A New Method of Channel Feedback Quantization for High Data Rate MIMO Systems

Transcription

1 A New Method of Channel eedback Quantization for High Data Rate MIMO Systems Mehdi Ansari Sadrabadi, Amir K. Khandani and arshad Lahouti Coding & Signal Transmission Laboratorywww.cst.uwaterloo.ca) Dept. of Elec. and Comp. Eng., University of Waterloo, Waterloo, ON, Canada, N2L 3G1 mehdi, khandani, Tel: , ax: Abstract In this work, we study a Multiple-Input Multiple-Output wireless System where the channel state information is partially available at the transmitter through a feedback link. We present a technique, which simultaneously, facilitates low decoding complexity for high data rate MIMO systems, and provide significant performance improvement through a low-rate feed back of partial channel state information. Based on Singular Value Decomposition, the MIMO channel is split into independent channels, which allows separate, and therefore efficient, decoding of the transmitted data signal. Effective feed back of the required spatial channel information entails efficient encoding/quantization of a Haar unitary matrix. The parameter reduction of an unitary matrix to its basic parameters is performed through Givens rotation decomposition. We prove that Givens rotation matrices of a Haar unitary matrix are statistically independent, and subsequently derive the probability distribution function of the matrices elements. Next, based on these analyses, an efficient quantization scheme is proposed. Performance evaluation is provided for a scenario, where the rates allocated to each independent channel is adapted according to its corresponding gain. Results indicate 1-2 db performance improvement in comparison to MIMO systems without feedback at the cost of a very low-rate feedback link 8 bits). The rate adaptation information, also part of the feedback, is encoded using a trellis structure. I. INTRODUCTION Multiple-Input Multiple-Output MIMO) communication systems have drawn considerable attention in response to the increasing requirements of high spectral efficiency in wireless communications. In fact, the capacity of MIMO systems equipped with transmit and receive antennas scales up almost linearly with the minimum of and in flat Rayleigh fading environments [1] [2]. In recent years, much work has been done in studying the transmission strategies for MIMO systems in different scenarios in which the transmitter and/or the receiver have full or partial knowledge of the Channel State Information CSI). In [3], it has been shown that the achievable bit rate with perfect CSI available at the transmitter and the receiver is significantly higher as compared to the case that only the receiver has access to CSI. Due to practical confinements such as imperfect channel estimation and feedback data rate limit, CSI is not perfect at the transmitter. However unlike single antenna systems, where exploiting CSI at the transmitter does not significantly enhance the capacity, for multiple antenna systems the capacity improvement through even partial CSI can be substantial [4]. When CSI is available at the transmitter, beamforming can be used to exploit transmit diversity through spatial match filtering. In the context of Multiple- Input Single-Output MISO) systems, several quantization schemes have been suggested to feedback instantaneous channel state information to the transmitter. A simple and effective scheme is suggested for a 3G wireless standard [5]. The authors in [6] design a codebook of beamformer vectors, with the objective of minimizing the outage probability. A similar work has been reported in [7] where the codebook design criterion is derived to maximize the received SNR. In the case of MIMO systems the problem of quantizing the channel state information is more involved when compared to the MISO scenarios. In [8], assuming partial CSI is available at the transmitter of a MIMO system, a criterion is presented to design a precoder based on capacity maximization. However, [8] does not provide a practical approach to design such a precoder when the number of receive antennas are more than one. In general, optimum Maximum Likelihood decoding in a MIMO system is equivalent to a lattice decoding problem, which incurs significant complexity. Lower complexity decoding algorithms can be devised by proper design of a transmit strategy e.g., the Bell Labs Layered Space Time system [1]. However, this in turn is achieved at the cost of degraded performance [9] [1]. In this work, we present a technique to simultaneously address both practical feedback scheme for a

2 MIMO system as opposed to MISO) and achieving a low complexity decoding algorithm in a unified framework. Consider a scenario that MIMO channel is split into several independent subchannels by means of Singular Value Decomposition SVD). This allows independent decoding of subchannels and results in low decoding complexity. On each of subchannels the modulation format is adapted to the subchannel SNR at the receiver. We assume the availability of a feedback link between the transmitter and the receiver. In this scheme, the spatial information of the channel and the rate index of each subchannel is needed at the transmitter. We develop an algorithm to quantize the spatial information of the channel based on minimizing the interference between subchannels. The rate allocation strategy is determined at the receiver and fed back to the transmitter using an