layer error characteristics of lattice-reduction aided ... - Semantic Scholar
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
LAYER ERROR CHARACTERISTICS OF LATTICE-REDUCTION AIDED V-BLAST DETECTORS Tien Duc Nguyen University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
Xuan Nam Tran University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
A BSTRACT Recently, lattice reduction aided (LRA) detectors have been introduced into Vertical Bell-Labs Layered Space-Time (VBLAST) systems to obtain nearly optimal bit error rate (BER) performance for only small additional complexity. In this paper, the layer error characteristics of LRA-V-BLAST detectors are analyzed and compared with those of conventional VBLAST ones. Two important conclusions are drawn for the LRA-V-BLAST detectors. First, the variation of their mean square error (MSE) within each detection iteration is not as large as in conventional V-BLAST detectors. Second, thanks to lattice reduction there exists an inherent sub-optimal detection order from the last to the first layer. These conclusions allow LRA-V-BLAST detectors to avoid optimal ordering to further reduce the complexity. LRA-V-BLAST detectors without optimal ordering are shown to obtain almost the same BER performance of LRA-V-BLAST detector with optimal ordering. I.
I NTRODUCTION
Recent research on information theory of wireless communications has shown that enormous capacity increase can be achieved on a multiple-input multiple-output (MIMO) system in a rich scattering environment [1]. Aiming at the MIMO capacity, spatial division multiplexing (SDM) is combined with an Among MIMO-SDM detectors, the Vertical Bell-Labs Layered Space-Time (V-BLAST) detector [2] was known as the most promising one in terms of having good trade-off between the bit error rate (BER) and computational complexity. The main principle of the V-BLAST detector is to combine a linear detector based on zero-forcing or minimum mean square error (MMSE) criterion with a successive interference canceler (SIC). Detection for transmitted symbols from transmit antennas is carried out iteratively based on an optimal order decided by the computed mean square errors (MSE). Recently, there has been a great interest in applying lattice reduction to linear and V-BLAST detectors to obtain better BER performance at a cost of only small additional complexity [4]–[6]. While lattice reduction requires only small additional complexity, the optimal ordering process needs to compute the norms of all the nulling vectors and thus requires a complexity order proportional to the third power of the number of transmit antennas, i.e., O(N 3 ), where N denotes the number of transmit antennas and is assumed equal to the number of receive antennas. The open problem is how to reduce the complexity associated with optimal ordering without degrading BER performance significantly. We have carried out detailed analyses c 1-4244-0330-8/06/$20.002006 IEEE
Tadashi Fujino University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
on the layer error characteristics of the LRA-V-BLAST detectors and found two important observations. First, the variation MSE within each detection iteration of a LRA-V-BLAST detector is not as large as in a conventional V-BLAST detector. As a result, optimal ordering can be avoided without significant degradation of the BER performance. Second, thanks to lattice reduction the probability that the lower layers have the minimum MSE is particularly higher than upper layers1 such that in almost all the cases optimal ordering can be readily defined as from the last to the first layer. We shall refer this inherent order as the sub-optimal ordering throughout this paper. Once again, it is clear that optimal ordering can be avoided to reduce further the complexity of the LRA-V-BLAST detectors while BER performance is still kept almost the same as that of the case with optimal ordering. In fact, simulations of a 4 × 4 MIMO shows that the degradation of the signal to noise ratio (SNR) is only about less than 0.5dB and 1dB for the case of LRA-V-BLAST with sub-optimal and random ordering, respectively. To the best knowledge of the authors both of these observations have not been realized before. The remainder of the paper is organized as follows. In Sect. II., we present the signal model and detection methods for V-BLAST systems. Principle of lattice reduction aided VBLAST detectors are summarized in Sect. III.. Detailed analyses on the layer error characteristics of the LRA-V-BLAST detectors is given in Sect. IV.. Simulation results are shown in Sect. V.. Finally, the paper is summarized and concluded in Sect. VI.. II.
V-BLAST S YSTEMS
A. System Model We consider a MIMO-SDM with N transmit and M receive antennas signalling over a rich scattering flat fading channel as shown in Fig. 1. It is assumed that N ≤ M . Each single channel from the nth transmit antenna to the mth receive antenna is represented by the complex gain hmn with n = 1, 2, . . . , N , and m = 1, 2, . . . , M . It is common to model these hmn by Gaussian random variables with zero mean and unit variance, i.e., hmn ∼ Nc (0, 1). The assumption of rich scattering ensures that hmn are mutually independent. We assume that the noise samples zm is independent complex Gaussian random variables with zero mean and variance σz2 , i.e., zm ∼ Nc (0, σz2 ). The system model of the considered system is expressed in matrix form as y = Hs + z
(1)
1 We refer the N th transmit antenna as the lowest (last) layer and the 1st antenna as the highest (first) layer as plotted in Fig. 1.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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In the next parts, we shall present the derivations of the ZF and MMSE weight matrix used in the linear detector for nulling and then summarize the V-BLAST algorithm.
