Full Rank Spatial Channel Estimation at Millimeter Wave Systems

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1 Full Rank Spatial Channel Estimation at Millimeter Wave Systems siao-lan Chiang, Wolfgang Rave, Tobias Kadur, and Gerhard Fettweis Vodafone Chair for Mobile Communications Technische Universität Dresden elmholtzstrasse 8, Dresden, Germany {hsiao-lanchiang, rave, tobiaskadur, Abstract Spatial channel estimation (CE) for hybrid beamforming architecture is one of the crucial challenges for millimeter wave (mmw) links The limited beamwidth makes the CE incapable of resolving multiple channel paths from the same cluster when the angular spread is less than or equal to the beamwidth Therefore, how to efficiently obtain reliable links which provide maximum spatial degree-of-freedom gain for data transmission is the key technology To this end, a novel CE method to estimate full rank array response matrices is presented The full rank array response matrices lead to the full rank reconstructed channel matrix, which provides the maximum spatial degree-of-freedom gain of the sparse mmw channels Compared with the previously proposed CE methods which use transmit or receive diversity rather than spatial degrees of freedom, the proposed one shows significant increase of data rates as SNR is getting larger I INTRODUCTION With the rapid increase of data rates in wireless communication, bandwidth shortage is getting more critical As a result, there is a growing interest in using millimeter wave (mmw) for future wireless communications taking advantage of the enormous amount of available spectrum [] Measurements of most large and small scale parameters for mmw backhaul and access links have been presented in [][3] It seems that path loss in such environment is very severe, and in order to improve capacity and service quality, mmw together with large antenna array is a promising approach [4] At mmw systems, a combination of analog beamforming (ABF, operating in passband) [][6] and digital beamforming (DBF, operating in baseband) is one of the low-cost solutions to higher data rate transmission This combination is commonly denoted as hybrid beamforming (BF) [7] ABF is typically implemented at each antenna using a finite set of possible phase shifts, where one entry can be selected from a codebook to produce a beam pattern with a specific main lobe direction A hybrid beam can be regarded as a linear combination of multiple analog beams with the coefficients defined in the DBF, which are to be optimally combined according to some criteria, eg, to maximize mutual information The problem of spatial channel estimation (CE) is complicated by the fact that only steering angles can be changed by phase shifters equipped in ABF structure For this reason, to obtain full channel information means to obtain the coupling of ABFs across the channel and therefore exhaustive analog beam training is required in the beginning [8] Typically, the problem of CE can be formulated as an l -norm optimization problem [9] since mmw channels consist of a quite small number of significant paths, meaning that angles of departure and arrival (AoDs and AoAs) are sparse in the angular domain Orthogonal matching pursuit (OMP) [] offers a solution to l -norm optimization problems of CE The columns in the codebook used in OMP is widely assumed to be mutual incoherent or orthogonal, and each column is the coupling of one predefined departure array response vector and one predefined arrival array response vector In general, it is difficult for these predefined array response vectors to realize multiple paths from the same cluster if the angular spreads are small, for example, less than the half-power beamwidth (PBW) In such a case, either the estimated departure or the estimated arrival array response vectors