A Correlation Tensor-Based Model for Time Variant

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 channel 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. 2. Definition of the channel tensor. The 2–dimensional t ⌊ T ⌋ T W k = ⌊ Nt ⌋ W subtensor H(f0,t0), and the 3–dimensional subtensor TW H(t1) are depicted. Nf where (·)H denotes the Hermitian transpose. The core ten- Fig. 3. Definition of the non-overlapping stationarity win- sor fulfills some special properties, especially the property dows. Each window has TW time samples. In the other of all orthogonality, which means that the rows of all un- dimensions, each window is of same size as H. folding matrices S[n] are orthogonal, c.f. [4]. With the help of the Kronecker product it is possible to write equation (6) in terms of matrix operations. where •4 denotes the 4–mode inner product. Please note that n is the time index within the k–th window. To get in- (n) A[n] = U · S[n]· sight into the spatial structure of the channel, we decompose T this correlation tensor via the following HOSVD U (N) ⊗···⊗ U (n+1) ⊗ U (n−1) ⊗···⊗ U (1) . (8) Rˆ = S × U (1)× U (2)× U (3)× U (4)× U (5)× U (6). Please note that this formulais only valid for the MATLAB– k k 1 k 2 k 3 k 4 k 5 k 6 k (13) like unfolding defined in Section 2.2. (n) The matrices Uk contain the left singular vectors of the 3. TENSOR CHANNEL MODEL ˆ ˆ n–th unfolding matrices (Rk)[n] of Rk. The symmetries of the correlation tensor are also reflected by its HOSVD. As already mentioned, we represent the channel coefficients Therefore, we can choose a HOSVD such that the following in form of the tensor equations hold CMR×MT×Nf ×Nt H ∈ , (9) ∗ (1) (4) Uk = Uk , where MR and MT are the number of antennas at the trans- ∗ (2) (5) mitter and receiver, and Nf and Nt are the number of sam- Uk = Uk , (14) ∗ ples taken in frequency and time, respectively. Please note (3) (6) U = U . that the frequency domain of the channel is connected to its k k delay time τ via a Fourier Transform. In this case, equation (13) can be simplified to Similarly to [3, 5], we now define the channel correlation tensor as Rˆ S (1) (2) (3) k = k ×1 Uk ×2 Uk ×3 Uk M M N M M N ∗ ∗ ∗ (15) R =E{H(t) ◦ H(t)∗} ∈ C R× T× f × R× T× f , (1) (2) (3) ×4 U ×5 U ×6 U , (10) k k k H CMR×MT×Nf where (t) ∈ is the frequency selective and the core tensor Sk, according to Section 2.4, can be MIMO channel at time snapshot t. calculated via Assuming that the channel is block–wise stationary in H H H time, we define an averaging window of size T , so that S Rˆ (1) (2) (3) W k = k ×1 Uk ×2 Uk ×3 Uk the channel within the –th window, denoted with H , can T T T (16) k k (1) (2) (3) be assumed stationary (see Figure 3). The over–all channel ×4 Uk ×5 Uk ×6 Uk . matrix H is then defined as The proposed channel model consists in computing the ˆ Nt H H H H Nt correlation tensors Rk for all windows, i.e., ∀k =1 . . . ⌊ ⌋, = 1 4 1 4 · · · 4 ⌊ ⌋ , (11) TW TW as they describe exhaustively the correlation properties of where 4 denotes the concatenation of the tensors Hk the channel, seen as a temporal block stationary stochastic along the 4–th dimension, as introduced in [5]. We com- process. pute an estimate of the k–th correlation tensor by averaging In the following we give a brief description of two appli- in time, as cations of the proposed channel model, namely the genera-

