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IEICE TRANS. COMMUN., VOL.E89–B, NO.3 MARCH 2006
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PAPER
Computational Complexity Reduction of MLD Based on SINR in MIMO Spatial Multiplexing Systems Katsunari HONJO†a) , Student Member and Tomoaki OHTSUKI†† , Member
SUMMARY MIMO spatial multiplexing systems are attracting a lot of attention because of their high spectral effiencies. Maximum Likelihood Detection (MLD) is known to be the optimal signal detection method for MIMO spatial multiplexing systems in terms of bit error rate (BER). However, the main drawback of MLD is its high complexity. In this paper, to reduce the computational complexity of MLD and to attain good BER in MIMO spatial multiplexing systems, we propose the minimum mean square error (MMSE)-MLD that combines MMSE detection and MLD according to the estimated SINR from each transmit antenna. We also propose the ordered successive MMSE detection (OSD)-MLD that combines OSD and MLD according to the estimated SINR from the transmit antennas. Simulation results show that the proposed MMSE-MLD and OSDMLD can attain almost identical BER to that of MLD but with less complexity. key words: MIMO, MLD, SINR
1.
Introduction
Recently, multiple-input multiple-output (MIMO) systems with multiple antennas at both transmitter and receiver that realize a high bit rate data transmission have drawn much attention for high spectral efficiencies [1]–[10]. Various detection schemes in MIMO spatial multiplexing systems have been proposed for MIMO systems, which can be divided roughly into the linear approach or the nonlinear approach. The zero-forcing (ZF) detection and the minimum mean square error (MMSE) detection that employ the linear approach are the most basic signal detection schemes. Meanwhile, Vertical Bell Laboratories Layered Space-Time (V-BLAST) detection [3], [4], which is often called Ordered Successive Detection (OSD), and Maximum Likelihood Detection (MLD) [7] that employ the nonlinear approach have been shown to outperform the detection schemes by the linear approaches. In the V-BLAST detection, each signal is detected by a successive interference cancellation that nulls the interferes by linearly weighting the received signal vector with a zero-forcing nulling vector. However, in the VBLAST detection its performance suffers great degradation due to the error propagation. An adaptation of original VBLAST to the MMSE criterion was presented in [5], [6]. MLD is another nonlinear approach, and generates the best performance in terms of bit error rate (BER). However, the Manuscript received November 8, 2004. Manuscript revised August 24, 2005. † The author is with the Graduate School of Electrical Engineering, Tokyo University of Science, Noda-shi, 278-8510 Japan. †† The author is with the Department of Information and Computer Science, Keio University, Yokohama-shi, 223-8522 Japan. a) E-mail: [email protected] DOI: 10.1093/ietcom/e89–b.3.914
computational complexity of the MLD increases exponentially with a constellation size and the number of transmit antennas. In this paper, to reduce the computational complexity of MLD and to attain good BER in MIMO spatial multiplexing systems, we propose two reduced complexity detection schemes. In the proposed detection schemes, first we estimate the signal to interference plus noise ratio (SINR) for each transmit signal and set the threshold according to the estimated SINR. The threshold of the estimated SINR is determined in advance according to the required performance and the computational complexity. One proposed detection scheme combines the MMSE detection and MLD according to the threshold of the estimated SINRs. The other proposed detection scheme combines OSD and MLD according to the threshold of the estimated SINR. We refer to these detection schemes as MMSE-MLD and OSD-MLD, respectively. We show that the proposed MMSE-MLD and OSD-MLD can attain almost identical performance to that of MLD but with less computational complexity by setting the threshold of the estimated SINR appropriately. We also compare the proposed MMSE-MLD and OSD-MLD and show that when QPSK is used, MMSE-MLD is superior to OSD-MLD, and vice versa when 16QAM is used. The remainder of this paper is organized as follows. In Sect. 2, the MIMO system model is introduced. In Sect. 3, the conventional detection schemes in MIMO spatial multiplexing systems are reviewed. The proposed detection schemes referred to as MMSE-MLD and OSD-MLD are described in Sect. 4. Section 5 shows some simulation results. Finally, the brief conclusion is given in Sect. 6. Superscripts (·)T , (·)H represent transpose and complex conjugate transpose, respectively. 2.
System Model
We consider the MIMO spatial multiplexing system with Nt transmit antennas and Nr (Nt ≤ Nr ) receive antennas shown in Fig. 1. An incoming information bit is partitioned into Nt signals, and each signal is mapped into symbols and sent through a different transmit antenna. Each receive antenna receives the signals from all Nt transmit antennas over rich-scattering wireless channels. Let x = [x1 . . . xNt ]T denote the Nt × 1 transmitted signal vector, then the corresponding Nr × 1 received signal vector r = [r1 . . . rNr ]T is given by
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
HONJO and OHTSUKI: COMPUTATIONAL COMPLEXITY REDUCTION OF MLD BASED ON SINR IN MIMO SYSTEMS
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Detect the signal xˆimax according to the ordering. yimax = Gimax · r (1) , xˆimax = Q(yimax )
(5)
where Gimax denotes the imax -th row of the MMSE nulling matrix G in (3). Fig. 1 Model of a MIMO spatial multiplexing system with Nt transmit and Nr receive antennas.
r = Hx + n.
Canceling The detected signal is subtracted from the last modified received vector.
(1)
In (1), n = [n1 . . . nNr ] denotes white Gaussian noise of variance σ2 observed at Nr receive antennas, E[nnH ] = σ2 INr . The average transmit power of each antenna is normalized to unity, E[xxH ] = INt . The Nr × Nt channel transfer matrix H is modeled as an independent complex Gaussian random variable with zero mean and unit variance, and its element hi j is the channel gain from the transmit antenna j to the receive antenna i.
r (2) = r (1) − xˆimax himax .
T
3.
Conventional Detection Schemes
The modified channel matrix is expressed as H (2) = [h1 , . . . , himax −1 , himax +1 , . . . , hNt ].
