A CORRELATION TENSOR–BASED MODEL FOR TIME VARIANT FREQUENCY SELECTIVE MIMO CHANNELS Martin Weis, Giovanni Del Galdo, and Martin Haardt Ilmenau University of Technology - Communications Research Laboratory P.O. Box 100565, 98684 Ilmenau, Germany [email protected], {giovanni.delgaldo, martin.haardt}@tu-ilmenau.de ABSTRACT this paper, we introduce a more general tensor–based chan- nel model, which truly captures the nature of MIMO chan- In this contribution we present a new analytical channel nels. The generalized Higher Order Singular Value Decom- model for frequency selective, time variant MIMO systems. position The model is based on a correlation tensor, which allows a (HOSVD) [4] gives us the possibility to analyze the eigen- natural description of multi–dimensional signals. By apply- structure of the channel along more dimensions, i.e., along ing the Higher Order Singular Value Decomposition (HO- space and frequency. SVD), we gain a better insight into the multi–dimensional The paper is organized as follows: Section 2 gives a eigenstructure of the channel. Applications of the model brief introduction of the relevant tensor algebra, which is include the denoising of measured channels and the possi- needed to understand the proposed model. Section 3 intro- bility to generate new synthetic channels displaying a given duces the tensor–based channel model and its applications. correlation in time, frequency, and space. The proposed Moreover, this section shows the applicability and validity model possesses advantages over existing 2–dimensional ei- of the model on channel measurements. In Section 4 the genmode–based channel models. In contrast to them, the conclusions are drawn. tensor–based model can cope with frequency and time se- lectivity in a natural way. 2. BASIC TENSOR CALCULUS 1. INTRODUCTION 2.1. Notation Multiple Input Multiple Output (MIMO) schemes offer the To facilitate the distinction between scalars, vectors, matri- chance to fulfill the challengingrequirements for future com- ces and higher–ordertensors, we use the following notation: munication systems, as higher data rates can be achieved scalars are denoted by lower–case italic letters (a, b, ...), by exploiting the spatial dimension. To investigate, design, vectors by boldface lower–case italic letters (a, b, ...), ma- and test new techniques, it is crucial to use realistic channel trices by boldface upper–case letters (A, B, ...), and ten- models. sors are denoted as upper–case, boldface, calligraphic let- We propose a tensor–based analytical channel model ters (A, B, ...). This notation is consistently used for lower– which, in contrast to traditional models, can cope with non– order parts of a given structure. For example, the entry with stationary time and frequency selective channels. The lat- row index i and column index j in a matrix A is symbol- ter are particularly relevant for wireless communications. ized by ai,j . Furthermore, the i–th column vector of A is We represent the frequency selective, time variant MIMO denoted as ai. As indices, mainly the letters i, j, k, and n channel as a 4–dimensional tensor H ∈ CMR×MT×Nf ×Nt , are used. The upper bounds for these indices are given by where MR and MT are the number of antennas at the trans- the upper–case letters I, J, K, and N, unless stated other- mitter and receiver, whereas Nf and Nt are the number of wise. samples taken in frequency and time, respectively. To visualize the spatial structure of the channel, eigen- 2.2. n–mode vectors and tensor unfoldings mode–based models have been introduced, such as [1, 2]. However, these models use a 2–dimensional correlation ma- In the (2–dimensional) matrix case we distinguish between trix which considers one dimension only. Alternatively, by a row vectors and column vectors. As a generalization of this cumbersome stacking of the channel coefficients, as in [2], idea, we build the n–mode vectors {a} of an N–th order I1×I2×···×IN it is possible to consider more dimensions. Moreover, by tensor A ∈ C by varying the index in of the following this approach, it is not possible to investigate the elements {ai1,...,in,...,iN } while keeping the other indices eigenmodes of different dimensions separately, whereas the fixed. In Figure 1, this is shown for a 3–dimensional ten- proposed tensor–based channel model allows this. sor. Please note