could be repeated Therefore, the reconstructed channel matrix may only provide transmit or receive diversity owever, neither provides fewer data rates than spatial degrees of freedom [], which exploits multiple resolvable channel paths for data transmission This paper presents a novel CE method to obtain maximum spatial degree-of-freedom gain By enforcing the rank of the estimated array response matrices, the reconstructed channel by these estimated matrices can ensure to be full rank Consequently, the reconstructed channel matrix can offer the maximum spatial degree-of-freedom gain Compared with the non-full rank channel matrices, the simulation results by the proposed full rank CE show significant increase of data rates This paper is organized as follows: Section II describes the systems and the mmw channel models Section III introduces the non-full rank CE, the motivation of the full rank CE, and how to obtain it Simulation results are presented in Section IV, and we conclude our work in Section V We use the following notations throughout this paper a is a scalar, a is a vector, and A is a matrix a i is the i th entry of a; a i,j or (A) i,j is the (i, j) th entry of A; a(i) is the i th column vector of A; and A( : N) denotes the first N column vectors [a(),, a(n)] of A A, A, and A T denote complex conjugate, ermitian transpose and transpose of A, respectively a is the l -norm of a []; a is the -norm of a; A and A F are the determinant and Frobenius norm of A, respectively diag(a) is the vector formed by the diagonal elements of A;vec(A) is vectorization /6/$3 6 European Union 4

2 Figure The beamforming system diagram, where each RF chain includes digital-to-analog converter (DAC) at the transmitter and analog-to-digital converter (ADC) at the receiver, and each ABF consists of multiple phase shifters connecting to one RF chain of A; rank(a) is the (column) rank of A; and [A B] denotes horizontal concatenation I N is the N N identity matrix mmw channels are different to Rayleigh or Rician fading channel models, which are often assumed for centimeter wave communications One key difference is its sparsity in spatial frequency domain [3][4] Thus, the channel matrix in () can be modeled as the sum of outer products of the array response vectors associated with the N P paths [] N P = α p a A (p)a D(p) p= =[a A (),, a A (N P )] }{{} A A α } {{ α NP } D a D () a D (N P ) } {{ } A D = A A DA D, (3) II SYSTEM MODELS In the assumed mmw systems [3], uplink and downlink transmission timing between a transmitter and a receiver is synchronized Fig shows a single link where the transmitter with N T antennas communicates N S data streams to the receiver with N R antennas The transmitter is equipped with a precoder, which consists of an N T N RF ABF matrix F RF and an N RF N S DBF matrix F BB [7] The precoder steers N S hybrid beams, and each hybrid beam is formed by a weighted combination (defined in DBF) of the N RF analog beams The N RF analog beams at the transmitter are selected from a codebook F = { f RF (l) =the l th column of F RF,l =,,N F }, where F RF C NT NF is given by F RF = NT e jπ NF N jπ F e N F NT (NF )(NT ) jπ e N jπ F e N F () The power constraint on each hybrid beam is enforced by F RF f BB (n s ) =,n s =,,N S, where f BB (n s ) is the n th s column of F BB At the receiver, the codebook W = { w RF (k) =the k th column of WRF,k =,,N W } is used in the ABF, and W RF C NR NW has the similar structure as () The same number N RF of the RF devices and the same number N S of the data streams are assumed at the receiver The received signal after the combiner W RF W BB can be written as r = W BBW RF F RF F BB s + W BBW RF z, () where r C NS is the combined received data vector, s C NS is the transmitted data vector satisfying E[ss ]= N S I NS, and z is an N R additive