G = arg min Ex − Gr2 G
SINRi =
(2)
where · denotes the vector norm. We can obtain the solution to the problem, which can be given by G = H H (HH H + σ2 INr )−1 .
y = [y1 , y2 , . . . , yNt ]T = G · r xˆ = Q(y)
(4)
where Q(·) denotes the hard decision operation corresponding to the constellation in use. 3.2 Ordered Successive MMSE Detection (OSD) [6] The OSD algorithm consists of three steps: ordering, nulling, and interference canceling. The basic detection algorithm for OSD is described as follows. (·)(k) (1 ≤ k ≤ Nt ) denotes the detection stage. Initialization At the first detection stage, r (1) and H (1) are set to r and H, respectively. Nulling
(GH)ii 2 E[|xi |]2 Nt
.
(8)
(GH)il 2 E[|xl |2 ] + σ2 (G)i 2
l=1, li
3.3 Maximum Likelihood Detection (MLD) It is known that MLD is an optimal method for signal detection in terms of BER. MLD considers all the possible symbol combinations x, and chooses the symbol combination that minimizes the squared Euclidean distance.
(3)
We multiply the received vector r by the weight matrix G to get the decision statistic y and make the hard decision.
(7)
Ordering When the cancellation of the detected signal is employed, the order in which the components of x are detected is important to the overall system performance. OSD calculates SINRi for each signal, and detects and cancels the component of xˆimax that maximizes SINRi .
3.1 MMSE Detection It is known that compared to MLD, the MMSE detection provides the performance degradation, but much less computational complexity. We generate the weight matrix G that minimizes the error power between the received signals and the desired signals.
(6)
xˆ = arg min r − Hx2 . x
(9)
Assuming a constellation size of M, MLD needs to evaluate all the M Nt symbol combinations. It follows that the computational complexity of MLD increases exponentially with a constellation size and the number of transmit antennas. 4.
Proposed Detection Schemes
In this section, to reduce the computational complexity of MLD, we propose two signal detection schemes, MMSEMLD that combines MMSE detection and MLD according to the estimated SINR, and OSD-MLD that combines OSD and MLD according to the estimated SINR. 4.1 MMSE-MLD Figure 2 shows the flow chart of the signal detection process of the proposed MMSE-MLD. First, SINR is estimated at the receiver for each signal from (8). A set of indexes of the signals with which SINRs exceed the threshold ζ is set to
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Fig. 2
A detection process of the MMSE-MLD.
A and a set of indexes of the signals with which SINRs are smaller than the threshold ζ is set to B. If the signals with which SINRs exceed the threshold ζ exist, these signals in A are detected by the MMSE detection. yA = GA · r, xˆA = Q(yA )
(10)
where GA denotes all the rows contained in A of the MMSE weight matrix G. The threshold ζ is determined in advance according to the required performance and the computational complexity. Afterwards, the detected signals xˆ A are subtracted from the received signal vector r as the interference, resulting in the modified received signal vector r . r = r − xˆ A HA
(11)
where HA denotes all the columns contained in A of H. The modified channel matrix H is represented by H = HA
(12)
where HA denotes the matrix that removes all the columns contained in A of H. The signals included in B are detected by MLD using the modified received signal vector r and the modified channel matrix H . When the signals with which SINRs exceeding the threshold ζ do not exist, all the signals are detected by MLD. For all the signals, MMSEMLD does not always need to perform MLD. Therefore, the proposed MMSE-MLD can reduce the computational complexity compared to MLD.
Fig. 3
A detection process of the OSD-MLD.
4.2 OSD-MLD Figure 3 shows the flow chart of the signal detection process of the proposed OSD-MLD. First SINR is estimated at the receiver for each signal. If the signals with which the SINRs exceed the threshold ζ exist, only a signal xˆimax with which SINR is maximum is detected by the MMSE detection scheme. (k) (k) y(k) imax = Gimax · r , (k) xˆ(k) imax = Q(yimax )
(13)
where (·)(k) denotes the detection stage. Afterwards, the detected signal xˆ(k) imax is subtracted from the received signal vector r (k) as the interference, resulting in the modified received vector r (k+1) . (k) r (k+1) = r (k) − xˆ(k) imax himax
(14)
where h(k) imax denotes the imax th column of H. The modified channel matrix H (k+1) is represented by H (k+1) = H (k) . imax
(15)
where H (k) denotes the matrix that removes the imax th imax
columns of H (k) . Then SINRs are computed again to the signals other than the detected signals. Note that SINRs are improved as an interference is reduced. Similarly, the same
HONJO and OHTSUKI: COMPUTATIONAL COMPLEXITY REDUCTION OF MLD BASED ON SINR IN MIMO SYSTEMS
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procedure is repeated until the signals with which SINRs exceed the threshold ζ do not exist. The signals with which SINRs smaller than the threshold ζ are detected by MLD using the modified received signal vector r (k+1) and the modified channel matrix H (k+1) . For all the signals, since OSDMLD does not always need to perform MLD, it can reduce the computational complexity compared to MLD. 5.