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Figure 1: System model of a MIMO-SDM system. where s = [s1 , s2 , . . . , sN ]T is the transmit signal vector, y = [y1 , y2 , . . . , yM ]T is the receive signal vector, z = [z1 , z2 , . . . , zM ]T is the noise vector and h11 h12 . . . h1N h21 h22 . . . h2N (2) H = . .. .. .. .. . . . hM1
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is the M ×N MIMO channel matrix. Some further assumptions about the transmit power constrain, noise and signal correlation are as follows: E{ssH } = ζs2 I N = N1 I N , and E{sz H } = 0, where I N and 0 denote a N × N identity matrix and a zero matrix of appropriate size, respectively. Here E{•} and (•)H represent the expected and complex transpose operation, respectively. B.
V-BLAST Detectors
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to detect it using a linear combination (i) s¯ki = Q wH ki y
(3)
where H † denotes the pseudo-inverse of H. The nulling vector for the nth layer is corresponding to the nth column of W or H wn = H † (7) n
where [·]n denotes the nth column of the matrix inside the brackets. Estimates of the transmitted symbol s by the ZF detector are given by sˆ = W H y = H † y
(8)
Thus MSE of associated with detecting sn is given by MSEn = E {[sn − sˆn ][sn − sˆn ]∗ }
(9)
= σz2 wn 2 2) MMSE Method
−1 W = H H H H + ρI N where ρ = σz2 /ζs2 . Similar to [8] if we define H y ¯ H= √ and y ¯= ρI N 0
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where we use the notation aκ to denote a vector formed from the κth column of matrix A. Thus the detection operation of VBLAST detectors can be described simply as successive nulling and cancellation [2].
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then the MMSE method can be expressed in the similar form of the ZF as ¯ y ¯ sˆ = W H y = H
where wki is derived based on the zero-forcing (ZF) or the minimum mean square error (MMSE) method and Q{•} represents the quantization (decision) function. This detection operation is also referred to as nulling in [2] due to the similarity with the operation of adaptive antennas. Next, the detected signal is fed back to the linear detector to be cancelled from the receive signal vector in the next stage, i.e. y (i+1) = y (i) − s¯ki hki
(6)
The weight matrix using the MMSE method is given by [8]
The principle of V-BLAST detectors [2] is summarized as follows. Detection of transmitted signals is carried out iteratively according to an optimal order. Denote {k1 , k2 , . . . , kN } as a permutation of the set of transmit antenna indices {1, 2, . . . , N } specifying the order in which components of s is detected. During the ith detection iteration or detection stage, the detector computes MSEs of all transmitted signals sn and selects the signal with the minimum MSE, i.e. ki = arg min {MSEj , j = 1, 2, . . . , N − 1 + i}
1) ZF Method
(12)
¯ † , then MSE associated with deLet w ¯ n be the nth row of H tecting sn using MMSE method is given by MSEn = E {[sn − sˆn ][sn − sˆn ]∗ } ¯ n 2 = σz2 w
(13)
¯ y Interestingly, if we replace H, y and w by H, ¯ and w, ¯ respectively, then the MMSE detector can be described equivalently as a ZF detector. In the following parts we shall restrict our explanation for the case of ZF. The application of MMSE, where appropriate, can be done by the above replacement. 3) V-BLAST Algorithm The V-BLAST algorithm is summarize in Table 1, where the notation H κ = [ ] means removing the κth column of H.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
Table 1: The V-BLAST detection algorithm (1) (2) (3)
For i = 1 : N H W = H† ki = arg min wj 2 , j = 1, 2, . . . , N − i + 1
(4) (5) (6) (7) (8)
(i) sˆki = wH ki y s¯ki = Q{ˆ ski } y (i+1) = y (i+1) − H ki s¯ki H ki = [ ] End
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III.