white Gaussian noise (AWGN) vector, where the elements are complex Gaussian random variables, z nr CN(,σz), n r =,,N R which consists of three factors: the arrival array response matrix A A C NR NP NP NP, the diagonal matrix D C with N P complex attenuation coefficients {α p }, and the departure array response matrix A D C NT NP Each array response vector in A D can be expressed as [6] a D (p) = [ T,e jπλ sinφd,pδ d,,e jπλ d] sinφd,p(nt )Δ, NT (4) where φ D,p stands for the p th AoD, Δ d = λ is the distance between two antennas, and λ is the wavelength at the carrier frequency Each array response vector in A A has the similar form as (4) Assume Gaussian signaling [4], given F BB, F RF, W RF, W BB, the system throughput is shown as C =log I NS + N S R z (W BBW RF F RF F BB ) ( W BB W RF F RF F BB ), () where R z = σ zw BB W RF W RF W BB is the noise covariance matrix after combining III SPATIAL CANNEL ESTIMATION To simplify the CE problem, let us assume that the DBFs at the transmitter and the receiver initially operates with F BB = I and W BB = I To obtain the observations for CE, all the combinations of the columns in the codebooks F and W are trained by using a known training sequence {s,,s T } of length T (the coherence time of the channel is assumed to be sufficiently long) After removing the training signals, the average of the noisy observations for the coupling 43

3 coefficient with a single combination of f RF (l) and w RF (k) can be written as y k,l = T ( ) w RF (k) f RF (l) s t + w RF (k)z(t) T s t t= = w RF (k) f RF (l)+ T = w RF (k) f RF (l)+z k, T t= k =,,N W,l =,,N F, s t w RF (k)z(t) (6) where z k is the averaged noise coupling with w RF (k) Collecting the observations associated with all the combinations of w RF (k) and f RF (l) in a matrix, we obtain y, y,nf Y = y NW, y NW,N F = W RF F RF + Z, (7) where Z is the N W N F AWGN matrix coupling with W RF Vectorizing (7) and using rules for Kronecker product, it becomes y V = vec(y) =( F T RF W RF )vec()+vec(z ) =( F T RF W RF )vec(a A DA D)+vec(Z ) (8) =( F T RF W RF )(A D A A ) vec(d)+vec(z ) }{{} Φ = Φ(A D A A ) vec(d)+vec(z ), where y V C NF NW is the measurement vector A Non-Full Rank Channel Estimation The CE problem in (8) can be formulated as a least squares problem subject to a sparsity constraint as [8] (ÂD, ÂA, ˆD) =arg min y V Φ(A D A A ) vec(d), A D,A A,D a D (p ) A D,p =, N P, st a A (p ) A A, (9) vec(d) = N P, where N P (N P N P ) is the number of the estimated channel paths A D = {ã D (i d )=the i th d column of ÃD,i d =,,N D } is the codebook consisting of N D candidates of the departure array response vectors, and ÃD NT ND C is presented in a given realization à D = NT e jπ ND N jπ D e N D NT (ND )(NT ) jπ e N jπ D e N D () The codebook A A = {ã A (i a )=the i th a column of ÃA,i a =,,N A } consists of N A candidates of the arrival array Algorithm : Non-Full Rank CE by OMP [8] Input: y V, à D, à A Output: ÂD,  A, ˆD  D = empty matrix,  A = empty matrix, ˆΨ = empty matrix y R = y V 3 Ψ = Φ(à D ÃA) 4 for p =:N P g = Ψ y R 6 (î D, î A) = arg max ( ) gg i,i i D {,,N D } A A i {,,N } where i =(i D ) N A + i A 7  D =[ÂD ãd(î D )] and ÂA =[ÂA ãa(î A )] 8 ˆΨ =[ˆΨ Φ(ãD(î D) ã A(î A))] 9 y R =(I NF N W ˆΨ( ˆΨ ˆΨ) ˆΨ )yv end diag( ˆD) =(ˆΨ ˆΨ) ˆΨ yv response vectors, and ÃA C NR NA has the similar structure as () The diagonal matrix D C N P N P has N P non-zero entries We assume that N D = N T and N A = N R, so that à D and ÃA are IDFT matrices (9) can be solved by OMP [8][], and the pseudocode is shown in Algorithm OMP is an iterative greedy algorithm that selects the column of the codebook which is most correlated with the current residual (y R in Algorithm ) at each step In Algorithm Step 8, the OMP leads the current selected column orthogonal to the previous selected ones, which means that