Transmit power allocation for a modified V-BLAST system

power allocation V-BLAST

1074IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 2004Transmit Power Allocation for a Modified V-BLAST SystemSeung Hoon Nam, Oh-Soon Shin, and Kwang Bok (Ed) Lee, Member, IEEEAbstract—In this letter, we present a modification of Vertical Bell Laboratories Layered Space–Time (V-BLAST), and propose an effective transmit power allocation (TPA) scheme for the modified system. The proposed TPA scheme minimizes the uncoded biterror rate (BER) averaged over all detection stages, and requires small feedback overhead. Simulation results show that the modified V-BLAST system with the proposed TPA scheme provides a significant reduction in the uncoded BER compared with the conventional V-BLAST system. When the minimum mean-square error nulling is adopted, the modified V-BLAST system is found to achieve the uncoded BER performance comparable to that of the maximum-likelihood detection for the conventional V-BLAST architecture. Index Terms—Bit-error rate (BER), detection ordering, multiple-input multiple-output (MIMO) system, transmit power allocation (TPA), Vertical Bell Laboratories Layered Space–Time (V-BLAST).I. INTRODUCTIONMULTIPLE-INPUT multiple-output (MIMO) systems can provide enormous capacities through appropriate space–time processing [1]–[3]. A spatio-temporal processing architecture studied in [3] is a means to approach the theoretical capacity limit of MIMO systems. This system requires a feedback mechanism and complex processing at both the transmitter and receiver. On the other hand, Diagonal Bell Laboratories Layered Space–Time (D-BLAST) in [4] eliminates the need for feedback. However, this system requires a complex coding structure that makes the detection procedure complicated. In [5], Vertical BLAST (V-BLAST) has been proposed as a simplified version of D-BLAST. This system does not need a complex coding structure, unlike a D-BLAST. Simple coding and detection structures make the V-BLAST attractive. There have been some attempts to improve the original V-BLAST system [6]–[10]. One such approach is to incorporate an effective transmit power allocation (TPA) into the original V-BLAST, where equal power is assigned to transmit antennas [5]. Transmit power is adapted according to thePaper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received March 1, 2003; revised May 27, 2003; October 13, 2003; and November 7, 2003. This work was supported in part by the Brain Korea 21 Project. This paper was presented in part at the 13th IEEE Symposium on Personal, Indoor and Mobile Radio Communications, Lisbon, Portugal, September 2002. S. H. Nam was with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea. He is now with Samsung Advanced Institute of Technology, Suwon 440-600, Korea (e-mail: seunghoon.nam@). O.-S. Shin was with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea. He is now with the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 USA (e-mail: osshin@fas.harvard.edu). K. B. Lee is with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: klee@snu.ac.kr). Digital Object Identifier 10.1109/TCOMM.2004.831368channel condition, which necessitates a feedback mechanism. However, a feedback overhead needed for TPA is relatively small, compared with the system in [3]. In [6]–[8], TPA schemes with small feedback overhead have been investigated to increase the capacity of V-BLAST systems through rate adaptation [10]. However, there have been few attempts to improve the error-rate performance of MIMO systems for a given fixed rate. In this letter, we present a modification of V-BLAST that incorporates TPA, and propose an effective TPA scheme to improve the bit-error rate (BER) performance. In the V-BLAST detection algorithm, a diversity order for an earlier detection stage is less than that for a later one [2], [5], [9]. Considering this characteristic and the effects of error propagation [9], it is apparent that early detection stages limit the overall performance. Although a detection-ordering scheme in [5] mitigates this problem, it may not sufficiently compensate for low diversity orders of early detection stages [9]. Hence, TPA for the modified V-BLAST may offer further improvement of early stages, enhancing the overall performance. We develop a TPA scheme that minimizes the BER. The transmit power determined at the receiver is transferred to the transmitter. The feedback overhead for the proposed TPA is relatively small, compared with the system in [3], since the feedback information contains only the transmit power rather than full channel state information. II. MODIFIED V-BLAST SYSTEM A. System Description The modified V-BLAST system is equipped with transmit ) receive antennas. In the transmitter, a data stream and ( is demultiplexed into independent substreams, and then each substream is encoded into transmit symbols using the same modulation scheme. Based on the feedback information, is assigned to the data symbol , and the transmit power the symbol is transmitted through the th transmit antenna. The receiver estimates the transmit symbols from the received receive antennas, and determines the transmit signals at ( ). power The baseband equivalent of the -dimensional received at sampling instants signal vector may be expressed as (1) where denotes the transmit symbol vector with each element having the unit average power, and denotes the channel matrix, whose element at the th row and th column is the channel gain from the th transmit antenna to the th receive antenna, and they are assumed to be independent and identically distributed (i.i.d.)0090-6778/04$20.00 © 2004 IEEE

