CP decomposition approach to blind separation for DS-CDMA system using a new performance index

2.1 Notations and definitions

Let us first introduce some essential notation. Scalars are denoted by lowercase letters, e.g., a. Vectors are denoted by boldface lowercase letters, e.g., a; matrices are denoted by boldface capital letters, e.g., A. Higher-order tensors are denoted by boldface Euler script letters, e.g., . The p th column of a matrix A is denoted a _p , the (i,j) entry of a matrix A is denoted by A _ij , and element (i,j,k) of a third-order tensor is denoted by T _ijk . 1 will represent a vector containing ones, and I the identity matrix.

Definition 2.1

The scalar product of two same-sized tensors , $\mathcal{Y}\epsilon {\mathbb{C}}^{I_1\times I_2\times \cdots \times I_N}$ is defined as:

$〈\mathcal{X},\mathcal{Y}〉=\sum _{i_1=1}^{I_1}\cdots \sum _{i_N=1}^{I_N}{X}_{i_1,\cdots ,i_N}{Y}_{i_1,\cdots ,i_N}^{\ast }.$

(1)

where $X_{i_1,\cdots ,i_N}$ is the (i₁,⋯, i _N ) elements of the N ^{t h} order of tensor.

2.2 Preliminaries

A tensor of order d is an object defined on a product between d linear spaces. Once the bases of these spaces are fixed, a d t h order tensor can be represented by a d-way array (a hyper matrix) of coordinates [4]. The order of a tensor thus corresponds to the number of indices of the associated array. We are interested in decomposing a third-order tensor as:

$\mathcal{T}=\sum _{r=1}^R{\lambda }_r\mathcal{D}\left(r\right).$

(7)

where A _ir (resp. B _jr and C _kr ) denote the entries of vector a _r (resp. b _r and c _r ). The above decomposition is called the canonical polyadic decomposition (CP) of tensor . The model (7) can be written in a compact form using the Khatri-Rao product, as:

$\begin{array}{lcr}{\mathbf{\text{T}}}_1^{I,\mathit{\text{KJ}}}& =& \mathbf{A}\mathit{\Lambda }{\left(\mathbf{C}\odot \mathbf{B}\right)}^T,\\ {\mathbf{\text{T}}}_2^{J,\mathit{\text{KI}}}& =& \mathbf{B}\mathit{\Lambda }{\left(\mathbf{C}\odot \mathbf{A}\right)}^T,\\ {\mathbf{\text{T}}}_3^{K,\mathit{\text{JI}}}& =& \mathbf{C}\mathit{\Lambda }{\left(\mathbf{B}\odot \mathbf{A}\right)}^T.\end{array}$

where ${\mathbf{\text{T}}}_1^{I,\mathit{\text{KJ}}}\left(\text{resp.}{\mathbf{\text{T}}}_2^{J,\mathit{\text{KI}}}\text{and}{\mathbf{\text{T}}}_3^{K,\mathit{\text{JI}}}\right)$ is the matrix of size I × K J (resp. J × K I and K × J I) obtained by unfolding the array of size I × J × K in the first mode (resp. the second mode and the third mode), and Λ is the R × R diagonal matrix defined as Λ= D i a g{λ₁,…, λ _R }; see [5] for further details on matrix unfoldings.

The explicit handwriting of decomposable tensors as given in (8) is subject to scale indeterminacies. In the tensor literature, optimization of the CP decomposition (8) has been carried out without isolating the scaling factor Λ, which is generally included in one of the loading matrices, so that Λ= I. In [5], Kolda and Bader proposed to reduce the indeterminacies by normalizing the vectors and storing the norms in Λ. Our first proposal is to pull the factors λ _r outside the product and calculate the optimal value of the scaling factor, which permits to monitor the conditioning of the problem. Scaling indeterminacies are then clearly reduced to unit modulus but are not completely fixed, hence the difficulty in estimating the identification error of loading matrices A, B and C. Our second proposal (Section 5) is to calculate the 3R complex phases (reducing to signs in the real case).