any two columns of ˆΨ, ˆψ(i) and ˆψ(j),i j, are distinct owever, it cannot guarantee that the corresponding array response vectors are different as well Let ˆψ(i) = Φ(ã D (î D ) ã A (î A )) and ˆψ(j) =Φ(ã D (ĵ D ) ã A (ĵ A )) We can find that ˆψ(i) ˆψ(j) holds if one of the following three conditions happens () ã D (î D ) ã D (ĵ D ) and ã A (î A )=ã A (ĵ A ), () ã D (î D )=ã D (ĵ D ) and ã A (î A ) ã A (ĵ A ), (3) ã D (î D ) ã D (ĵ D ) and ã A (î A ) ã A (ĵ A ), where the first two conditions include the repeated array propagation vectors Therefore, the estimated array response matrix could be rank deficient, ie, rank(âd) < N P or rank(âa) < N P, which leads to a rank deficient channel matrix, rank(ĥ) min(rank(âd), rank(âa)) <N P B Proposed Full Rank Channel Estimation We hope that rank(ĥ) = N P for the reason that the reconstructed channel matrix can provide N P links for parallel data transmission in order for higher data rates Accordingly, two constraints, rank(a D )=N P and rank(a A )=N P, are considered in the CE problem formulation, which becomes a 44

4 Table I AN EXAMPLE OF TE GIVEN AND TE SELECTED AODS ANDAOAS WIT N P =3AND N P =BY TE NON-FULL RANK AND TE FULL RANK CE The given/selected angle The given/selected angle The given/selected angle wrt the st strongest path wrt the nd strongest path wrt the3 rd strongest path Given AoDs φ D, =48 o φ D, = o φ D,3 = 3 o - Estimated AoDs by the non-full rank CE (Algorithm ) ˆφD, =489 o ˆφ D, =489 o - â D () = â D () Estimated AoDs by the full rank ˆφD, =489 o ˆφD, = 3 o CE (Algorithm ) (close to the 3 rd given AoD) - â D () =â D () Given AoAs φ A, = 9 o φ A, = 4 o φ A,3 = o - Estimated AoAs by the non-full rank CE (Algorithm ) ˆφA, = 3 o ˆφ A, = o - â A () =â A () Estimated AoAs by the full rank ˆφA, = 3 o ˆφA, =94 o CE (Algorithm ) (close to the 3 rd given AoA) - â A () =â A () Note Algorithm : Proposed Full Rank CE by OMP Input: y V, Ã D, Ã A Output: ÂD, Â A, ˆD Â D = empty matrix, Â A = empty matrix, ˆΨ = empty matrix, I D =, I A = Figure A typical example of mmw channels with N P =3, where Path and belong to the same cluster so that φ D, φ D, and φ A, φ A, least squares problem subject to one sparsity and two full rank constraints as (ÂD, ÂA) =arg min y V Φ (A D A A ) vec(d), A D,A A a D (p ) A D, a A (p ) A A, st vec(d) = N P, () rank(a D )=N P, rank(a A )=N P () can also be solved by OMP with some modifications, and the pseudocode is shown in Algorithm Compared with Algorithm Step 6, the different part is that, in Algorithm Step 6, the indices in the sets I D and I A (consisting of all the previous selected indices) will not be selected again to ensure that there are no repeated vectors in ÂD and ÂA A simple example of the selected AoDs and AoAs by Algorithm and Algorithm is shown in Fig and Table I with N P =3, N P =, N T = N R = N D = N A =3, T =, SNR =db, and α > α > α 3, where the two strongest paths from the same cluster have nearly the same y R = y V 3 Ψ = Φ(Ã D ÃA) 4 for p =:N P g = Ψ y R 6 (î D, î A) = arg max ( ) gg i,i i D {,,N D }\I D A A A i {,,N }\I where i =(i D ) N A + i A 7 Â D =[ÂD ãd(î D )] and ÂA =[ÂA ãa(î A )] 8 ˆΨ =[ˆΨ Φ(ãD(î D) ã A(î A))] 9 y R =(I NF N W ˆΨ( ˆΨ ˆΨ) ˆΨ )yv I D = I D {î D} and I A = I A {î A} end diag( ˆD) =(ˆΨ ˆΨ) ˆΨ yv AoDs and AoAs The capacity of the channels by these two algorithms is introduced in terms of the rank of the channel ) rank(ĥ) by the non-full rank CE (Algorithm ): From Table I, we can find that the two selected vectors â D () and â D () are repeated Since â D () = â D () and â A () â A (), the reconstructed channel becomes [ ] ˆα Ĥ = ÂA ˆDÂ D =[â A (), â A ()] â ˆα D (), and because ÂA has full column rank, rank(ĥ) =rank(âd) =, which provides receive diversity but no spatial degree-offreedom gain [] The predefined 3 angles in the codebooks A D and A A are { o, ±38 o, ±78 o, ±8 o, ±448 o, ±8 o, ± o, ±94 o, ±3 o, ±343 o, ±3868 o, ±4343 o, ±489 o, ±434 o, ±6 o, ±694 o, 9 o (or 9 o )}