proposed detection schemes are almost identical to that of MLD as the optimum threshold. The optimum threshold ζ is 13 dB in Fig. 4 and 16 dB in Fig. 5. Figures 6, 7 also show a threshold ζ versus BER for QPSK and 16QAM when SNR per receive antenna is 22 dB. The optimum threshold
Simulation Results
In our computer simulations, the conventional detection schemes, that is, the MMSE detection, OSD and MLD, and the proposed detection schemes, that is, MMSE-MLD and OSD-MLD, are compared. We assume a MIMO spatial multiplexing system with Nt = Nr = 4 using QPSK and 16QAM, quasi static flat Rayleigh fading channels, and the perfect knowledge of the channel responses at the receiver. 5.1 Bit Error Rate (BER) Fig. 6
BER of each detection scheme. (QPSK, SNR = 22 dB)
Figures 4, 5 show a threshold ζ versus BER for QPSK and 16QAM when SNR per receive antenna is 20 dB. We observe that the BERs of the proposed MMSE-MLD and OSDMLD approach that of MLD as the threshold ζ becomes higher. We refer to the threshold ζ that the BERs of the
Fig. 7
Fig. 4
BER of each detection scheme. (QPSK, SNR = 20 dB)
Table 1 (QPSK)
BER of each detection scheme. (16QAM, SNR = 22 dB) Optimum thresholds ζ in the proposed detection schemes. SNR
MMSE-MLD
OSD-MLD
15 dB
11 dB
11 dB
18 dB
12 dB
12 dB
20 dB
13 dB
13 dB
22 dB
13 dB
13 dB
Table 2 Optimum thresholds ζ in the proposed detection schemes. (16QAM)
Fig. 5
BER of each detection scheme. (16QAM, SNR = 20 dB)
SNR
MMSE-MLD
OSD-MLD
18 dB
15 dB
15 dB
20 dB
16 dB
16 dB
22 dB
17 dB
17 dB
25 dB
18 dB
18 dB
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ζ is 13 dB in Fig. 6 and 17 dB in Fig. 7. From these figures, we can see that in QPSK the optimum threshold ζ does not change between SNR = 20 dB and 22 dB. On the other hand, when SNR is 22 dB, the optimum threshold is higher than that when SNR is 20 dB in 16QAM. Our proposed detection schemes have the different optimum thresholds for different SNR values. Tables 1, 2 show the optimum thresholds ζ for the proposed detection schemes in QPSK and 16QAM. We observe that the optimum threshold becomes higher as the increase of SNR in QPSK and 16QAM. This is because when SNR changes, SINR also changes. 5.2 Computational Complexity We consider the computational complexities of the proposed MMSE-MLD and OSD-MLD. Note that the computational complexities greatly depend on the number of signals with which MLD is performed. We consider only the complex multiplication operation to evaluate the computational complexity of the detection schemes as same as [9]. Although there are a few evaluation methods of computational complexity for MLD [10], we evaluate the computational complexity supposing the normal MLD [9] in this paper. Table 3 denotes the number of multiplications for the conventional detection schemes and estimation of SINR. Note that we assume Nt = 4 in OSD. In the proposed detection schemes, when m signals are detected by MMSE/OSD and n signals are detected by MLD, we denote the detection by (m, n) Table 3 The numbers of multiplications for the conventional detection schemes and estimation of SINR. M Nt {Nr (Nt + 1)}
MLD
Nr3 + Nr Nt (2Nr + 1)
MMSE OSD
4Nr3
where m + n = Nt . When i SINRs for the signals are calculated and i signals are detected by MLD, MMSE detection, and OSD, we denote the numbers of multiplications by SINR(i), MLD(i), MMSE(i), and OSD(i), respectively. Tables 4 and 5 show the number of multiplications for each detection (m, n) in MMSE-MLD and OSD-MLD. As a result, the total number of multiplications for each detection (m, n) in the proposed detection schemes for QPSK and 16QAM are given by Tables 6 and 7, respectively. The total number of multiplications for MLD is 5120 for QPSK modulation and 1310720 for 16QAM. Note that the reason why the number of multiplications of detection (0, 4) in the proposed detection schemes is higher than that of MLD is because we calculate the SINRs for all the signals. From these Tables, we observe that as the number of signals detected by MLD becomes larger, the total number of multiplications of the proposed detection schemes also becomes larger. Figures 8–11 show a threshold ζ versus the probability that the signals are detected with the detection (m, n) in the proposed detection schemes in [QPSK|16QAM]. From these figures, we observe that compared to MLD, the proposed detection schemes, MMSE-MLD and OSD-MLD, can reduce the probability of detecting all the signals by MLD at the optimum thresholds. Next we compare Fig. 8 and Fig. 10 to Fig. 9 and Fig. 11, respectively. Although in MMSE-MLD and OSDMLD, the probabilities that the signals are detected with (0, 4) are the same, the probabilities for the detections (1, 3), (2, 2) in OSD-MLD are lower than those in MMSE-MLD, respectively. This is because in OSD-MLD SINRs are improved by reducing the interference. Figures 12, 13 show a threshold ζ versus the number of multiplications normalized by the total number of multiplications of MLD for the proposed detection schemes with
+ Nr Nt (2Nr + Nt + 1) + Nt2 + 8Nr
+(Nt − 1){2Nr2 + (Nr + 1)(Nt − 1) + Nr }
Table 6 The number of multiplications for each detection (m, n) in the proposed detection schemes for QPSK.
+(Nt − 2){2Nr2 + (Nr + 1)(Nt − 2) + Nr } +(Nt − 3){2Nr2 + (Nr + 1)(Nt − 3) + Nr } SINR
(0,4)
Nr3 + Nr Nt (2Nr + Nt + 1) + Nt2
Table 4 The number of multiplications for each detection (m, n) in MMSE-MLD. (0,4) SINR(4) + MLD(4) (1,3) SINR(4) + MMSE(1) + MLD(3) (2,2) SINR(4) + MMSE(2) + MLD(2) (3,1) SINR(4) + MMSE(3) + MLD(1) (4,0) SINR(4) + MMSE(4)
Table 5 (0,4)
(1,3)
(2,2)
(3,1)
(4,0)
MMSE-MLD
5408
1320
496
344
304
OSD-MLD
5408
1537
869
822
794
Table 7 The number of multiplications for each detection (m, n) in the proposed detection schemes for 16QAM. MMSE-MLD OSD-MLD
(0,4) 1311008 1311008
(1,3) 65832 66049
