Downlink MU-MIMO with THP Combined with Pre- and Post-processing and Selection of the Processing Vectors for Maximization of Per Stream SNR

Transcription

1 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 Downlink MU-MIMO with THP Combined with Pre- and Post-processing and Selection of the Processing Vectors for Maximization of Per Stream SNR Nanda Kishore Chavali, Kiran Kuchi and V Umapathi Reddy Abstract In this paper, we consider a downlink multiuser multiple-input multiple-output MU-MIMO system with multiple antennas at the transmitter and multiple antennas at each user, where the transmitter can send one or more data streams to each user We propose a non-iterative method by combining Tomlinson Harashima precoding THP with pre- and post-processing and, selecting processing vectors based on the maximization of instantaneous signal to noise ratio SNR of the data stream at the input of the detector of each user The post-processing vectors for all users are found to be eigenvectors corresponding to the maximum eigenvalue of a certain matrix involving the channel matrix of the user The transmitter computes the vectors of linear processing matrix through orthonormalization procedure in a single step The feedback matrix at the transmitter is then obtained from the effective channel matrix and the linear processing matrix We express the instantaneous SNR of all data streams in terms of eigenvalues of a Wishart matrix, obtained from the channel matrix of the user and find the diversity order for each data stream Considering multiple scenarios, we find outage probability of instantaneous SNR for all data streams and the cumulative distribution function cdf of the sum-rate capacity for all users using the proposed method and compare the results with those of recently proposed methods that provide closed form solution [], [] and also with that of Block Diagonalization [3], channel inversion and THP [4] in scenarios where they are applicable Index Terms Multiuser multiple-input and multiple-output MU-MIMO system, Tomlinson-Harashima Precoding THP, pre- and post-processing I INTRODUCTION The multiuser multiple-input multiple-output MU-MIMO technology enhances the communication capabilities of users connected to a base station BS in a wireless communication system [5], [6] In the downlink MU-MIMO, transmitter employs multiple antennas to allocate one or more data streams to each user and users are expected to receive data streams without interuser and inter-stream interference To Copyright c 5 IEEE Personal use of this material is permitted However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieeeorg Nanda Kishore Chavali is with Uurmi Systems Pvt Ltd, Hyderabad, India and is an external research scholar at Department of Electrical Engineering, Indian Institute of Technology, Hyderabad, India nanda@uurmicom Kiran Kuchi is with the Department of Electrical Engineering, Indian Institute of Technology, Hyderabad, India kkuchi@iithacin V Umapathi Reddy is an Honorary Professor, Department of Electrical Engineering, Indian Institute of Technology, Hyderabad, India vur@iithacin get rid of the interuser and the inter-stream interference and to achieve the sum-rate capacity [7], a precoding based on dirty paper coding DPC has been proposed [8], [9] The complexity of DPC is very high and practical methods for achieving similar performance have been explored since then The linear precoding techniques based on channel inversion and regularized channel inversion are of low complexity and applicable when the number of transmit antennas is greater than or equal to the total number of antennas of all users [3], [] A non-linear pre-equalization technique called Tomlinson Harashima precoding THP is proposed in the downlink MU- MIMO with single antenna users to provide interferencefree communication in [4] This precoding technique achieves better performance than linear techniques since it constrains the transmitted power while the complexity is comparable to them In the downlink MU-MIMO, the sum-rate capacity can be improved with multiple antennas at users [7] With modulo operation both at the transmitter and the receiver, the authors in [] proposed a scheme called eigenmode THP to bring down the peak to average power ratio PAPR that resulted in eigenmode DPC for multi-antenna MU-MIMO broadcast channel The expressions for the transmit and the receive filters are provided only for the case of two users, each with two antennas and a single data stream is allocated for each user For users having multiple antennas, an antenna selection method is proposed in [] to solve the dimensionality problem in zero-forcing ZF pre-processing based on the maximization of sum-rate capacity A Block Diagonalization based precoding scheme for canceling interuser interference and a method for maximizing the sum information rate under a power constraint have been proposed in [3] The cost of this approach is the increased number of antennas at the transmitter The authors in [3] have also proposed a coordinated transmit-receive iterative processing to relax the constraint on the number of transmitting antennas at the BS A precoding scheme, named successive zero-forcing SZF with iterative water-filling algorithm for obtaining precoding matrices for users having multiple antennas, is proposed in [3] Iterative solutions, based on the minimization of mean square error MSE between transmit and receive signal vectors have been proposed for selecting the feedback filter, the feed-forward filters and the receive filter in [4] - [8] An iterative coordinated THP design has been proposed in [9] to support

2 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 multiple data streams in downlink MU-MIMO with multiple antennas at the users Starting with random post-processing vectors initially, they proposed an iterative algorithm for obtaining the processing vectors by reducing the multiuser interference MUI after THP in each step In the transceiver design of the downlink MU-MIMO with users having multiple antennas, [], [] offer a non-iterative closed form solution In [], the authors have proposed a nonlinear joint transmitter-receiver processing algorithm where linear preprocessing matrix corresponding to each user is constrained to be in the null space of augmented channel matrix, constructed from channel matrices of other users The feedback matrix and post-processing filter at the receiver are then obtained by minimizing the maximum value of the post detection noise variance The authors have demonstrated the performance improvement of this algorithm over other joint transmit receive processing algorithms known in the literature This method assumes that the number of data streams for each user to be equal to the number of receive antennas of that user Inspired by