LRA-V-BLAST D ETECTORS
The motivation of applying LR to linear detectors is to improve the decision region. The idea of LRA detectors was first introduced in [4] and further studied in [5, 6]. The application of LR to the V-BLAST detectors is summarized below. Viewing the noiseless received signal Hx as a lattice generated by H, the idea is to transform the system representation to another representation via a unimodular transform matrix T, as y = Hs + z = (HT )(T −1 s) + z = H u + z,
(14)
where H = HT and u = T −1 s are the new complex channel matrix and transmitted signal vector, respectively. Using the new model in (14), a LRA-V-BLAST detector uses its algorithm to estimate u instead of s. Since T −1 contains integer elements, if s is taken from a complex integer set then u is a complex integer vector. Therefore, for LRA-V-BLAST detectors, in each detection iteration the quantization operation of imagthe estimates u ˆn is identical to rounding off the real and √ un } + I{ˆ un }, where = −1. inary part as u¯n = R{ˆ ˆ = Tu ¯ , and Finally, the original signal vector is recovered as s ˆ is then quantized to the signal constelthe decision estimate s lation as s¯ = Q{ˆ s}. IV.
L AYER E RROR C HARACTERISTICS OF LRA-V-BLAST D ETECTORS
A. MSE Variation In this section, we analyze the layer characteristics of the LRAV-BLAST detectors. Note that MSE for the nth layer is given by ˆn 2 } = σz2 (w H (15) MSEn = E{un − u n wn ) If all layers have the same MSE then it is apparent that optimal ordering is not necessary. The detection then can be performed in a predefined order such as from the last to the first layer or in a random order. Unfortunately, due to fading the channel is varied in a random fashion, and thus MSEs also vary accordingly. Clearly, the variation of MSEs significantly affects the error performance of V-BLAST detectors without optimal ordering due to the effect of error propagation. The more MSEs vary, the worse BER is. In order to gain insight into the error characteristics of the LRA-V-LBAST detectors, without loss of generality we have performed a simulation for a 4 × 4 MIMO system, and computed the cumulative distribution function (CDF) of
MSE for all layers. In the simulation, 104 realizations of the complex Gaussian channel matrix H were generated and used for analysis. For each realization of H the V-BLAST detectors take a random order to detect transmitted symbols sn , then compute the nulling weight vectors wn for the ZF detector and the associated MSE using (15). Figure 2 compares CDFs of the maximum and minimum MSE of the LRA-V-BLAST and the conventional V-BLAST detectors in all 4 detection iterations. The first observation immediately realized from the figure is that for the case of LRA-V-BLAST detector, the variation between MSEmax and MSEmin is small in all iterations. In the first detection layer, the average variation is about 0.5. As the number of iteration increases, MSEmin moves close to MSEmax causing the variation to decrease gradually. For the case of the conventional V-BLAST detector, the variation of MSE is much larger, particularly in the beginning iterations. Although it is also reduced as the number of iteration increases, except for the last iteration, there is still a large variation of MSE. This observation can be seen clearer in Fig. 3, where we plot CDFs of the ratio of MSEmax over MSEmin . Since the degree of variation of MSE will affect bit error performance of the V-BLAST detectors, we can conclude that optimal ordering may not as important for LRA-V-BLAST as for the conventional V-BLAST detector. As a result, optimal ordering can be avoided to reduce further complexity of the LRA-V-BLAST detector. In fact, our simulation results in the next section show that the difference in BER between the case of without ordering (random ordering) and the optimal ordering is very small and can be negligible for a MIMO system with a small number of antennas. The second observation is that the values of MSEmax and MSEmin obtained in the ending iterations reduces to zero very quickly due to the increased diversity. This explains the advantage of V-BLAST detectors. However, since the V-BLAST detectors suffer from the problem of error propagation, correct detection in the beginning iteration is of particular importance. As a result, the LRA-V-BLAST detectors are less sensitive the error propagation and has better BER performance. B. Inherent Sub-Optimal Order of LRA-V-BLAST Detectors In the previous section, we have shown that optimal ordering may not necessary for the LRA-V-BLAST detector and thus can be avoided to reduce further complexity. In this section, we shall consolidate this conclusion and show further that for the LRA-V-BLAST detector there is in fact an inherent suboptimal detection order from the last to the first layer. In order to do so we carried out an analysis based on the histogram of the index of the selected layer using the optimal ordering in all detection iterations. Similar to the previous section and without loss of generality, a 4 × 4 MIMO system was used for simulation. The simulation was run for 104 realizations of H and the ZF detector. Again, for each realization of H the optimal ordering based on MSE in (15) was used to locate the detection layer. Note from Fig. 4 that after the first run, the percentage that the last, third, second and first layer was selected is 66.72%, 18.71%, 9.75%, 4.82%, respectively. Thus the recommended detection order is from the last to the first layer. In
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The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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Figure 4: Histogram of the index of the selected column for the LRA-V-BLAST detector.
Figure 2: CDFs of MSEmax and MSEmin in 4 detection layers for V-BLAST and LRA-V-BLAST detectors.
