Comparison of MMSE SDMA with Orthogonal SDMA Approach

Transcription

1 Comparison of MMSE SDMA with Orthogonal SDMA Approach Semester Project Name: Guftaar Ahmad Sardar Sidhu Majors: Communication Systems and Electronics Supervisor: Prof. Dr. Werner Henkel Tutor: Khaled Shawky Hassan, MSc. Semester: Fall 2008 Institute: Jacobs University Bremen, Germany 1

2 Contents 1 Introduction MIMO systems Multiple Access Systems Space Division Multiple Access (SDMA Multi-User MIMO systems Introduction SVD based SDMA Channel Block Diagonalization based SDMA Iterative algorithm based SDMA Orthogonal SDMA System Model Transmit/Receive Filter Design and Channel Block Diagonalization MMSE SDMA Transmit/Receive filter Design Simulation Results Orthogonal SDMA MMSE SDMA Conclusion and Future Work Conclusion Future Work

3 Chapter 1 Introduction 1.1 MIMO systems Multiple Input Multiple Output (MIMO systems use multiple antennas at both the transmitter and the receiver ends. These systems have gained considerable importance because of there potential for dramatic gain in capacity, inreased diversity and reliability. Figure 1.1 shows a typical MIMO system with two antennas on both the transmitter and the receiver sides. Figure 1.1: MIMO system For a fixed power and bandwidth, the capacity of a MIMO channel increases linearly with the number of antennas. This is the MIMO multiplexing gain. MIMO can also be used to increase SNR, to improve the reliability of data, by sending the data from more then one antennas. This is called MIMO diversity gain. Space-time codes are an example of systems benefiting from the diversity gain. MIMO channel is represented by a channel matrix H, each co- 3

4 efficient in the channel matrix represents the gain between each antenna-pair, i.e, the transmitter and the [ receiver. ] For the above shown MIMO system, h11 h the channel matrix H = 12, where h h 21 h 11, h 12, h 21, and h 22 represent 22 the channel gain between the first transmit antenna Tx 1 and first receive antenna Rx 1, second transmit antenna Tx 2 and Rx 1, Tx 1 and second receive antenna Rx 2, and Tx 2 and Rx 2 respectively. Multiple antennas can be used both for the single user systems and the multi-user systems. In single user systems, the multi-antenna base station (BS communicates with a multiantenna user terminal, while in the second case the BS transmits to several user equipped with more then one receive antennas. In this work we shall focus on the multi-user multi-antenna systems, known as multi-user MIMO systems. Multi-user MIMO can be generalized into two categories, based on the direction of communication: MIMO downlink model, in which a single transmitter, the (BS transmits data to multiple receivers, and MIMO uplink model in which multilple senders send data to a single receiver, i.e more then one user-terminals send to the BS. 1.2 Multiple Access Systems The term multiple access means the sharing of the resource amongst users where multiple access systems coordinate access between multiple users. There exist different ways of multiple access. The frequency division multiple access (FDMA, the time division multiple access (TDMA, and the code division multiple access (CDMA have been widely used in wireless communications in the last decades due to their perfect orthogonal division multiplexing. FDMA gives users an individual allocation of one or several frequency bands, allowing them to utilize the allocated spectrum without interfering with each other. Multiple access is achieved by assigning subsets of orthogonal subcarriers to individual users. In TDMA instead of allocating different frequencies, users are separated by time, this means that the users use the same frequencies, but at different times. In CDMA users are separated by means of different codes. CDMA does not divide up the channel in time (as in TDMA, or frequency (as in FDMA. Instead, it encodes data with a special code associated with each user. The users use the same frequencies and the same time slots but with different codes. Therefore, users are allowed to share the same resources (frequency and time. Multiple antenna systems have brought up another multiple access scheme, which is the space division multiple access. This scheme isolates different users transmission by transmitting orthognal or partial orthogonal beams in the space. 4

