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Generalized generating function with tucker decomposition and alternating least squares for underdetermined blind identification
EURASIP Journal on Advances in Signal Processing volume 2013, Article number: 124 (2013)
Abstract
Generating function (GF) has been used in blind identification for realvalued signals. In this paper, the definition of GF is first generalized for complexvalued random variables in order to exploit the statistical information carried on complex signals in a more effective way. Then an algebraic structure is proposed to identify the mixing matrix from underdetermined mixtures using the generalized generating function (GGF). Two methods, namely GGFALS and GGFTALS, are developed for this purpose. In the GGFALS method, the mixing matrix is estimated by the decomposition of the tensor constructed from the Hessian matrices of the GGF of the observations, using an alternating least squares (ALS) algorithm. The GGFTALS method is an improved version of the GGFALS algorithm based on Tucker decomposition. More specifically, the original tensor, as formed in GGFALS, is first converted to a lowerrank core tensor using the Tucker decomposition, where the factors are obtained by the left singularvalue decomposition of the original tensor’s mode3 matrix. Then the mixing matrix is estimated by decomposing the core tensor with the ALS algorithm. Simulation results show that (a) the proposed GGFALS and GGFTALS approaches have almost the same performance in terms of the relative errors, whereas the GGFTALS has much lower computational complexity, and (b) the proposed GGF algorithms have superior performance to the latest GFbased baseline approaches.
1. Introduction
Blind identification (BI) of linear mixtures has recently attracted intensive research interest in many fields of signal processing including blind source separation (BSS). This work is devoted to BI of underdetermined mixtures with complex sources. Underdetermined mixtures are commonly encountered in many practical applications, such as in the radio communication context, where the reception of more sources than sensors becomes increasingly possible with the growth in reception bandwidth. In these applications, one often has to also deal with complex sources. One reason is that the communication signals are usually complexvalued such as the quadratureamplitude modulation (QAM) signal, and minimumshift keying (MSK) signal. Another reason is that frequency domain methods are often used for blind separation or identification from convolutive mixtures due to its computational efficiency [1, 2], while the objective functions used in the frequency domain are usually defined on complexvalued variables.
A large number of methods for BI of underdetermined mixtures start from the assumption that the sources are sparse by nature (i.e., in its own domain such as the time domain) or could be made sparse in another domain (e.g., a transform domain). A predefined transform such as shorttime Fourier transform (STFT) or a learned transform using, e.g., simultaneous codeword optimization (SimCO), is usually applied to sparsify the data [3, 4] if the signal by nature is not sparse. Due to the sparsity of the sources, the scatter plot typically shows high signal values in the directions of the mixing vectors, which can be localized by using some clustering techniques [5, 6]. It should be noted that although some signals such as speech signals have some degree of sparsity in one domain or another, many other signals such as the majority of communication signals do not possess such a property. Hence, it is necessary to develop BI methods for the underdetermined mixtures that do not impose any sparsity constraint on the sources.
To this aim, many methods for BI of underdetermined mixtures turn to the use of various decomposition methods based on different data structures such as correlation [7, 8] and higherorder cumulant [9–14] matrices. The main idea of these algorithms is to construct a tensor based on the cumulants of the observations and then to estimate the mixing matrix by the decomposition of such a tensor. This is notably the case for secondorder blind identification of underdetermined mixtures (SOBIUM) [7], fourthorder blind identification of underdetermined mixtures (FOBIUM) [9], fourthorderonly blind identification (FOOBI) [10], FOOBI2 [10], and blind identification of mixtures of sources using Redundancies in the daTa Hexacovariance matrix (BIRTH) [11, 12] algorithms, which use secondorder statistics tensors and fourth and sixthorder cumulant tensors, respectively. A family of the methods named blind identification of overcomplete mixtures of sources (BIOME) is proposed in [13], based on the evenorder cumulants of the observations. However, all the methods proposed in [7–14] exploit only the statistical information contained in the data measured by secondorder or higherorder statistics.