The number of multiplications for each detection (m, n) in OSD-MLD. SINR(4) + MLD(4)
(1,3)
SINR(4) + OSD(1) + SINR(3) + MLD(3)
(2,2)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + MLD(2)
(3,1)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + OSD(1) + SINR(1) + MLD(1)
(4,0)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + OSD(1) + SINR(1) + OSD(1)
(2,2) 3376 3749
(3,1) 440 918
(4,0) 304 794
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Fig. 8 A threshold versus the probability that signals are detected with (m, n) in MMSE-MLD. (SNR = 20 dB)
Fig. 9 A threshold versus the probability that signals are detected with (m, n) in OSD-MLD. (SNR = 20 dB)
Fig. 10 A threshold versus the probability that signals are detected with (m, n) in MMSE-MLD. (SNR = 22 dB)
QPSK and 16QAM when SNR = 20 and 22 dB, respectively. From Fig. 12, when SNR is 20 dB, the normalized numbers of multiplications of MMSE-MLD and OSD-MLD are about 27.5 and 30.4 % compared to the number of the multiplications of the MLD at the optimum thresholds, respectively. Furthermore, when SNR is 22 dB, the normalized numbers of multiplications of MMSE-MLD and OSDMLD are about 18.8 and 23.4 % at the optimum thresh-
Fig. 11 A threshold versus the probability that signals are detected with (m, n) in OSD-MLD. (SNR = 22 dB)
Fig. 12 The number of multiplications normalized by the number of multiplications in MLD in the proposed detection schemes. (QPSK, SNR = 20, 22 dB)
Fig. 13 The number of multiplications normalized by the number of multiplications in MLD in the proposed detection schemes. (16QAM, SNR = 20, 22 dB)
olds, respectively. On the other hand, From Fig. 13, when SNR is 20 dB, the normalized numbers of multiplications of MMSE-MLD and OSD-MLD are about 46.2 and 45.7 % compared to the number of the multiplications of the MLD at the optimum thresholds, respectively. Furthermore, when SNR is 22 dB, the normalized numbers of multiplications
IEICE TRANS. COMMUN., VOL.E89–B, NO.3 MARCH 2006
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MMSE-MLD
OSD-MLD
15 dB
0.479
0.491
18 dB
0.321
0.343
20 dB
0.275
0.304
22 dB
0.188
0.234
Table 9 The number of multiplications normalized by the number of multiplications in MLD for the optimum thresholds ζ in the proposed detection schemes. (16QAM) SNR 18 dB 20 dB 22 dB 25 dB
MMSE-MLD 0.573 0.462 0.368 0.208
OSD-MLD 0.569 0.457 0.361 0.207
of MMSE-MLD and OSD-MLD are about 36.8 and 36.1 % at the optimum thresholds, respectively. Consequently, the proposed MMSE-MLD and OSD-MLD can achieve almost the identical BER to that of MLD but with less complexity. The number of multiplications of the proposed detection schemes for the SNR of 22 dB is smaller than that of the proposed detection schemes for the SNR of 20 dB at the optimum thresholds. This is because the probabilities for the detections (0,4) and (1,3) are lower than those of the detections (0,4) and (1,3) compared to 20 dB. Tables 8, 9 show the number of multiplications normalized by the number of multiplications in MLD for the optimized threshold ζ in the proposed detection schemes at different SNR values. From these Tables, we also observe that the proposed detection schemes can reduce the number of multiplications more as SNR becomes higher. When QPSK is used, we compare the number of multiplications of MMSE-MLD to that of OSD-MLD. From Table 8, we can see that MMSE-MLD needs smaller number of multiplications than OSD-MLD at the optimum threshold. Therefore, MMSE-MLD is superior to OSD-MLD. Furthermore, when 16QAM is used, we compare the number of multiplications of MMSE-MLD to that of OSD-MLD. As shown in Table 7, the proposed detection schemes need more multiplications for the detection (0, 4) than the other detections, that is, (3, 1), (2, 2), (1, 3) and (4, 0). In addition in the proposed detection schemes the probabilities that the signals are detected with (0, 4) are the same. Therefore, the number of multiplications of MMSEMLD and OSD-MLD are almost equal, as shown in Table 9. On the other hand, the BER of OSD-MLD is always superior to that of MMSE-MLD in Figs. 5, 7. Since the number of multiplications of MMSE-MLD and OSD-MLD are almost equal and the BER of OSD-MLD is superior to that of MMSE-MLD, we can say that OSD-MLD is superior to MMSE-MLD. Figures 14 and 15 show the SNR versus BER for QPSK and 16QAM, respectively. In Fig. 14, the thresholds ζ of
Fig. 14
Fig. 15
SNR versus BER. (QPSK)
SNR versus BER. (16QAM)
11 and 13 dB are optimum for the target BER of 10−3 and 10−5 , respectively. Similarly, in Fig. 15 the thresholds ζ of 15 and 17 dB are optimum for the target BER of 10−2 and 10−3 , respectively. From Figs. 14 and 15, we can see that the proposed detection schemes can attain almost identical BER to that of MLD with less complexity when the SNR is smaller than the required SNR for the target BER. However, the BERs of the proposed detection schemes are worse than that of MLD when the SNR is higher than the required SNR. This is because the BER of MMSE/OSD cannot be better than that of MLD. In addition the optimum threshold becomes higher as the required SNR becomes higher. Moreover, we can set a threshold ζ arbitrarily in the proposed detection schemes. If we think that BER is more important than the computational complexity, we set a threshold ζ high. Conversely, if we think that the computational complexity is more important than BER, we set a threshold ζ low. In this way we can obtain the desired BER and the desired computational complexity easily. Note that our proposed detection schemes can achieve almost identical BER to that of MLD with less complexity. Thus, the proposed detection schemes are useful for a reduction of power consumption at the receiver.
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6.