TW tion of new channel realizations and the subspace–based de- ˆ 1 ∗ Rk ≈ Rk = · (Hk) ◦ (Hk) noising of a channel measurement. Then we apply the cor- T i4=n i4=n W nX=1 (12) relation tensor–based channel model to channel measurements, and compare its performance with the tensor–based 1 H H∗ = · k •4 k , channel model presented in [3]. TW 3.1. Channel Synthesis S T Here unvecI×I ( k)[7] denotes the inverse matrix vec(·) In the 2–dimensional case [1, 2], the joint spatial correlation operation applied to the 7–th unfolding of the core tensor S S T matrix is deﬁned as k. Therefore, unvecI×I ( k)[7] is a square matrix of ˆ R H H H (17) same size as Rk with I = MRMTNf . The correlation k =E{vec( k) · vec( k) } . ˆ matrix Rk can be calculated from the HOSVD of the tensor ˆ Similarly to (12), we can compute an estimate of Rk, de- R via ˆ k noted by Rk with ˆ e T eH Rk = U · unvecI I (Sk) · U (25) TW k × [7] k 1 H Rˆ H H (18) k = vec ( k)i4=n ·vec ( k)i4=n . T T W n=1 S X Please note that unvecI×I ( k)[7] is not a diagonal ma- ˆ trix. Therefore, we have to perform an additional SVD for From the information given in the correlation matrix Rk, it ˜ the calculation of Xk, as follows is possible to construct a new random synthetic channel Hk, displaying the same spatio–frequency correlation as H (t) k S T Σ˜ H via unvecI×I ( k)[7] = Vk · k · Vk . ˜ vec Hk = Xk · g , (19) Now the matrix Xk can be calculated as where the entries of the vector g are i.i.d. zero mean com- 1 plex Gaussian random numbers with unit variance, and (3) (2) (1) Σ˜ 2 Xk = Uk ⊗ Uk ⊗ Uk · Vk · k . (26) 1 ˆ 2 Xk = Rk . (20) ˜ Especially in cases, where Σk is of low rank it is compu- Rˆ The matrix Xk is computed via a 2–dimensional eigenvalue tationally cheaper to calculate Xk directly from k using decomposition of equation (26).

Rˆ = U · Σ · U H , (21) k k k k 3.2. Denoising a Measured Channel ˆ where the matrix Uk contains all eigenvectors of Rk and In order to reduce the noise in measured channels, we ex- Σ k is the diagonal matrix containing the eigenvalues. Then tend an idea proposed in [2] to the correlation tensor–based we deﬁne Xk as follows channel model. For every window k, as deﬁned above, we 1 1 Z ˆ 2 construct a tensor k(t), calculated for every time snapshot R = U · Σ 2 = X . (22) k k t, using the following equation

By computing the matrix Xk we can generate new random H H H Z H (1) (2) (3) synthetic channels by means of equation (19). In the follow- k(t)= k(t) ×1 Uk ×2 Uk ×3 Uk , (27) ing we show that it is also possible to compute the matrix Rˆ (n) Xk from the HOSVD of the correlation tensor k. where the matrices Uk are computed from the correlation ˆ With the help of the MATLAB–like unfoldingfrom Sec- tensor Rk given in (12). With the help of the tensor Zk(t), tion 2, the connection between the estimated 2D correlation the channel can be reconstructed exactly via the synthesis ˆ ˆ matrix Rk and the estimated correlation tensor Rk is given equation by (1) (2) (3) Hk(t)= Zk(t) ×1 U ×2 U ×3 U . (28) ˆ ˆ T ˆ T ˆ k k k vec Rk = Rk = Rk = vec Rk . (23) [3] [7] Denoising the measurement tensor Hk(t) is possible by sim- (n) To compute the matrix Xk, we can either compute a SVD ply considering only the first Ln singular vectors of Uk , ˆ ˆ ˆ of Rk, or apply a HOSVD on Rk. This relation between corresponding to the Ln largest singular values of Rk. Ln the 2–dimensional model and the correlation tensor–based should be determined with the help of the singular value model follows from the following derivation. For the reason spectra of the HOSVD. This is similar to the well known of simplicity we first define the following unitary matrix: low–rank approximationof a matrix via the 2D SVD. Thereby, we assume that the omitted singular vectors span the noise e (3) (2) (1) U = U ⊗ U ⊗ U . (24) ′ CL1×L2×L3 k k k k space. Thus, we obtain the tensor Zk(t) ∈ and