In order to exploit statistical information more effectively, a family of BI approaches was proposed in [15–18] by exploiting the statistical information with the characteristic function (CAF) or generating function (GF). In these works, the authors showed that the mixing matrix can be estimated up to trivial scaling and permutation indeterminacies by decomposing the tensor composed of partial derivatives of the GF. It is worth mentioning that the algorithms in [15–17] have been only applied to BI problems involving realvalued sources. In [18], the CAF approach was extended to the case of mixtures of complexvalued sources, which often occurs in digital communications. However, extra effort is required to obtain the correct real and imaginary combination of the mixing matrix since the real and imaginary parts of the mixing matrix are treated separately, leading to an increased computational cost due to the increased dimension of the matrix that needs to be processed. In this paper, we propose the Generalized Generating Function (GGF) to exploit the statistical information carried on the complex random variable. We show that the proposed GGF can exploit the statistical information carried on complex random variables in a more effective way than the GF presented in [18] due to the algebraic structure adopted by GGF (as detailed in Algebraic structure based on generalized generating function). Furthermore, a simple method for the mixing matrix estimation is derived based on tensor decomposition where the tensor is composed of the Hessian matrices of the GGF of the observations.
The remainder of this paper is organized as follows. In Problem formulation the BI problem is formulated and relevant assumptions are presented. In Algebraic structure based on generalized generating function we firstly generalize the definition of the GF for complexvalued random variables and then derive the corresponding core equation for BI. In Blind identification based on tensor decomposition, the GGFALS and GGFTALS approaches are developed for the estimation of the mixing matrix. In the GGFALS algorithm, the mixing matrix is estimated by directly decomposing the tensor, constructed from the Hessian matrices of the second GGF of the observations, using the alternating least squares (ALS) algorithm. In the GGFTALS algorithm, the Tucker decomposition is firstly applied to convert the original tensor to a lowerorder core tensor, then the mixing matrix is obtained by decomposing the core tensor with the ALS algorithm. Furthermore, the factors of Tucker decomposition are obtained by the left singular vectors of the original tensor’s mode3 matrix. Computer simulations are used to illustrate the performance of the proposed GGF approaches in Simulations and analysis. Finally, the paper is concluded in Conclusions.
2. Problem formulation
Considering the following linear mixture model
where the stochastic vector z(t) ∈ C ^{Q} represents the observation signals, s(t) ∈ C ^{P} contains the unobserved source signals, and w(t) ∈ C ^{Q} denotes additive noise. From now on, the noise w(t) is simply ignored for convenience, except when running computer experiments. The unknown mixing matrix A ∈ C ^{Q × P} characterizes the way that the sources are acquired by the sensors. BI aims to estimate the mixing matrix from the observations based on the assumption that the source signals are statistically independent. The mixing matrix obtained may in turn be used to estimate the original source signals from the observations. In addition, we make the following assumptions:

(i)
The mixing matrix A is of full (row) rank.

(ii)
The number P of sources is known.

(iii)
The number of sensors is smaller than the number of sources, i.e., Q < P.
3. Algebraic structure based on generalized generating function
3.1. Core equation based on generalized generating function
For a real stochastic vector x ∈ R ^{Q}, the GF ϕ _{ x }(u) obtained by dropping the term of the square root of (−1) in the exponent of a CAF is defined as
where u ∈ R ^{Q} is an arbitrary vector referred to as a processing point [15], and E[ ] denotes an expectation operator. Nevertheless, both the observation vector z and the mixing matrix A discussed in this paper belong to the complex field. Hence, a definition of GF for complex variables is required. One such definition has been presented in [18] as
where $\mathfrak{R}\left(\xb7\right)$ and $\mathfrak{I}\left(\xb7\right)$ denote taking the real and imaginary parts from their arguments (i.e., complexvalued vectors) to form a realvalued vector of the same dimension. It is actually defined by assimilating C to R^{2}. Thus the GF of a complex variable in (3) is defined as a function of the real and imaginary parts. In this paper, we generalize the definition of GF for real stochastic vector in (2) to the following complex form
Note that the statistical information exploited by GF/GGF is related to the number of processing points. Theoretically, a complete statistical description of the probability density function requires the evaluation of the GF/GGF at all (infinitely many) possible processing points. However, this often becomes computationally infeasible. In practice, such statistical information is obtained approximately by the evaluation of GF/GGF at a finite number of processing points. Hence, in comparison with the GF presented in [18], the GGF defined in (4) can exploit the statistical information carried on the complex variables more effectively when the number of the processing points stays the same, thanks to the incorporation of the imaginary part of the exponent to the function. Furthermore, as compared with the use of the GF in (3), using the GGF in (4) offers a simpler way for the estimation of the mixing matrix due to the exploitation of an elegant algebraic structure.