3.1 Low-rank approximation

By convention (A,B,C).Λ denotes the tensor of rank R of coordinates $\sum _{r=1}^R{\lambda }_r{A}_{\mathit{\text{ir}}}{B}_{\mathit{\text{jr}}}{C}_{\mathit{\text{kr}}}$. Minimizing error (10) means finding the best rank-R approximate of and its CP decomposition. The cost function (10) can also be written in three equivalent compact forms with respect to the three loading matrices:

$\begin{array}{lcr}{\rm Y}\left(\mathbf{A},\mathbf{B},\mathbf{C},\mathit{\Lambda }\right)& =& {∥{\mathbf{\text{T}}}_1^{I,\mathit{\text{KJ}}}-\mathbf{A}\mathit{\Lambda }{\left(\mathbf{C}\odot \mathbf{B}\right)}^T∥}_F^2,\\ =& {∥{\mathbf{\text{T}}}_2^{J,\mathit{\text{KI}}}-\mathbf{B}\mathit{\Lambda }{\left(\mathbf{C}\odot \mathbf{A}\right)}^T∥}_F^2,\\ =& {∥{\mathbf{\text{T}}}_3^{K,\mathit{\text{JI}}}-\mathbf{C}\mathit{\Lambda }{\left(\mathbf{B}\odot \mathbf{A}\right)}^T∥}_F^2.\end{array}$

3.2 Conditioning of the problem

In view of matrix G, we can see that coherences play a role in the conditioning of the problem. From Equations 11 to 12, and since diagonal entries of G are equal to 1, it is indeed clear that imposing cross scalar products of the form ${\mathbf{a}}_p^H{\mathbf{a}}_q$ to have a modulus strictly smaller than 1 will lead with greater chances to an acceptable conditioning. Also note that scalar products do not appear individually in (12) but through their products, since entries of G can also be written as $G_{\mathit{\text{pq}}}={\mathbf{a}}_p^H{\mathbf{a}}_q{\mathbf{b}}_p^H{\mathbf{b}}_q{\mathbf{c}}_p^H{\mathbf{c}}_q$. This statement has deeper implications, particularly in existence and uniqueness of the solution to Problem (10), as subsequently elaborated.

3.3 Existence

According to the results in [15, 16], the infimum of (10) may not be reachable. In fact, the set of tensors of rank at most ξ is not closed if ξ > 1. Examples of the lack of closeness have been provided in the literature [15, 16], which suffice to prove it. In other words, it may happen that for a given tensor, and for any rank-r approximation of it, there always exists another better rank-r approximation.

the infimum of (10) is reached. It may be seen that this condition already gives a quantitative bound to the conditioning of (11) because coherences bound extra-diagonal entries of matrix G, which has ones on its diagonal.

3.4 Uniqueness

The uniqueness of the tensor decomposition can be ensured by using a sufficient condition based on Kruskal’s theorem (9), previously mentioned.

4.1 Alternating least squares algorithm

The ALS algorithm was proposed for CP computation by Carroll and Chang in [20] and Harshman in [3] and still stays the workhorse algorithm today, mainly owing to its ease of implementation [21]. ALS is hence the classical solution to minimize the cost function (10), despite its lack of convergence proof. This iterative algorithm alternates among the estimation of matrices A, B, and C.

where T^{I, K J} is the unfolding matrix of size I × J K defined in Section 2.2, and () ^† is the Moore-Penrose pseudo inverse. By symmetry, the expressions are similar for $\hat{\mathbf{B}}$ and $\hat{\mathbf{C}}$.

4.2 Proposed algorithms

Therefore, we need to find three matrices A, B, and C of unit norm columns which minimize (16). Stack these three matrices in a I+J+K by R matrix denoted by X. The objective can now be also written Υ(X,Λ), for the sake of convenience.

where M of size R × R and N of size I × R. The gradient of Υ with respect to B and C is similar, taking into account the fact that matrices M and N need to be defined accordingly (for the gradient of Υ with respect to B and C, the dimension of matrix N is J × R and K × R, respectively, while the dimension of M is always R × R).

The difficulty that arises in constrained optimization is to make sure that the move remains within the feasible set, , defined by the constraints. In the following subsections, we propose two versions of our algorithm, with two different ways of calculating scale factor Λ.

Descent algorithms are also determined by the steps that will be executed in the chosen direction. There are various methods for the step selection, and the most widely used are Backtracking and Armijo[22]. To compute the stepsize ℓ(k) in Algorithm 1 and Algorithm 2, we use a popular inexact line search method, very simple and quite effective, which is the backtracking line search. It depends on two fixed constants α, β with 0 < α < 0.5 and 0 < β < 1.

4.2.1 Algorithm 1

In the recent work on CP tensor decomposition, optimization of the objective function (10) was made without explicitly considering the factor Λ. More precisely in most contributions, the scaling factor is integrated into loading matrices, so that Λ may be set to the identity. The first solution we propose is to use a projected gradient algorithm while calculating Λ as the product of normalizing factors of matrices A, B, and C:

$\mathit{\Lambda }\left(k\right)=\mathit{\Lambda }(k-1)⊡{\mathit{\Lambda }}_A⊡{\mathit{\Lambda }}_B⊡{\mathit{\Lambda }}_C.$

(22)

where ⊡ is the Hadamard product, Λ _A =Diag{∥a₁∥,⋯,∥a _R ∥}, and similar definitions for Λ _B and Λ _C . This approach, which we call ‘Algorithm 1’ can be described by the pseudo-code below:

4.2.2 Algorithm 2

The other approach is to consider Λ as an additional variable. By canceling the gradient of Υ(X,Λ) with respect to Λ, one obtains Equation 11, which can be solved for Λ when X is fixed. This gives the algorithms below.