With the Kronecker version of the HOSVD (8) it follows [L1] H Z′ (1) from the latter relation ( k)denoised (t)= k(t) ×1 Uk (29) T [L ] [L ] ˆ Rˆ Rˆ (2) 2 (3) 3 vec Rk = vec k = k ×2 Uk ×3 Uk , [7] ∗ e e T S [Ln] = Uk ⊗ Uk · ( k)[7] (n) where Uk indicates the matrix containing the ﬁrst T H e S e Ln singular vectors along the n-th dimension and for the = vec Uk · unvecI×I ( k)[7] · Uk . k-th window. n(t)

Matrix È 2 ematrix Denoising |·| Approach -

Generate Synthetic Channel TX n(t) RX

Tensor - È 2 etensor traj Denoising |·| Approach X[m] Y [m] Fig. 4. Block diagramm of the performance test for the 2 denoising approaches. Fig. 6. Synthetic IlmProp scenario characterized by rich Noisy channel 2D channel model (H1) scattering (H1) 0.6 Tensor channel model (H1) 2D channel model (H2) Tensor channel model (H2) 0.5

tensor 0.4 e , 0.3 TX matrix

e 0.2 RX 0.1

0 X[m] Y [m] −20 −15 −10 −5 0 5 10 15 20 Signal to Noise Ratio [dB]

Fig. 5. Reconstruction error for two synthetic noisy channels: a rich multi–path channel H1; a 2–path channel H2. The tensor–based denoising outperforms the 2D approach Fig. 7. Synthetic IlmProp scenario characterized by 2 paths for richer channels. The green curve represents the error for (H2) the noisy channels.

channels. Figure 5 shows the reconstruction error for two noisy synthetic channels, namely H and H , and two denois- 1 2 3.3. Validation on Measurements ing approaches. The channels are created with the IlmProp, a flexible geometry based channel model capable of gen- Next, we apply the proposed correlation tensor–based model erating frequency selective time variant multi–user MIMO on measurements gathered from the main train station in channels displaying realistic correlation in frequency, time, Munich, Germany. The multi-dimensional RUSK MIMO space, and between users, cf. [6]. The Figures 7 and 6 show channel sounder employed a 16 × 8 antenna architecture the geometries of the synthetic channels. The first chan- with a 16–elementuniformcircular array (UCA) at the trans- nel H1, is richer in multi-path components than the second mitter and an 8–element uniform linear array (ULA) at the H2. The reconstruction error (power) is defined as the Eu- receiver. The antenna spacing at both arrays was about λ/2. clidian distance between the noiseless channel and the re- The bandwidth was 120 MHz at a carrier frequency of 5.2 constructed (denoised) channel GHz. The frequency spacing was 3.125 kHz which yields a total of 385 frequency bins. The receiver measured a com- Nf MRMT 2 plete channel response every 18.432 ms. A total of 9104 ex = vec (Hsynthetic) − vec (Hdenoised) , n n time snapshots were taken. The mobile terminal was mov- nX=1 (30) ing in a Non Line-Of-Sight (NLOS) regime. The environ- for x = {matrix, tensor}, as also depictedFigure 4. In Fig- ment was particularly rich in multi-path components. ure 5, the thick lines represent the error obtained via the pro- For the calculation of the channel model we consider posed tensor–based method. The thin lines show the recon- only 25 adjacent frequencybins aroundthe center frequency, struction error achieved by applying a low–rank approxi- thus spanning a bandwidth of 7.5 MHz. We divide the chan- ˆ mation directly on R. We can observe that the tensor–based nel into windows of TW = 25 samples in the time domain, approach leads to a better subspace estimate, because of the as in Figure 3. additional singular vectors in the frequency direction. This We first consider 10 adjacent time windows of the mea- translates to a lower reconstruction error. The gain with re- sured channel. To assess the behavior of the channel in spect to the 2D approach becomes more relevant for richer the frequency domain, we compare the proposed correlation tensor–based channel model with the model presented realizations we calculate the parameters for both models and in [3] by means of the ergodic capacity compare the modeled capacities to CMEAS. The capacity of the proposed model is denoted by CCTB, as for Corre- ρ H lation Tensor–Based (CTB), whereas the structured model C =E log2 det IMR + Hf,t · Hf,t . MT from [3] is denoted by Cstruct. The results are given in Fig- ure 10. Theoretically, the reference capacity CMEAS and the Numerically, the equation above is computed for subbands capacity obtained from the proposed model Cstruct must be of 5 frequency samples each and by averaging in time and equal, as they are calculated with the same channel model. frequency in every window, so that As we take a finite number of samples, the two capacities still differ slightly, i.e., the Cstruct are not on the diagonal. 1 1 ρ H C = log det IM + Hf,t · H , Also in this case, the proposedchannel model shows a better F T 2 R M f,t W Xf,t T performance than the structured model. From the results shown in Figures 8 and 10, we can see where F =5, and TW = 25. For the observed 10 windows, that the proposed method achieves a higher accuracy, al- we obtain a total of 5 · 10 = 50 capacity estimates. though the gap in capacity to the structured model is not From each window, using all 25 frequencybins, we com- large. As these figures refer to particular time snapshots of a pute the correlation tensor as in equation (12) and the pa- specific channel measurement, we now investigate how the rameters for the model proposed in [3], which we refer to performances change with different propagation conditions. as the structured model. Please note that we do not com- To do so, we divide once more the measurement tensor H pare our model with existing 2–dimensional ones (such as into windows of 25 time and frequency samples. Then, for Kronecker [7] or Weichselberger [1]), because they are not each window, we compute the distance between the corre- suitable for modeling the channel in the frequency dimen- lation tensors RCTB (correlation tensor–based model) and sion. With the tensor–based models we generate new chan- Rstruct (structured model), by means of a straightforward nel tensors with exactly the same sizes as the measured one. extension of the metric proposed in [8]. It is defined as We then compute the capacities in the way just mentioned and we plot the results in Figure 8 hvec (R ) , vec (R )i D =1 − CTB struct , (31) kvec (RCTB) kF ·kvec (Rstruct) kF