5 However, in the later case, the interference can not be completely mittigated. 1.3 Space Division Multiple Access (SDMA Multi-user MIMO exploits the availability of multiple independent radio terminals in order to enhance the communication capabilities of each individual terminal. In contrast, single-user MIMO only considers the access to the multiple antennas that are already physically connected to the terminal. SDMA allows this terminal to transmit (or receive multiple signals to (or from multiple users in the same band simultaneously. Space division multiple access enables creating parallel spatial pipes next to higher capacity pipes through spatial multiplexing and/or diversity, by which it is able to offer superior performance in radio multiple access communication systems. By using smart antenna technology, SDMA techniques offer attractive performance enhancements. The radiation pattern of the base station, both in transmission and reception, is adapted to each user to obtain highest gain in the direction of that user. 5

6 Chapter 2 Multi-User MIMO systems 2.1 Introduction Use of spatial domain, in wireless communications, using multiple antennas at the transmitter and the receiver can increase both the reliability and throughput of the system. Now a days the research on MIMO systems has shifted to provide services to more then one users, so called Multi-user MIMO systems. Multi-user MIMO can leverage multiple users as spatially distributed transmission resources, at the cost of somewhat more complex signal processing. The main limiting factor on the performance of a multi-user MIMO system is the presence of multi-user interference, therefore the key challange in such a system is to design the transmit(tx and receive (Rx filters, while keeping in mind the inter-user interference along with the inter-stream interference of the same user. There are different approaches which are used to design the optimum Tx and Rx filters SVD based SDMA In many wireless applications channel state information (CSI can be known at both ends of the transceiver. In frequency division duplex (FDD channels, the CSI is fedback to the transmitter, while in time division duplex (TDD channels, the transmitter itself estimates the CSI in the receive mode. The well known SVD approach along with water-filling power allocation scheme gives the best peroformance for the known CSI. This approach decouples the MIMO chasnnel into eigen subchannels (eigen modes and allocates power to these subchannels according to the water filling strategy. However, SVD scheme for the best performance requires adaptive modulation accross differen eigen subchannels, which inreases the complexity of the system. 6

7 2.1.2 Channel Block Diagonalization based SDMA The second approach uses the block diagonalization, also known as channel inversion or zero-forcing (ZF approach. This approach finds the optimal transmitter and receiver matrices such that all inter-user interference is cancelled out leaving with only the inter-stream interference due to channel uncertainty. In this work, we call this approach Orhtogonal SDMA. Orthogonal SDMA, however, imposes a strict condition on the dimension of the Multi-user MIMO system, such that number of transmit antennas at the BS must be greater than or equal to the total number of receive antennas on all users. This condition makes this strategy impractical in some scenarios Iterative algorithm based SDMA This approach removes the limitation on the number of transmit antennas and jointly optimizes the power allocation and transceiver filters, using an iterative algorithm. This approach tries to minimize the mean squared error (MSE in received symbol which have both inter-user interference and inter-stream interference. We call this approach Minimum Mean Squared Error(MMSE SDMA. 2.2 Orthogonal SDMA As a first step we consider the design of trnasmit (Tx and recieve (Rx filters in multi-user MIMO systems using orhogonal SDMA approach. Orthogonal SDMA requires the CSI, both at the transmitter and the recievers. We look into a solution which decouples the multi-user MIMO Tx-Rx optimization into single user MIMO Tx-Rx optimizations [1]. This approach proposes a transmit filter based on null space constraint which block diagonalizes the channel matrix. This approach leeds to complete cancelation of inter-user interference, however, it requires that the total number of transmit antennas at BS are greater than or equal to the total number of receive antennas on all users System Model We consider the downlink transmission over flat fading channels with OFDM multicarrier transmission, where flat fading condition is assumed over each sub-carrier. The system consists of a base station, equipped with M antennas and K users, each equipped with N k antennas, such that the total number of receive antennas N = K k=1 N k, and M N. The base station with M 7