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the next run, the lats column was removed and simulation was run again. Again, the suggestion is to detect transmitted signals from the third to the first layer. Similar observation can also be made for the next simulation run. This leads us to the conclusion that for the LRA-V-BLAST detector there exits a sub-optimal order from the last to the first layer. Simulation results show that using sub-optimal ordering does not reduce the BER performance significantly compared with the case of optimal ordering while we can save a large amount of complexity for optimal ordering. Note from Fig. 5 that the inherent sub-optimal order does not exist for the conventional V-BLAST detector. Clearly, the LRA-V-BLAST detector has advantage over the conventional V-BLAST detector in terms of layer error characteristics. V.
S IMULATION R ESULTS
In order to support our conclusions in Sect.IV., we have performed simulations for BER performance. Simulation was run
Figure 5: Histogram of the index of the selected column for the conventional V-BLAST detector. for a 4 × 4 MIMO system. The channel was assumed the flat Rayleigh fading channel with its complex gain generated using complex Gaussian random variables. The total transmit power was normalized to one and 4-QAM was used for modulation. In each system, both ZF and MMSE nulling weight matrices were included in the LRA-V-BLAST detectors. A. Results BERs of the LRA-V-BLAST detectors are shown in Fig. 6 for 4 × 4 MIMO systems, respectively. In each figure, we compare BERs obtained using random, sub-optimal, and optimal ordering when both ZF and MMSE nulling matrices are used. It can be clearly seen from the figures that using random or the suboptimal order does not reduce BER performance of the LRAV-BLAST detectors significantly. The largest loss in Eb /N0 recorded is only about 1dB for the case of MMSE nulling with random ordering. For ZF nulling or MIMO systems with a small number of antennas, BERs are almost the same for all three methods of ordering. Also clear is that ZF nulling has worse BER performance than MMSE due to noise amplification problem. The reference BER bound by the maximum like-
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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Figure 7: BER gap in logarithmic scale between detectors with and without detection ordering .
lihood (ML) detection in Fig. 6 confirms the merit of using lattice reduction. As concluded in the previous works [4, 5, 6], the LRA-V-BLAST detectors allow us to achieve the same diversity order of the ML detector with only small degradation in Eb /N0 . Next, we investigate the BER gap in logarithmic scale between the case of optimal ordering with the case of random ordering and the case with the sub-optimal ordering. The value of Eb /N0 was fixed at 10dB and the gap in the logarithm scale was measured for MIMO systems with M = N = {2, 3, ..., 10}. As can be seen from Fig. 7 the BER gap increases as the number of antennas increases, particularly for the case of MMSE nulling and with random ordering. This observation suggests the use of sub-optimal ordering and possibly with ZF nulling in MIMO systems with large number of antennas.
LRA-V-BLAST detectors there is actually a sub-optimal detection order from the last to the first layer. We have also shown that without optimal ordering we can save a complexity order of O(N 3 ) while do not reduce BER performance significantly compared with the case of optimal ordering. Therefore, LRAV-BLAST detectors with the sub-optimal ordering can provide the best trade-off between BER performance and complexity. It is particularly suitable for MIMO systems with a small number of antennas.
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Complexity Note
In this section, we shall show how much complexity we can save by removing optimal ordering step in the LRA-V-BLAST detectors. The complexity of ordering in step (3) of the algorithm in Table 1 includes that for computing the norm of N columns of the weight matrix W . The size of W for the case of ZF nulling is M × N . As there are N iterations and after one iteration the number of columns in W is reduced by one, the complexity is given by N
iM = M
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(16)
Assume that M = N the complexity order of the optimal ordering process is O(N 3 ). VI.
C ONCLUSION
In this paper we have analyzed the layer error characteristics of LRA-V-BLAST detectors. We have shown that thanks to lattice reduction optimal ordering can be avoided to reduce further complexity in LRA-V-BLAST detectors. Moreover, in
R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecom., vol.10, no.6, pp.585-595, November/December, 1999. [2] G. D. Golden, C. J. Foschini, R. A. Valenzuela and P.W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture”, Electronics Letters, vol. 35, no. 1, pp. 14-16, January 1999. [3] B. Vucetic and J. Yuan,“Space-Time Coding,” Wiley 2003. [4] H. Yao and G. W. Wornell, “Lattice-reduction-aided detectors for MIMO communication systems,” Proc. GLOBECOM’02, vol. 1, pp. 424–428, Nov. 2002. [5] C. Windpassinger, R. F. H. Fischer, “Low-complexity nearmaximum-likelihood detection and precoding for MIMO systems using lattice reduction,” Proc. IEEE ITW’03, France, 2003. [6] D. W¨ubben, R. Bohnke, V. Kuhn, and K. D. Kammeyer, “Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice reduction,” Proc. IEEE ICC’04, vol. 2, pp. 798–802, June 2004. [7] L. Hanzo, M, M¨unster, B.J. Choi and T. Keller, “OFDM and MC-CDMA for Broadband Multi-user Communications, WLANs and Broadcasting,” Wiley & Sons, 2003. [8] B. Hassibi,“An Efficient Square-Root Algorithm for BLAST,” in Proc. IEEE ICASSP’00, Istanbul, Turkey, pp. 5-9, 2000.