20 where h·, ·i denotes the scalar product, and k·kF is the Capacity,correlationtensorbasedmodel(CTB) 19.5 Capacity,Kroneckerlikestructure(struct)[3] Frobenius norm. The latter ranges in the interval [0, 1], and becomes larger, the more the signal spaces of the two cor- 19 relation tensors overlap. The metric becomes zero for non–

18.5 overlapping, orthogonal signal subspaces. Figure 11 depicts the results of this comparison(top plot) and the power of the 18 channel (bottom one). Simulations show that for lower val- 17.5 ues of the metric, the structured model performs worse, as it is incapable of mimicking the full statistics of the chan- 17

Modelcapacity[Bits/s/Hz] nel. We can observe that there is a correlation between the 16.5 power of the channel and the metric D, and thus, between

16 the powerand the performanceof the modelproposedin [3]. This can be interpreted as follows. For smaller powers, the 15.5 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20 channel is richer in multi-path, and the eigenstructure of the Measuredcapacity[Bits/s/Hz] correlation tensor can be better represented by a Kronecker– like structure. Fig. 8. Capacity comparison between the proposed model The validity of this observation needs to be investigated and the structured model [3]. The closer the points are to further with the help of additional channel measurements. the diagonal, the more precisely the modeled capacity ﬁts In particular, the capacity might not be the best metric to the capacity of the channel. compare these models, as it is computed independently for each frequency bin, and therefore, does not account for any We can see that the proposed correlation tensor–based correlation along this dimension. model performs better than the structured model. Since the The portion of the channel used for the capacity com- estimate of the ergodic capacity is computed from very few parison from the Figures 8 and 10 are highlighted in Figure samples, its variance is presumably large. To avoid this 11. In the time windows for which the metric D is high, problem and eliminate any inﬂuence of the estimation er- both models show the same performance with respect to the rors on the performance of the two models, we follow the capacity. processing scheme depicted in Figure 9. From one measurement window we compute a reference correlation tensor 4. CONCLUSIONS ˆ Rref and from it, using equation (19), we generate 30000 new channel realizations. The latter are used to compute the We present a novel correlation tensor–based channel capacity CMEAS which we take as reference. From the new model which is applicable for frequencyselective, time vari- 0.8 ˆ 30000new Rstruct realizations CCTB 0.7