5 Parameter Table II SIMULATION PARAMETERS Value Carrier frequency [Gz] 6 Training time (T in (6)), 4, 36 Number of paths (N P ) Number of the estimated paths (N P ) Number of RF chains (N RF ) Number of data streams (N S ) Number of Tx antennas (N T ) 3 Number of Rx antennas (N R ) 3 Number of columns in ÃD (N D ) 3 Number of columns in ÃA (N A ) 3 α Path attenuation coefficient (α p C, N P p) p= αp = and α > > α NP ) rank(ĥ) by the full rank CE (Algorithm ): Since â D () â D () and â A () â A (), the reconstructed channel matrix can be written as [ ][ ] ˆα Ĥ = ÂA ˆDÂ âd () D =[â A (), â A ()] ˆα â D (), and due to the fact that rank(âa) =and rank(âd) =, we have rank(ĥ) =, which provides two spatial degrees of freedom [], and each link supports one data stream transmission and reception IV NUMERICAL RESULTS Table II lists all the simulation parameters Assume that the two strongest paths are from the same cluster and the angular spread is less than or equal to PBW ( π π N T ( or N R ) = 6 o ), therefore the first two AoDs and AoAs are defined as { φd, U( π, π ) φ D, = φ D, +Δφ { and φa, U( π, π ) φ A, = φ A, +Δφ, () where Δφ D is the angular spread modeled as a uniform distribution, Δφ U( 6o, 6o ) Except the angles defined in (), others are uniformly distributed over the interval ( π, π ) Two CE methods are simulated: the non-full rank CE, see Algorithm, and the proposed full rank CE, see Algorithm A Mean Absolute Error of the CE As introduced in Subsection III-B, when φ D, φ D, and/or φ A, φ A,, the second selected angles ˆφ D, and ˆφ A, by Algorithm are mostly close to φ D,3 and φ A,3, respectively ence, the mean absolute error (MAE) between the given angles and the selected ones by the non-full rank and the full rank CE methods shown in Fig 3 is given by Mean absolute error (degree) Non full rank CE (Alg ), T = Proposed full rank CE (Alg ), T = Figure 3 The MAE between the given AoDs/AoAs and the selected ones by the non-full rank and the full rank CE methods with T = Mean absolute error (degree) db 8 db Non full rank CE (Alg ), T = Non full rank CE (Alg ), T = 4 Non full rank CE (Alg ), T = 36 db error floor ( o ) Figure 4 The MAE between the given AoDs/AoAs and the selected ones by the non-full rank CE method with T =, 4, 36 Mean absolute error (degree) db Proposed full rank CE (Alg ), T = Proposed full rank CE (Alg ), T = 4 Proposed full rank CE (Alg ), T = 36 4 db error floor (3 o ) db Figure The MAE between the given AoDs/AoAs and the selected ones by the full rank CE method with T =, 4, 36 and MAE non-full rank = ( ) N P p= ˆφD,p φ D,p + ˆφA,p φ A,p E N P (3) [ ( MAE full rank = E N P ˆφD, φ D, + ˆφA, φ A, + ˆφ )] D, φ D,3 + ˆφA, φ A,3 (4) 46

6 Note that only two estimated paths (N P =) are considered in (4) In Fig 3, Algorithm shows worse MAE because the two selected paths are mainly the first and the third paths (p =, 3) and the path loss of the third path is larger than the first two paths When the number T of the observations increases, see Fig 4 and, the MAE by these two methods decrease significantly because the averaged noise power becomes smaller In Fig 4, the error floor of the non-full rank CE method by T =, 4, 36 occurs when SNR =, 8, 98 db, and we can find that when the length of the training signals are doubled and tripled (T = 4, 36), there are 3 and 48 db improvements in SNR Moreover, in Fig, the error floor of the proposed full rank CE method by T =, 4, 36 occurs when SNR =, 4, 8 db Similar to the cases in Fig 4, when the length of the training signals are doubled and tripled, there are 3 and 48 db improvements in SNR as well B Achievable Data Rate Based on the estimated ÂA, ÂD, and ˆD obtained in (9) and (), the