4.2.2.0 Stopping criterion

The convergence is usually considered to be obtained at the k th iteration when the error between tensor , and the tensor reconstructed from the estimated loading matrices, does not significantly change between iterations k and k + 1.

However, in the complex case, the phase of the entries of loading matrices found at the end of the algorithm – as defined by the stopping criterion above - is different from the original. To remedy this problem, we propose a new performance criterion in order to minimize the distance between the original and the estimated matrices. Although this criterion is not usable when actual loading matrices are unknown, it still permits to assess the performances effectively attained.

6.1 Example 1: random loading matrices

In order to illustrate the behavior and performances of the proposed algorithms, we first report simulation results run on random tensors. In a first stage, we compare ‘Algorithm 2’, ‘Algorithm 1’, and the gradient descent algorithm named herein ‘Algorithm 0’. We will analyze the errors of matrix estimation obtained through the performance criterion proposed in Section 5. Two scenarios are analyzed using random tensors: (i) one tensor of rank 2 with dimensions 3 × 3 × 3 and (ii) another tensor of rank 5 with dimensions 8 × 8 × 8. Loading matrices are initialized with random-valued columns and the two tensors are corrupted by an additive Gaussian noise.Figures 1 and 2 depict matrix estimation errors implied in (25) as a function of SNR. As expected, it can be seen that the results using Algorithm 0 show a poor performance when compared to the results obtained with Algorithms 1 and 2 (curves with diamonds and circles). This supports the idea that our algorithms isolating the scaling matrix are attractive. Furthermore, we check that when the phase constraints are neglected in the calculation of the performance measure, the results are significantly more optimistic, particularly at high SNR (curves of the same color), which supports the interest of using our performance index defined in (5).In order to show the significant difference between Algorithms 1 and 2, we will examine the convergence speed of the tensor reconstruction according to the number of iterations, since the final error is about the same (cf. Figures 1 and 2). We consider the case of a 3 × 3 × 3 tensor of rank 2. The latter was constructed from three 3 × 2 Gaussian matrices drawn randomly. The results are presented in Figure 3, and show that in all our experiments, Algorithm 2 converged faster in terms of number of iterations, while Algorithm 1 and Algorithm 0 require more iterations to reach convergence.

6.2 Example 2: application To DS-CDMA system

In this example, we place ourselves in a blind context. We assume that the receiver has no knowledge neither on the spreading codes nor on symbol sequences. Classically, telecommunications blind techniques are based on some a priori knowledge, such as temporal properties of transmitted signals or the spatial properties of the receiver [25–27].

Recently, algebraic tensor methods have received considerable attention in signal processing [2]. It also turns out that multilinear algebra tools are often more powerful than their matrix equivalent. Sidiropoulos et al. are the first to adopt tensor approaches in the telecommunications field in 2000 [12]. They observed that the samples of a CDMA signal received by an array of antennas can be stored in a cube, each dimension corresponding to a diversity (coding diversity, temporal diversity, and spatial diversity). Thus, they showed that the deterministic blind separation problem of CDMA signals can be solved by the CP decomposition [28].

In this example, we propose to apply the CP decomposition algorithms as detailed in Section 4 with the new performance criterion on the DS-CDMA technique. A comparison with the ALS algorithm is then made.

6.2.1 Tensor modeling

We consider R users with one transmitting antenna, transmitting simultaneously their signals to an array of K receiving antennas. In other words, we consider a communication system of type ‘multiuser SIMO’.

For example, assume the user r transmits the symbols s _r of size J. These symbols are spreaded by c _r CDMA code of length I, uniquely allocated to user r and assumed to be unknown at the receiver. These codes are binary sequences taking values from {−1,1} and could be non-orthogonal. The spreading sequence propagates along a single path in a memoryless channel and is received by the antenna array under angle of arrival θ _r . Each of the K antennas receives a signal Y _k (t) of size J × I. Our approach for detection and separation of the received signals is to exploit the multilinear algebraic structure of these signals using the new performance criterion. We observe these signals during a time span of length J T _s , where T _s is the symbol period. The Y _k (t) signals are sampled at the chip period T _c = T _s /I, where I is the spreading factor. Therefore, each antenna provides a set of IJ samples that can be ordered in a tensor of order 3, denoted $\mathcal{Y}\in {\mathbb{C}}^{I\times J\times K}$. The Y _ijk component of this tensor that corresponds to the sample of the overall signal received by the k th antenna at time i of the j th symbol period can be written as follows:

$Y_{\mathit{\text{ijk}}}=\sum _{r=1}^R{A}_{\mathit{\text{kr}}}{S}_{\mathit{\text{jr}}}{C}_{\mathit{\text{ir}}}i\in \left[1,I\right]j\in \left[1,J\right]k\in \left[1,K\right].$

(26)

where the complex scalar A _kr = β _r a _k (θ _r ), with β _r is the fading coefficient of the r th user and a _k (θ _r ) the response of the antenna k at the angle of arrival θ _r .