Conclusion
In this paper, we propose the reduced complexity signal detection schemes, the minimum mean square error (MMSE)Maximum Likelihood Detection (MLD) and the ordered successive MMSE detection (OSD)-MLD, that combine MMSE/OSD and MLD according to the estimated SINR. We showed that the proposed MMSE-MLD and OSD-MLD can attain almost identical BER to that of MLD but with less complexity. We also showed that when QPSK is used, MMSE-MLD is superior to OSD-MLD, and vice versa when 16QAM is used. References [1] G.J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Tech. J., vol.1, no.2, pp.41–59, Autumn 1996. [2] G.J. Foschini and M.J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol.6, no.3, pp.315–335, 1998. [3] 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,” Proc. IEEE ISSSE 1998, pp.295–300, Sept. 1998. [4] G.J. Foschini, G.D. Golden, R.A. Valenzuela, and P.W. Wolnianski, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,” IEEE J. Sel. Areas Commun., vol.17, no.11, pp.1841–1852, Nov. 1999. [5] M. Debbah, B. Muquet, M. Courville, M, Muck, S. Simoens, and P. Loubaton, “A MMSE successive interference cancellation scheme for a new adjustable hybrid spread OFDM system,” Proc. IEEE VTC’00 Spring, vol.2, pp.745–749, Tokyo, May 2000. [6] A. Benjebbour, H. Murata, and S. Yoshida, “Comparison of ordered successive receivers for space-time transmission,” Proc. IEEE VTC’01 Fall, vol.4, pp.2053–2057, Atlantic City, USA, Sept. 2001. [7] R. Van Nee, A.V. Zelst, and G. Awater, “Maximum likelihood decoding in a space division multiplexing system,” Proc. IEEE VTC’00 Spring, vol.2, pp.6–10, Tokyo, May 2000. [8] X. Li, H.C. Huang, A. Lozano, and G.J. Foschini, “Reducedcomplexity detection algorithms for system using multi-element arrays,” Proc. IEEE GLOBECOM’00, pp.1072–1076, SF, CA, Nov. 2000. [9] H. Sung, K.B. Lee, and J.W. Kang, “A simplified maximum likelihood detection scheme for MIMO systems,” Proc. IEEE VTC’03 Fall, vol.1, pp.419–423, Orlando, USA, Oct. 2003. [10] A. Beck, E.C. Burg, A.P. Djuknic, G.M. Gvoth, T.G. Haessig, D. Manji, S. Milbrodt, M.A. Rupp, M. Samardzija, D. Siegel, A.B. Sizer, T. Sizer, II, C. Walker, S. Wilkus, S.A. Wolniansky, and W. Peter, “Prototype experience for MIMO BLAST over thirdgeneration wireless system,” IEEE J., Sel. Areas Commun., vol.21, no.3, pp.440–451, April 2003.
Katsunari Honjo was born in Chiba, Japan in 1980. He received the B.S. degree in Electrical Engineering from Tokyo University of Science, Chiba, Japan in 2004. He is currently working towards M.E. degree in Graduate School of Electrical Engineering, Tokyo University of Science, Chiba, Japan. His current research interest includes deta detection in MIMO systems. He is a student member of IEEE.
Tomoaki Ohtsuki received the B.E., M.E., and Ph.D. degrees in Electrical Engineering from Keio University, Yokohama, Japan in 1990, 1992, and 1994, respectively. From 1994 to 1995 he was a Post Doctoral Fellow and a Visiting Researcher in Electrical Engineering at Keio University. From 1993 to 1995 he was a Special Researcher of Fellowships of the Japan Society for the Promotion of Science for Japanese Junior Scientists. From 1995 to 2005 he was with Tokyo University of Science. From 1998 to 1999 he was with the department of electrical engineering and computer sciences, University of California, Berkeley. He is now an Associate Professor at Keio University. He is engaged in research on wireless communications, optical communications, signal processing, and information theory. Dr. Ohtsuki is a recipient of the 1997 Inoue Research Award for Young Scientist, the 1997 Hiroshi Ando Memorial Young Engineering Award, Erricson Young Scientist Award 2000, 2002 Funai Information and Science Award for Young Scientist, and IEEE the 1st Asia-Pacific Young Researcher Award 2001. He is a senior member of the IEEE and a member of the SITA.
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PAPER
Computational Complexity Reduction of MLD Based on SINR in MIMO Spatial Multiplexing Systems Katsunari HONJO†a) , Student Member and Tomoaki OHTSUKI†† , Member
SUMMARY MIMO spatial multiplexing systems are attracting a lot of attention because of their high spectral effiencies. Maximum Likelihood Detection (MLD) is known to be the optimal signal detection method for MIMO spatial multiplexing systems in terms of bit error rate (BER). However, the main drawback of MLD is its high complexity. In this paper, to reduce the computational complexity of MLD and to attain good BER in MIMO spatial multiplexing systems, we propose the minimum mean square error (MMSE)-MLD that combines MMSE detection and MLD according to the estimated SINR from each transmit antenna. We also propose the ordered successive MMSE detection (OSD)-MLD that combines OSD and MLD according to the estimated SINR from the transmit antennas. Simulation results show that the proposed MMSE-MLD and OSDMLD can attain almost identical BER to that of MLD but with less complexity. key words: MIMO, MLD, SINR
1.
Introduction
Recently, multiple-input multiple-output (MIMO) systems with multiple antennas at both transmitter and receiver that realize a high bit rate data transmission have drawn much attention for high spectral efficiencies [1]–[10]. Various detection schemes in MIMO spatial multiplexing systems have been proposed for MIMO systems, which can be divided roughly into the linear approach or the nonlinear approach. The zero-forcing (ZF) detection and the minimum mean square error (MMSE) detection that employ the linear approach are the most basic signal detection schemes. Meanwhile, Vertical Bell Laboratories Layered Space-Time (V-BLAST) detection [3], [4], which is often called Ordered Successive Detection (OSD), and Maximum Likelihood Detection (MLD) [7] that employ the nonlinear approach have been shown to outperform the detection schemes by the linear approaches. In the V-BLAST detection, each signal is detected by a successive interference cancellation that nulls the interferes by linearly weighting the received signal vector with a zero-forcing nulling vector. However, in the VBLAST detection its performance suffers great degradation due to the error propagation. An adaptation of original VBLAST to the MMSE criterion was presented in [5], [6]. MLD is another nonlinear approach, and generates the best performance in terms of bit error rate (BER). However, the Manuscript received November 8, 2004. Manuscript revised August 24, 2005. † The author is with the Graduate School of Electrical Engineering, Tokyo University of Science, Noda-shi, 278-8510 Japan. †† The author is with the Department of Information and Computer Science, Keio University, Yokohama-shi, 223-8522 Japan. a) E-mail: [email protected] DOI: 10.1093/ietcom/e89–b.3.914
computational complexity of the MLD increases exponentially with a constellation size and the number of transmit antennas. In this paper, to reduce the computational complexity of MLD and to attain good BER in MIMO spatial multiplexing systems, we propose two reduced complexity detection schemes. In the proposed detection schemes, first we estimate the signal to interference plus noise ratio (SINR) for each transmit signal and set the threshold according to the estimated SINR. The threshold of the estimated SINR is determined in advance according to the required performance and the computational complexity. One proposed detection scheme combines the MMSE detection and MLD according to the threshold of the estimated SINRs. The other proposed detection scheme combines OSD and MLD according to the threshold of the estimated SINR. We refer to these detection schemes as MMSE-MLD and OSD-MLD, respectively. We show that the proposed MMSE-MLD and OSD-MLD can attain almost identical performance to that of MLD but with less computational complexity by setting the threshold of the estimated SINR appropriately. We also compare the proposed MMSE-MLD and OSD-MLD and show that when QPSK is used, MMSE-MLD is superior to OSD-MLD, and vice versa when 16QAM is used. The remainder of this paper is organized as follows. In Sect. 2, the MIMO system model is introduced. In Sect. 3, the conventional detection schemes in MIMO spatial multiplexing systems are reviewed. The proposed detection schemes referred to as MMSE-MLD and OSD-MLD are described in Sect. 4. Section 5 shows some simulation results. Finally, the brief conclusion is given in Sect. 6. Superscripts (·)T , (·)H represent transpose and complex conjugate transpose, respectively. 2.