Now, replacing z by its model and neglecting the noise contribution yield
Defining φ _{ z }(u) = log ψ _{ z }(u), which is often referred to as the ‘second’ GGF, and using the source independence property, the second GGF of the observations can be rewritten as
Consequently, by calculating the derivative of the conjugate gradient of φ _{ z }(u) with respect to u (more details can be found in Appendix 1), we can obtain the following core equation for the Hessian matrix ψ _{ z }(u),
with
where (·)^{*} denotes the conjugate operator. It is necessary to point out that ψ _{ s }(A ^{H} u) is a diagonal matrix (more details can be found in Appendix 2).
3.2. Estimating ψ_{ z }(u)
In this subsection, we discuss how to consistently estimate the Hessian matrix ψ _{ z }(u). Under the ergodicity assumption, the mean value of a random variable can be estimated by a time average. Hence, we can estimate the GGF of the observation vector as
The conjugate gradient ς_{ z }(u) = ∂ ψ _{ z }(u)/∂ u* which is a Q × 1 vector can be estimated by
Similarly, the gradient ζ_{ z }(u) = ∂ ς_{ z }(u)/∂ u ^{T} which is a Q × Q matrix can be estimated by
Based on the above analysis, the Hessian matrix ψ _{ z }(u) of the second GGF φ _{ z }(u) can be obtained as
4. Blind identification based on tensor decomposition
In this section, the GGFALS and GGFTALS algorithms are developed for the estimation of the mixing matrix. In the GGFALS algorithm, the mixing matrix is estimated by decomposing the tensor which is formed of the Hessian matrices of the GGF of the observations. An improved version of the GGFALS algorithm, i.e., the GGFTALS algorithm, is also developed, where the Tucker decomposition is firstly employed to convert the original tensor as used in the GGFALS algorithm to a lowerrank core tensor, and the mixing matrix is then estimated by decomposing the core tensor with the ALS algorithm.
4.1. The GGFALS algorithm
Evaluating the Hessian matrices ψ _{ z }(u) of GGF at a series of processing points u _{1}, u _{2},..., u _{ K } and using Equation (7), one can obtain the following joint diagonalization (JD) problem
in which ψ _{ s }(A ^{H} u _{ k }) is diagonal, k =1,…,K. The problem we need to address is to estimate the mixing matrix A based on the set {ψ _{ z }(u _{1}), ⋯, ψ _{ z }(u _{ K })}. For the determined/overdetermined case, it is obvious that JD methods, such as the ACDC method [19], can be used to estimate the mixing matrix. However, this method does not work when Q < P i.e., in the underdetermined case.
As shown in [7], the JD problem (9) can be seen as a particular case of the parallel factor (PARAFAC) decomposition, also known as canonical decomposition (CAND), of the thirdorder tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$ built by stacking the K matrices ψ _{ x }(u _{ k }) along the third mode. Specifically, the tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$ is built by stacking ψ _{ z }(u _{1}), ψ _{ z }(u _{2}),..., ψ _{ z }(u _{ K }) as follows: ${\left(\mathcal{M}\right)}_{\mathit{ijk}}={\left({\mathbf{\psi}}_{z}\left({\mathbf{u}}_{k}\right)\right)}_{\mathit{ij}}$, i =,…,Q, j = 1,…,Q, k = 1,…,K. Define a matrix D ∈ C ^{K × P} by (D)_{ kl } = (ψ _{ s }(A ^{H} u _{ k })_{ ll }, l = 1,…,P, k = 1,…,K. Then we have
which we write as
where º denotes the tensor outer product, and a _{ l } and d _{ l } represent the l th column of A and D _{,} respectively. In this way, the mixing matrix A can be estimated by solving the following problem. Given the thirdorder tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$, we can compute its CAND with P components of the rankone tensors that best approximates $\mathcal{M}$, i.e.,
where ‖‖_{F} is the Frobenius norm.