power allocation V-BLAST

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 20041075complex Gaussian random variables with zero mean and unit variance. The elements of the -dimensional noise vector are assumed to be i.i.d. complex Gaussian random variables with zero mean and variance of . It is assumed that the channel estimation at the receiver is perfect, and that the receiver determines the transmit power ( ) for transmit antennas with the total power constraint , and sends the power to the transmitter through an error-free feedback channel. A diagonal in (1) represents the matrix transmit power. For the conventional V-BLAST system without for all , and thus, is equal to the feedback, identity matrix . B. Detection Algorithm for the Modified V-BLAST System For a conventional V-BLAST system, a detection algorithm has been presented in [5]. The algorithm uses the linear nulling and successive interference cancellation processes to estimate transmit symbols. This algorithm can easily be modified for the modified V-BLAST system. 1) Detection Algorithm for ZF Nulling: With a TPA matrix being introduced, the detection algorithm in [5] can be modified as follows. Initialization: (2a) (2b) (2c) (2d) Recursion: (2e) (2f) (2g) (2h) (2i) (2j) (2k) where denotes the Moore–Penrose pseudoinverse, is is the th row of a matrix, is the norm of a vector, is a matrix formed by zethe th column of a matrix, and roing the th columns of the argument matrix. is the slicing operator associated with a modulation scheme, is the estimated value of . In (2a)–(2k), is the and symbol index detected at the th stage, and thus the sequence is the detection order of transmit symbols. Note that this detection order is determined based on the signal-to-interference-plus-noise ratio (SINR) of transmit sym, since not but is used in (2d) and (2j). bols with The rationale for this is that the TPA, to be described in Section III, is determined under the assumption that the detection . procedure follows the detection order forThe nulling vector in (2f) can be rewritten as (3) corresponds to the where the vector . We denulling vector of the th detection stage, when fine the postdetection SINR, which determines the performance in (2f). of each stage, as the SINR of the decision statistic The interference component is zero for zero-forcing (ZF), but nonzero for minimum mean-square error (MMSE). From (2f) for the th symbol can be and (3), the postdetection SINR calculated as (4) is a random variable and is related to the weight vector and transmit power for the th symbol. The average performance of the th detection stage is determined from the prob. The effects of on the distribution ability distribution of may be characterized in terms of a diversity order [9]. It of in (4) provides the distribution of with a is known that [2], [9]. This indicates that later dediversity order of tection stages achieve higher diversity orders, when error propon agation is not considered. The effects of transmit power and the average performance will be discussed in Section IV. 2) Detection Algorithm for MMSE Nulling: Unlike the ZF nulling that removes the interference components completely but results in noise enhancement, the MMSE nulling compromises interference suppression and noise enhancement, such that the mean-square error (MSE) between the transmit symbol and estimate of the receiver is minimized. In the case of MMSE, in (2c) and (2i) should be changed as the nulling matrix(5) denotes the conjugate transpose. Furthermore, the where detection order should be determined from the SINR for the MMSE nulling. Correspondingly, (2d) and (2j) are modified to(6) is the nulling matrix in the case of , and denotes the element of the matrix at the th row and th column. Except for determinations of and described in (5) and (6), the detection algorithm for the MMSE case is the same as that for the ZF case in (2a)–(2k). For the for the th symbol is MMSE case, the postdetection SINR calculated as (7) where

