations may correspond to identical condition numbers, so long as their DOAs result in the same value of cosθ i cosφ i (i = 1,2). More specifically, for fixed β, D and (θ 1 ,φ 1 ), the mixing matrix would be singular so long as: Since cosθ 2 cosφ 2 P [0, 1], the value of possible m should be within [´βcosθ 1 cosφ 1 , (β/D)βcosθ 1 cosφ 1 ]. Suppose that the set of valid m is {m k , k = 1,2, . . . }, then for each of its element there are correspondingly a group of (θ 2k ,φ 2k ) satisfying cosθ 2k cosφ 2k = (Dcosθ 1 cosφ 1 + m k¨D /β), and this is further depicted by Figure 5. In Figure 6, condition numbers under varied DOAs have been shown while representative values of β and D are set for the sake of illustrating conclusions obtained in this section. In Figure 6a, the condition number becomes extremely large for a group of (θ 2 ,φ 2 ) distributed in a circle centered on the origin.  Figure 6g,h, there exist more circles consisting of singular points, since the set of valid m are {´3,´2,´1} and {´2,´1,0}, respectively. In Figure 6c,d, β is set to be smaller than 1, and Dcosθ 1 cosφ 1 > 1, thus it can be seen that singular cases have been effectively avoided. In e and f, the value of D is increased remarkably, and hence, original singular cases are successfully removed. For example, in Figure 6e βcosθ 1 cosφ 1 R Z and D > (β/βcosθ 1 cosφ 1 ,´t βcosθ 1 cosφ 1 u) = 4, thus according to the previous analysis, there would not be singular points. To conclude, it has been demonstrated in Figure 6 the validity of conclusions obtained in this section.

Countermeasure for Singular Mixtures
For most applications, the occurrence of (nearly) singular mixtures would be disastrous, and the intricate actual scenarios have made it somehow inevitable. And here, we attempt to propose certain countermeasure against such situations to avoid extremely poor separation performance.
According to Equation (20), reducing the power of noise could be helpful, possibly via means of band pass filtering. Hence, more effective actions may be taken in terms of changing the condition number.
One straightforward idea is that: when the mixing process is (nearly) singular, certain extra phase delay Ψ could be introduced into the observations. This could be effective since when the mixtures are (nearly) singular, 2π(β/D)¨(cosθ 2 cosφ 2 -Dcosθ 1 cosφ 1 ) equals or is quite close to 2mπ, mPZ, and the value of the condition number after the introduction of the extra phase delay would be: Thus, by setting the proper value of Ψ the revised condition number would be more beneficial for the separation, and the ideal value of Ψ would be (2m + 1)π, mPZ. Notice that this could not be realized via directly shifting the phase of the mixtures, since it can be deduced from Equations (8) and (10) that this actually would not change the condition number. In practice, Ψ could be introduced through shifting the mixtures along the axis of the observation time.
More specifically, consider the following scheme depicted by Figure 7. The idea here is that the introduction of extra phase delay in Equation (22) is achieved with one of the observations undergoing an artificial time-lag. For instance, if the observation from the second sensor is shifted backward by k points, then the equivalent phase delay introduced is: in which T s is the sample interval. It should be stated that the proposed scheme is only designed to combat the situations of (nearly) singular mixtures. In fact, only when originally 2π(β/D)¨(cosθ 2 cosφ 2 -Dcosθ 1 cosφ 1 ) is close to 2mπ, mPZ can it be ensured that the extra phase delay is constructive rather than destructive. One of the actual problems is judging the singularity of the current observations and deciding whether the proposed procedure is needed. Since the condition number of the mixing matrix and the SINR at the output is generally hard to obtain, the correlation coefficient ρ can act as one practical criterion [34]. For random variables x(ξ) and y(ξ), their correlation coefficient is defined as: in which c xy is the cross covariance of x(ξ) and y(ξ), σ 2 x and σ 2 y are the variances of them. The idea is that when the original mixing matrix is (nearly) singular, it can be seen from Equation (8) that the two observations would be (approximately) coherent, and the module value of the correlation coefficient between the two observations would approach 1. More specifically, since the sources are presumed to be zero-mean and independent, it can be calculated from Equations (8) and (38) that the module value of the correlation coefficient between the two observations (denoted as ρ x1x2 ) equals:ˇρ in which σ 2 s 1 and σ 2 s 2 are the variances of the sources. Hence, it can be seen that while cond(A) = 8,ˇρ ; if σ 2 s 1 " σ 2 s 2 , then for cond(A) = 2,ˇˇρ x 1 x 2ˇ" 0, which means then the two observations are orthogonal.