reconstructed channel Ĥ = ÂA ˆDÂ D has a singular value decomposition (SVD): Ĥ = UΛV, () where the columns of V C NT N P and U C N R N P are, respectively, the right and the left singular vectors of Ĥ, and the diagonal entries of Λ = diag(λ,,λ N P ) R N P N P, λ λ N P >, are the singular values of Ĥ The design criterion that can be used for the precoder reconstruction is to minimize the Frobenius norm of the error between the precoder and the right singular vectors of Ĥ [7] (ˆF RF, ˆF BB )=arg min V( : N S ) F RF F BB F, F RF,F BB st f RF (n rf ) { f RF (l),l =,,N F },n rf =,,N RF, F RF f BB (n s ) = v(n s ),n s =,,N S,N S N P (6) (6) can be solved by OMP [] The same design criterion leads to the desired combiner Two cases of the given AoDs and AoAs simulated in Fig 6 are listed as follows: Case : all the AoDs and AoAs are independent and uniformly distributed over the interval ( π, π ) except the first two strongest paths whose AoDs and AoAs belong to the same cluster and the values are defined in () Case : all the AoDs and AoAs are independent and uniformly distributed over the interval ( π, π ) In Fig 6 Case, Algorithm shows better results than Algorithm because it can always obtain spatial degree-offreedom gain even if its MAE is worse, while Algorithm may provide transmit or receive diversity mostly When the number T of the observations increases, the data rates do not show obvious improvement since the decrease of MAE by more observations occurs when SNR < db In Case, the data rates by Algorithm approach the results by Algorithm Data rate (bit/s/z) Case : Non full rank CE (Alg ), T = Case : Proposed full rank CE (Alg ), T = Case : Non full rank CE (Alg ), T = 4 Case : Proposed full rank CE (Alg ), T = 4 Case : Non full rank CE (Alg ), T = Case : Proposed full rank CE (Alg ), T = Figure 6 The data rates by two different CE methods and the same BF reconstruction method because the given AoDs and AoAs in this case are all from different clusters; therefore, Algorithm provides not only transmit or receive diversity but also spatial degree-of-freedom gain as Algorithm V CONCLUSION Owing to sparse mmw channels and limited beamwidth, exploiting the maximum spatial degrees of freedom is an essential technology in order to increase the data rates This paper presents a solution to obtain it by the estimated full rank array response matrices From the simulation results, it shows that the proposed method achieves higher data rates than the previously proposed ones especially when SNR increases ACKNOWLEDGMENT The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/7-3) under grant agreement n 6963 (MiWaveS) REFERENCES [] T Rappaport, R W eath Jr, T Daniels, and J Murdock, Millimeter Wave Wireless Communications, Prentice all, 4 [] T A Thomas, et al, 3D mmwave Channel Model Proposal, IEEE VTC Fall, pp 6, 4 [3] T S Rappaport, G R MacCartney, M K Samimi, and S Sun, Wideband Millimeter-Wave Propagation Measurements and Channel Models for Future Wireless Communication System Design, IEEE Transactions on Communications, vol 63, no 9, pp 39 36, Sept [4] T S Rappaport, et al, Millimeter Wave Mobile Communications for G Cellular: It Will Work!, IEEE Access, Vol, pp , May 3 [] A ajimiri, et al, Integrated Phased Array Systems in Silicon, IEEE Proceedings, Vol 93, pp 637 6, Sep [6] J C Liberti, et al, Smart Antennas for Wireless Communications: IS-9 and Third Generation CDMA Applications, Prentice all, NJ, 999 [7] O E Ayach, et al, Low Complexity Precoding for Large Millimeter Wave MIMO Systems, IEEE International Conference on Communications (ICC), pp , Jun [8] R Mendez-Rial, C Rusu, et al, Channel estimation and hybrid combining for mmwave: Phase shifters or switches? Information Theory and Applications Workshop (ITA), pp 9 97, 47

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