System Model
We consider the MIMO spatial multiplexing system with Nt transmit antennas and Nr (Nt ≤ Nr ) receive antennas shown in Fig. 1. An incoming information bit is partitioned into Nt signals, and each signal is mapped into symbols and sent through a different transmit antenna. Each receive antenna receives the signals from all Nt transmit antennas over rich-scattering wireless channels. Let x = [x1 . . . xNt ]T denote the Nt × 1 transmitted signal vector, then the corresponding Nr × 1 received signal vector r = [r1 . . . rNr ]T is given by
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
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Detect the signal xˆimax according to the ordering. yimax = Gimax · r (1) , xˆimax = Q(yimax )
(5)
where Gimax denotes the imax -th row of the MMSE nulling matrix G in (3). Fig. 1 Model of a MIMO spatial multiplexing system with Nt transmit and Nr receive antennas.
r = Hx + n.
Canceling The detected signal is subtracted from the last modified received vector.
(1)
In (1), n = [n1 . . . nNr ] denotes white Gaussian noise of variance σ2 observed at Nr receive antennas, E[nnH ] = σ2 INr . The average transmit power of each antenna is normalized to unity, E[xxH ] = INt . The Nr × Nt channel transfer matrix H is modeled as an independent complex Gaussian random variable with zero mean and unit variance, and its element hi j is the channel gain from the transmit antenna j to the receive antenna i.
r (2) = r (1) − xˆimax himax .
T
3.
Conventional Detection Schemes
The modified channel matrix is expressed as H (2) = [h1 , . . . , himax −1 , himax +1 , . . . , hNt ].
G = arg min Ex − Gr2 G
SINRi =
(2)
where · denotes the vector norm. We can obtain the solution to the problem, which can be given by G = H H (HH H + σ2 INr )−1 .
y = [y1 , y2 , . . . , yNt ]T = G · r xˆ = Q(y)
(4)
where Q(·) denotes the hard decision operation corresponding to the constellation in use. 3.2 Ordered Successive MMSE Detection (OSD) [6] The OSD algorithm consists of three steps: ordering, nulling, and interference canceling. The basic detection algorithm for OSD is described as follows. (·)(k) (1 ≤ k ≤ Nt ) denotes the detection stage. Initialization At the first detection stage, r (1) and H (1) are set to r and H, respectively. Nulling
(GH)ii 2 E[|xi |]2 Nt
.
(8)
(GH)il 2 E[|xl |2 ] + σ2 (G)i 2
l=1, li
3.3 Maximum Likelihood Detection (MLD) It is known that MLD is an optimal method for signal detection in terms of BER. MLD considers all the possible symbol combinations x, and chooses the symbol combination that minimizes the squared Euclidean distance.
(3)
We multiply the received vector r by the weight matrix G to get the decision statistic y and make the hard decision.
(7)
Ordering When the cancellation of the detected signal is employed, the order in which the components of x are detected is important to the overall system performance. OSD calculates SINRi for each signal, and detects and cancels the component of xˆimax that maximizes SINRi .
3.1 MMSE Detection It is known that compared to MLD, the MMSE detection provides the performance degradation, but much less computational complexity. We generate the weight matrix G that minimizes the error power between the received signals and the desired signals.
(6)
xˆ = arg min r − Hx2 . x
(9)
Assuming a constellation size of M, MLD needs to evaluate all the M Nt symbol combinations. It follows that the computational complexity of MLD increases exponentially with a constellation size and the number of transmit antennas. 4.
Proposed Detection Schemes
In this section, to reduce the computational complexity of MLD, we propose two signal detection schemes, MMSEMLD that combines MMSE detection and MLD according to the estimated SINR, and OSD-MLD that combines OSD and MLD according to the estimated SINR. 4.1 MMSE-MLD Figure 2 shows the flow chart of the signal detection process of the proposed MMSE-MLD. First, SINR is estimated at the receiver for each signal from (8). A set of indexes of the signals with which SINRs exceed the threshold ζ is set to
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Fig. 2
A detection process of the MMSE-MLD.
A and a set of indexes of the signals with which SINRs are smaller than the threshold ζ is set to B. If the signals with which SINRs exceed the threshold ζ exist, these signals in A are detected by the MMSE detection. yA = GA · r, xˆA = Q(yA )
(10)
where GA denotes all the rows contained in A of the MMSE weight matrix G. The threshold ζ is determined in advance according to the required performance and the computational complexity. Afterwards, the detected signals xˆ A are subtracted from the received signal vector r as the interference, resulting in the modified received signal vector r . r = r − xˆ A HA
(11)
where HA denotes all the columns contained in A of H. The modified channel matrix H is represented by H = HA
(12)
where HA denotes the matrix that removes all the columns contained in A of H. The signals included in B are detected by MLD using the modified received signal vector r and the modified channel matrix H . When the signals with which SINRs exceeding the threshold ζ do not exist, all the signals are detected by MLD. For all the signals, MMSEMLD does not always need to perform MLD. Therefore, the proposed MMSE-MLD can reduce the computational complexity compared to MLD.