the idea of [], the authors in [] have come up with a transceiver processing algorithm based on block successive zero-forcing BSZF and non-linear THP and relaxed the constraint on the number of data streams allocated to a user They have demonstrated the improvement in sumrate capacity over the previously proposed methods using simulations However, the BSZF requires that the linear preprocessing matrix for each user to be in the null space of a matrix constructed by augmenting channel matrices of all users considered previously Therefore, this method is not applicable to scenarios where the number of transmitter antennas is less than the total number of receive antennas of a subset of users In [], the authors have outlined a method to serve an arbitrary number of streams to the users with multiple antennas However, no criteria is proposed for the selection of processing vectors In this paper, by extending the idea of [], we propose a non-iterative method for the selection of processing vectors based on the maximization of the instantaneous SNR of the data streams at the detector input of each user We find the expressions for the instantaneous SNR of all the data streams and express them in terms of eigenvalues of a Wishart matrix, obtained from the channel matrix of the user and find the diversity order for each data stream Considering a single transmitter base station, we determine the outage probability and the cumulative distribution function cdf of sum-rate capacity of the proposed method and compare with other methods that offer closed form solutions [] and [], channel inversion, Block Diagonalization [3], THP [4], under several scenarios We may mention here that while comparing with [], we did not consider inter-cluster interference It is shown that our method addresses all scenarios of the downlink MU- MIMO with users having multiple antennas with the condition that the total number of data streams of all users is less than or equal to the number of transmit antennas The remainder of this paper is organized as follows In Section II, the system model is given and in Section III, we present the method for the selection of the transmit and receive filters based on the maximization of per stream SNR We give the diversity analysis of the proposed method in Section IV Section V provides simulation results and compares the performance of the proposed method with other methods mentioned above The numerical comparison of complexities for obtaining the processing vectors using different methods is provided in Section VI In Section VII, we summarize the results of the paper Notation: Column vectors and matrices are denoted by lowercase and uppercase boldface letters, respectively, a denotes the Euclidean norm of the vector a, T and H denote, respectively, the transpose and Hermitian transpose of, < a, b > denotes the inner product of the vectors a and b, defined as a H b, I denotes identity matrix of appropriate dimension The superscript denotes complex conjugate II SYSTEM MODEL We consider the downlink MU-MIMO system with N t antennas at the transmitter for transmitting N s -length data stream vector a = [a T a T a T K] T to K users, where elements of k th user vector a k = [a k, a k,, a klk ] T are drawn from a constellation A of average power σ a The K users with N, N,, N K antennas are served with L, L,, L K data streams respectively, with the condition that L k N k, k =,,, K and the total number of data streams transmitted is N s = K i= L i N t The matrix of channel between pairs of transmit and receive antennas of size N r N t is represented as H = [H T H T H T K ]T where H i denotes the i th user channel matrix of size N i N t and N r = K i= N i The elements of the channel matrix H are assumed to be independent and identically distributed iid complex Gaussian variables with zero-mean and unit variance The channel matrix is assumed to be known at the transmitter The transmitted signal vector x = [x x x Nt ] T is obtained using THP [4] as shown in Fig and is passed through MU-MIMO channel The received signal vector for all users can be represented as y = Hx + n where y = [y T y T yk T ]T is vector of size N r with y k = [y k y k y knk ] T of size N k denoting the received signal vector of k th user and n = [n T n T n T K ]T noise vector corresponding to all users with n k = [n k n k n knk ] T of size N k, modeled as a zero-mean circularly symmetric complex white Gaussian random vector with covariance matrix I The variation of noise among users is due to different path loss coefficients for different users in the downlink We apply post-processing filter Wk H of size L k N k for k th user, k =,,, K see Fig After post-processing, the received signal vector for all users can be represented as ỹ = W H Hx + ñ where W is the block diagonal matrix W W W =, 3 W K

3 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 Fig Downlink MU-MIMO system with THP and post-processing ỹ = [ỹ T ỹt ỹ T K ]T with ỹ k = W H k y k, ñ = W H n and W k = [w k, w k,, w klk ] with each element as N k vector III SELECTION OF PROCESSING VECTORS W k, k =,,, K In this section, we provide a procedure for the selection of the processing vectors that maximize the instantaneous SNR of data streams at the detector input of the user For the system as described by, the equivalent channel matrix is defined as H = W H H 4 where H is of size N s N t Combining 3 and 4, we express h H wh H h H wh H h H L w H L H H = = h H 5 ck h H wkh H K ck+ wkh H K h H w H KL K H K Ns where k ck = L i 6 for k =,,, K and c = We note from 5 that row index of H corresponding to the post-processing vector for l th data stream of k th user is given by ck + l By performing orthonormalization of rows of H [] and arranging vectors in the matrix form, we can write H = i= s s s } s Ns s Ns s Ns3 {{ s NsNs } S f H f H f H Ns } {{ } F H where F is of size N t N s The matrix F has orthonormal columns and will be an unitary matrix when N s = N t Note that the equivalent channel matrix H is not necessarily 7 square The transmitter applies THP [4] See Fig with the equivalent channel matrix decomposed as H = SF H and the feedback filter B = [b ij ] Ns N s selected as B = GS 8 where G = diag/s, /s,, /s NsNs and S is the lower left triangular matrix given in 7 Note here that B in the feedback path is the lower left triangular matrix with unity diagonal elements b ij = for i < j, b ii = and i, j N s and I the identity matrix of size N s N s While applying the precoding for the l th data stream of k th user k =,, K and l =,, L k, assuming that a kl s are drawn from a M ary square constellation A = {a I + ja Q a I, a Q {±, ±3,, ± M }}, the modulo operation reduces the symbols x ck+l into the boundary region of A, where ck is given in 6 This is equivalent to adding integer multiples of M to the real and imaginary parts of x ck+l [4] The symbols at the output of the precoder are given by ck+l x ck+l = a kl + d kl i= b ck+li where d kl { Md I + jd Q d I, d Q Z} are implicitly selected by the modulo operation Thus, the effective data symbols ã kl = a kl + d kl, k =,, K, l =,, L k 9 are assumed to be passed through the feedback structure without the modulo operation Let ã = [ã T ãt ã T K ]T, ã k = [ã k ã k ã klk ] T, d = [d T d T d T K ]T, d k = [d k d k d klk ] T and x = [ x x x Ns ] T Then, the precoder with equivalent linear interpretation of modulo operation can be represented as shown in Fig [4] From Fig THP with equivalent linear interpretation Fig, we note that the precoded signal x at the output of feedback filter is related to ã and B as which can be written as ã ã = ã K x i ã = B x s s s Ns s NsNs s Ns s NsNs The transmitted signal vector is then given by x x x Ns x = F x 3