8 transmit antennas is transmitting L = K k=1 L k data streams to the given K users, where L k are the data streams received by user k. This general setup is shown in the figure 2.1. Figure 2.1: SDMA MIMO System The symbols of each user are collected in data vector x k = [x k1, x k2,..., x klk ] T and the global data vector is x = [ x T 1,xT 2,...,xT K] T. The flat fading channel between the transmitter and user k is represented by N k M matrix H k. The overall N M channel matrix H is given by H = [ H T 1,HT 2,...,HT K] T. The global data vector x is pre-processed by the global transmit filter F which is obtained by stacking K transmit filters F = [F 1,F 2,...,F K ]. Based on this model, user k receives the vector y k = H k.f.x + n k, (2.1 where n k represents the additive white Gaussian noise (AWGN at the k th user terminal. User k post-processes the received vector y k with its L k N k 8

9 receive filter G k, to decode its L k symbols resulting in r k = G k.h k.f.x + G k.n k. (2.2 The global data vector x contains the symbols of all users, so to avoid interuser interference completely, H.F must be block diagonalized Transmit/Receive Filter Design and Channel Block Diagonalization Our first aim is to find the optimal transmit filter F such that all multi-user interference (MUI is cancelled, hence resulting in H.F a block diagonalized matrix. By looking into our system model, carefully, the zero forces constranint forces F k to lie in the null space of Hc k, where Hc k is obtained by removing the corresponding rows of user k from the channel matrix H, and can be represented as Hc k = [ H T 1,H T 2,...,H T k 1,HT k+1,...ht K] T. We use the matrices F and G as defined in [1]. F = B.E, where E k is an N k L k optimized power allocation matrix and the matrix B k, an orthogonal basis for the null space of Hc k, and B = [B 1,B 2,...,B K ]. This means that matrix F ensures both the orthogonality and optimized power allocation. The post-processing matrix G k is an L k N k matrix. Both matrices, E k and G k, are completely defined in the following text. To obtain the optimal E and G, here we use the joint Tx - Rx optimization as proposed in [1], where the optimization is performed over channel H k.b k for user k. min E [ x k r k 2]. (2.3 E k,g k s.t trace ( F H k.f k = Pk. Using the langrange multiplier technique and Karush-Kuhen-Tucker (KKT conditions, the above probelm can be seen as the minimization of Lagrangian L (µ k,e k,g k = E [ x k r k 2] + µ k [ trace ( F H k.f k Pk ]. (2.4 Using 2.2 and F = B.E in 2.4 L (µ k,e k,g k = E [ x k G k.h k.b k.e k.x k + G k.n k 2] [ ( +µ k trace E H k.bh k.b ] ke k Pk (2.5 where µ k is the langragian multiplier and must be selected to satisfy the power constraint. The following KKT conditions are necessary and sufficient for optimality: F k and G k are optimal if and only if there is a µ k that together with F k and G k satisfy the conditions F L (µ k,f k,g k = 0 (2.6 9

10 G L (µ k,f k,g k = 0 (2.7 µ k 0, trace ( F k F H k Pk 0 (2.8 Using the SVD of the channel H k B k, and using the power allocation scheme similar to [2], we obtain the following Tx and Rx filters F k and G k F k = B k.e k (2.9 where G k = (Σ Ek Σ HBk 1 (U HBk (2.10 E k = V HBk Σ Ek (2.11 where σ 2 is the noise varience. H k B k = U HBk Σ HBk V H HB k (2.12 (Σ Ek 2 = σ µk (Σ HBk 1 σ 2 (Σ HBk 2 (2.13 Computation of Langrange Multiplier We now see how to obtain a µ k > 0 that satisfies the power constraint Using 2.11 in 2.9 Which gives Using this value in equation 2.14 trace ( F H k.f k = Pk. (2.14 F k = B k.v HBk.Σ Ek (2.15 F H k.f k = Σ H E k.v H HB k B H k.b k.v HBk Σ Ek (2.16 F H k.f k = (Σ Ek 2 (2.17 trace (Σ Ek 2 = P k (2.18 Using value of Σ Ek from 2.13 ( σ trace (Σ HBk 1 σ 2 (Σ HBk 2 P k = 0 (2.19 µk 10