Fig. 3
A detection process of the OSD-MLD.
4.2 OSD-MLD Figure 3 shows the flow chart of the signal detection process of the proposed OSD-MLD. First SINR is estimated at the receiver for each signal. If the signals with which the SINRs exceed the threshold ζ exist, only a signal xˆimax with which SINR is maximum is detected by the MMSE detection scheme. (k) (k) y(k) imax = Gimax · r , (k) xˆ(k) imax = Q(yimax )
(13)
where (·)(k) denotes the detection stage. Afterwards, the detected signal xˆ(k) imax is subtracted from the received signal vector r (k) as the interference, resulting in the modified received vector r (k+1) . (k) r (k+1) = r (k) − xˆ(k) imax himax
(14)
where h(k) imax denotes the imax th column of H. The modified channel matrix H (k+1) is represented by H (k+1) = H (k) . imax
(15)
where H (k) denotes the matrix that removes the imax th imax
columns of H (k) . Then SINRs are computed again to the signals other than the detected signals. Note that SINRs are improved as an interference is reduced. Similarly, the same
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procedure is repeated until the signals with which SINRs exceed the threshold ζ do not exist. The signals with which SINRs smaller than the threshold ζ are detected by MLD using the modified received signal vector r (k+1) and the modified channel matrix H (k+1) . For all the signals, since OSDMLD does not always need to perform MLD, it can reduce the computational complexity compared to MLD. 5.
proposed detection schemes are almost identical to that of MLD as the optimum threshold. The optimum threshold ζ is 13 dB in Fig. 4 and 16 dB in Fig. 5. Figures 6, 7 also show a threshold ζ versus BER for QPSK and 16QAM when SNR per receive antenna is 22 dB. The optimum threshold
Simulation Results
In our computer simulations, the conventional detection schemes, that is, the MMSE detection, OSD and MLD, and the proposed detection schemes, that is, MMSE-MLD and OSD-MLD, are compared. We assume a MIMO spatial multiplexing system with Nt = Nr = 4 using QPSK and 16QAM, quasi static flat Rayleigh fading channels, and the perfect knowledge of the channel responses at the receiver. 5.1 Bit Error Rate (BER) Fig. 6
BER of each detection scheme. (QPSK, SNR = 22 dB)
Figures 4, 5 show a threshold ζ versus BER for QPSK and 16QAM when SNR per receive antenna is 20 dB. We observe that the BERs of the proposed MMSE-MLD and OSDMLD approach that of MLD as the threshold ζ becomes higher. We refer to the threshold ζ that the BERs of the
Fig. 7
Fig. 4
BER of each detection scheme. (QPSK, SNR = 20 dB)
Table 1 (QPSK)
BER of each detection scheme. (16QAM, SNR = 22 dB) Optimum thresholds ζ in the proposed detection schemes. SNR
MMSE-MLD
OSD-MLD
15 dB
11 dB
11 dB
18 dB
12 dB
12 dB
20 dB
13 dB
13 dB
22 dB
13 dB
13 dB
Table 2 Optimum thresholds ζ in the proposed detection schemes. (16QAM)
Fig. 5
BER of each detection scheme. (16QAM, SNR = 20 dB)
SNR
MMSE-MLD
OSD-MLD
18 dB
15 dB
15 dB
20 dB
16 dB
16 dB
22 dB
17 dB
17 dB
25 dB
18 dB
18 dB
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ζ is 13 dB in Fig. 6 and 17 dB in Fig. 7. From these figures, we can see that in QPSK the optimum threshold ζ does not change between SNR = 20 dB and 22 dB. On the other hand, when SNR is 22 dB, the optimum threshold is higher than that when SNR is 20 dB in 16QAM. Our proposed detection schemes have the different optimum thresholds for different SNR values. Tables 1, 2 show the optimum thresholds ζ for the proposed detection schemes in QPSK and 16QAM. We observe that the optimum threshold becomes higher as the increase of SNR in QPSK and 16QAM. This is because when SNR changes, SINR also changes. 5.2 Computational Complexity We consider the computational complexities of the proposed MMSE-MLD and OSD-MLD. Note that the computational complexities greatly depend on the number of signals with which MLD is performed. We consider only the complex multiplication operation to evaluate the computational complexity of the detection schemes as same as [9]. Although there are a few evaluation methods of computational complexity for MLD [10], we evaluate the computational complexity supposing the normal MLD [9] in this paper. Table 3 denotes the number of multiplications for the conventional detection schemes and estimation of SINR. Note that we assume Nt = 4 in OSD. In the proposed detection schemes, when m signals are detected by MMSE/OSD and n signals are detected by MLD, we denote the detection by (m, n) Table 3 The numbers of multiplications for the conventional detection schemes and estimation of SINR. M Nt {Nr (Nt + 1)}
MLD
Nr3 + Nr Nt (2Nr + 1)
MMSE OSD
4Nr3
where m + n = Nt . When i SINRs for the signals are calculated and i signals are detected by MLD, MMSE detection, and OSD, we denote the numbers of multiplications by SINR(i), MLD(i), MMSE(i), and OSD(i), respectively. Tables 4 and 5 show the number of multiplications for each detection (m, n) in MMSE-MLD and OSD-MLD. As a result, the total number of multiplications for each detection (m, n) in the proposed detection schemes for QPSK and 16QAM are given by Tables 6 and 7, respectively. The total number of multiplications for MLD is 5120 for QPSK modulation and 1310720 for 16QAM. Note that the reason why the number of multiplications of detection (0, 4) in the proposed detection schemes is higher than that of MLD is because we calculate the SINRs for all the signals. From these Tables, we observe that as the number of signals detected by MLD becomes larger, the total number of multiplications of the proposed detection schemes also becomes larger. Figures 8–11 show a threshold ζ versus the probability that the signals are detected with the detection (m, n) in the proposed detection schemes in [QPSK|16QAM]. From these figures, we observe that compared to MLD, the proposed detection schemes, MMSE-MLD and OSD-MLD, can reduce the probability of detecting all the signals by MLD at the optimum thresholds. Next we compare Fig. 8 and Fig. 10 to Fig. 9 and Fig. 11, respectively. Although in MMSE-MLD and OSDMLD, the probabilities that the signals are detected with (0, 4) are the same, the probabilities for the detections (1, 3), (2, 2) in OSD-MLD are lower than those in MMSE-MLD, respectively. This is because in OSD-MLD SINRs are improved by reducing the interference. Figures 12, 13 show a threshold ζ versus the number of multiplications normalized by the total number of multiplications of MLD for the proposed detection schemes with
+ Nr Nt (2Nr + Nt + 1) + Nt2 + 8Nr
+(Nt − 1){2Nr2 + (Nr + 1)(Nt − 1) + Nr }
Table 6 The number of multiplications for each detection (m, n) in the proposed detection schemes for QPSK.