In practice, a threshold associated with the noise level and the number of samples can be set based on experience, and when the module value of the correlation coefficient between the observations exceeds the threshold, the mixtures can be regarded as (nearly) singular. And then the proposed scheme should be adopted. During our simulations, it is found that generally speaking k "1 or 2 should be suitable under most scenarios.

Simulations
In this section, experiments are carried out using Intel(R) Core (TM) 2 Duo CPU E6550 @ 2.33 GHz and MATLAB 7.12.0(R2011a). Experiment 1: Separation performance of ICA algorithms under varied noise levels and the singularity of the mixing matrix.
In this experiment, quadrature phase shift keying (QPSK) and binary phase shift keying (BPSK) signals are adopted as sources with carrier frequencies 0.38, 0.127 MHZ, and symbol rates 2, 1 Kbps, respectively. The sample rate is set as 1 MHz, and the observation length is 1024. The receiving array consists of two sensors with the element spacing being two times the wavelength of source 1. The DOA of source 1 is (25.1˝, 40.5˝), while the elevation angle of source 2 is 27.9˝.
The SNR at the input of the separator and the SINRM adopted here as the measurement for separation performance are defined as in [35]. Typical ICA algorithms tested include: FastICA [22], EASI [23] and Infomax [24]. The initialization of the demixing matrix is generated randomly. The non-linear function selected for FastICA is G(y) = log(0.1 + y), with more options listed in [22]; the adaptation step for EASI is 0.01, and the nonlinearity is chosen as g i (y) = |y i | 2 y i for 1 ď i ď P; the updating step for Infomax is 0.1, and the nonlinearity is selected in the form of g i (y) = -|y i | α i -1 sign(y), 1 ď i ď P, with α i = 2. Results are obtained with 500 times of Monte Carlo simulations. Figure 8 shows curves of SINRM 1 varying with the input SNR, in which the azimuth angles of source 2 is´81.1˝, and thus the condition number is 2.56. It can be seen that for these representative ICA algorithms, their performances measured by the SINR increase along with input SNR. More specifically, curves of the three algorithms are roughly parallel with the theoretical upper limit indicated by (d), which means that their output SINRs vary almost linearly with the input SNR. Take the FastICA as an example, while the SNR increases from 19 dB to 21 dB, the output SINR increases from 11.04 dB to 13.06 dB. Figure 9 demonstrates the SINRM 1 under varied condition numbers, and the different condition numbers are controlled by the azimuth angle of source 2. It can be seen that the SINRs measuring the performance decrease along with the increase of the condition number. Curves of their performance are all below the theoretical line based on Equation (20), and are varying in a nearly parallel way with the curve (d). For instance, in curve (a), the output SINR decreases by 2.21 dB (from 9.93 dB to 7.72 dB) when the condition number increases from 6.17 to 7.79. This is close to the result obtained with Equation (20), since 20log(7.79/6.17) « 2.03.
In general, results obtained in this experiment have demonstrated well the influence of the environmental noise and the condition number on the separation performance of typical ICA algorithms, and do not conflict with the theoretical limitation indicated by Equation (20).  In this experiment, the same sources have been adopted, except that the carrier frequency of source 2 is set according to different values of D tested. The DOA of source 1 is (30˝, 30˝) in Figure 10a,c,e, and (´45˝, 45˝) in Figure 10b,d,f. The performances have been demonstrated with representative values of β and D, as well as for source 2 from all possible directions. The input SNR is set at 30 dB. Since it has been demonstrated in Figures 8 and 9 that performances of typical ICA algorithms are all influenced by the condition number in ways alike, only results from the FastICA algorithm as described in Experiment 1 are presented, while similar results have been obtained with other ICA algorithms mentioned in Experiment 1 during our simulations. Figure 10 demonstrates the SINRM 1 under varied practical factors simulating possible actual scenarios. It can be seen that in Figure 10a,b, the separation performance endures dramatic deterioration (at least 15 dB) for source 2 from certain groups of directions. This is because in Figure 10a Figure 10b). In Figure 10c,d, β is reduced to be smaller than 1. For Figure 10c, since Dcosθ 1 cosφ 1 > 1, the singular cases have been completely avoided for source 2 from all possible directions; while in Figure 10d, Dcosθ 1 cosφ 1 < 1, thus the mixture would turn singular for cosθ 2 cosφ 2 = Dcosθ 1 cosφ 1 , though with β < 1. In Figure 10e,f, the value of D is set according to Equation (34). In Figure 10e, since βcosθ 1 cosφ 1 R Z, D is set to be greater than (β/βcosθ 1 cosφ 1 ,-t βcosθ 1 cosφ 1 u) = 4; in Figure 10f, βcosθ 1 cosφ 1 = 1, thus D > β > 2 should be enough for the prevention of singular cases and we set D = 3; and it should be noticed herein that the mixture would still become singular for cosθ 2 cosφ 2 = 0, which is straightforward from Equation (22). To conclude, the experiment has validated the conclusions obtained in Section 3.2 about the influence of fundamental practical factors on the separation performance, especially those concerned the cases of singular mixtures. Experiment 3: Effectiveness of the proposed countermeasure against singular mixtures.