LAYER ERROR CHARACTERISTICS OF LATTICE-REDUCTION AIDED V-BLAST DETECTORS Tien Duc Nguyen University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
Xuan Nam Tran University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
A BSTRACT Recently, lattice reduction aided (LRA) detectors have been introduced into Vertical Bell-Labs Layered Space-Time (VBLAST) systems to obtain nearly optimal bit error rate (BER) performance for only small additional complexity. In this paper, the layer error characteristics of LRA-V-BLAST detectors are analyzed and compared with those of conventional VBLAST ones. Two important conclusions are drawn for the LRA-V-BLAST detectors. First, the variation of their mean square error (MSE) within each detection iteration is not as large as in conventional V-BLAST detectors. Second, thanks to lattice reduction there exists an inherent sub-optimal detection order from the last to the first layer. These conclusions allow LRA-V-BLAST detectors to avoid optimal ordering to further reduce the complexity. LRA-V-BLAST detectors without optimal ordering are shown to obtain almost the same BER performance of LRA-V-BLAST detector with optimal ordering. I.
I NTRODUCTION
Recent research on information theory of wireless communications has shown that enormous capacity increase can be achieved on a multiple-input multiple-output (MIMO) system in a rich scattering environment [1]. Aiming at the MIMO capacity, spatial division multiplexing (SDM) is combined with an Among MIMO-SDM detectors, the Vertical Bell-Labs Layered Space-Time (V-BLAST) detector [2] was known as the most promising one in terms of having good trade-off between the bit error rate (BER) and computational complexity. The main principle of the V-BLAST detector is to combine a linear detector based on zero-forcing or minimum mean square error (MMSE) criterion with a successive interference canceler (SIC). Detection for transmitted symbols from transmit antennas is carried out iteratively based on an optimal order decided by the computed mean square errors (MSE). Recently, there has been a great interest in applying lattice reduction to linear and V-BLAST detectors to obtain better BER performance at a cost of only small additional complexity [4]–[6]. While lattice reduction requires only small additional complexity, the optimal ordering process needs to compute the norms of all the nulling vectors and thus requires a complexity order proportional to the third power of the number of transmit antennas, i.e., O(N 3 ), where N denotes the number of transmit antennas and is assumed equal to the number of receive antennas. The open problem is how to reduce the complexity associated with optimal ordering without degrading BER performance significantly. We have carried out detailed analyses c 1-4244-0330-8/06/$20.002006 IEEE
Tadashi Fujino University of Electro-Communications Chofugaoka 1-5-1, Chofu-shi, Tokyo Japan
on the layer error characteristics of the LRA-V-BLAST detectors and found two important observations. First, the variation MSE within each detection iteration of a LRA-V-BLAST detector is not as large as in a conventional V-BLAST detector. As a result, optimal ordering can be avoided without significant degradation of the BER performance. Second, thanks to lattice reduction the probability that the lower layers have the minimum MSE is particularly higher than upper layers1 such that in almost all the cases optimal ordering can be readily defined as from the last to the first layer. We shall refer this inherent order as the sub-optimal ordering throughout this paper. Once again, it is clear that optimal ordering can be avoided to reduce further the complexity of the LRA-V-BLAST detectors while BER performance is still kept almost the same as that of the case with optimal ordering. In fact, simulations of a 4 × 4 MIMO shows that the degradation of the signal to noise ratio (SNR) is only about less than 0.5dB and 1dB for the case of LRA-V-BLAST with sub-optimal and random ordering, respectively. To the best knowledge of the authors both of these observations have not been realized before. The remainder of the paper is organized as follows. In Sect. II., we present the signal model and detection methods for V-BLAST systems. Principle of lattice reduction aided VBLAST detectors are summarized in Sect. III.. Detailed analyses on the layer error characteristics of the LRA-V-BLAST detectors is given in Sect. IV.. Simulation results are shown in Sect. V.. Finally, the paper is summarized and concluded in Sect. VI.. II.