4 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 Since F H F = I, there is no increase in the signal power at the transmitter due to linear processing by F The increase in M signal power due to precoding is M, which is negligible for moderate values of M [4] From, the received signal component corresponding to the l th data stream of k th user, after performing the postprocessing, is given by ỹ kl = w H klh k x + w H kln k 3 Substituting in 3 and using w H kl H k = sck+l f H + s ck+l f H + + s ck+lck+l f H ck+l from 5 and 7, we can write 3 as ỹ kl = sck+l f H + s ck+l f H + + s ck+lck+l f H ck+l [f f f ck+l f Ns ] x x x Ns + wh kln k 4 As f, f,, f Ns are orthonormal, 4 can be simplified as ỹ kl = s ck+l x + s ck+l x + + s ck+lck+l x ck+l +w H kln k 5 The term in the brackets of 5 is ck + l th row of right hand side of multiplied by s ck+lck+l We can then write 5 as ỹ kl = s ck+lck+l ã kl + w H kln k 6 A scaling with /s ck+lck+l and a modulo operation are employed at the user to find the estimate of the transmitted symbol a kl Thus, the downlink MU-MIMO channel with users having multiple antennas is decoupled into parallel single data stream channels The multiuser interference MUI is completely eliminated after post processing at each user Neglecting the transmit power increase caused by modulo operation for large QAM constellation, this penalty is very small [4], from 9 and 6, the instantaneous signal to noise ratio SNR corresponding to the l th data stream of k th user can be written as γ kl = s ck+lck+l w kl 7 where ck is as given in 6, is the noise variance at user-k We now find w kl, k =,,, K, l =,,, L k, based on the maximization of the instantaneous SNR of the associated data stream at the input of the detector of the user Performing orthonormalization of rows of H given in 5 [], the resultant vectors can be arranged in matrix form as given below H = where u <u, h > u u <u, h Ns > u <u, h Ns > u u Ns u i = h i i l= f H f H f H Ns 8 < u l, h i > u l u l 9 for i =,,, N s and its normalized version is f i = u i u i Comparing 8 and 7, we note that u i = s ii We can then express γ kl, given in 7 as γ kl = u ck+l w kl The u i, for i =,,, N s, is computed as follows i u i < u l, = h i h H i > u l u l l= h i i < u m, h i > u m u m The expansion of is given in 3 at the bottom of the page Canceling 3 rd and 4 th terms given in the second step of 3, and noting that u l and u m for l m are mutually orthogonal, we can simplify u i as u i = h H i h i h H i Combining and 4, we obtain u i = h H i I i i u m u H m u m h i 4 f m f H m h i 5 From 5 and 5, we can express the instantaneous SNR corresponding to l th layer of k th user given in as wkl HH k I ck+l f m fm H H H k w kl γ kl = w kl σ k 6 u i = h H h i h H i i i u lu H l u l= l = h H h i h H i i i u lu H l u l= l h i h i i i l= u H h m i h H i um i u m + l= u H l h i h H i u l i h H i u l + u lu H l l= i h H i u lu H h m i u H l um u l u m h i u l + i i l=,m l h H i u lu H m h i u H l um u l u m 3 4

5 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 From 6, it is clear that w kl as the eigenvector corresponding to the largest eigenvalue of H k I ck+l f m fm H H H k maximizes γ kl The procedure for finding the processing vectors successively is given below For the user-k for k =,,, K, compute ck using 6 For data streams l =,, L k and k =,,, K compute w kl as the eigenvector corresponding to the largest eigenvalue of H k I ck+l f m fm H H H k Then, compute h ck+l using 5 and f ck+l using 9 and After completing step, we find the linear preprocessing matrix F as F = [f f f Ns ] 7 3 Since F H F = I, using 7, the lower triangular matrix S is obtained as S = HF 8 where H is found from 3 and 4 or from 5 4 The matrix in feedback-path B is then computed using 8 Note that all the processing vectors are computed at the transmitter and the post-processing vectors w kl have to be sent to user-k IV DIVERSITY ANALYSIS In this section, we first derive the expression for instantaneous SNR for all the data streams in terms of singular values of a certain matrix H k that is obtained from k th user channel matrix When the elements of the user channel matrices are zero-mean, iid complex Gaussian distributed, the SNR of each stream can be expressed in terms of the eigenvalues of a Wishart matrix W = H k HH k Then, following steps of [], we obtain the diversity order for each data stream Recall the instantaneous SNR corresponding to l th data stream of k th user given in 6 which we repeat below for convenience γ kl = w H kl H k I ck+l f m fm H H H k w kl w kl 9 First, we express the γ kl in terms of singular values of a matrix obtained from the user channel matrix as stated in the following Lemma Lemma The maximum instantaneous SNR of l th data stream of k th user is determined by l th singular value of a matrix obtained by selecting N t ck columns of the matrix H k F where Hk is the k th user channel matrix, F is an orthonormal matrix of size N t N t obtained by extending columns of the linear processing matrix F, ck = k i= L i and L i is the number of data streams allocated to user-i Proof of Lemma From 9, we can write wkl HH k I ck f mfm H ck+l m=ck+ f mfm H γ kl = w kl H H k w kl 3 Now, from Theorem 8 of [3, Chapter ], the set of column vectors f, f,, f Ns of F can be extended to form a basis of vector space C Nt Let the set of extended vectors be {f Ns+, f Ns+,, f Nt } Now, by augmenting these vectors to columns of F, we form an augmented orthonormal matrix F of size N t N t as F = {f, f,, f Ns, f Ns+,, f Nt } 3 Then, the first summation in the bracketed term of 3 can be written as where ck f m f H m = F cknt FH 3 ij = diag,,,,,,, 33 }{{}}{{} i times j i times for integers i, j; i j The second summation in the bracketed term of 3 can be expressed as where ck+l m=ck+ f m f H m = F ckl Nt FH 34 ije = diag,,,,,,,,,,, 35 }{{}}{{}}{{} i times j times e i j times for integers i, j, e and i + j e Combining 3 and 34, and noting that F F H = I, we can write 3 as γ kl = wh kl H k F I cknt ckl Nt FH H H k w kl w kl σ a 36 From 33 and 35, it is easy to see that cknt ckl Nt = ckl Nt cknt = and hence, we can write the terms in the numerator of 36 as I cknt ckl Nt = and I cknt I ckl Nt I cknt H 37 H k F I cknt ckl Nt FH H H k = H k FI cknt I ckl Nt I cknt H FH H H k 38 In 38, the affect of post multiplication of H k F = [ hk hk hknt ] 39 of size N k N t with I cknt makes the first ck columns of H k F as zeros vectors Therefore, H k F I cknt = [ } {{ } ck times h kck+ hkck+ hknt ] 4 We discard the first ck columns of H k F given in 39 and represent the remaining N t ck columns as a matrix H k = [ hkck+ hkck+ ] hknt 4 5