efficient low rate approach. The system model is described in Section II. In Section III, parametrization and statistics of right singular matrix is discussed. The unitary matrices are decomposed into Givens matrices. Statistical properties of Givens rotation matrices are theoretically derived. eedback design is developed based on these properties and decoding strategy at the receiver. In Section IV, simulation results are presented. Section V concludes the paper. II. SYEM MODEL We consider an independent and identically distributed block fading channel model. or a multiple transmit antenna system with transmit antennas and receive antennas, this model leads to the following complex baseband representation of the received signal, 1) where x is the vector of transmitted symbols, is the channel matrix, n is is zero mean Gaussian noise vector with autocorrelation I identity matrix) and y is the received signal. The power constraint of the transmitted signal is defined as I stands for the expectation.) The elements of the channel matrix H are circularly symmetric complex Gaussian distributed with zero mean and unit variance. Singular Value Decomposition of matrix as [11] is defined H V U 2) where V and U are unitary matrices and is a diagonal matrix. If matrix U is available at the transmitter, it can be used as a precoder to project data signal onto the channel direction. Therefore, at the receiver, the received signal is given by! " $#%&! ' ) 3) The receiver filters the received matrix y by V. r V y n 4) Therefore, MIMO channel with transmit antennas and receive antennas is transformed to rank * H+ parallel subchannels. Parallelizing the channel results in low decoding complexity. In transition from 3) to 4), we use the fact that elements of n are statistically independent and rotating n by the unitary matrix V does not change the distribution of the noise. As can be seen in 4), the subchannels provide different gains corresponding to matrix. In order to establish a reliable high data rate system, we consider a scenario of transmitting and receiving data separately in each subchannels with different rates and equal energy. It can be shown that using equal energy maximizes the rate under the assumption of continuous approximation. The main goal of this method is to achieve a data rate as high as possible meeting a certain target bit error rate. By this assumption, the transmitter needs the rate information of each subchannel in addition to the right singular matrix of the channel. The modulation format to each of subchannels are adapted. The higher the power gain of a subspace channel is, the more complex the modulation scheme can be chosen. III. EEDBACK DESIGN In the scenario described in previous section, the transmitter needs to know the right singular matrix of the channel. We assume availability of a noiseless feedback link from the receiver to the transmitter. By SVD of H at the receiver side, the unitary matrix U is computed, quantized and sent for the transmitter. The strategy of the receiver is to minimize the interference between parallel subchannels in order to detect the data in each channel, independently. Assuming quantization error, # for U, the received signal is, r #, #% n ) 5) The variance of the interference signal is as follows, Tr.-/ #, #% -/ Tr1 #, #%, # # 3254 Tr #, #%6, # #7 2 4 Tr8, #, #% Tr9-1, # - :) 6) In the above derivation, we use the property that the singular values of a Gaussian matrix are independent with the corresponding singular vectors [12] and ;< = 254 I. Also, we use the equality Tr*?@A+. Tr stands for trace function). As a result, minimizing the mean of the interference power leads to the minimization of the robenius norm of B U. In order to minimize the interference, the unitary matrix U should be quantized based on the expression in 6). In the following, we study the algebraic and statistical properties of unitary matrices.

3 * A. Statistics of Singular Matrix of a Random Gaussian Matrix In almost all analytic study of MIMO antenna systems, the channel between the transmitter and the receiver is considered as a Rayleigh fading channel. This means the entries of the channel matrix are statistically independent and identically distributed, and have complex Gaussian random variables with zero mean. We are interested in the probability distribution of the singular matrices of the mentioned channel matrix in the space of M* + The set M* + of = unitary matrices forms a group with respect to the matrix multiplication). It is shown that such a random unitary matrix takes its values uniformly from M* + in the sense of the following property. or all V M* + and for a matrix U M* +, the distribution of U and VU are the same. Such a distribution is called Haar distribution and unitary matrix U is called Haar unitary matrix [13]. We refer to the above property as right invariant property. A complex matrix can be described by real parameters. However, the definition of a unitary matrix implies a dependency between the elements of the matrix. In fact, a unitary matrix U can be described by independent parameters. Here, for the purpose of matrix decomposition using SVD, out of parameters are also redundant in the sense that the decomposition is unique up to parameters. or instance, the singular value decomposition can be done such that the diagonal elements of U in 2) set to be real. Several different approaches can be taken to parameterize an unitary matrix U in its independent parameters. In this work, we consider decomposition using Givens rotation matrices [11]. In the followings, we derive probability distribution of the elements of Givens matrices of a Haar unitary matrix. Matrix U is the product of Givens rotations [11], i.e., U G* + 7) where each G* + is an unitary matrix with two parameters and given in the following expression, G* +! "! #$"&% ') 8) where is in the position * + and * +, is in * + and +, is in * 5+,.