V-BLAST S YSTEMS
A. System Model We consider a MIMO-SDM with N transmit and M receive antennas signalling over a rich scattering flat fading channel as shown in Fig. 1. It is assumed that N ≤ M . Each single channel from the nth transmit antenna to the mth receive antenna is represented by the complex gain hmn with n = 1, 2, . . . , N , and m = 1, 2, . . . , M . It is common to model these hmn by Gaussian random variables with zero mean and unit variance, i.e., hmn ∼ Nc (0, 1). The assumption of rich scattering ensures that hmn are mutually independent. We assume that the noise samples zm is independent complex Gaussian random variables with zero mean and variance σz2 , i.e., zm ∼ Nc (0, σz2 ). The system model of the considered system is expressed in matrix form as y = Hs + z
(1)
1 We refer the N th transmit antenna as the lowest (last) layer and the 1st antenna as the highest (first) layer as plotted in Fig. 1.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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In the next parts, we shall present the derivations of the ZF and MMSE weight matrix used in the linear detector for nulling and then summarize the V-BLAST algorithm.
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Figure 1: System model of a MIMO-SDM system. where s = [s1 , s2 , . . . , sN ]T is the transmit signal vector, y = [y1 , y2 , . . . , yM ]T is the receive signal vector, z = [z1 , z2 , . . . , zM ]T is the noise vector and h11 h12 . . . h1N h21 h22 . . . h2N (2) H = . .. .. .. .. . . . hM1
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V-BLAST Detectors
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to detect it using a linear combination (i) s¯ki = Q wH ki y
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where H † denotes the pseudo-inverse of H. The nulling vector for the nth layer is corresponding to the nth column of W or H wn = H † (7) n
where [·]n denotes the nth column of the matrix inside the brackets. Estimates of the transmitted symbol s by the ZF detector are given by sˆ = W H y = H † y
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Thus MSE of associated with detecting sn is given by MSEn = E {[sn − sˆn ][sn − sˆn ]∗ }
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−1 W = H H H H + ρI N where ρ = σz2 /ζs2 . Similar to [8] if we define H y ¯ H= √ and y ¯= ρI N 0
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then the MMSE method can be expressed in the similar form of the ZF as ¯ y ¯ sˆ = W H y = H
where wki is derived based on the zero-forcing (ZF) or the minimum mean square error (MMSE) method and Q{•} represents the quantization (decision) function. This detection operation is also referred to as nulling in [2] due to the similarity with the operation of adaptive antennas. Next, the detected signal is fed back to the linear detector to be cancelled from the receive signal vector in the next stage, i.e. y (i+1) = y (i) − s¯ki hki
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The weight matrix using the MMSE method is given by [8]
The principle of V-BLAST detectors [2] is summarized as follows. Detection of transmitted signals is carried out iteratively according to an optimal order. Denote {k1 , k2 , . . . , kN } as a permutation of the set of transmit antenna indices {1, 2, . . . , N } specifying the order in which components of s is detected. During the ith detection iteration or detection stage, the detector computes MSEs of all transmitted signals sn and selects the signal with the minimum MSE, i.e. ki = arg min {MSEj , j = 1, 2, . . . , N − 1 + i}
1) ZF Method
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¯ † , then MSE associated with deLet w ¯ n be the nth row of H tecting sn using MMSE method is given by MSEn = E {[sn − sˆn ][sn − sˆn ]∗ } ¯ n 2 = σz2 w
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¯ y Interestingly, if we replace H, y and w by H, ¯ and w, ¯ respectively, then the MMSE detector can be described equivalently as a ZF detector. In the following parts we shall restrict our explanation for the case of ZF. The application of MMSE, where appropriate, can be done by the above replacement. 3) V-BLAST Algorithm The V-BLAST algorithm is summarize in Table 1, where the notation H κ = [ ] means removing the κth column of H.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
Table 1: The V-BLAST detection algorithm (1) (2) (3)
For i = 1 : N H W = H† ki = arg min wj 2 , j = 1, 2, . . . , N − i + 1
(4) (5) (6) (7) (8)
(i) sˆki = wH ki y s¯ki = Q{ˆ ski } y (i+1) = y (i+1) − H ki s¯ki H ki = [ ] End
j
III.
LRA-V-BLAST D ETECTORS
The motivation of applying LR to linear detectors is to improve the decision region. The idea of LRA detectors was first introduced in [4] and further studied in [5, 6]. The application of LR to the V-BLAST detectors is summarized below. Viewing the noiseless received signal Hx as a lattice generated by H, the idea is to transform the system representation to another representation via a unimodular transform matrix T, as y = Hs + z = (HT )(T −1 s) + z = H u + z,
(14)
where H = HT and u = T −1 s are the new complex channel matrix and transmitted signal vector, respectively. Using the new model in (14), a LRA-V-BLAST detector uses its algorithm to estimate u instead of s. Since T −1 contains integer elements, if s is taken from a complex integer set then u is a complex integer vector. Therefore, for LRA-V-BLAST detectors, in each detection iteration the quantization operation of imagthe estimates u ˆn is identical to rounding off the real and √ un } + I{ˆ un }, where = −1. inary part as u¯n = R{ˆ ˆ = Tu ¯ , and Finally, the original signal vector is recovered as s ˆ is then quantized to the signal constelthe decision estimate s lation as s¯ = Q{ˆ s}. IV.