6 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 of size N k N t ck After removing first ck zero column vectors from ckl Nt, we get a matrix l Nt ck of size N t ck N t ck see 33 and 35 Then, using 4, 4 and 38, we can rewrite 36 as γ kl = wh kl H k I l Nt ck H H k w kl w kl By singular value decomposition SVD of H k, we have 4 H k = Ūk Σ k VH k 43 where Ūk = [ū k ū k ū knk ] is an orthonormal matrix of size N k N k, Vk = [ v k v k v knt ck] is an orthonormal matrix of size N t ck N t ck and Σ k is of size N k N t ck with singular values [ λ k λk λkl λkp ] on its principle diagonal We assume here that p L k, which is true when elements of H k are complex Gaussian iid Substituting 43 in 4 and using V k H V k = I, we have γ kl = wh klūk Σ k ΣH k Ū H k w kl w kl wh Σ klūk k VH k V l Nt ck k ΣH k Ū H k w kl w kl 44 The problem of finding the maximum value of γ kl in 44 is equivalent to the problem stated in the proposition given below Proposition Let A be a complex p q matrix of rank p > and let fz = z H AA H z, g z = z H z and g i z = z H u i for i =,,, l, l p and l, p are integers, where u i is the i th left singular vector of A Then, l th left singular vector of A as z, maximizes fz subject to constraints g z = and g i z =, i =,,, l, l p Proof The proof of the proposition is an extension of Example 66 given in [3, Chapter 6] Substitute A = Ūk Σ k and z = w kl Then, from 43 u i = ū ki and w kl = ū kl The constraint g i w kl = wklūki H = for i =,,, l, and l Nt ck see 33 make the second term of 44 as zero As both H k and A = Ūk Σ k have identical left singular vectors and singular values, w kl as l th left singular vector ū kl of H k maximizes the γ kl and the corresponding maximum value is given by Hence, Lemma is proved γ max kl = λ kl 45 When the elements of k th user channel matrix H k are zeromean, unit variance iid complex Gaussian, the elements of H k are also zero-mean, unit variance iid complex Gaussian since F is a unitary matrix Then, we can express the SNR of each data stream see 45 in terms of the eigenvalues of the Wishart matrix W = H k HH k Using the probability density function pdf of eigenvalues of the Wishart matrix, the diversity analysis of a single user MIMO channel with single and multiple data streams is given in [] Following the steps given in [], the achievable diversity order for l th data stream of k th user is given by α kl = N t ck l + N k l + 46 where ck is given in 6 For user-, ck = and H HH = H H H and the achievable diversity order for l th data stream can be written as α l = N t l + N l + 47 From 47, the achievable diversity orders for all data streams of user- are equal to that of a single user MIMO beamforming [] with the equivalent channel matrix of same size Note that the diversity gain for a data stream of a user can be increased either by increasing the number of receiver antennas of that user or by increasing the number of transmitting antennas Assuming equal power for all data streams at the transmitter, the instantaneous value of the sum-rate capacity for user-k is defined as L k r k = l= log + γ max kl 48 We use this metric for evaluating the sum-rate capacity using simulations V SIMULATION RESULTS We conducted simulations using Matlab to evaluate the outage probability and the cdf of the sum-rate capacity for each user assuming elements of channel matrices of all users as zero-mean iid complex Gaussian random variables with unit variance We also assumed equal power allocation to all data streams for each user in the simulations and same noise variance for all the users Considering a single base station transmitting multiple data streams to different users, we compute the outage probability and the CDF of sum-rate capacity of the proposed method and methods of [], [], THP [4], channel inversion and Block Diagonalization [3] in the following subsections We consider two categories of scenarios - A Scenarios with equal number of antennas and equal number of data streams for users, B Scenarios with different number of antennas and different number of data streams for users While comparing the results with method [], we did not consider the inter-cluster interference However, we implemented all other steps of the algorithm In all scenarios, we varied σ a from db to 4 db and for each value of σ k σ a, we used realizations of channel matrices In all the simulations, the data symbols are drawn from 6-QAM constellation In all the methods that use THP, the same Mod operation is used At the transmitter, the symbol ã kl is obtained implicitly by Mod operation as a periodic extension of a kl At the detector input of the receiver, ã kl is mapped back to a kl through the use of the same Mod operation Hence, we used in the SNR calculations The increase in signal power due to M Mod operation in THP precoding is given by M, which is negligible for moderate values of M [] For 6-QAM modulation scheme, the increase in transmit power is 8 db 6

7 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 TABLE I DIFFERENT SCENARIOS CONSIDERED IN SIMULATIONS Category Scenario Description N =, N =, N 3 =, L =, L =, L 3 =, and Nt = 6 A N =, N =, N 3 =, L =, L =, L 3 =, and Nt = 5 3 N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, and Nt = B 4 N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, and Nt = 6 Pγ kl < γ 3 Stream of channel inv Stream of channel inv Stream of prop method Stream of prop method Stream of method [] Stream of method [] Stream of method [] Stream of method [] Stream of THP [4] Stream of THP [4] Stream of BLK DIAG [3] Stream of BLK DIAG [3] TABLE II DIVERSITY ORDERS OF DIFFERENT STREAMS OF USERS IN [] AND PROPOSED METHOD IN SCENARIO- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 6 4 User Data Stream Diversity order with index index Method [] Proposed method in db σ k Fig 3 Outage probability obtained using different methods for the first user in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 6 and γ = 5 db and for 64-QAM modulation scheme, the increase is 684 db In the simulations, method [] is included in the scenarios where L i = N i for i =,, 3 and N r = N t Channel inversion requires the total number of antennas of all users less than or equal to the number of transmitting antennas, ie, N r N t, and each receiving antenna is treated as a singleantenna user thereby requiring N s = N r The same holds in THP [4] too In channel inversion, there is a transmitter power penalty However, in the simulations, the transmitter power is kept same as in other methods by appropriate scaling Pγ kl < γ 3 Stream of channel inv Stream of channel inv Stream of method [] Stream of method [] Stream of prop method Stream of prop method Stream of method [] Stream of method [] Stream of THP [4] Stream of THP [4] Stream of BLK DIAG [3] Stream of BLK DIAG [3] A Category - A Under Category-A, we considered Scenarios- and - with equal number of antennas and equal number of data streams for users as described in Table I Scenario-: In Scenario-, we considered K = 3, N = N = N 3 = and N t = 6 We allocated two data streams to each user L = L = L 3 = All the methods - the proposed, [], [], THP [4], channel inversion and Block Diagonalization [3] can be fitted into the frame work of Scenario- For each channel realization, we computed the instantaneous SNR for all data streams of users using all the methods From the instantaneous SNRs, we obtained the outage probability P γ kl < γ as a function of σ a for a fixed γ = 5 db for all data streams for all the methods The results are plotted in Figs 3 and 4 for user- and 3, respectively For user-, the results are similar except the shift in the σ a axis The diversity orders for the data streams of all users, evaluated using 38 in [] and using 46 for the proposed method, are tabulated in Table The diversity analysis for other methods is not provided in [], [4] and [3] From the results of Figs 3, 4 and Table II, we note the following The outage probabilities for both data streams of a particular user computed using channel inversion are in db σ k Fig 4 Outage probability obtained using different methods for the third user in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 6 and γ = 5 db identical The outage probabilities for both data streams computed using Block Diagonalization [3] are different THP [4] performs better than channel inversion and Block Diagonalization [3] The outage probabilities computed using the method [] for all data streams of a particular user are equal as all the data streams of the user have the same diversity order due to the equal-diagonal QR decomposition employed in this method The results in figures and table agree with each other The outage probabilities for different data streams of a particular user obtained using the method [] and the proposed method, are different as they have different diversity orders The results in figures and table agree with each other for the proposed method The outage probabilities for the proposed method and method [] are identical The proposed method and [] 7

8 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 CDF Channel inv for all users BLK DIAG [3] for all users User 3 User User Method [] Method [] Prop method THP [4] Channel inv BLK DIAG [3] Sum rate in bits/s/hz Fig 5 Cdf of sum-rate capacity obtained using different methods in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 6 and = db σ k Pγ kl < γ 3 Method [] Prop method BLK DIAG [3] in db σ k Fig 7 Outage probability obtained using method [], Block Diagonalization [3] and the proposed method for the first user in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 5 and γ = 5 db CDF Method [] Method [] Prop method THP [4] Channel inv BLK DIAG [3] Sum rate in bits/s/hz Fig 6 Cdf of the total sum-rate capacity of all users obtained using different methods in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 6 and σ a σ k = db perform better over method [] for the first data stream and, method [] performs better for the second data stream of each user The proposed method and [] perform better than channel inversion or Block Diagonalization [3] for the first data stream of user-3 and, channel inversion and THP [4] perform slightly better for the second data stream of user- 3 An alternative way of comparing the performance of these methods is by using the cdf of the sum-rate capacity of each user From the instantaneous values of the sum-rate capacity obtained using 48 at σ a = db in all realizations, we computed the cdf for all users and plotted the results in Fig 5 and Fig 6 We note from Fig 5 that method [] performs better than channel inversion and Block Diagonalization [3] The proposed method and method [] perform consistently better than method [], channel inversion and Block Diagonalization [3] for all users and the improvement increases with user index The performance of the proposed method, THP [4] and method [] are identical in Scenario- Further, considering the total sum-rate capacity of all users, plots of Fig 6 show that the performance of THP [4] is same as that of the proposed method and [], and these methods perform slightly better than [] and much better than channel inversion and Block Diagonalization [3] Scenario-: In Scenario-, we considered K = 3, N = N = N 3 =, L = L = L 3 = and N t = 5 In this scenario, method [], THP [4], channel inversion are not applicable Note that method [] requires at least five transmitting antennas to have precoding matrices for all users see 5 in [] and hence, we set N t = 5 to compare the performance, though the proposed method can work with N t 3 Note that the Block Diagonalization [3] is applicable in this scenario We obtained the outage probability P γ kl < γ for a fixed γ = 5 db using the proposed method, Block Diagonalization [3] and method [] and plotted results in Figs 7 and 8 corresponding to user- and 3, respectively For user-, the results are similar to that of user-3 except that the gap between the curves decreases From 46, we note that the diversity orders of the data streams of user-, user- and user-3 obtained using the proposed method are, 8 and 6, respectively in Scenario- We have also computed the cdf of sum-rate capacity at σ a = db and plotted results in Fig 9 and Fig From the results in Figs 7, and 8, we note that the performance of the proposed method and method [] is identical for user- The performance of the Block Diagonalization [3] and [] is identical for user-3 The performance of the proposed method is consistently better than that of Block Diagonalization [3] for all users In the plots of Fig 9, we note that the curves with descriptors and + are almost overlapping indicating that the performance of user-3 with the 8