11 Trace of Σ HBk is the sum of eigen values, hence σµ 1/2 k Which gives the langrangian multiplier (λ j 1 σ 2 (λ j 2 P k = 0 (2.20 j=1 j=1 µ 1/2 k = 2.3 MMSE SDMA σ j=1 (λ j 1 P k + σ 2 j=1 (λ j 2 (2.21 The block diagonalization approach described above, eliminates the interuser interference leaving each user to deal with interference among its own data streams. As described above, the inter stream interference problem can be solve by minimizing the MSE for each user. One of the draw back of orthogonal SDMA is, it requires the number of Tx antennas to be greater than or equal to the total number of Rx antennas. We now disscuss MMSE SDMA which removes the constraint on number of antennas. This approach uses an iterative algorithm that repeatedly goes through power control and optimizing the transmit and receive filters. we use the joint transceiver design proposed by [3], which solves the optimizing problem under the constraint of total power for all users which gives the better dynamical power allocation than per-user power allocation [3]. The complexity of this algorithm is reduced by introducing MMSE receivers as a function of transmit filters, resulting in optimizing only the transmit filters Transmit/Receive filter Design We use the similar system model as used for orthogonal case but without any condition on number of antennas. ( The transmitter with the M transmit antennas transmits up to L = K k=1 L k symbols to the K existing users, each having N k receive antennas, on a M N flat fading channel, where L k are the number of parallel data streams trnasmitted to user k, and N = K k=1 N k. Each user receives a vector y k, contains a combination of the symbols from all users. The received vector is then filtered through L k N k receive filter G H k in order to estimate the L k 1 vector r k r k = G H k H k Fx + G H k n k (

12 where F = [F 1,F 2,...,F K ] and n k is the AWGN vector with zero mean and varience σ 2. The above expression can also be expressed as ( K r k = G H H k F i x i + n k. (2.23 i=1 We want to design optimum transmit and receive filters to minimize the symbol estimation error under total power constraint. Therefore, min [MSE k] (2.24 F,G ( K s.t trace F H k.f k = P. k=1 where MSE k denotes the mean squared error of user k s symbols and defined as [MSE] k = E r k x k 2 = trace (E k, (2.25 where E k is the error covarience matrix, defined as [ E k = E (r k x k (r k x k H], (2.26 Using 2.23 into 2.26, we get E k = ( G H k H kf k I ( F H k HH k G k I +G H k R n k G k +G H k K i=1,i k H k F i F H i HH k G k, (2.27 where R nk = σ 2 I is the noise covarience matrix. We assumed that the individual datastreams are independent i.e, E ( xx H = I. E k = G H k H k F k F H k H H k G k G H k H k F k F H k H H k G k +I+G H k R nk G k +G H k Simpifying the above equation, we get ( K E k = G H k H k F k F H k H H k + R nk + E k = G H k i=1,i k H k F i F H i H H k K i=1,i k (2.28 H k F i F H i H H k G k, G k +I G H k H k F k F H k H H k G k, ( (2.29 Hk F k F H k H H k + R (n+ik Gk + I G H k H k F k F H k H H k G k, (

13 where R (n+ik is the noise and interference covarience matrix seen by user k, and expressed as R (n+ik = R nk + K i=1,i k H k F i F H i HH k. (2.31 Using MMSE crterion the optimum transceiver matrices are obtained when the mean square error of the system is minimized. For a given optimum F the optimization problem becomes min G k (MSE k. (2.32 In order to solve this optimization problem, we put the gradient of MSE k equal to zero, therefore which can be written as G k = ( H k F k F H k H H k + R (n+ik 1 Hk F k, (2.33 G k = R 1 (n+i k H k F k ( I + F H k H H k + R 1 (n+i k H k F k 1. (2.34 which is the well known MMSE receiver. Expression 2.34 shows how to calculate the optimum receive filters for the given optimum transmit filters. Now we need to find the optimum transmit filters. The optimum problem can be seen as min (MSE k. (2.35 F k ( K s.t trace F H k.f k P. k=1 Using the value of G k from 2.34 in Equ. 2.30, we get where E k = ( I + F H k HH k + R 1 (n+i k H k F k 1, (2.36 E k = ( I + F H k R Hk F k 1, (2.37 R Hk = H H k R 1 (n+i k H k. (2.38 Using the eigen value decomposition of matrix R Hk, the optimum transmitter is given by [3] F k = V k Σ k (

