Determining the Number of Coherent/Correlated Sources Using FBSS-based Methods

Determining the number of sources from observed data, is a fundamental problem in array signal processing. In this paper, first we focus on two popular estimators based on information theoretic criteria, AIC and MDL. Then another algorithm based on eigenvalue grads, namely EGM is presented. The computer simulation results prove the effective performance of the EGM for non-coherent signals but in the small differences between the incident angles of non-coherent sources, MDL and AIC have a much better detection performance than EGM. These methods can detect only non-coherent signals, and the performance of them will be sharply declined even signals are coherent and/or correlated. So, first forward/backward spatial s moothing (FBSS) method is used as a pre-processing step to solve the coherency/correlation, and then MDL, AIC and EGM algorithms are run to estimate the number of signals. Numerical results show that FBSS-based EGM offers higher detection probability rather than FBSS-based MDL and AIC in the case of coherent sources as well as correlated ones. Also, the higher detection probability can be achieved for correlated case compared to coherent one.


Introduction
Wireless direction finding is the p rocedure for determining signal sources by observing signal direction of arrivals (DOAs). Its history came back to the beginning of wireless commun ications. This technology is used in many fields such as radar, sonar, mobile co mmunicat ions, astronomy, seismo logy and etc. So, DOA estimation using sensor arrays and direction finding, are important subjects in signal processing [1]. Most of the DOA estimation algorith ms, assume that the number of sources is known a priori and may give mislead ing results if the wrong number of sources is used. So, determin ing the number of signals is an important problem in field o f array signal processing [2].
In order to determine the number of sources to satisfy the requ iremen t o f DOA est imato r, man y paramet ric and nonparametric detection methods have been proposed [3]. A number of related methods have been widely studied in [4].
On e o f the mo s t wid ely -us ed ap p roaches is th at o f informat ion theoretic criteria, wh ich introduced by Wax and Kailath for the first time. They proposed an approach to this problem, based on the Akaike's informat ion criterion (AIC) and minimu m description length (MDL). According to these two algorith ms, the number of signal sources is determined as the value for wh ich the AIC or M DL criterion is minimized [5].
However, despite these methods are efficient in detection point of view, they involve the estimation of a covariance matrix and its eigenvalue decomposition which introduce generally high comp lexity. In order to reduce the computational co mplexity and attain accurate detection performance, a low-co mp lexity M DL method is developed in [6]. This method employs the training data of the desired signal, to quickly partit ion the array data into two orthogonal components in signal and noise subspaces.
To enhance the probability of detection, an accurate minimu m description length (MDL) method was devised in [7]. Since the training sequence of the desired signal is used to calculate the minimu m mean square error (MMSE), the performance of proposed method is significantly superior to the traditional MDL method at low SNRs and small number of snapshots.
Luo Jing Qing [8] proposed a set of eigenvalue gradient methods (EGMs), wh ich like AIC and MDL methods, it also determines the number of non-coherent sources according to the eigenvalues of auto-correlation matrix.
Another algorith m that considers a bound or threshold for eigenvalues is investigated in [9]. This approach is a simp le and efficient way to increase detection probability in the case of unequal power sources.
Above discussed techniques assume that noise is spatially uncorrelated. In practice, signals experience colored noise which causes decreasing the performance of conventional Sources Using FBSS-based M ethods methods, rapidly. To overco me this problem, a signal enumerator for spatially correlated noise has been proposed in [10]. They co mbine the co mbined informat ion theoretic criteria and eigenvalue correction to estimate the nu mber of coherent signals.
Li Jing-hua and Li Rong [11] established a novel effect ive detection algorith m in spatial colo r noise by using the canonical correlat ion coefficients of the joint covariance matrix. Co mpared with the other algorithms, this algorithm has a better performance and lower co mplexity In the real environ ment, the coherent source is common, and the above-mentioned methods are suitable for non-coherent signals, and their performance will be sharply declined even signals are coherent or correlated [12]. A partial solution to this problem, applicable to coherent or correlated sources, was proposed in [13]. The method uses the MDL principle and decomposes data into signal and noise components. The MDL descriptor is then computed for signal and noise co mponents separately, and the results are added to obtain the total MDL cost. Another way is to use spatial smoothing (SS) method to solve the coherency/correlation, and then estimate the number of signals. Spatial s moothing scheme partitions the total array of sensors into sub-arrays and then generates the average of the sub-array output covariance matrices [14,15]. The second method is more applicable because SS is a pre-processing required for both determining the number of sources and DOA estimat ion.
The main purpose of this paper is to determine the number o f sources in three types of sources, non-coherent, coherent, and correlated, based on MDL, A IC and EGM methods. First, conventional MDL, A IC and EGM algorith ms are applied to detect the number of non-coherent sources. Then, by applying spatial s moothing as a pre-processing part, above mentioned algorithms are used for detecting the number of coherent/correlated sources.
The paper is organized as follows. After the statement and formulat ion of the problem in Section 2, Sect ion 3 presents the derivation of the three detection criteria for non-coherent sources (MDL, AIC and EGM). In addit ion, two experiments and associated simulation results are presented. Section 4 presents forward/backward spatial smoothing (FBSS) technique for determin ing the nu mber of coherent/correlated sources. Moreover, it illustrates simu lation results of two experiments, coherent and correlated sources. Finally, Sect ion 5 concludes this research.