9 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 Method [] Prop method BLK DIAG [3] Method [] Prop method BLK DIAG [3] Pγ kl < γ CDF in db σ k Sum rate in bits/s/hz Fig 8 Outage probability obtained using method [], Block Diagonalization [3] and the proposed method for the third user in Scenario- K = 3, N = N = N 3 =, L = L = L 3 =, N t = 5 and γ = 5 db Fig Cdf of the total sum-rate capacity of all users using method [], Block Diagonalization [3] and the proposed method in Scenario- K = 3, N =, N =, N 3 =, L = L = L 3 =, N t = 5 and σ a σ k db = Stream of prop method Stream of prop method Stream of method [] Stream of method [] CDF User of method [] User of method [] User 3 of method [] User of prop method User of prop method User 3 of prop method User of BLK DIAG [3] User of BLK DIAG [3] User 3 of BLK DIAG [3] Sum rate in bits/s/hz Fig 9 Cdf of sum-rate capacity for all users using method [], Block Diagonalization [3] and the proposed method in Scenario- K = 3, N =, N =, N 3 =, L = L = L 3 =, N t = 5 and σ a σ k db = Pγ kl < γ in db σ k Fig Outage probability obtained using method [], Block Diagonalization [3] and the proposed method for the first user in Scenario-3 K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, N t = and γ = 5 db proposed method is almost same as that of user- with method [] and, the curves with descriptors and are identical indicating that the performance of user- is same with both methods The performance of the Block Diagonalization [3] is same for all users The plots of Fig 9 and show the performance improvement of the proposed method over other methods B Category - B Under Category-B, we consider Scenarios-3, and -4 with different number of antennas and different number of data streams for users as described in Table I Scenario-3: In Scenario-3, we considered K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3 and N t = Method [], THP [4], channel inversion are not applicable in this scenario Note that method [] requires at least ten transmitting antennas to derive precoding matrices for all users see 5 in [] and hence, we set N t = though the proposed method can work with N t 6 Block Diagonalization [3] is not applicable as the dimension of the null space of the augmented channel matrix H see 7 in [3] for user- is and does not support two data streams for user- We obtained the outage probability P γ kl < γ for a fixed γ = 5 db using the proposed method and method [] and plotted results in Figs and corresponding to users and 3, respectively We have also computed the cdf of sum-rate capacity at σ a = db using these methods and plotted results in Fig 3 and Fig 4 From the results in Figs, and, we note that the performance of the proposed method is the same as that of method [] for user- The 9

10 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 Pγ kl < γ Stream of prop method Stream of prop method Stream 3 of prop method Stream of method [] Stream of method [] Stream 3 of method [] CDF Method [] Prop method in db σ k Sum rate in bits/s/hz Fig Outage probability obtained using method [] and the proposed method for the third user in Scenario-3 K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, N t = and γ = 5 db Fig 4 Cdf of the total sum-rate capacity of all users using method [] and the proposed method in Scenario-3 K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, N t = σ a σ k = db Stream of User Stream of User Stream of User Stream of User 3 Stream of User 3 Stream 3 of User 3 CDF User of method [] User of method [] User 3 of method [] User of prop method User of prop method User 3 of prop method Pγ kl < γ Sum rate in bits/s/hz Fig 3 Cdf of sum-rate capacity for all users using method [] and the proposed method in Scenario-3 K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, N t = σ a σ k = db in db σ k Fig 5 Outage probability obtained using the proposed method in Scenario- 4 K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3, N t = 6 and γ = 5 db performance of the proposed method is better for the first and second data streams of user-3 and the performance of both methods is identical for the third data stream of user-3 In the plots of Fig 3, note that the curves with descriptors and are overlapping indicating that the performance of user- is same with the proposed and method [] The performance of the proposed method is better for users and 3 Further, considering the total sum-rate capacity of all users, the plots of Fig 4 show the improved performance of the proposed method over method [] Scenario-4: In this scenario, we considered K = 3, N = 3, N = 4, N 3 = 5, L =, L =, L 3 = 3 and N t = 6 Method [], THP [4], Channel inversion are not applicable in this scenario Block Diagonalization [3] is also not applicable in this scenario as null space does not exist for the augmented channel matrix H k see 7 in [3] For method [], user- and user- precoding matrices F and F can be computed see 5 in [] However, the precoding matrix F 3 for user-3 does not exist and hence method [] is not applicable in this scenario For each channel realization, we computed the instantaneous SNR for all data streams of users using the proposed method and obtained the outage probability P γ kl < γ as a function of σ a for a fixed γ = 5 db, and the results are plotted in Fig 5 We may mention here that Scenario-4 under Category-B is included to emphasize that the proposed method is applicable if the number of antennas at the transmitter is greater than or equal to the total number of data streams, irrespective of the number of antennas of users From simulation results presented in Scenarios- to -4, we summarize the following The total sum-rate performance of the proposed method is better than that of channel inversion and method []

11 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 TABLE III DIFFERENT SCENARIOS CONSIDERED UNDER CATEGORY-B FOR THE COMPARISON OF COMPUTATIONAL COMPLEXITIES OF THE PROPOSED METHOD AND [] N t N N N 3 L L L Number of Flops x Prop method Method [] Method [] THP [4] in Scenario- Channel inversion and method [] are not applicable in Scenarios-, -3, and -4 The total sum-rate performance of the proposed method is better than that of Block Diagonalization [3] in Scenarios-, and - Block Diagonalization [3] is not applicable in Scenarios-3, and -4 The total sum-rate performance of the proposed method is identical to that of THP [4] in Scenario- THP [4] is not applicable in Scenarios-, -3 and -4 The total sum-rate performance of the proposed method is identical to that of method [] in Scenario- and better than that of [] in Scenarios- and -3 Method [] is not applicable in Scenario-4 From the simulation results of these scenarios, it follows that the proposed method is applicable for all the scenarios VI COMPARISON OF COMPUTATIONAL COMPLEXITY OF DIFFERENT METHODS In this section, we provide the numerical comparison of complexities of different non-linear methods that offer noniterative closed form solutions - the proposed, [], [] and THP [4] in terms of approximate number