L AYER E RROR C HARACTERISTICS OF LRA-V-BLAST D ETECTORS
A. MSE Variation In this section, we analyze the layer characteristics of the LRAV-BLAST detectors. Note that MSE for the nth layer is given by ˆn 2 } = σz2 (w H (15) MSEn = E{un − u n wn ) If all layers have the same MSE then it is apparent that optimal ordering is not necessary. The detection then can be performed in a predefined order such as from the last to the first layer or in a random order. Unfortunately, due to fading the channel is varied in a random fashion, and thus MSEs also vary accordingly. Clearly, the variation of MSEs significantly affects the error performance of V-BLAST detectors without optimal ordering due to the effect of error propagation. The more MSEs vary, the worse BER is. In order to gain insight into the error characteristics of the LRA-V-LBAST detectors, without loss of generality we have performed a simulation for a 4 × 4 MIMO system, and computed the cumulative distribution function (CDF) of
MSE for all layers. In the simulation, 104 realizations of the complex Gaussian channel matrix H were generated and used for analysis. For each realization of H the V-BLAST detectors take a random order to detect transmitted symbols sn , then compute the nulling weight vectors wn for the ZF detector and the associated MSE using (15). Figure 2 compares CDFs of the maximum and minimum MSE of the LRA-V-BLAST and the conventional V-BLAST detectors in all 4 detection iterations. The first observation immediately realized from the figure is that for the case of LRA-V-BLAST detector, the variation between MSEmax and MSEmin is small in all iterations. In the first detection layer, the average variation is about 0.5. As the number of iteration increases, MSEmin moves close to MSEmax causing the variation to decrease gradually. For the case of the conventional V-BLAST detector, the variation of MSE is much larger, particularly in the beginning iterations. Although it is also reduced as the number of iteration increases, except for the last iteration, there is still a large variation of MSE. This observation can be seen clearer in Fig. 3, where we plot CDFs of the ratio of MSEmax over MSEmin . Since the degree of variation of MSE will affect bit error performance of the V-BLAST detectors, we can conclude that optimal ordering may not as important for LRA-V-BLAST as for the conventional V-BLAST detector. As a result, optimal ordering can be avoided to reduce further complexity of the LRA-V-BLAST detector. In fact, our simulation results in the next section show that the difference in BER between the case of without ordering (random ordering) and the optimal ordering is very small and can be negligible for a MIMO system with a small number of antennas. The second observation is that the values of MSEmax and MSEmin obtained in the ending iterations reduces to zero very quickly due to the increased diversity. This explains the advantage of V-BLAST detectors. However, since the V-BLAST detectors suffer from the problem of error propagation, correct detection in the beginning iteration is of particular importance. As a result, the LRA-V-BLAST detectors are less sensitive the error propagation and has better BER performance. B. Inherent Sub-Optimal Order of LRA-V-BLAST Detectors In the previous section, we have shown that optimal ordering may not necessary for the LRA-V-BLAST detector and thus can be avoided to reduce further complexity. In this section, we shall consolidate this conclusion and show further that for the LRA-V-BLAST detector there is in fact an inherent suboptimal detection order from the last to the first layer. In order to do so we carried out an analysis based on the histogram of the index of the selected layer using the optimal ordering in all detection iterations. Similar to the previous section and without loss of generality, a 4 × 4 MIMO system was used for simulation. The simulation was run for 104 realizations of H and the ZF detector. Again, for each realization of H the optimal ordering based on MSE in (15) was used to locate the detection layer. Note from Fig. 4 that after the first run, the percentage that the last, third, second and first layer was selected is 66.72%, 18.71%, 9.75%, 4.82%, respectively. Thus the recommended detection order is from the last to the first layer. In
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The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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the next run, the lats column was removed and simulation was run again. Again, the suggestion is to detect transmitted signals from the third to the first layer. Similar observation can also be made for the next simulation run. This leads us to the conclusion that for the LRA-V-BLAST detector there exits a sub-optimal order from the last to the first layer. Simulation results show that using sub-optimal ordering does not reduce the BER performance significantly compared with the case of optimal ordering while we can save a large amount of complexity for optimal ordering. Note from Fig. 5 that the inherent sub-optimal order does not exist for the conventional V-BLAST detector. Clearly, the LRA-V-BLAST detector has advantage over the conventional V-BLAST detector in terms of layer error characteristics. V.