Signal Model
Consider narrowband signals emitted fro m the far field imp inging on array of sensors ( < ) . The × 1 measurements of the output of the array corrupted by additive noise can be expressed as where ( ) = [ 1 ( ), 2 ( ), … , ( )] is the matrix of array manifo ld, ( ) = [ 1 ( ), 2 ( ). . . , ( ) ] is the vector of signal, and ( ) is the additive noise. The additive noise is assumed to be co mp lex, ergodic Gaussian vector stochastic process, independent of the signals, with zero mean and covariance matrix g iven by 2 [16]. The covariance mat rix for array is is the correlat ion matrix of signals. In fact, considering ergodic processes, correlation matrix is acquired fro m limited data, that is where is the number of snapshots.
To solve the problem, determining the number of sources fro m � , we consider the following assumptions [13].
1. The array manifold, defined as the set of steering vectors, ( ) is known.
2. Any subset of steering vectors fro m the array man ifold is linearly independent.
3. The nu mber of sensors is greater than the number of sources, namely, > .

Determining the Number of Non-coherent Sources
In this section, first, three well-known algorithms, M DL, AIC and EGM , are described for non-coherent signals. Consequently, related simu lation results that show the performance of these algorithms are illustrated with mo re details.

Esti mating the Number of Signals with Information Theoretic Criteria
The information theoretic criteria for model selection, introduced by Akaike Schwartz and Rissanen, address the following general problem: Given a set of observation = { ( 1 ) , ( 2 ) , … , ( )} and a family of models, that is, a parameterized family of probability densities ( | ) selects the model that best fits the data.
Akaike's proposal was to select the model which gives the minimu m AIC, defined by = −2 � � �� + 2 (4) where � is the maximu m likelihood estimate of the parameter vector , and is the nu mber of free adjusted parameters in . Inspired by Akaike's pioneering work, Schwart z and Rissanen approached the problem fro m quite different points of view. Rissanen's approach is based on informat ion theoretic arguments. Since each model can be used to encode the observed data, Rissanen proposed to select the model that y ields the min imu m code length. It turns out that in the large-sample limit, both Schwart z's and Rissanen's approaches yield the same criterion, given by To apply the information theoretic criteria to detect the number of signals, we can say that [16] � ( 1 ) , … , ( ) Taking the logarith m and o mitting terms that do not depend on the parameter vector ( ) , we find the log-likelihood function ( ( ) ) as The ma ximu m-likelihood estimate is the value of ( ) that maximizes (7). These estimates are Substituting the maximu m likelihood estimates (8) in the log-likelihood (7), with some straightforward manipulat ions, we obtain The form of AIC for this problem is therefore given by equation (10).
while the MDL criterion is given by equation (11).
The number o f signals ̂ is determined as the value of ∈ {0,1, … , − 1}, for which either the AIC or the MDL is minimized [16].