of floating point operations flops required for computing the processing vectors The complexity analysis of obtaining processing vectors using these methods is given in the Appendix The number of flops for a particular method is obtained by adding the number of flops counted in all steps described in the Appendix for all users All these methods require a modulo operation at the transmitter and at the receiver For comparing the computational complexity of different methods, we consider many scenarios of Categories-A and - B, considered in Sec V Under Category-A, we consider the scenarios with N r = N s = N t, N = N = N 3 = Nr 3 and L = L = L 3 = Ns 3 The number of flops computed using the proposed method, [], [] and [4] under these scenarios is plotted in Fig 6 Under Category-B, we consider the scenarios with dissimilar number of antennas and data streams for users as given in Table III corresponding to different number of transmit antennas N t The number of flops computed using the proposed method and [] under these scenarios is plotted in Fig 7 We note here that methods [] and [4] are not applicable in scenarios considered under Category-B We note from Figs 6 and 7 that the number of flops required for THP [4] is less than that of other methods in Number of transmitting antennas Nt Fig 6 Comparison of computational complexities of different methods in different scenarios under Category-A N r = N s = N t, N = N = N 3 = N r 3 and L = L = L 3 = Ns 3 Number of Flops x 6 Prop method Method [] Number of transmitting antennas Nt Fig 7 Comparison of computational complexities of different methods in different scenarios listed in Table-III corresponding to different number of transmit antennas under Category-B scenarios that are applicable to THP [4] The number of flops required for methods [] and [] is same under Category-A The number of flops required for the proposed method is less than that of [] or [] in scenarios that are applicable to them VII CONCLUSION For the downlink MU-MIMO with users having multiple antennas, we proposed a non-iterative method by combining THP with pre- and post-processing The transmit and receive filters are obtained by maximizing the instantaneous SNR of data streams at detector input for each user We derived expressions for the instantaneous SNR of all the data streams of k th user and expressed them in terms of eigenvalues of a Wishart matrix obtained from the k th user channel matrix We then used the results in the literature for determining the diversity orders of different data streams with the proposed method We have performed simulations to evaluate the outage probability and the cdf of the sum-rate capacity for all users using the

12 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 proposed method and compared with those of other methods We have also compared the computational complexity of the proposed method with that of other methods that offer noniterative closed form solutions We may mention here that the proposed method is applicable if the number of transmitting antennas is at least equal to the total number of data streams, irrespective of the number of antennas of the users APPENDIX COMPLEXITY ANALYSIS OF OBTAINING PROCESSING VECTORS USING DIFFERENT METHODS We analyze the complexity of different methods by counting the number of flops required in all steps of the algorithm while obtaining processing vectors using the proposed method, methods of [], [] and THP [4] It has been shown in [4] that there exists an equivalence between the operations involving the complex matrices or vectors and that of the real matrices or vectors obtained by mapping the complex matrices or vectors A complex vector z and a complex matrix Z can be mapped to a real vector ẑ and a real matrix Ẑ as [4] [ ] [ Rz RZ IZ ẑ = and Ẑ = Iz IZ RZ ] 49 We count the number flops for a particular operation by considering the mapped real vectors and real matrices of double the size as described in 49 For example, the multiplication of two real matrices A of size m n and B of size n p requires mnp flops [, Sec 4] while the multiplication of two complex matrices C of size m n and D of size n p requires 6mnp flops For all the methods, the number of flops in each step are expressed in terms of the number of transmitting antennas N t, the number of receiving antennas of users N i, i =,,, K, the number of data layers allocated to users L i, i =,,, K and the total number of antennas of all users N r, with the notation followed in this paper A Proposed method The following steps are required in the computation of processing vectors using the proposed method For obtaining the post-processing vectors w kl, l =,,, L k for user-k for k =,,, K, we need to compute Eigenvectors of H k I ck+l f m fm H H H k As the matrix P k = I ck+l f m fm H satisfies Pk = P k, the processing vectors can be obtained by computing the left singular vectors of H k P k of size N k N t a Multiplication of H k and P k requires 6N k Nt flops b The number of flops required for computing left singular vectors of H k P k is 3Nk N t + 4Nt 3 c Computing h ak+ to h ak+lk requires multiplication of Wk H = [w k, w k,, w klk ] H and H k and takes 6L k N k N t flops For obtaining f i, i =,,, N s and S, we need to perform orthonormalization of rows of H of size N s N t and this requires 6N s Nt flops [, Sec 58] 3 For obtaining G, we need to perform N s scalar divisions 4 Finally, for obtaining B, we need to multiply G and S and this requires 8Ns flops B Method [] The following steps are involved in the computation of processing vectors for method [] For user-k, k =,,, K a The computation of linear processing matrix F k satisfying 3 of [] requires computation of right singular vectors of a complex matrix H k see 5 in [] of size N k N t where N k = K i=k+ N i This requires 6 N k Nt + 88Nt 3 flops [, Sec 545] As H K is assumed to be a zero matrix, the number of flops is not counted in this step for k = K b In the computation of post-processing vector and components of feedback matrix, 33 of [] requires Geometric mean decomposition GMD of H k N k of size N k N t N k The multiplication of H k and N k requires 6N k N t N t N k flops The GMD requires [N k + N t N k ]N k flops over and above those required for SVD [5] Thus, the total number of flops required for GMD is 3Nk N t N k + 76N t N k 3 + [N k + N t N k ]N k Note that for counting the number of flops for SVD, we took the number given in the fifth row of the table in [, Sec 545] using R-SVD and mapped to the complex case c The computation of G k given in 33 of [] requires N k flops d The computation of post-processing matrix in 333 of [] requires 8Nk flops e The computation of diagonal submatrix B kk in 334 of [] requires 8Nk flops Finally, non-diagonal submatrices of the matrix B = DHF in the feedback path requires computation of submatrices D k H k F i for i = k + K and k =,,, K The submatrix D k H k F i requires 6Nk N t+6n k N t N i flops Therefore, the total number of flops required for obtaining all non-diagonal subma- ] trices is C Method [] [ K k= K i=k+ 6N k N t + 6N k N t N i The following operations are performed in the computation of processing vectors using the method [] For user-k, k =,,, K a The computation of linear processing matrix F k satisfying 6 of [] requires computation of right singular vectors of a complex matrix H eq,k of size N k N t where N k = k i= N i This requires 6 N k N t + 88N 3 t flops [, Sec 545] As H eq,

