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A correlation tensor-based model for time variant frequency selective MIMO channels

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A correlation tensor-based model for time variant frequency selective MIMO channels
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  A CORRELATION TENSOR–BASED MODEL FOR TIME VARIANT FREQUENCYSELECTIVE MIMO CHANNELS  Martin Weis, Giovanni Del Galdo, and Martin Haardt  Ilmenau University of Technology - Communications Research LaboratoryP.O. Box 100565, 98684 Ilmenau, Germanymartin.weis@stud.tu-ilmenau.de,  { giovanni.delgaldo, martin.haardt } @tu-ilmenau.de ABSTRACT In this contribution we present a new analytical channelmodel for frequencyselective, time variant MIMO systems.The model is based on a correlation tensor, which allows anatural description of multi–dimensionalsignals. By apply-ing the Higher Order Singular Value Decomposition (HO-SVD), we gain a better insight into the multi–dimensionaleigenstructure of the channel. Applications of the modelinclude the denoising of measured channels and the possi-bility to generate new synthetic channels displaying a givencorrelation in time, frequency, and space. The proposedmodelpossessesadvantagesoverexisting2–dimensionalei-genmode–based channel models. In contrast to them, thetensor–based model can cope with frequency and time se-lectivity in a natural way. 1. INTRODUCTION Multiple Input Multiple Output (MIMO) schemes offer thechancetofulfillthechallengingrequirementsforfuturecom-munication systems, as higher data rates can be achievedby exploiting the spatial dimension. To investigate, design,and test new techniques, it is crucial to use realistic channelmodels.We propose a tensor–based analytical channel modelwhich, in contrast to traditional models, can cope with non–stationary time and frequency selective channels. The lat-ter are particularly relevant for wireless communications.We represent the frequency selective, time variant MIMOchannel as a 4–dimensional tensor H ∈  C M  R × M  T × N  f  × N  t ,where  M  R  and  M  T  are the number of antennas at the trans-mitter and receiver, whereas  N  f   and  N  t  are the number of samples taken in frequency and time, respectively.To visualize the spatial structure of the channel, eigen-mode–based models have been introduced, such as [1, 2].However,these modelsuse a 2–dimensionalcorrelationma-trixwhichconsidersonedimensiononly. Alternatively,byacumbersome stacking of the channel coefficients, as in [2],it is possible to consider more dimensions. Moreover, byfollowing this approach, it is not possible to investigate theeigenmodes of different dimensions separately, whereas theproposed tensor–based channel model allows this.In [3], a tensor–based channel model was introduced.Thelatter is howevera tensorextensionof[1], andthereforeassumes a Kronecker like structure of the eigenmodes. Inthis paper, we introduce a more general tensor–based chan-nel model, which truly captures the nature of MIMO chan-nels. The generalized Higher Order Singular Value Decom-position(HOSVD) [4] gives us the possibility to analyze the eigen-structure of the channel along more dimensions, i.e., alongspace and frequency.The paper is organized as follows: Section 2 gives abrief introduction of the relevant tensor algebra, which isneeded to understand the proposed model. Section 3 intro-duces the tensor–based channel model and its applications.Moreover, this section shows the applicability and validityof the model on channel measurements. In Section 4 theconclusions are drawn. 