Esti mating the Number of Signals Using EGM
Just like tradit ional AIC and MDL methods, EGM family also determine the number of non-coherent sources according to the eigenvalues of auto-correlation matrix [8]. The first step is calculating the spatial auto-correlation matrix of the output data ( ) of the sensor array by (3). The second step is applying eigen-decomposition on auto-correlation , and arranges the eigenvalues in descending order, There is a significant d ifference between and +1 . So, the number of signal k can be determined by checking the difference between neighbour eigenvalues. This is the main idea of the set of EGM methods. A common used checking method named EGM 1 [8] is cited as follows: 1. Define the average grads of all eigenvalues by

Calculate the gradients of all neighbor eigenvalues as
3. Find out all satisfying∆ ≤ � to construct the set 4. Take the 0 that is the first one of the last continuous block of in the set { } , and the estimated nu mber of signals is ̂ = 0 − 1.

Simulation of Non-Coherent Signals
Co mputer simulations have been carried out to examine the effectiveness of above mentioned algorith ms in the case of non-coherent sources. Detection probability is the performance metric to evaluate and also compare MDL, AIC and EGM algorithms.
The performance of determining the number of non-coherent sources with MDL, A IC and EGM under different SNRs is evaluated in the first experiment. In this experiment, the uniform linear array with 10 sensors is used, element spacing is half-wavelength and SNR is define as Two non-coherent signals with equal powers impinge on the array at 0°and 10°, SNR changes from -15 to 30 with step size 1 and the number of snapshots is 100. For each SNR, 1000 Monte Carlo trials are run to find probability of success.
In Fig. 1, the probabilit ies of detection for M DL and AIC are compared with EGM in the case of non-coherent sources. According to this Figure, for SNRs lower than −10 , the performance of EGM is better than both MDL and AIC methods. For higher SNRs, greater than −5 , MDL and EGM show 100% success but AIC offers detection probability a b it lo wer than 100% . In the second experiment, the performance of determin ing the number of non-coherent sources using MDL, AIC and EGM under different d istances between source angles is evaluated. In this case, the uniform linear array with 8 sensors is used. Two non-coherent signals with equal powers imp inge on the array, the angular distance changes fro m 1 to 10 degrees and the number of snapshots is 100.
The simulat ion results of experiment 2 are shown in Fig.  2. As depicted in this figure, in s mall differences between the incident angles (lower than 6°), M DL and AIC have a much better detection performance than EGM. In addit ion, for d ifferences lower than 6°, EGM performance degrades drastically. In the other hand, MDL and AIC performance will be decreased for the angular differences lo wer than 2°. Moreover, for the differences greater than 7°, EGM is a much better than MDL and also AIC.

Determining the Number of Coherent/Correlated Sources
A major problem with above methods (MDL, A IC and EGM) is that, they are not applicable to the case of fully correlated signals, referred to as the coherent signals. This case appears especially in mu ltipath propagation and therefore it is of great practical importance.
In this investigation, we focused on spatial smoothing method to solve the coherency/correlation, and then the number of coherent sources is estimated considering the pre-processed output signal of array antenna.
The spatial smoothing scheme first suggested by Evans and extensively studied by Shan [14]. Their solution is based on a pre-processing scheme that divides the total array of sensors into overlapped sub-arrays and then, generates the average of the sub-array output covariance matrices. This forward-only smoothing scheme makes use of a larger number of sensor elements than the conventional ones, and in particu lar requires 2 sensor elements to estimate any directions of arrival. In [15], it is proved that by simu ltaneous use of a set of forward and co mplex conjugated backward sub-arrays, it is always possible to estimate any directions of arrival using at most[ 3 /2] sensor elements. So, in this research work, we pre-processed coherent signals using FBSS as the first step of the process of determining the number of coherent sources.