In this experiment, the carrier frequencies of the QPSK and BPSK sources are set at 0.4 MHz and 0.2 MHz respectively, thus D = 2. The DOA of source 1 is (56˝, 63˝). The sampling frequency is 1.2 MHz and the length of observation is 5120. The input SNR is 30 dB. After numerous experiments, the threshold of the correlation coefficient for judging the singular cases is chosen as 0.99. In Figure 11, the correlation coefficient between source 1 and its corresponding restitution component is adopted as other measurement of performance. In Figure 11a-c, β = 0.75; and in Figure 11d-f, β = 4.5. Figure 11a,d demonstrate the separation performance, without the proposed countermeasure. It can be seen that the restitution quality is unsatisfactory while source 2 is from certain directions (the distribution of such DOAs of source 2 can be seen clearer in the subplot northeast of Figure 11a,d). In Figure 11b,e, the proposed countermeasure has been adopted and k = 1. Thus, the extra phase delay introduced for these singular cases is Ψ 1 = 2πkT s (f 2´f 1 ) = π/3; and since k = 2 in Figure 11c,f, the extra delay therein is Ψ 2 = 2π/3. It can be seen that restitution qualities in the originally singular points have been dramatically improved, and the failure of separation has been effectively avoided. Thus, the effectiveness of the proposed countermeasure for the singular mixtures has been validated. In Figure 12, the waves of the sources and their estimations are demonstrated to show the improvement of the restitution quality in a more explicit way. Figure 12a presents the original sources in Figure 11a-c. In Figure 12b, the recovered components for DOA of source 2 being (´11.5˝, 61.2˝) (the corresponding condition number is 46.7) have been presented, and we can see that the distortion in the wave forms is unbearable. In Figure 12c,d, the proposed scheme is applied for the same situation, and it is obvious that the restitution quality has been improved significantly. Similar analysis can be done for Figure 12e

Conclusions
In this paper, the performance of ICA algorithms in sensor arrays is fully discussed. First, the analytic connection between the environment noise variance, the source variance, the condition number of the mixing matrix and the optimal signal to interference-plus-noise ratio is deduced in Equation (20). It indicates quantitatively how the separation performance would be determined by the environment noise level and the singularity of the mixing process. Next, the relationships between several fundamental practical factors in sensor array receiving and the condition number are analyzed in detail. Factors concerned include the element spacing of the array (depicted by the parameter β), the frequencies (depicted by the parameter D) and locations of the sources (depicted by the DOAs of sources). Generally speaking, it is found that: (1) the condition number varies with the element spacing in a periodical way, with the period is determined by D and DOAs; (2) the condition number also varies with the reciprocal of D periodically, and the period is determined by β and DOAs; (3) as for the locations of sources, we discover that sources from varied locations may lead to identical condition numbers, so long as their DOAs result in the same value of cosθ i cosφ i (i = 1,2). Considering its significance for applications, the situations where the mixtures become singular have been paid special attention to. Main conclusions concerning the prevention of singular cases include: (1) the cases of singular mixtures can be well prevented for sources with diverse carriers and from all possible directions, so long as the element spacing of the array is smaller than wavelengths of the sources. And the only possible exception exists when cosθ 2 cosφ 2´D cosθ 1 cosφ 1 = 0, then the mixture would be singular regardless of β; (2) for certain β and (θ i ,φ i )(i = 1,2), the cases of singular mixtures could be avoided by sufficiently large values of D, the reciprocal of which is smaller than min(1/D) max in Equation (34). Moreover, a countermeasure for the singular cases has been proposed, based on the introduction of extra phase delay. And in reality, the scheme is realized with one of the observations undergoing artificial time-shifting. Conclusions and the effectiveness of the proposed countermeasure are validated by succeeding experiments. As a whole, conclusions and results obtained in this paper could be instructive while applying ICA algorithms in sensor arrays or designing the receiving array, especially for the prevention of critical failure in separation. Future works may focus on analyzing the effects of more complicated practical factors such as the multi-path effect or array calibration imperfection, which would involve more complex model of the array output and coupling among different kinds of factors. Also, the effectiveness of the proposed scheme in Section 3.3 could be further promoted, possibly based on the interpolation of the observations to allow better control on the value of the extra phase delay by changing the sample interval. Since the desired phase delay is (2m + 1)π, m P Z in theory.