that in general there are (I1 · I2 · · · In−1 · In [3], a tensor–based channel model was introduced. In+1 · · · IN ) such vectors. In the 2–dimenionalcase the col- The latter is however a tensor extension of [1], and therefore umn vectors are equal to the 1–mode vectors, and the row assumes a Kronecker like structure of the eigenmodes. In vectors are equal to the 2–mode vectors. The n–th unfold- I 3 2.3.2. The outer product I 3 I 3 We now define the outer productbetween 2 tensors. Assume an N–th order tensor A and a K–th order tensor B. Then, I 1 I 1 I 1 the outer product, denoted as (A ◦ B), is a (N + K)–th dimensional tensor whose entries are given by I I I 2 2 2 A B ( ◦ )i1,i2,...,iN ,j1,j2,...,jK = ai1,i2,...,iN · bj1,j2,...,jK , Fig. 1. Mode 1, 2, and 3 vectors of a 3–dimensional tensor. for all possible values of the indices. Therefore, the outer product creates a tensor with all combinations of possible pairwise element–products. In×(I1I2···In−1In+1···IN ) ing matrix A(n) ∈ C is the matrix consisting of all n–mode vectors. In [4], the ordering of the 2.3.3. The n–mode inner product n–mode vectors was defined in a cyclic way. In contrast to the definition in [4] we define the n–th unfolding matrix as The n–mode inner product is denoted as A = B •n C. The follows: resulting tensor A has order N + K − 2, where N and K are the orders of B ∈ CI1×···×IN and C ∈ CJ1×···×JN , A CIn×(I1I2···In−1In+1In+2···IN ) [n] = {aj,k} ∈ , respectively. It is related to the outer product and implies an additional summation over the n–th dimension of both with j = in and tensors. Therefore, we define the n–mode inner product as N l−1 In k =1+ (i − 1) · I . L q A = Bin=l ◦ Cjn=l , (4) l=1X,l=6 n q=1Y,q=6 n Xl=1 This definition assures that the indices of the n–mode vec- where Bin=l is the (N − 1)–th dimensional subtensor of B tors vary faster in the following ascending order which we obtain when we set the index along the dimension n equal to l. The tensor Cjn=l is defined in an analogous i1,i2,...,in−1,in+2,...,iN . (1) way. Please note that the tensors B and C must be of same size along the n–th dimension, and therefore In = Jn. This ordering becomes particularly important for our later derivations, especially for equation (23). Please note that 2.3.4. The vec(·) operator for tensors this unfolding definition is also consistent with the MATLABr command reshape. Therefore, we will refer to this un- The vec(·) operator stacks all elements of a tensor into a folding as the MATLAB–like unfolding. vector. Thereby the indices in of an N–dimensional tensor A vary in the following ascending order 2.3. Tensor operations i1,i2,...,iN−1,iN . 2.3.1. The n–mode product Please note that the unfolding definition in Section 2.2 en- To perform a generalized Higher Order Singular Value De- sures that the vec(·) operation for an N–dimensional tensor composition (HOSVD), it is necessary to transform the n– is equal to the transpose of its (N + 1)–th unfolding mode vector space of a tensor. This can be done with the A AT n–mode product between a tensor and a matrix. Let us as- vec( )= [N+1] . (5) A CI1×I2×···×IN sume a tensor = {ai1,i2,...,iN } ∈ and a matrix U ∈ CJn×In . Then the n–mode product, denoted 2.4. Higher Order Singular Value Decomposition by A×n U,isa (I1 ×I2 ×···×In−1 ×Jn ×In+1 ×···×IN ) I ×I ×···×IN tensor, whose entries are given by Every N–th order complex tensor A ∈ C 1 2 can be decomposed into the form (A ×n U) = i1,i2,...,in−1,jn,in+1,...,iN (1) (2) (N) A = S ×1 U ×2 U ···×N U , (6) In (2) n ai1,i2,...,in−1,in,in+1...,iN · ujn,in , in whichthe matricesof the n–mode singular vectors U ( ) = iXn=1 (n) (n) (n) CIn×In u1 , u2 ,..., uIn ∈ are unitary, and the core h i for all possible values of the indices. With the help of the tensor S ∈ CI1×I2×···×IN is a tensor of the same the size unfolding definition from above we can write the n–mode as A. The basis matrices U (n) contain the left singular product also in terms of matrix operations. Then, the n–th (n) (n) (n) vectors u1 , u2 ,..., uIn of the matrix unfoldings A[n]. unfolding of the resulting tensor B can be calculated as The core tensor S can be calculated with the equation B = U · A . (3) (1)H (2)H (N)H [n] [n] S = A ×1 U ×2 U ···×N U , (7) M ×M H(f0, t0) ∈ C R T frequency H CMR×MT×Nf ×TW 1 ∈ T k =1 W H 2 TW f0 k =2 M ×M ×N Nt H(t1) ∈ C R T f time t0 t1 H N Fig.