S IMULATION R ESULTS
In order to support our conclusions in Sect.IV., we have performed simulations for BER performance. Simulation was run
Figure 5: Histogram of the index of the selected column for the conventional V-BLAST detector. for a 4 × 4 MIMO system. The channel was assumed the flat Rayleigh fading channel with its complex gain generated using complex Gaussian random variables. The total transmit power was normalized to one and 4-QAM was used for modulation. In each system, both ZF and MMSE nulling weight matrices were included in the LRA-V-BLAST detectors. A. Results BERs of the LRA-V-BLAST detectors are shown in Fig. 6 for 4 × 4 MIMO systems, respectively. In each figure, we compare BERs obtained using random, sub-optimal, and optimal ordering when both ZF and MMSE nulling matrices are used. It can be clearly seen from the figures that using random or the suboptimal order does not reduce BER performance of the LRAV-BLAST detectors significantly. The largest loss in Eb /N0 recorded is only about 1dB for the case of MMSE nulling with random ordering. For ZF nulling or MIMO systems with a small number of antennas, BERs are almost the same for all three methods of ordering. Also clear is that ZF nulling has worse BER performance than MMSE due to noise amplification problem. The reference BER bound by the maximum like-
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
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lihood (ML) detection in Fig. 6 confirms the merit of using lattice reduction. As concluded in the previous works [4, 5, 6], the LRA-V-BLAST detectors allow us to achieve the same diversity order of the ML detector with only small degradation in Eb /N0 . Next, we investigate the BER gap in logarithmic scale between the case of optimal ordering with the case of random ordering and the case with the sub-optimal ordering. The value of Eb /N0 was fixed at 10dB and the gap in the logarithm scale was measured for MIMO systems with M = N = {2, 3, ..., 10}. As can be seen from Fig. 7 the BER gap increases as the number of antennas increases, particularly for the case of MMSE nulling and with random ordering. This observation suggests the use of sub-optimal ordering and possibly with ZF nulling in MIMO systems with large number of antennas.
LRA-V-BLAST detectors there is actually a sub-optimal detection order from the last to the first layer. We have also shown that without optimal ordering we can save a complexity order of O(N 3 ) while do not reduce BER performance significantly compared with the case of optimal ordering. Therefore, LRAV-BLAST detectors with the sub-optimal ordering can provide the best trade-off between BER performance and complexity. It is particularly suitable for MIMO systems with a small number of antennas.
B.
Complexity Note
In this section, we shall show how much complexity we can save by removing optimal ordering step in the LRA-V-BLAST detectors. The complexity of ordering in step (3) of the algorithm in Table 1 includes that for computing the norm of N columns of the weight matrix W . The size of W for the case of ZF nulling is M × N . As there are N iterations and after one iteration the number of columns in W is reduced by one, the complexity is given by N
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(16)
Assume that M = N the complexity order of the optimal ordering process is O(N 3 ). VI.
C ONCLUSION
In this paper we have analyzed the layer error characteristics of LRA-V-BLAST detectors. We have shown that thanks to lattice reduction optimal ordering can be avoided to reduce further complexity in LRA-V-BLAST detectors. Moreover, in
R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecom., vol.10, no.6, pp.585-595, November/December, 1999. [2] G. D. Golden, C. J. Foschini, R. A. Valenzuela and P.W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture”, Electronics Letters, vol. 35, no. 1, pp. 14-16, January 1999. [3] B. Vucetic and J. Yuan,“Space-Time Coding,” Wiley 2003. [4] H. Yao and G. W. Wornell, “Lattice-reduction-aided detectors for MIMO communication systems,” Proc. GLOBECOM’02, vol. 1, pp. 424–428, Nov. 2002. [5] C. Windpassinger, R. F. H. Fischer, “Low-complexity nearmaximum-likelihood detection and precoding for MIMO systems using lattice reduction,” Proc. IEEE ITW’03, France, 2003. [6] D. W¨ubben, R. Bohnke, V. Kuhn, and K. D. Kammeyer, “Near-maximum-likelihood detection of MIMO systems using MMSE-based lattice reduction,” Proc. IEEE ICC’04, vol. 2, pp. 798–802, June 2004. [7] L. Hanzo, M, M¨unster, B.J. Choi and T. Keller, “OFDM and MC-CDMA for Broadband Multi-user Communications, WLANs and Broadcasting,” Wiley & Sons, 2003. [8] B. Hassibi,“An Efficient Square-Root Algorithm for BLAST,” in Proc. IEEE ICASSP’00, Istanbul, Turkey, pp. 5-9, 2000.