13 IEEE Transactions on Vehicular Technology, DOI: 9/TVT65739 is assumed to be a zero matrix, the number of flops is not counted in this step for k = b The computation of post-processing vector and components of feedback matrix using 39 to 4 of [], requires the computation of generalized triangular decomposition GTD of H eq,k N k of size N k N t N k The multiplication of H eq,k and N k requires 6N k N t N t N k flops The GTD requires [N k + N t N k ]N k flops over and above those required for SVD [6] Thus, the total number of flops required for GTD is 3Nk N t N k +76N t N k 3 +[N k +N t N k ]N k Note that for counting the number of flops for SVD, we took the number given in the fifth row of the table in [, Sec 545] using R-SVD and mapped to complex case c The computation of diagonal submatrix of C kk in 4 of [] requires 8L k flops d The computation of Λ k in 4 of [] requires L k flops e The computation of post-processing matrix R k in 4 of [] requires 8L k flops Finally, non-diagonal submatrices of feedback matrix RH eq F requires computation of submatrices R k H k F i for i = k and k =, 3,, K The submatrix R k H k F i requires 6N k N t L i + 6L k N k L i flops Therefore, the total number of flops required [ for computing all non-diagonal submatrices is K ] k k= i= 6N kn t L i + 6L k N k L i D THP [4] The following steps are required in the computation of processing vectors using THP [4] This method is applicable when N t N r and N s = N r The LQ-decomposition of the complex channel matrix H see 4 in [4] of size N r N t requires 6N r Nt flops [, Sec 58] The computation of G requires N r scalar divisions 3 The computation of matrix in the feedback path B = GS requires 8Nr flops REFERENCES [] J Liu and W Krzymien, A novel nonlinear joint transmitter-receiver processing algorithm for the downlink of multiuser MIMO systems, IEEE Transactions on Vehicular Technology, vol 57, no 4, pp 89 4, July 8 [] L Sun and M Lei, Adaptive joint nonlinear transmit-receive processing for multi-cell MIMO networks, in IEEE Global Communications Conference GLOBECOM, Dec, pp [3] Q Spencer, A Swindlehurst, and M Haardt, Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels, IEEE Transactions on Signal Processing, vol 5, no, pp 46 47, Feb 4 [4] C Windpassinger, R Fischer, T Vencel, and J Huber, Precoding in multiantenna and multiuser communications, IEEE Transactions on Wireless Communications, vol 3, no 4, pp 35 36, July 4 [5] G Caire and S Shamai, On the achievable throughput of a multiantenna gaussian broadcast channel, IEEE Transactions on Information Theory, vol 49, no 7, pp 69 76, July 3 [6] Q Spencer, C Peel, A Swindlehurst, and M Haardt, An introduction to the multi-user MIMO downlink, IEEE Communications Magazine, vol 4, no, pp 6 67, Oct 4 [7] S Vishwanath, N Jindal, and A Goldsmith, Duality, achievable rates, and sum-rate capacity of gaussian MIMO broadcast channels, IEEE Transactions on Information Theory, vol 49, no, pp , Oct 3 [8] M Costa, Writing on dirty paper corresp, IEEE Transactions on Information Theory, vol 9, no 3, pp , May 983 [9] W Yu and J Cioffi, Sum capacity of gaussian vector broadcast channels, IEEE Transactions on Information Theory, vol 5, no 9, pp , Sept 4 [] H Lee, K Lee, B Hochwald, and I Lee, Regularized channel inversion for multiple-antenna users in multiuser MIMO downlink, in IEEE International Conference on Communications, 8, May 8, pp [] M Noda, M Muraguchi, T Khanh, K Sakaguchi, and K Araki, Eigenmode tomlinson-harashima precoding for multi-antenna multiuser MIMO broadcast channel, in International Conference on Information, Communications Signal Processing, Dec 7, pp 5 [] Y Wu, J Zhang, H Zheng, X Xu, and S Zhou, Receive antenna selection in the downlink of multiuser MIMO systems, in IEEE 6nd Vehicular Technology Conference, VTC-5-Fall, 5, vol, Sept 5, pp [3] A Dabbagh and D Love, Precoding for multiple antenna gaussian broadcast channels with Successive Zero-forcing, IEEE Transactions on Signal Processing, vol 55, no 7, pp , July 7 [4] A Mezghani, R Hunger, M Joham, and W Utschick, Iterative THP transceiver optimization for multi-user MIMO systems based on weighted Sum-MSE minimization, in IEEE 7th Workshop on Signal Processing Advances in Wireless Communications, July 6, pp 5 [5] P Tejera, W Utschick, G Bauch, and J Nossek, Subchannel allocation in multiuser multiple-input multiple-output systems, IEEE Transactions on Information Theory, vol 5, no, pp , Oct 6 [6] W Ho and Y C Liang, Efficient power minimization for MIMO broadcast channels with BD-GMD, in IEEE International Conference on Communications, June 7, pp [7] S Shi, M Schubert, and H Boche, Downlink MMSE transceiver optimization for multiuser MIMO systems: Duality and Sum-MSE minimization, IEEE Transactions on Signal Processing, vol 55, no, pp , Nov 7 [8] P Ubaidulla and A Chockalingam, Robust THP transceiver designs for multiuser MIMO downlink with imperfect CSIT, EURASIP J Adv Signal Process, vol 9, Feb 9 [9] K Zu, B Song, M Haardt, and R de Lamare, Coordinated tomlinsonharashima precoding design algorithms for overloaded multi-user MIMO systems, in International Symposium on Wireless Communications Systems ISWCS, Aug 4, pp [] N K Chavali, V U Reddy, and K Kuchi, Combining pre- and postprocessing with tomlinson-harashima precoding in downlink MU-MIMO with each user having arbitrary number of antennas, in International Conference on Signal Processing and Communications SPCOM, July 4, pp 4 [] G H Golub and C F Van Loan, Matrix Computations 3rd Ed Baltimore, MD, USA: Johns Hopkins University Press, 996 [] E Sengul, E Akay, and E Ayanoglu, Diversity analysis of single and multiple beamforming, IEEE Transactions on Communications, vol 54, no 6, pp , June 6 [3] H Dym, Linear Algebra in Action, nd Edition American Mathematical Society, 3 [4] I E Telatar et al, Capacity of multi-antenna gaussian channels, European transactions on telecommunications, vol, no 6, pp , 999 [5] Y Jiang, W W Hager, and J Li, The geometric mean decomposition, Linear Algebra and Its Applications, vol 396, pp , 5 [6] Y Jiang, W Hager, and J Li, The generalized triangular decomposition, Mathematics of computation, vol 77, no 6, pp 37 56, 8 3