Forward/B ack ward S pati al Smoothi ng Techni que for Coherent/Correlated Signal Identificati on
FBSS scheme starts by dividing a uniform linear array with sensors into uniformly overlapping sub-arrays of size 0 (see Fig. 3). Let ( ) stands for the output of the th sub-array for = 1,2, … ≡ − 0 + 1 where denotes the total number of these forward sub-arrays [15].  (16) Then, the covariance matrix o f the th sub-array is given by = � ( ) ( ( ) ) † � (17) Forward spatially smoothed covariance matrix as the mean of the forward sub-array covariance matrices is The covariance matrix of the th backward sub-array is given by = � ( ) ( ( ) ) † � (19) In the same way as (18), the spatially s moothed backward sub-array covariance matrix is = 1 ∑ =1 (20) Finally, the forward/backward smoothed covariance matrix � as the mean of and is given by Since the smoothed covariance matrix � in (21) has exactly the same form as the covariance matrix for so me non-coherent situations, the eigenstructure-based techniques can be applied to this smoothed covariance matrix to successfully estimate the nu mber of coherent sources [15].

Simulation of Coherent/Correlated Sources
Co mputer simulations have been carried out to examine the effectiveness of above mentioned algorith ms in coherent case. An uniform linear array with 9 sensors is used, SNR changes from -10 to 10 with step size 1 d B and the number of snapshots is 100 . Fo r each SNR, 1000 Monte Carlo trials are run.
As the experiment 3, the performance of FBSS-based MDL, AIC and EGM algorith ms under different SNRs for coherent sources is investigated. Three coherent signals with equal powers imp inge on array at 0°,10°and 20°are considered with the path coefficients (0.7 + 0.7 ) , (0.6 + 0.5 ) and (0.2 + 0.4 ), respectively. As depicted in Fig. 4, for SNRs greater than 2 , the performance of EGM is better than MDL and AIC methods but in SNRs lower than 0 , none of methods can detect the number of coherent sources. In experiment 4, signal sources are considered as correlated non-coherent ones. Three correlated signals with equal powers imp inge on array at 0°, 10°and 20°. Correlated signals are obtained by filtering the signal through a first order auto-recursive (AR1) filter, given by ( ) = ( − 1 ) + ( ) (22) where α ∈ [0,1] is the correlation coefficient. In this experiment = 0.8 is assumed. Simulat ion results are shown in Fig. 5. In this case, FBSS-based methods can effectively detect the number of sources and offer high detection probabilit ies even much better than experiment 3. According to Fig. 5, EGM offers higher performance with respect to MDL and AIC methods. As a highlighted re mark, EGM method shows high detection probability for low SNRs (lower than −4 ).

Conclusions
In this paper, we focused on the performance evaluation of three popular methods, AIC, MDL as well as EGM. As shown in simulat ion results, these algorithms are appropriate for determin ing the number of non-coherent signals and the performance of them will be decreased when signals are coherent and/or correlated. A solution to this problem, applicable to coherent as well as correlated non-coherent sources, is to use spatial s moothing method to solve the coherency or correlation, and then estimating the number of signals.
Co mputer simulations showed that, in the case of non-coherent signals, in low SNRs, EGM is better than MDL and AIC. Also, in small angular differences, MDL and AIC algorith ms offer a higher detection performance with respect to EGM.
Moreover, simu lation results for FBSS-based algorith ms, in the case of coherent sources as well as correlated ones, showed the higher detection probability for EGM co mpared to AIC and MDL.
In this research we co mpared M DL, AIC and EGM algorith ms in additive white Gaussian noise (AWGN) channel. Eigen incre ment threshold (EIT) method which is proposed as a method to determine the number of sources can be researched in AW GN channel and co mpared to AIC, MDL and EGM algorithms in both non-coherent and coherent cases. In addition, these methods can be investigated in colored noise scenario and in the case that