power allocation V-BLAST

1076IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 2004Note that the postdetection SINR depends on the transmit power for other symbols as well as that for the th symbol, in contrast to the ZF case. III. TRANSMIT POWER ALLOCATION In Section III-A, we derive a TPA scheme that minimizes the BER. In the derivations, we assume that ZF is used for nulling, and the MMSE case is discussed in brief. In Section III-B, the results of Section III-A are applied to an uncoded quadrature amplitude modulation (QAM). A. Derivation of TPA Scheme To derive a TPA scheme, we first express the BER of each transmit symbol as a function of transmit power , and find that minimizes the overall BER. If the cancellations of previously detected symbols are perfect, the of the th transmit symbol for a given channel state BER may be represented as a function of the postdetection SINR in (4) (8) where the function is determined by a specific modulation scheme. Taking the effects of error propagation into considera( ) of each detection tion, the actual BER stage for a given channel state may be approximated aswhere is the Lagrange multiplier, and the total transmit power constraint is given as (12) From , a set of equations is found as(13) Solving simultaneous equations in (12) and (13), we can obtain a set of transmit power that minimizes the BER in (10). As discussed in Section II, the postdetection SINR in (7) for the MMSE case depends on the transmit power for other symbols, as well as that for the corresponding symbol. In this case, the derivation of the TPA scheme is not tractable in a manner described above, and this necessitates an approximation of (7). Note that the postdetection SINR in (4) for the ZF case is expressed as a product of the transmit power of the corresponding symbol and the postdetection SINR obtained with equal TPA. This implies that the transmit power for each symbol affects the postdetection SINR as a scaling factor. We use this relationship to approximate (7), on the basis that the fundamental operation of MMSE is the same as that of ZF, except for the consideration in (7) is approxiof noise. Hence, the postdetection SINR mated as (14)where (9) where denotes the probability of bit error at the th de) previous detectection stage, given that stages out of ( tion stages are erroneously detected. The approximations in (9) result from neglecting second and higher order terms of and , which may be justified at high signal-to-noise ratio (SNR). Since the transmit symbols are independent of one anmay be calculated as an arithmetic other, the overall BER mean of the BER for every symbolis the transmit power,is the postdetec, and is the weight vector for tion SINR for . It should be noted that the approximation might not be justified at low SNR. With the approximation in (14), the transmit power can be calculated in a similar manner as the ZF and should be separately calculated case. Note that to determine the transmit power and to estimate the transmit symbols, respectively, since there is no explicit relationship and , in contrast to the ZF case. This may between increase the computational requirements of the receiver compared with the case of the conventional V-BLAST system. B. Application to Uncoded QAM When uncoded -ary QAM is employed for all symbols, the BER for the th symbol can be tightly approximated by an as [11] exponential function of (15) and from (4) and (14), is defined as ZF (16) MMSE .(10) We use the Lagrange multiplier method to find transmit that minimizes the overall BER in (10) under power the total transmit power constraint. The cost function may be expressed as (11)