Several algorithms exist for the computation of tensor decomposition. The standard way for computing the tensor decomposition is by using an ‘ALS’ algorithm [20]. Several improved versions, such as the enhanced line search (ELS) [21] and extrapolating search direction (ESD) [22], are proposed to accelerate the rate of convergence of the ALS. Hence, the ALS is chosen here to compute the CAND.
To a large extent, the practical importance of tensor decomposition stems from its uniqueness properties. It is clear that the tensor decomposition can only be unique up to a permutation of the rank1 terms and scaling of the factors of the rank1 terms. Therefore, we consider the tensor decomposition (11) as essentially unique if any other matrix pair A ^{’} and D ^{’} _{,} that satisfies (11) is related to A and D via
with Δ _{1}, Δ _{2} ∈ C ^{P×P} being diagonal matrices, satisfying Δ _{1} Δ _{1} Δ _{2} = I, and P ∈ R ^{P×P} being a permutation matrix.
The krank. The Kruskal rank or krank of a matrix A, denoted by κ _{A}, is the maximal number λ such that any set of λ columns of A is linearly independent.
Theorem 1. The tensor decomposition of (11) is essentially unique if[23]
We call a property generic when it holds with probability one. Generically, the mixing matrix is of full rank and of full krank when the parameters it involves are drawn from continuous probability densities. Hence, in practice, κ _{ A } = min(Q, P) and κ _{ D } = min(K, P).
In summary, we come to the following conclusion: when Q ≥ P, P ≥ 2, then the generic essential uniqueness is guaranteed for K ≥ 2; when Q < P and if K ≥ P, then the generic essential uniqueness is guaranteed for P ≤ 2Q − 2, if K < P, then the generic essential uniqueness is guaranteed for P < Q − 1 + K/2.
4.2. The GGFTALS algorithm
In order to ensure the GGFALS algorithm has robust performance, a large value is often chosen for K in the tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$. Nevertheless, this will lead to a heavy computational load. To reduce the computational complexity, the Tucker decomposition [23–25] is firstly applied to represent the tensor $\mathcal{M}$ as a lowerrank core tensor, whose size is much smaller than the tensor $\mathcal{M}$. Then the ALS algorithm is used to perform the CAND of the core tensor. In this way, the computational complexity can be reduced dramatically.
Matricization. Matricization, also known as unfolding or flatting, is the process of turning an Nway tensor into a matrix. The moden matricization of a tensor $\mathcal{F}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{1}\times {I}_{2}\cdots \times {I}_{N}}$ is denoted by F _{(n)} which arranges the moden fibers to be the columns of the resulting matrix, that is, the tensor element (i _{1}, ⋯, i _{ N }) is mapped to the matrix element (i _{ n }, j) where
with
The nrank . Let $\mathcal{F}$ be an N thorder tensor of size I _{1} × I _{2} ⋯ × I _{ N }. Then the nrank of $\mathcal{F}$, denoted by ${\mathrm{rank}}_{n}\left(\mathcal{F}\right)$, is the column rank of F _{(n)}. In other words, if we let ${r}_{n}={\mathrm{rank}}_{n}\left(\mathcal{F}\right)$ for n = 1,…,N, then we can say that $\mathcal{F}$ is a rank r _{1}, ⋯, r _{ N } tensor.