0.6 Measurement ˆ 30000new R CMEAS Window ref realizations 0.5

0.4 ˆ 30000new R Cstruct CTB realizations Correlationtensordistance 0.3 0 50 100 150 200 250 300

Fig. 9. Capacity comparison out of the reference tensor, 0.02 taken from the above measurements. 0.015

19.4 0.01

Power Capacity,correlationtensorbasedmodel(CTB) 19.2 Capacity,Kroneckerlikestructure(struct)[3] 0.005

19 0 0 50 100 150 200 250 300 Windownumber 18.8

18.6 Fig. 11. Distance of the correlation tensors (after Herdin) for the Kronecker like model after [3] and the correlation 18.4 tensor–based model. The lower part of the picture draws

Modelcapacity[Bits/s/Hz] the power proﬁle of the channel. 18.2

18 REFERENCES 17.8 17.8 18 18.2 18.4 18.6 18.8 19 19.2 19.4 [1] W. Weichselberger, M. Herdin, H. Ozcelik,¨ and E. Bonek, “A stochas- Measuredcapacity[Bits/s/Hz] tic MIMO channel model with joint correlation of both link ends,” IEEE Trans. on Wireless Comm., vol. 5, no. 1, pp. 90–100, Jan. 2006. Fig. 10. Results of the capacity comparison [2] G. Del Galdo, M. Haardt, and M. Milojevic, “A subspace-based channel model for frequency selective time variant MIMO channels,” in in Proc. 15th Int. Symp. on Personal, Indoor, and Mobile Radio Commu- nications (PIMRC), Barcelona, Spain, Sept. 2004. ant MIMO channels. We assume temporal block-wise sta- [3] N. Costa and S. Haykin, “A novel wideband MIMO channel model tionarity. The tensor calculus allows us to analyze the chan- and McMaster’s wideband MIMO SDR,” Asilomar Conf. on Signals, nel eigenmodes also along the frequencydimension, permit- Systems, and Computers, Nov. 2006., to be published. ting us to cope with frequency selective channels as well. [4] L. de Lathauwer, B. de Moor, and J. Vandewalle, “A multilinear sin- The model lets us generate new random channels which gular value decomposition,” SIAM J. Matrix Anal. Appl., vol. 21, no. 4, 2000. display given correlation properties in space and frequency. [5] F. Römer, M. Haardt, and G. Del Galdo, “Higher order SVD based Moreover, the tensor–based approach yields an efficient de- subspace estimation to improve multi-dimensional parameter estima- noising application for channel measurements, due to an im- tion algorithms,” in 40th Asilomar Conf. on Signals, Systems, and proved subspace estimate. Moreover, the applicability and Computers, Pacific Grove, CA, US, Nov. 2006. validity of the proposed model is proven on channel mea- [6] G. Del Galdo, M. Haardt, and C. Schneider, “Geometry-based surements. channel modelling of MIMO channels in comparison with channel sounder measurements,” Advances in Radio Science - Kleinheubacher Berichte, pp. 117–126, October 2003, more information on the model, as well as the source code and some exemplary scenarios can be found at http://tu-ilmenau.de/ilmprop. ACKNOWLEDGEMENTS [7] C.-N. Chuah, J. M. Kahn, and D. Tse, “Capacity of multi-antenna array systems in indoor wireless environment,” IEEE Global Telecom- munications Conference, Sydney, Australia, 1998, vol. 5, pp. 1894– A part of this work has been performed in the framework of 1899, 1998. the IST project IST-4-027756 WINNER II, which is funded [8] M. Herdin, “Non – stationary indoor MIMO radio channels,” Ph.D. by the European Union. The authors also gratefully ac- Dissertation, Technische Universität Wien, Austria, 2004. knowledge the support of the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under contract no. HA 2239/1-2. The authors also would like thank Prof. Thomä(Elec- tronic Measurement Research Lab, TU Ilmenau) for pro- viding the measurement data. Those have been performed in the framework of the WIGWAM project, funded by Ger- man Federal Ministry of Research and Education (BMBF).