power allocation V-BLAST

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 20041077In this case, (13) can be rewritten as(17) By solving simultaneous equations in (12) and (17), we can find the solution for the transmit power as(18) ( ) are required in As shown in (16) and (18), calculating , and they are calculated using the nulling vectors ( ). These nulling vectors can be calculated . Note using the detection algorithm in (2a)–(2k) with ( ) need that the conditional probabilities to be predetermined to calculate the transmit power. If we set ( ) to ignore error propagations in the overall BER equation in (10), the TPA in (18) may be simplified toFig. 1. Effects of BER expressions for TPA on the uncoded BER performance of the modified V-BLAST for a (4, 4) MIMO system.(19) IV. NUMERICAL RESULTS In this section, the performance of the modified V-BLAST system with the proposed TPA is evaluated and compared with that of the conventional V-BLAST system with equal TPA. The performance of the maximum-likelihood (ML) detection with equal TPA is also presented for comparison purposes. Although the ML detection does not need feedback and provides the best performance among detection schemes of the conventional V-BLAST [12], it generally requires larger complexity than the V-BLAST detection. Quaternary phase-shift keying (QPSK) modulation ( ) is assumed to be employed, and the transmit power for the modified V-BLAST system is calculated using (19), unless explicitly specified. We use ( , ) notation to represent a MIMO configuration with transmit and receive antennas. Fig. 1 shows the effects of BER expressions for TPA on the overall uncoded BER performance of the modified V-BLAST for a (4, 4) MIMO system. For both ZF and MMSE nulling schemes, two BER curves are obtained through simulations using TPA in (18) and (19), respectively. It is observed that the two BER curves are almost indistinguishable for both the ZFFig. 2. Effects of detection ordering and proposed TPA on the uncoded BER performance of each detection stage for a (4, 4) MIMO system with ZF nulling, when error propagation is ignored.and MMSE cases. This indicates that it hardly affects the performance of TPA, whether the error propagation is considered or not in the BER equation. Based on this observation, we use the simpler (19) rather than (18) in the subsequent results. In Fig. 1, we also verify that the semianalytic results agree well with the simulation results, especially for the MMSE case. A slight difference may be due to the approximations in (9) and Gaussian approximation of interference in (15). Fig. 2 shows the effects of the detection ordering and proposed TPA on the uncoded BER performance of each detection stage for a (4, 4) MIMO system with ZF nulling. Note that only the detection ordering is used for the conventional V-BLAST system, whereas both the detection ordering and proposed TPA are used for the modified V-BLAST system. Error propagation is ignored in calculating the BER for each stage, and the results are obtained through semianalytic methods using (15) and (16). Without the detection ordering and TPA, the BERs for earlier detection stages are shown to decrease more slowly with SNR per receive antenna ( ) than the BERs for later stages. This verifies that the earlier detection stage has the lower diversity

power allocation V-BLAST

1078IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 2004Fig. 3. Effects of detection ordering and proposed TPA on the uncoded BER performance of each detection stage for a (4, 4) MIMO system with MMSE nulling, when error propagation is ignored.Fig. 4. Uncoded BER performance for a (4, 4) MIMO system.order. The use of detection ordering is found to decrease the BER of the first and second detection stages, and increases that of the fourth stage. It is interesting to note that the detection ordering does not change the diversity order, but increases the average SINR for early detection stages, since the detection ordering just shifts the BER curves without changing the slope. When the proposed TPA is incorporated, further improvement for the first and second stages is observed. The major impact of the proposed TPA scheme is also shown to increase the average SINR for early detection stages. Since the overall BER performance is limited by early detection stages, the proposed TPA scheme is expected to decrease the overall BER. Furthermore, more reliable decisions in early detection stages may reduce the effects of error propagation. Fig. 3 shows the effects of the detection ordering and proposed TPA for the MMSE case under the same conditions as in Fig. 2. In the absence of TPA, the effect of detection ordering is shown to be similar to the ZF case. However, the improvement in low detection stages with the MMSE nulling is shown to be much more significant than with the ZF nulling. When both the detection ordering and proposed TPA schemes are employed, we can observe further significant improvement for all the detection stages. It is remarkable that the performance of all the detection stages is very similar to one another, and it is better than that of the last detection stage without TPA at high SNR region, unlike the ZF case. The reason for this is that disparities in the interference-plus-noise power among detection ’s in (16), are much smaller with the MMSE than stages, with the ZF nulling, due to reduced noise enhancement. The ’s make the TPA the more effective in smaller disparities in compensating for the worst detection stage, so that it no longer limits the overall performance. In Fig. 3, it should also be noted that the last detection stage without TPA achieves the same diversity order of as the ML detection scheme. This implies that the TPA with the MMSE nulling can realize a greater diversity order than the ML detection.1 This is because the TPA reduces deep fades in the received signal, increasing the diversity order.1This diversity advantage over the ML detection may diminish and even disappear if the input is space–time coded.The effect of TPA on diversity order is more remarkable for the ’s in MMSE than for the ZF, due to smaller disparities in (16). Hence, the modified V-BLAST system with the MMSE nulling achieves a form of diversity due to TPA as well as inherent receive diversity, realizing a greater diversity order than the ML detection that achieves only receive diversity. In Fig. 4, the overall uncoded BER performance of the (4, 4) modified V-BLAST system with the proposed TPA scheme is compared with that of the conventional V-BLAST system. The effects of error propagation are not neglected, and simulations are used to obtain the actual performance. Comparing Fig. 4 with Figs. 2 and 3, it can be seen that the overall BER is mainly determined from the BER for the first detection stage. The modified V-BLAST system is shown to significantly outperform the conventional one. When the ZF nulling is used, the SNR gain is . In the case of the MMSE, the about 4.0 dB at BER of modified V-BLAST system provides 2.5 dB SNR gain at BER . Furthermore, the modified V-BLAST system with the of MMSE nulling is shown to provide almost the same performance as the ML detection at low SNR values, and outperform the ML detection for the SNR greater than 14 dB. This is because the modified V-BLAST system with the MMSE nulling realizes a greater diversity order than the ML detection, as explained previously. V. CONCLUSIONS In this letter, we have presented a modification of V-BLAST, and proposed an effective TPA scheme that minimizes the BER averaged over all detection stages. The feedback overhead for the proposed TPA scheme is relatively small. It has been found that the TPA scheme in combination with detection ordering improves the performance of lower detection stages, resulting in significant reduction in the overall uncoded BER. Simulation results have shown that the modified V-BLAST system with the proposed TPA scheme achieves 2.5–4.0 dB of SNR gain over the conventional V-BLAST system at uncoded BER of . With the MMSE nulling, the modified V-BLAST system has been found to achieve a diversity gain as well as the average SNR gain, resulting in performance comparable to or better than that of complex ML detection, which does not require any feedback.