- ) of the above matrix. The other diagonal elements of the matrix /* + are and the rest of the elements are zero. Since /* + is a unitary matrix, then 1 where is a real number and is a complex number. In the following, the statistical properties of Givens matrices corresponding to a Haar unitary matrix U will be derived. It is necessary to know the statistics of Givens rotations components of unitary U in order to determine the strategy of quantizer. The key point of codebook design for a Haar unitary matrix is the following result. Theorem 1: Assume that U is an 2 unitary matrix with Haar distribution which is decomposed into Givens matrices as follows, U G* ) 9) The set of Givens matrices G* for are statistically independent of each other. Moreover, probability distribution function of the elements of G* is : * &; <+ <: * + : * ; <+ = A CB + = =ED ) 1) The proof is presented in [14]. B. Quantization of Unitary Matrices Based on the criterion presented for designing quantizer in 6), the distortion measure of the quantizer is defined as follows, *HG + AIKJ.* - U +?LU-/1+ ) 11) In section III-A, we introduced Givens rotation matrices as basic components of a unitary matrix. It means that, the minimum number of elements constructing uniquely a unitary matrix is determined by Givens matrices. According to the result of Theorem 1 these components are statistically independent of each other. Therefore, in order to quantize a unitary matrix, we simply quantize Givens matrices independent of each other. We change the notation in 7) and rewrite the expression as follows, U M NOQPR GS ) 12) We define the distortion measure for a general format of a Given Rotation matrix, G as follows, */ + AIKJ.* - G + LG- + 13) where LG is the quantized version of G. The expression in 11) can be simplified as follows, *HG + ; * U +?LU+/* U +?LU + Tr I +CU ULU :) 14)

4 The first order approximation of *G + in terms of */ S + A7 7? * + <+ is, *HG + Tr I +<U NOQP R NO45P R GS LG : NOQPR Tr U GS * GS + LGS + N 5P R Tr * I +<U GS LGS + NOQP R * / S + 15) In the derivation of 15), we have applied the fact that Tr AB A Tr BA. Therefore, in order to minimize the distortion measure of U, we minimize the distortion measure of each Givens matrix. The basic parameters of a Givens matrix in 8) named and ; are quantized as L and L. The transmitter uses L and L to construct LG as follows, L L 16) + L L where L + L for the sake of brevity, we only displayed the variable part of a Givens matrix). According to the construction scheme, LG is also unitary. It can be easily shown that the first order approximation of, G is, -1, G-/ - G + LG-/ + + L + L ) 17) By the above approximation, we separate the distortion caused by and which makes the design of quantizer easy. By applying 17), the distortion defined in 13) is, * / + ; + + L ; + L By applying 15) and 17) we have, :) 18) 4NO5P R *HG + SO + S S + L S ; S + L S 19) where S and S are the parameters of / S, and L S and L S are the parameters of / S. We can interpret 19) in terms of Euclidean distance by modifying the probability of the parameter : * + "!$# & 7 7&% *'%) 1+ otherwise 2) where * is a constant and determined by normalizing the probability distribution function in 2). Therefore, the equation 19) will be as follows, NOQP R *HG +,+ S + L S $- S + L S 21) Since the parameters in 21) are statistically independent according to Theorem 1, we use rate allocation method for quantization presented in [15]. C. Encoding of Rate Allocation Information As discussed, besides the quantized right singular matrix of the channel that is fed back to the transmitter, information pertaining the rate to be allocated to each eigen-channel is also fed back. This indicates a set of indices from a set of./ predetermined rates, e.g., different modulation schemes. Obviously, the total rate is bounded and since we can always perform SVD of the channel matrix such that the singular values become ordered, the indices correspond to an ordered set of increasing rates. To encode this information, we can use a trellis diagram with.1/ states and stages plus an initial stage). The states correspond to the set of possible rates in an increasing fashion, and there is a branch from each state to another state in the next stage only if the entering state is located at the same or a lower level position. Each path in the trellis then corresponds to a set of eigen-channel rates, whose index would be chosen by the receiver and fed back to the transmitter. The trellis structure exploits the ordering property of the rates, and therefore allows their efficient coding at a rate of [16] ;:=<?.1/ + ).1/ + The complexity of this algorithm is very low and is in fact, proportional to the number of states. Similar structures have been used to address the points of a block-based trellis quantizer in [16] or a pyramid vector quantizer in [?]. This method of enumeration is also similar in spirit to the earlier works of [17] and [18] in addressing the points of a signal constellation shell mapping). IV. PERORMANCE EVALUATION In this section, we present the performance results of the system described in section II including the effect of feedback link in the system. We assume that precoding is performed by quantized version of right singular matrix of the channel by applying the quantization method presented in section III-B. or different eigen-channels, we use different modulation schemes. The process of selecting the appropriate modulation scheme for each subchannel is accomplished at the receiver. We restrict the system to transmit data with the power 5 on each transmit antenna. At the receiver, the channel state information