The nmode product. The nmode (matrix) product of a tensor $\mathcal{F}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{1}\times {I}_{2}\cdots \times {I}_{N}}$ with a matrix ${\mathbf{U}}^{J\times {I}_{n}}$ is denoted by $\mathcal{F}{\times}_{n}\phantom{\rule{0.25em}{0ex}}\mathbf{U}$ and is of size I _{1} × ⋯ × I _{ n − 1} × J × I _{ n + 1} × ⋯ I _{ N }. Elementwise, we have
Tucker decomposition is a form of higherorder principal component analysis (PCA). It decomposes a tensor into a core tensor multiplied by a matrix along each mode as shown in Figure 1. Thus, in the threeway case where $\mathcal{F}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{1}\times {I}_{2}\times {I}_{3}}$, we have
where $\mathcal{G}\in {C}^{{J}_{1}\times {J}_{2}\times {J}_{3}}$ is the core tensor; ${\mathbf{A}}^{\left(1\right)}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{1}\times {J}_{1}},{\mathbf{A}}^{\left(2\right)}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{2}\times {J}_{2}},{\mathbf{A}}^{\left(3\right)}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{3}\times {J}_{3}}$ are the factor matrices (which are usually column unitary for realvalued data) and can be thought of as the principal components in each mode; and $\mathcal{U}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{{I}_{1}\times {I}_{2}\times {I}_{3}}$ represents errors or noise.
With the help of Tucker decomposition, the tensor $\mathcal{M}$, which is composed of the Hessian matrices ψ _{ z }(u _{ k }) of the second GGF of the observations, can be compressed. Since the mixing matrix A is of full rank and the Hessian matrices ψ _{ s }(A ^{H} u _{ k }) of the second GGF of the sources are diagonal, the moden (n = 1, 2, 3) matrices of tensor $\mathcal{M}$ are M _{(1)} ∈ C ^{Q × QK}, M _{(2)} ∈ C ^{Q × QK}, and M _{(3)} ∈ C ^{K × QQ} respectively. Meanwhile, the nrank of the M _{(1)} and M _{(2)} is
On the other hand, if we assume ${\mathrm{rank}}_{3}\left(\mathcal{M}\right)=L$ and L ≤ K, then $\mathcal{M}$ is a rank(Q,Q,L) tensor. Thus, the Tucker decomposition of tensor $\mathcal{M}$ is the socalled Tucker1 decomposition [23]
where $\mathcal{T}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times L}$ is the core tensor, I ∈ R ^{Q × Q} is an identity matrix, and G ∈ C ^{K × L} is a columnunitary matrix. This is equivalent to a standard twodimensional PCA since
where T _{(3)} ∈ C ^{L × QQ} is the mode3 matrix of core tensor $\mathcal{T}$. It is obvious that (18) corresponds to the PCA of M _{(3)}. Therefore, G ∈ C ^{K × L} consists of the Lleading left singular vectors of M _{(3)}. Since K > L and G is a columnunitary matrix, it is straightforward to derive
Therefore, the core tensor can be obtained by
Because the first and second factors of the Tucker decomposition in (17) are identity matrices, the core tensor $\mathcal{T}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times L}$ is also a symmetric tensor [23], as for the tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$ and the CAND of the core tensor is as follows
where a _{ l } and d _{ l } are the l th column of A and D, respectively. The CAND process can also be implemented by the ALS algorithm. Nevertheless, our goal is to estimate the mixing matrix A, whereas the CAND of core tensor $\mathcal{T}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times L}$ only conduces A. Hence, it is necessary to derive the mixing matrix A based on A. Since the first and second factors of Tucker decomposition (17) are identity matrices, it is straightforward to derive
4.3. Computational analysis
In this subsection, we aim at giving an insight into the numerical complexity of the proposed algorithm. For the GGFALS algorithm, the ALS algorithm is directly used to decompose the tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$; therefore, the computational complexity is O(3PKQ ^{2} + QKP ^{2} + Q ^{2} P ^{2}) per iteration. For the GGFTALS algorithm, the computational complexity is dominated by the Tucker decomposition of the original tensor $\mathcal{M}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times K}$ (realized by the singularvalue decomposition (SVD) of the mode3 matrix M _{(3)}) and the decomposition of the core tensor $\mathcal{T}\phantom{\rule{0.25em}{0ex}}\in \phantom{\rule{0.25em}{0ex}}{C}^{Q\times Q\times L}$ using the ALS algorithm. Therefore, the computational complexities for these two operations per iteration are O(Q ^{6}) and O(3PLQ ^{2} + QLP ^{2} + Q ^{2} P ^{2}), respectively. Table 1 shows the computational complexity for the GGFALS, and GGFTALS methods.