2. BASIC TENSOR CALCULUS2.1. Notation To facilitate the distinction between scalars, vectors, matri-ces andhigher–ordertensors, we use the followingnotation:scalars are denoted by lower–case italic letters  ( a,b,... ) ,vectors by boldface lower–case italic letters  ( a , b ,... ) , ma-trices by boldface upper–case letters  ( A , B ,... ) , and ten-sors are denoted as upper–case, boldface, calligraphic let-ters ( A  , B ,... ) . Thisnotationisconsistentlyusedforlower–order parts of a given structure. For example, the entry withrow index  i  and column index  j  in a matrix  A  is symbol-ized by  a i,j . Furthermore, the  i –th column vector of   A  isdenoted as  a i . As indices, mainly the letters  i ,  j ,  k , and  n are used. The upper bounds for these indices are given bythe upper–case letters  I  ,  J  ,  K  , and  N  , unless stated other-wise. 2.2.  n –mode vectors and tensor unfoldings In the (2–dimensional) matrix case we distinguish betweenrow vectors and column vectors. As a generalization of thisidea, we build the  n –mode vectors  { a }  of an  N  –th ordertensor  A   ∈  C I  1 × I  2 ×···× I  N  by varying the index  i n  of theelements  { a i 1 ,...,i n ,...,i N  }  while keeping the other indicesfixed. In Figure 1, this is shown for a 3–dimensional ten-sor. Please note that in general there are  ( I  1  ·  I  2  ··· I  n − 1  · I  n +1  ··· I  N  )  suchvectors. Inthe2–dimenionalcasethecol-umn vectors are equal to the 1–mode vectors, and the rowvectors are equal to the 2–mode vectors. The  n –th unfold-  1  I  2  I  2  I  1  I  3  I  2  I  1  I  3  I  3  I  Fig. 1 . Mode 1, 2, and 3 vectors of a 3–dimensional tensor.ing matrix A  ( n )  ∈  C I  n × ( I  1 I  2 ··· I  n − 1 I  n +1 ··· I  N  ) is the matrixconsisting of all  n –mode vectors. In [4], the ordering of the n –mode vectors was defined in a cyclic way. In contrast tothe definition in [4] we define the  n –th unfolding matrix asfollows: A  [ n ]  =  { a j,k } ∈ C I  n × ( I  1 I  2 ··· I  n − 1 I  n +1 I  n +2 ··· I  N  ) , with  j  =  i n  and k  = 1 + N   l =1 ,l  = n ( i L  −  1)  · l − 1  q =1 ,q  = n I  q  . This definition assures that the indices of the  n –mode vec-tors vary faster in the following ascending order i 1 ,i 2 ,...,i n − 1 ,i n +2 ,...,i N   .  (1)This ordering becomes particularly important for our laterderivations, especially for equation (23). Please note thatthisunfoldingdefinitionis alsoconsistentwiththe MATLAB  command  reshape . Therefore, we will refer to this un-folding as the MATLAB–like unfolding. 2.3. Tensor operations 2.3.1. The  n  –mode product  To perform a generalized Higher Order Singular Value De-composition (HOSVD), it is necessary to transform the  n –mode vector space of a tensor. This can be done with the n –mode product between a tensor and a matrix. Let us as-sume a tensor  A   =  { a i 1 ,i 2 ,...,i N  } ∈  C I  1 × I  2 ×···× I  N  and amatrix  U   ∈  C J  n × I  n . Then the  n –mode product, denotedby A  × n U  , is a  ( I  1 × I  2 ×···× I  n − 1 × J  n × I  n +1 ×···× I  N  ) tensor, whose entries are given by ( A  × n U  ) i 1 ,i 2 ,...,i n − 1 ,j n ,i n +1 ,...,i N  = I  n  i n =1 a i 1 ,i 2 ,...,i n − 1 ,i n ,i n +1 ...,i N   ·  u j n ,i n  , (2)for all possible values of the indices. With the help of theunfolding definition from above we can write the  n –modeproduct also in terms of matrix operations. Then, the  n –thunfolding of the resulting tensor B can be calculated as B [ n ]  =  U   · A  [ n ]  .  (3) 2.3.2. The outer product  We nowdefinetheouterproductbetween2tensors. Assumean  N  –th order tensor A  and a  K  –th order tensor B . Then,the outer product, denoted as  ( A  ◦ B ) , is a  ( N   +  K  ) –thdimensional tensor whose entries are given by ( A  ◦ B ) i 1 ,i 2 ,...,i N  ,j 1 ,j 2 ,...,j K =  a i 1 ,i 2 ,...,i N   ·  b j 1 ,j 2 ,...,j K  , for all possible values of the indices. Therefore, the outerproduct creates a tensor with all combinations of possiblepairwise element–products. 2.3.3. The  n  –mode inner product  The  n –mode inner product is denoted as A  = B • n C . Theresulting tensor A  has order  N   +  K   −  2 , where  N   and  K  are the orders of   B  ∈  C I  1 ×···× I  N  and  C  ∈  C J  1 ×···× J  N  ,respectively. It is related to the outer product and impliesan additional summation over the  n –th dimension of bothtensors. Therefore, we define the  n –mode inner product as A  = I  n  l =1 B i n = l  ◦ C j n = l  ,  (4)where B i n = l  is the  ( N   −  1) –th dimensional subtensor of  B which we obtain whenwe set the indexalongthe dimension n  equal to  l . The tensor  C j n = l  is defined in an analogousway. Please note that the tensors B and C must be of samesize along the  n –th dimension, and therefore  I  n  =  J  n . 2.3.4. The  vec( · )  operator for tensors The  vec( · )  operator stacks all elements of a tensor into avector. Thereby the indices  i n  of an  N  –dimensional tensor A  vary in the following ascending order i 1 ,i 2 ,...,i N  − 1 ,i N   . Please note that the unfolding definition in Section 2.2 en-sures that the  vec( · )  operation for an  N  –dimensional tensoris equal to the transpose of its  ( N   + 1) –th unfolding vec( A  ) = A  T[ N  +1]  .  (5) 2.4. Higher Order Singular Value Decomposition Every  N  –th order complex tensor A   ∈  C I  1 × I  2 ×···× I  N  canbe decomposed into the form A  = S  × 1 U  (1) × 2 U  (2) ··· × N   U  ( N  ) ,  (6)inwhichthematricesofthe n –modesingularvectors U  ( n ) =  u ( n )1  , u ( n )2  ,..., u ( n ) I  n   ∈  C I  n × I  n are unitary, and the coretensor S   ∈  C I  1 × I  2 ×···× I  N  is a tensor of the same the sizeas  A  . The basis matrices  U  ( n ) contain the left singularvectors u ( n )1  , u ( n )2  ,..., u ( n ) I  n of the matrix unfoldings A [ n ] .The core tensor S  can be calculated with the equation S   = A  × 1 U  (1) H × 2 U  (2) H ··· × N   U  ( N  ) H ,  (7)  f  0 t 0  t 1 frequencytime H  ( f  0 ,t 0 )  ∈  C M  R × M  T H ( t 1 )  ∈ C M  R × M  T × N  f  Fig. 2 . Definition of the channel tensor. The 2–dimensionalsubtensor  H  ( f  0 ,t 0 ) , and the 3–dimensional subtensor H ( t 1 )  are depicted.where  ( · ) H denotes the Hermitian transpose. The core ten-sor fulfills some special properties, especially the propertyof all orthogonality, which means that the rows of all un-folding matrices S  [ n ]  are orthogonal, c.f. [4]. With the helpof the Kronecker product it is possible to write equation (6)in terms of matrix operations. A  [ n ]  =  U  ( n ) · S  [ n ] ·  U  ( N  ) ⊗ ··· ⊗ U  ( n +1) ⊗ U  ( n − 1) ⊗ ··· ⊗ U  (1)  T .  (8)Pleasenotethatthisformulais onlyvalidfortheMATLAB–like unfolding defined in Section 2.2. 