Separating the received signals is then equivalent to decompose the tensor into a sum of R contributions, where R represents the number of active users in the system. To calculate the CP decomposition of , we resort to Algorithm 2 presented in Section 4 with performance index (25). The detection and separation of the matrix S of transmitted symbols will be made, using the following objective function:

${\rm Y}\left(\mathbf{A},\mathbf{C},\mathbf{S},\mathit{\Lambda }\right)={∥\mathcal{Y}-\sum _{r=1}^R{\lambda }_r{}_{\mathbf{a}{}_r{}^{\otimes }}{\mathbf{c}{}_r}_{{}^{\otimes }}{\mathbf{s}}_r∥}_F^2.$

(27)

where a _r , c _r , and s _r are the normalized vectors; the scaling ambiguities on the estimated symbols are eliminated by normalizing each symbol sequence by its norm and calculating the exact scaling factor Λ. As to correct the phases of the three estimated matrices, we will use the exact performance index (28) seen in Section 5.

$\begin{array}{ll}\mathcal{E}\left(\mathcal{T};\mathbf{A},\mathbf{C},\mathbf{S},\mathit{\Lambda }\right)& =\underset{\pi \in \Pi }{min}\sum _{r=1}^R\left(\underset{\phi ,\psi ,\chi }{min}\left\{{∥{\mathbf{a}}_r-e^{\mathrm{ȷ\phi }}{\widehat{\mathbf{a}}}_{\pi \left(r\right)}∥}^2\right.\right.\\ \left.\left.+{∥{\mathbf{c}}_r-e^{\mathrm{ȷ\psi }}{\widehat{\mathbf{c}}}_{\pi \left(r\right)}∥}^2+{∥{\mathbf{s}}_r-e^{\mathrm{ȷ\chi }}{\widehat{\mathbf{s}}}_{\pi \left(r\right)}∥}^2\right\}\right).\end{array}$

(28)

6.2.2 Simulation

In this experiment, we present the performance of the receiver algorithms (Algorithm 1 and Algorithm 2 with new performance criterion) which estimate blindly the symbol S.

We consider R = 4 users communicate simultaneously in the same bandwidth. Each user transmits a sequence J = 20 of consecutive symbols and is uniquely assigned a spreading sequence of length I = 10. The user symbols are generated from an i.i.d distribution and are modulated using a pseudo-random quaternary phase shift keying (QPSK) sequence. The signal is transmitted to a receiver of K = 4 antennas. The angles of arrival are described in Table 1.In Figure 4, we will illustrate the ability of the blind tensorial receiver Algorithm 2 using the new performance criterion and the ALS receiver for noisy data extraction. Using Monte Carlo simulations, we will represent the evolution of bit error rate (BER) according to the signal-to-noise ratio (SNR). These results shown in Figure 4 indicate that the performance of the proposed receiver Algorithm 2 is better since it converges faster than the ALS algorithm. That is tied to the normalization of the factor matrices and the exact calculation of the scaling factor. Moreover, we can see that BER of Algorithm 2 without the use of our criterion performance is very optimistic, which proves the interest of our study.

	Angles of arrival
Figure 4	(-60, -30, 0, 20)

The impact of factor $\frac{K}{R}$, which represents the number of receiving antennas per user is illustrated in Figure 5. In this figure, curve 1 represents the case where the number of antenna is K=4 and the number of users is R=2, K=4, and R=4 for the second curve, while for the third curve K=2 and R=5. The angles of arrival are described in Table 2. As a result, the overall system performance is enhanced when increasing the factor $\frac{K}{R}$, indicating the importance of spatial diversity.

	Angles of arrival
Curve 1	(-10, 20)
Curve 2	(-60, -40, 0, 20)
Curve 3	(-60,-50, 10, 40, 80)

Detailed Λ optimal calculation

Performance criterion details

Solving the sixth degree equation yields x. Replacing x in $siny=\frac{{\rho }_{\mathbf{a}}}{{\rho }_{\mathbf{b}}}sinx$ yields y.