5 and the instantaneous quantization noise power is known. or each subchannel, the probability of error is computed for different modulation schemes. The receiver selects a modulation scheme for each subchannel that achieves the target BER of the system and send the index of the corresponding modulation scheme for the transmitter through the feedback channel as described in section III-C. At the receiver side, we have, * ) where * ) + is the BER function of the modulation scheme with rate and is the target BER of the system and and are AWGN and quantization noise variance, respectively. We consider a set of QAM modulation formats. As derived in [19], the function * ) + in 23) is */ Bit Error Rate Average Rate: SNR: /. 2 V BLA Rate Allo. 5 bits feedback Rate Alloc. 8 bits feddback Rate Alloc. Perfect CSI at the receiver ) 24) SNRdB) ig. 1. Bit error rate for different schemes where and ig. 1 shows the bit error rate versus SNR for different MIMO systems with and. We use 5 bits and 8 bits for feedback link in each transmission block. The average bit rate of the system changes with the SNR. We compare the performance of this system with a V-BLA system which is proposed as a solution to overcome the complexity problem. The rate of V-BLA system for each SNR is equal to or less than the average rate of the system for a fair comparison. ig. 1 shows a significant improvement in comparison to V- BLA at the price of feedback. The performance of the system while perfect channel information is available at the transmitter is depicted where it can be achieved at a low rate feedback bits 8 bits). MIMO system. We developed an efficient algorithm for the quantization of unitary matrices. Also, we presented a low rate indexing of rate allocation information. In addition to the low decoding complexity of the system, the feedback rate is very low. The simulation results show a significant improvement in comparison to other schemes which are proposed to overcome the complexity problem. REERENCES [1] G. J. oschini and M. J. Gans, On the limits of of wireless communications in a fading environment, Wireless Pres. Commun., vol. 6, pp , Nov [2] E. Telatar, Capacity of multi-antenna gaussian channels, Bell Labs Journal, vol. 1, Nov/Dec [3] E. Biglieri, G. Caire, and G. Taricco, Limiting performance of block fading channels with multiple antennas, IEEE Trans. on Information Theory, vol. 47, pp , May 21. [4] E. Vistosky, U. Madhow, Space-Time Transmit Precoding with Imperfect eedback, IEEE Trans. Inform. Theory, vol. 47, pp , September 21. [5] 3GPP Technical Specification, Group Radio Access Network, Physical layer procedures fdd), vol. 5.6., Sept. 23. TS [6] K. K. Mukkavilli, A. Sabharwal, E. Erkipand, and B. Aazhang, On beamforming with finite rate feedback in multiple antenna systems, IEEE Transactions on In formation Theory, vol. 49, pp , Oct. 23. [7] David J. Love,Robert W. Heath, Jr.,and Thomas Strohmer, Grassmannian beamforming for Multiple-Input Multipe-Outpet systems, IEEE Transactions on In formation Theory, vol. 49, Oct. 23. [8] M. Skoglund, G. Jongren, On the capacity of a multiple-antenna communication link with channel side information, IEEE J. on Selected Areas in Comuunications, vol. 21, pp , April 23. [9] G. D. Golden, G. J. oschini, R. A. Valenzuela, and P. W. Wolniansky, Detection algorithm and initial laboratory results using v-blast space time communication architecture, Electron. Lett., vol. 35, pp , Jan [1] G. Ginis and J. M. Cioffi, On the relation between v-blast and the gdfe, IEEE Communications Letters, vol. 5, pp , Sep. 21. [11] G. H. Golub, C.. Van Loan, Matrix Computations. the Johns Hopkins University Press, third ed., [12] V. L. Girko, Theory of Random Determinants. Kluwer Academic Publishers, 199. [13]. Hiai, D. Petz, The Semicircle Law, ree Random Variables and Entropy, American Mathematical Society, vol. 77, 2. Mathematical Surveys and Monographs. [14] Mehdi Ansari and A. K. KHandani, A new method of channel feedback quantization for high data rate mimo systems, Technical repor, Dept. of ECE, University of Waterloo, Jan. 23, avaliable at [15] Allen Gersho, Robert M. Gray, Vector Quantization and Vector Compression. Kluwer Academic Publication, [16]. Lahouti and A. K. Khandani, Quantization of lsf parameters using a trellis modeling, IEEE Trans. Speech and Audio Proc., vol. 11, pp , Sept. 23. [17] G. R. Lang and. M. Longstaff, A leech lattice modem, IEEE J. Select. Areas Commun., vol. 7, pp , Aug [18] A. K. Khandani and P. Kabal, Shaping multi-dimensional signal spaces-part ii: Shell-addressed constellations, IEEE Trans. Inform. Theory, pp , Nov [19] J. G. Proakis, Digital Communication. McGraw-Hill, 4th ed., 2. V. CONCLUSION In this work, we presented efficient methods for channel information quantization used in a high data rate