3. TENSOR CHANNEL MODEL As alreadymentioned,we representthe channelcoefficientsin form of the tensor H ∈ C M  R × M  T × N  f  × N  t ,  (9)where M  R  and  M  T  are the numberof antennas at the trans-mitter and receiver, and  N  f   and  N  t  are the number of sam-ples taken in frequency and time, respectively. Please notethat the frequency domain of the channel is connected to itsdelay time  τ   via a Fourier Transform.Similarly to [3, 5], we now define the channel correla-tion tensor as R = E { H ( t )  ◦ H ( t ) ∗ } ∈ C M  R × M  T × N  f  × M  R × M  T × N  f  , (10)where H ( t )  ∈ C M  R × M  T × N  f  is the frequency selectiveMIMO channel at time snapshot  t .Assuming that the channel is block–wise stationary intime, we define an averaging window of size  T  W , so thatthe channel within the  k –th window, denoted with H k , canbe assumed stationary (see Figure 3). The over–all channelmatrix H is then defined as H =  H 1 4 H 1 4  ···  4 H ⌊  N tT  W ⌋   ,  (11)where  4  denotes the concatenation of the tensors  H k along the 4–th dimension, as introduced in [5]. We com-pute an estimate of the  k –th correlation tensor by averagingin time, as R k  ≈  ˆ R k  = 1 T  W · T  W  n =1 ( H k ) i 4 = n  ◦  ( H k ) ∗ i 4 = n = 1 T  W · H k  • 4 H ∗ k  , (12) T  W T  W T  W N  t N  f  H 1  ∈ C M  R × M  T × N  f  × T  W H 2 H ⌊  N  t T  W ⌋ k  = 1 k  = 2 k  =  ⌊  N  t T  W ⌋ Fig. 3 . Definition of the non-overlapping stationarity win-dows. Each window has  T  W  time samples. In the otherdimensions, each window is of same size as H .where  • 4  denotes the  4 –mode inner product. Please notethat  n  is the time index within the  k –th window. To get in-sight intothespatial structureof thechannel,wedecomposethis correlation tensor via the following HOSVD ˆ R k  = S  k × 1 U  (1) k  × 2 U  (2) k  × 3 U  (3) k  × 4 U  (4) k  × 5 U  (5) k  × 6 U  (6) k  . (13)The matrices  U  ( n ) k  contain the left singular vectors of the n –th unfolding matrices  (  ˆ R  k ) [ n ]  of   ˆ R k . The symmetriesof the correlation tensor are also reflected by its HOSVD.Therefore,we can choose a HOSVD such that the followingequations hold U  (1) k  =  U  (4) ∗ k  , U  (2) k  =  U  (5) ∗ k  , U  (3) k  =  U  (6) ∗ k  . (14)In this case, equation (13) can be simplified to ˆ R k  = S  k  × 1 U  (1) k  × 2 U  (2) k  × 3 U  (3) k × 4 U  (1) ∗ k  × 5 U  (2) ∗ k  × 6 U  (3) ∗ k  , (15)and the core tensor  S  k , according to Section 2.4, can becalculated via S  k  =  ˆ R k  × 1 U  (1) H k  × 2 U  (2) H k  × 3 U  (3) H k × 4 U  (1) T k  × 5 U  (2) T k  × 6 U  (3) T k  . (16)The proposed channel model consists in computing thecorrelationtensors  ˆ R k  forallwindows,i.e., ∀ k  = 1 ... ⌊  N  t T  W ⌋ ,as they describe exhaustively the correlation properties of the channel, seen as a temporal block stationary stochasticprocess.Inthe followingwe givea briefdescriptionoftwo appli-cations of the proposed channel model, namely the genera-tion of newchannelrealizationsandthe subspace–basedde-noising of a channel measurement. Then we apply the cor-relation tensor–based channel model to channel measure-ments, and compare its performance with the tensor–basedchannel model presented in [3].  3.1. Channel Synthesis In the 2–dimensionalcase [1, 2], the joint spatial correlationmatrix is defined as R  k  = E { vec( H k )  ·  vec( H k ) H }  .  (17)Similarly to (12), we can compute an estimate of   R  k , de-noted by  ˆ R  k  with ˆ R  k  = 1 T  W T  W  n =1 vec  ( H k ) i 4 = n  · vec  ( H k ) i 4 = n  H .  (18)From the information given in the correlation matrix  ˆ R  k , itis possibletoconstructanewrandomsyntheticchannel  ˜ H k ,displaying the same spatio–frequency correlation as H k ( t ) via vec   ˜ H k   =  X  k  · g ,  (19)where the entries of the vector g  are i.i.d. zero mean com-plex Gaussian random numbers with unit variance, and X  k  =  ˆ R  12 k  .  (20)The matrix X  k  is computedvia a 2–dimensionaleigenvaluedecomposition of  ˆ R  k  = U  k  · Σ k  · U  H k  ,  (21)where the matrix  U  k  contains all eigenvectors of   ˆ R  k  and Σ k  is the diagonal matrix containing the eigenvalues. Thenwe define X  k  as follows ˆ R  12 k  =  U   · Σ 12 =  X  k  .  (22)By computing the matrix X  k  we can generate new randomsyntheticchannelsbymeansofequation(19). Inthefollow-ing we show that it is also possible to compute the matrix X  k  from the HOSVD of the correlation tensor  ˆ R k .WiththehelpoftheMATLAB–likeunfoldingfromSec-tion 2, the connection between the estimated 2D correlationmatrix  ˆ R  k  and the estimated correlation tensor  ˆ R k  is givenby vec   ˆ R  k   =   ˆ R  k  T[3] =   ˆ R k  T[7] = vec   ˆ R k   .  (23)To compute the matrix  X  k , we can either compute a SVDof   ˆ R  k , or apply a HOSVD on  ˆ R k . This relation betweenthe 2–dimensional model and the correlation tensor–basedmodelfollows fromthe followingderivation. For the reasonof simplicity we first define the following unitary matrix: U  e k  =  U  (3) k  ⊗ U  (2) k  ⊗ U  (1) k  .  (24)With the Kronecker version of the HOSVD (8) it followsfrom the latter relation vec   ˆ R  k   = vec   ˆ R k   =   ˆ R k  T[7] =  U  e ∗ k  ⊗ U  e k  ·  ( S  k ) T[7] = vec  U  e k  ·  unvec I  × I   ( S  k ) T[7]  · U  e H k  . Here  unvec I  × I   ( S  k ) T[7]  denotes the inverse matrix  vec( · ) operation applied to the 7–th unfolding of the core tensor S  k . Therefore,  unvec I  × I   ( S  k ) T[7]   is a square matrix of same size as  ˆ R  k  with  I   =  M  R M  T N  f  . The correlationmatrix  ˆ R  k  can be calculated from the HOSVD of the tensor ˆ R k  via ˆ R  k  =  U  e k  ·  unvec I  × I   ( S  k ) T[7]  · U  e H k  (25)Please note that  unvec I  × I   ( S  k ) T[7]   is not a diagonal ma-trix. Therefore, we have to perform an additional SVD forthe calculation of  X  k , as follows unvec I  × I   ( S  k ) T[7]   =  V   k  ·  ˜Σ k  · V   H k  . Now the matrix X  k  can be calculated as X  k  =  U  (3) k  ⊗ U  (2) k  ⊗ U  (1) k  · V   k  ·  ˜Σ 12 k  .  (26)Especially in cases, where  ˜Σ k  is of low rank it is compu-tationally cheaper to calculate  X  k  directly from  ˆ R k  usingequation (26). 3.2. Denoising a Measured Channel In order to reduce the noise in measured channels, we ex-tend an idea proposed in [2] to the correlation tensor–basedchannel model. For every window  k , as defined above, weconstruct a tensor Z  k ( t ) , calculated for everytime snapshot t , using the following equation Z  k ( t ) = H k ( t )  × 1 U  (1) H k  × 2 U  (2) H k  × 3 U  (3) H k  ,  (27)where the matrices U  ( n ) k  are computed from the correlationtensor  ˆ R k  given in (12). With the help of the tensor Z  k ( t ) ,the channel can be reconstructed exactly via the synthesisequation H k ( t ) = Z  k ( t )  × 1 U  (1) k  × 2 U  (2) k  × 3 U  (3) k  .  (28)Denoisingthemeasurementtensor H k ( t ) ispossiblebysim-ply considering only the first  L n  singular vectors of   U  ( n ) k  ,corresponding to the  L n  largest singular values of   ˆ R  k .  L n should be determined with the help of the singular valuespectra of the HOSVD. This is similar to the well knownlow–rankapproximationofamatrixviathe2DSVD.Thereby,we assume that the omitted singular vectors span the noisespace. Thus, we obtain the tensor Z  ′ k ( t )  ∈ C L 1 × L 2 × L 3 and ( H k ) denoised ( t ) = Z  ′ k ( t )  × 1  U  (1) k  [ L 1 ] × 2  U  (2) k  [ L 2 ] × 3  U  (3) k  [ L 3 ] , (29)where  U  ( n ) k  [ L n ] indicates the matrix containing the first L n  singular vectors along the  n -th dimension and for the k -th window.  -- GenerateMatrixSyntheticChannelDenoising ApproachTensor Denoising Approach n ( t ) n ( t ) e matrix e tensor   | · | 2   | · | 2 Fig. 4 . Block diagramm of the performance test for the 2denoising approaches. −20 −15 −10 −5 0 5 10 15 2000.10.20.30.40.50.6       e    m   a    t   r    i   x     ,       e     t   e   n   s   o   r Signal to Noise Ratio [dB]Tensor channel model ( H 1 )2D channel model ( H 1 )Tensor channel model ( H 2 ) 2D channel model ( H 2 ) Noisy channel Fig. 5 . Reconstruction error for two synthetic noisy chan-nels: a rich multi–path channel H 1 ; a 2–path channel H 2 .The tensor–based denoising outperforms the 2D approachfor richer channels. The green curve represents the error forthe noisy channels.Figure 5 shows the reconstruction error for two noisysynthetic channels, namely  H 1  and  H 2 , and two denois-ing approaches. The channels are created with the IlmProp,a flexible geometry based channel model capable of gen-erating frequency selective time variant multi–user MIMOchannels displaying realistic correlation in frequency, time,space, and between users, cf. [6]. The Figures 7 and 6 showthe geometries of the synthetic channels. The first chan-nel H 1 , is richer in multi-path components than the second H 2 . The reconstruction error (power) is defined as the Eu-clidian distance between the noiseless channel and the re-constructed (denoised) channel e x  = N  f  M  R M  T  n =1  vec( H synthetic ) n  −  vec( H denoised ) n  2 , (30)for x  =  { matrix , tensor } , as alsodepictedFigure4. InFig-ure5, thethicklinesrepresenttheerrorobtainedviathepro-posed tensor–based method. The thin lines show the recon-struction error achieved by applying a low–rank approxi-mation directly on  ˆ R  . We can observe that the tensor–basedapproach leads to a better subspace estimate, because of theadditional singular vectors in the frequency direction. Thistranslates to a lower reconstruction error. The gain with re-spect to the 2D approach becomes more relevant for richer Y [m]X [m]RXTXtraj Fig. 6 . Synthetic IlmProp scenario characterized by richscattering ( H 1 ) Y [m]X [m]RXTX Fig. 7 . Synthetic IlmProp scenario characterized by 2 paths( H 2 ) channels. 3.3. Validation on Measurements Next,weapplytheproposedcorrelationtensor–basedmodelon measurements gathered from the main train station inMunich, Germany. The multi-dimensional RUSK MIMOchannel sounder employed a  16  ×  8  antenna architecturewitha16–elementuniformcirculararray(UCA)atthetrans-mitter and an 8–element uniform linear array (ULA) at thereceiver. The antenna spacing at both arrays was about  λ/ 2 .The bandwidth was 120 MHz at a carrier frequency of 5.2GHz. The frequency spacing was 3.125 kHz which yields atotal of 385 frequency bins. The receiver measured a com-plete channel response every 18.432 ms. A total of 9104time snapshots were taken. The mobile terminal was mov-ing in a Non Line-Of-Sight (NLOS) regime. The environ-ment was particularly rich in multi-path components.For the calculation of the channel model we consideronly25adjacentfrequencybinsaroundthecenterfrequency,thus spanninga bandwidthof 7.5MHz. We dividethe chan-nel into windows of   T  W  = 25  samples in the time domain,as in Figure 3.We first consider 10 adjacent time windows of the mea-sured channel. To assess the behavior of the channel inthe frequency domain, we compare the proposed correla-
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