power allocation V-BLAST

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 7, JULY 20041079Finally, it should be pointed out that the above results have been derived under ideal conditions. An important topic for future work is to investigate the impacts of practical conditions, such as imperfect channel estimation, feedback delay, quantization errors, and spatial correlation of MIMO channel. ACKNOWLEDGMENT The authors would like to thank the Editor and anonymous reviewers for their helpful comments. REFERENCES[1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, pp. 315–335, Mar. 1998. [2] V. Tarokh, A. Naguib, and N. Seshadri, “Combined array processing and space–time coding,” IEEE Trans. Inform. Theory, vol. 45, pp. 1121–1128, May 1999. [3] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, pp. 357–366, Mar. 1998. [4] G. J. Foschini, “Layered space–time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Tech. J., vol. 1, pp. 41–59, 1996.[5] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in Proc. URSI Int. Symp. Signals, Systems, Electronics, Pisa, Italy, Sept.–Oct. 1998, pp. 295–300. [6] M. F. Demirkol and M. A. Ingram, “Power-controlled capacity for interfering MIMO links,” in Proc. IEEE Vehicular Technology Conf., Atlantic City, NJ, Oct. 2001, pp. 187–191. [7] S. T. Chung, A. Lozano, and H. C. Howard, “Approaching eigenmode BLAST channel capacity using V-BLAST with rate and power feedback,” in Proc. IEEE Vehicular Technology Conf., Atlantic City, NJ, Oct. 2001, pp. 915–919. [8] K. J. Hwang and K. B. Lee, “Transmit power allocation with small feedback overhead for a multiple antenna system,” in Proc. IEEE Vehicular Technology Conf., Vancouver, BC, Canada, Sept. 2002, pp. 2158–2162. [9] W.-J. Choi, R. Negi, and J. M. Cioffi, “Combined ML and DFE decoding for the V-BLAST system,” in Proc. IEEE Int. Conf. Communications, New Orleans, LA, June 2000, pp. 1243–1248. [10] S. Catreux, P. F. Driessen, and L. J. Greenstein, “Data throughputs using multiple-input multiple-output (MIMO) techniques in a noise-limited cellular environment,” IEEE Trans. Wireless Commun., vol. 1, pp. 226–235, Apr. 2002. [11] S. T. Chung and A. J. Goldsmith, “Degree of freedom in adaptive modulation: A unified view,” IEEE Trans. Commun., vol. 49, pp. 1561–1571, Sept. 2001. [12] X. Zhu and R. D. Murch, “Multi-input multi-output maximum-likelihood detection for a wireless system,” in Proc. IEEE Vehicular Technology Conf., Rhodes, Greece, May 2001, pp. 137–141.

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