It is worth mentioning that a large value is usually chosen for the number of processing points K in order to accumulate sufficient statistical information from data, while the rank of the core tensor order L is often chosen to be much smaller than K. For example, K = 100, L = 8 are chosen typically in our simulations as shown in the next section. The SVD is required to be calculated only once, whereas the ALS algorithm needs multiple iterations to achieve convergence, for example, 1,000 iterations as in our experiments. Moreover, the number of sources P and the number of sensors Q are usually small, for example, P = 4, Q = 3 in our simulations. For these reasons, it can be readily derived from Table 1 that the computational cost of the GGFTALS algorithm is, in practice, much lower than that of the GGFALS algorithm (note that the complexity of SVD becomes negligible in this case).
5. Simulations and analysis
In this section, simulations are provided to illustrate the performance of the proposed GGF approaches for underdetermined mixtures of complex sources. The performance of the tested algorithms is evaluated and compared in terms of the relative error performance index (PI) versus the sample size and the signaltonoise ratio (SNR) of the observations. Here the relative error PI is defined as [7]$\mathrm{PI}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{E}\left\{\left\left\mathbf{A}\widehat{\mathbf{A}}\left\right/\left\right\mathbf{A}\right\right\right\}$, in which the norm is the Frobenius norm and Â represents the optimally ordered and scaled estimate of the mixing matrix A.
The experiments refer to the scenario that P = 4 narrowband source signals are received by a uniform circular array (UCA) with Q = 3 identical sensors of radius R _{ a } Considering a free space propagation model, the entries of the mixing matrix A are given by
where α _{ q } = (R _{ a }/λ)cos(2π(q − 1)/Q), β _{ q } = (R _{ a }/λ)sin(2π(q − 1)/Q), and $j=\sqrt{1}$. We have R _{ a }/λ = 0.55. The direction of arrival (DOA) of the different sources are given by θ _{1} = 3π/10, θ _{2} = 3π/10, θ _{3} = 2π/5, θ _{4} = 0 and ϕ _{1} = 7π/10, ϕ _{2} = 9π/10, ϕ _{3} = 3π/5, ϕ _{4} = 4π/5. The sources are unitvariance 4QAM with a uniform distribution, shaped by a raised cosine pulse shaping filter with a rolloff ρ = 0.3. All sources have the same symbol duration T = 4T _{ e }, where T _{ e } is the sample period. The observations are contaminated by additive zeromean complex Gaussian noise.
First, we compare the performance of the GGFALS algorithm with that of the GGFTALS algorithm and investigate the influence of the rank of the core tensor L on the performance of the GGFTALS algorithm. To this end, the following two simulation experiments are conducted. In the first simulation, we evaluate the performance of the GGFTALS algorithm with different core tensor rank L for a fixed number of processing points K, and compare it with GGFALS with the same number of processing points. In this simulation, K is chosen to be 100 for the GGFALS algorithm, both the real and imaginary parts of processing points are randomly drawn from [−1; 1], the SNR of the observations ranges from 0 to 25 dB, and the number of samples is 4,000. The core tensor rank for the GGFTALS algorithm is chosen as L = 10,8,5, respectively. The threshold value described in (12) to stop the ALS algorithm is 10^{−5}, and 100 Monte Carlo experiments are run.
The average performance of the GGFTALS algorithm versus the core tensor rank for a fixed number of processing points is shown in Figure 2. We can see that the performance curves of the GGFTALS and that of the GGFALS almost coincide when the rank of the core tensor is larger than 8, whereas the performance of the GGFTALS algorithm slightly deteriorates when the rank of the core tensor is less than 8. This indicates that the GGFTALS algorithm maintains the identification accuracy offered by GGFALS despite the fact that the core tensors used by GGFTALS have a much lower rank than that of the original tensor used by GGFALS.
In the second experiment, we investigate the performance of the GGFTALS algorithm with different number of processing points K when the rank of the core tensor L is fixed. The simulation conditions are the same as those in the previous simulation except the number of processing points for the GGFALS algorithm which are 20, 40, and 100 respectively, and the rank of the core tensor for the GGFTALS which is 8. The average performance of the GGFTALS versus the number of processing points for the fixed number of the core tensor rank is shown in Figure 3.