14 where the matrix Σ k is defined as [3] ( 1 Σ k = µ D 1/2 D 1 (2.40 and D = diag (Λ 1,Λ 2,...,Λ k, (2.41 Where Λ k is a diagonal matrix, contains the nonzero eigen values of matrix R Hk in descending order and V k contains the corressponding eigen vectors. The value of µ can be calculated from total power constraint ( K trace F H k.f k = P. (2.42 k=1 Now using Equ into Equ. 2.42, we get ( K trace Σ H k.vk H V k Σ k P = 0 (2.43 k=1 ( K trace Σ H k Σ k P = 0. (2.44 k=1 Using the results 2.40 and 2.41 into 2.44, we get K k=1 (Λ trace 1/2 k µ 1/2 = K k=1 trace ( (2.45 Λ 1 + P µ 1/2 = K k=1 Lk i=1 K k=1 Lk i=1 ( k λ 1/2 k,i ( λ 1 k,i + P (2.46 where λ k (s are the nonzero eigen values of matrix R Hk. Up to now, we have considered the perfect channel conditions. Now lets see what will happen if we have some channel uncertainties. Now assume the new channel after adding the error is given by H ke = H k + H k, where H k is the channel estimation error assumed to be Gaussian with zero mean and variance σ 2. The orthogonal SDMA algorithm, we disscussed, does not guarantee anything to deal with this problem. However, the MMSE algorithm tries iteratively to minimise the interference, considers this channel estimation error, a source of interference and tries to minimise it. Now the interference can be seen as K ( R (n+ik = R nk + (H k + H k F i F H i H H k + H H k (2.47 i=1,i k 14

15 which can be expressed as R (n+ik = R nk + K i=1,i k H k F i F H i HH k + R E. (2.48 Minimizing this channel error is the major advantage of MMSE SDMA over the orthogonal SDMA approaches. 15

16 Chapter 3 Simulation Results In this section we present some of our simulation results for the two discussed SDMA algorithms. In all simulations, flat Rayliegh fading channel is assumed for a single subcarrier. The elements of channel matrix H k are obtained from an i.i.d complex zero mean Gaussian distribution. The data streams at the BS are modulated using QPSK. TO generate the results, we assumed, a typical multi-user MIMO setup with a BS and 3 users, each eqipped with multiple antennas. In the following we shall use the notation T x,r x (L k in which T x represents the number of transmit antennas on BS, R x represents number of receive antennas used on each user side, and L k is the number of simultaneous data streams. Average symbol error rates (SER of these three users, for five thousand channel realizations, as a function of signal-to-noise ratio are presented to show the performance of the two SDMA approaches. 3.1 Orthogonal SDMA Figure 3.1 shows the performance of three different scenarios using the orthogonal SDMA approach. In each case, the user terminals are equipped with 2 receive antennas and the BS sends 2 simultaneous data streams to each user, the only difference in the three cases, is the number of transmit antennas at the BS. In first scenario, the BS is transmitting a total of 6 data streams simultaneously using 6 transmit antennas. This is the fully loaded case in a sence that we can not send more data streams without unavoidable multiuser interference. In the next two scenarios we increase the number of transmit antennas to 7 and 8 respectively, however the number of data streams are fixed to 6. Now we can see an improvement in the performance 16

17 10 0 Orthogonal SDMA ,2( SER ,2( ,2( SNR Figure 3.1: Orthogonal SDMA system which is due the diversity gain results from higher number of transmit antennas. The x-axis in figure is just normalized by a factor(1/3, means an SNR vale three corresponds to an actual value equal to nine. 3.2 MMSE SDMA Figures 3.2 shows the simulation results for MMSE SDMA algorithm. Here for this algorithm we couldn t get the results presented by [3]. Actually this algorithm does not work for the case where we have number of receive antennas on a particular user equal to the number of received simultaneous data streams. For this algorithm to work we must need at least double the Rx antennas than the number of data streams. We think the author of the [3] has hide some information that needs to be identify. However, we present some results. Figure 3.2 shows simulation results for [6,4(2],[7,4(2] and [8,4(2] systems. Again we can see the performance improvement due to incresing the number of transmit antennas. We are not comparing results with orthogonal system because the agorithm didn t work for the same system setup as orthogonal system. 17

18 10 1 MMSE SDMA 6,4(2 7,4(2 8,4( BER SNR Figure 3.2: MMSE-SDMA system 18

19 Chapter 4 Conclusion and Future Work 4.1 Conclusion In this work we looked into the problem of inter-user interference in multiuser MIMO systems. We disscussed two different SDMA MIMO schemes. The first scheme, the orthogonal SDMA, completely eliminats the inter-user interfere. It block diagonalizes the MIMO channel resulting in zero MUI, leaving each user to deal with its own interference. This scheme, however, puts a dimension limitation on MIMO system such that the total number of transmit antennas are equal to or great than the total number of receive antennas on all users. The other scheme, the MMSE SDMA, iteratively minimizes the mean squared error which contains both, the inter-user interference and inter-symbol interference. We looked into an algorithm [3] which divide the joint transmit-receive MMSE optimization problem into two sub-problems. It computes the optimum transmit filters with a constraint of total transmit power for all users, and then computes the receive filters from these optimized transmit filters. 4.2 Future Work A perfect CSI was assumed while finding the solutions of Orthogonal and MMSE SDMA. As we mentioned before, one can consider the channel uncetainties. These could be channel estmation errors, quantization errors or feed back errors. In this case we can not competely diagonalize the channel matrix so we have to consider the MMSE SDMA approach. As an extension to this work, we can look into an optimization problem that can take this channel uncertainty error as a source of interference and minimize it along with the other existing interferences. 19

20 The second thing, we propose as a future work, is an extension to our previous work [6]. We had looked into unequal error protection (UEP bit loading for different QoS users by multiplexing different users using FDMA and assumed an orthogonal multi-carrier transmission, i.e, OFDMA over OFDM. To realize UEP, we can multiplex the user along space, meaning that user can be multiplexed on different spatial channels. This could also nice to think of combinig FDMA with SDMA for UEP, which would be a three dimensional problem, i.e, SDMA over FDMA over OFDM. 20

21 Bibliography [1] A. Bourdoux and N. Khalid, Joint Tx-Rx Optimisation for MIMO- SDMA Based on a Null-space Constraint, in Proc. of IEEE VTC-02 Fall, Vancouver, Canada, Sept [2] H. Sampath, P. Stoica and A.Paulraj, Generalized Linear Precoder and Decoder Design for MIMO channels Using the Weighted MMSE Criterion, IEEE Transactions on Communications, Vol.49, No.12, December [3] D. Zhang and J. lu, Joint Transceiver Design Using Linear Processing for Downlink Multiuser MIMO Systems IEEE [4] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, Joint tx-rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization IEEE Transactions on Signal Processing, vol. 51, no. 9,pp , Sept [5] M. Schubert and H. Boche, Solution of the multi-user downlink beamforming problem with individual SINR constraints, IEEE Trans. on Vehicular Technology, vol. 53, no. 1, pp. 1-28, Jan [6] Hassan, K., Sidhu, G. and Henkel, W., Multiuser MIMO-OFDMA with Different QoS Using a Prioritized Channel Adaptive Technique, IEEE International Conference on Communications, ICC-2009, Dresden, Germany, Accepted. [7] H. Sampath and A. Paulraj, Joint transmit and receive optimization for high data rate wireless communications using multiple antennas, in Proc. Asilomar Conf. Signals, Systems and Computers, vol. 1, [8] M. Schubert, S. Shi, E. A. Jorswieck, and H. Boche, Downlink sum- MSE transceiver optimization for linear multi-user MIMO systems, in Proc. Asilomar Conf. on Signals, Systems and Computers, Monterey, CA, Sept

22 [9] S. Shi and M. Schubert, MMSE transmit optimization for multiuser multiantenna systems, in Proc. IEEE Internat. Conf. on Acoustics, Speech, and Signal Proc. (ICASSP, Philadelphia, USA, Mar