We can see that the average performance of GGFTALS is consistent with that of GGFALS when both algorithms use the same number of processing points. For instance, when the number of processing points is 100, the performance of GGFTALS is close to the performance of GGFALS, so is for K = 40. Therefore, the GGFTALS algorithm consistently offers similar performance to GGFALS when the number of processing points is varied.
Second, we compare the performance of the GGFALS and GGFTALS algorithms with that of the latest GFbased BI algorithm. Here, the LEMACFAC2 in [18] is chosen as the baseline algorithm. The number of processing points and the value of processing points for the GGFALS, GGFTALS, and LEMACFAC2 are the same. Specifically, the number of processing points is 100, both the real and imaginary parts of processing points are randomly drawn from [−1; 1], and the rank of the core tensor is 8. The threshold value described in (12) to stop the LM algorithm [26] exploited in LEMACFAC2 method is also 10^{−5}.
Figure 4 shows the PI of the tested algorithms as a function of the SNR when 4,000 samples are used. It can be seen from this figure that the performance of the GGFALS is consistent with that of the GGFTALS, and both perform better than LEMACFAC2. Figure 5 shows the PI of the tested algorithms as a function of the number of data samples N when the SNR is equal to 20 dB. Again, the LEMACFAC2 underperforms the GGFALS and the GGFTALS algorithms, and the GGFALS and the GGFTALS have almost the same performance. This confirms that the GGF algorithms perform better than the method based on the GF defined in (3) in exploiting the statistical information carried on the complex variables when using the same number of processing points.
6. Conclusions
We have presented two algorithms, namely, GGFALS and GGFTALS, for blind identification from underdetermined mixtures of complex sources using the second GGF of the observations. In the GGFALS algorithm, the mixing matrix is estimated by directly using the ALS algorithm to decompose the tensor constructed from the Hessian matrices of the second GGF of the observations. The GGFTALS algorithm is an improved version of the GGFALS algorithm, where the Tucker decomposition is first used to convert the original tensor into a lowerrank core tensor, and the mixing matrix is then estimated by applying the ALS algorithm to the core tensor. Simulation results have shown that (a) the proposed GGFALS and GGFTALS approaches have almost the same performances in terms of the relative errors, whereas the GGFTALS has a much lower computational complexity, and (b) the proposed GGF algorithms have superior performance to the latest GFbased BI approaches, since the GGF algorithms can exploit the statistical information carried on complex variables in a more effective way.
Appendix 1
In this appendix, we show the computational details of the core equation in Equation 7. First, the differentiation of (6) with respect to u ^{*} gives
Second, the differentiation of (23) with respect to u ^{T} gives
In a more compact form, we can obtain the core equation
Appendix 2
Let ${\varphi}_{s}\left(\tilde{\mathbf{u}}\right)$ denote the characteristic function of the source signals s(t). Due to the statistical independence of the elements of s(t) = [s _{1}(t), ⋯, s _{ P }(t)]^{T}, we have
where ${\varphi}_{{s}_{j}}\left({\tilde{u}}_{j}\right)=E\left[exp\left({\tilde{u}}_{j}^{*}{s}_{j}\right)\right]$, j = 1, ⋯, P. Therefore, for the second characteristic function φ _{ s }(ũ), we have
Consequently, the Hessian matrix ψ _{ s }(ũ) can be easily obtained as
where diag(·) represents an operator forming a diagonal matrix by assigning its arguments to the entries of the main diagonal.
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Acknowledgments
This work is supported in part by Natural Science Foundation of China under grant 61001106 and National Program on Key Basic Research Project of China under grant 2009CB320400, and Engineering and Physical Science Research Council of the UK under grant EP/H050000/1. The authors would like to thank the anonymous referees for their constructive comments for improving this paper and Dr. Mark Barnard for proofreading the manuscript.
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Gu, F., Zhang, H., Wang, W. et al. Generalized generating function with tucker decomposition and alternating least squares for underdetermined blind identification. EURASIP J. Adv. Signal Process. 2013, 124 (2013). https://doi.org/10.1186/168761802013124
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Keywords
 Blind identification
 Generalized generating function
 Tensor decomposition
 Tucker decomposition
 Underdetermined mixtures