+(Nt − 2){2Nr2 + (Nr + 1)(Nt − 2) + Nr } +(Nt − 3){2Nr2 + (Nr + 1)(Nt − 3) + Nr } SINR
(0,4)
Nr3 + Nr Nt (2Nr + Nt + 1) + Nt2
Table 4 The number of multiplications for each detection (m, n) in MMSE-MLD. (0,4) SINR(4) + MLD(4) (1,3) SINR(4) + MMSE(1) + MLD(3) (2,2) SINR(4) + MMSE(2) + MLD(2) (3,1) SINR(4) + MMSE(3) + MLD(1) (4,0) SINR(4) + MMSE(4)
Table 5 (0,4)
(1,3)
(2,2)
(3,1)
(4,0)
MMSE-MLD
5408
1320
496
344
304
OSD-MLD
5408
1537
869
822
794
Table 7 The number of multiplications for each detection (m, n) in the proposed detection schemes for 16QAM. MMSE-MLD OSD-MLD
(0,4) 1311008 1311008
(1,3) 65832 66049
The number of multiplications for each detection (m, n) in OSD-MLD. SINR(4) + MLD(4)
(1,3)
SINR(4) + OSD(1) + SINR(3) + MLD(3)
(2,2)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + MLD(2)
(3,1)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + OSD(1) + SINR(1) + MLD(1)
(4,0)
SINR(4) + OSD(1) + SINR(3) + OSD(1) + SINR(2) + OSD(1) + SINR(1) + OSD(1)
(2,2) 3376 3749
(3,1) 440 918
(4,0) 304 794
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Fig. 8 A threshold versus the probability that signals are detected with (m, n) in MMSE-MLD. (SNR = 20 dB)
Fig. 9 A threshold versus the probability that signals are detected with (m, n) in OSD-MLD. (SNR = 20 dB)
Fig. 10 A threshold versus the probability that signals are detected with (m, n) in MMSE-MLD. (SNR = 22 dB)
QPSK and 16QAM when SNR = 20 and 22 dB, respectively. From Fig. 12, when SNR is 20 dB, the normalized numbers of multiplications of MMSE-MLD and OSD-MLD are about 27.5 and 30.4 % compared to the number of the multiplications of the MLD at the optimum thresholds, respectively. Furthermore, when SNR is 22 dB, the normalized numbers of multiplications of MMSE-MLD and OSDMLD are about 18.8 and 23.4 % at the optimum thresh-
Fig. 11 A threshold versus the probability that signals are detected with (m, n) in OSD-MLD. (SNR = 22 dB)
Fig. 12 The number of multiplications normalized by the number of multiplications in MLD in the proposed detection schemes. (QPSK, SNR = 20, 22 dB)
Fig. 13 The number of multiplications normalized by the number of multiplications in MLD in the proposed detection schemes. (16QAM, SNR = 20, 22 dB)
olds, respectively. On the other hand, From Fig. 13, when SNR is 20 dB, the normalized numbers of multiplications of MMSE-MLD and OSD-MLD are about 46.2 and 45.7 % compared to the number of the multiplications of the MLD at the optimum thresholds, respectively. Furthermore, when SNR is 22 dB, the normalized numbers of multiplications
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MMSE-MLD
OSD-MLD
15 dB
0.479
0.491
18 dB
0.321
0.343
20 dB
0.275
0.304
22 dB
0.188
0.234
Table 9 The number of multiplications normalized by the number of multiplications in MLD for the optimum thresholds ζ in the proposed detection schemes. (16QAM) SNR 18 dB 20 dB 22 dB 25 dB
MMSE-MLD 0.573 0.462 0.368 0.208
OSD-MLD 0.569 0.457 0.361 0.207
of MMSE-MLD and OSD-MLD are about 36.8 and 36.1 % at the optimum thresholds, respectively. Consequently, the proposed MMSE-MLD and OSD-MLD can achieve almost the identical BER to that of MLD but with less complexity. The number of multiplications of the proposed detection schemes for the SNR of 22 dB is smaller than that of the proposed detection schemes for the SNR of 20 dB at the optimum thresholds. This is because the probabilities for the detections (0,4) and (1,3) are lower than those of the detections (0,4) and (1,3) compared to 20 dB. Tables 8, 9 show the number of multiplications normalized by the number of multiplications in MLD for the optimized threshold ζ in the proposed detection schemes at different SNR values. From these Tables, we also observe that the proposed detection schemes can reduce the number of multiplications more as SNR becomes higher. When QPSK is used, we compare the number of multiplications of MMSE-MLD to that of OSD-MLD. From Table 8, we can see that MMSE-MLD needs smaller number of multiplications than OSD-MLD at the optimum threshold. Therefore, MMSE-MLD is superior to OSD-MLD. Furthermore, when 16QAM is used, we compare the number of multiplications of MMSE-MLD to that of OSD-MLD. As shown in Table 7, the proposed detection schemes need more multiplications for the detection (0, 4) than the other detections, that is, (3, 1), (2, 2), (1, 3) and (4, 0). In addition in the proposed detection schemes the probabilities that the signals are detected with (0, 4) are the same. Therefore, the number of multiplications of MMSEMLD and OSD-MLD are almost equal, as shown in Table 9. On the other hand, the BER of OSD-MLD is always superior to that of MMSE-MLD in Figs. 5, 7. Since the number of multiplications of MMSE-MLD and OSD-MLD are almost equal and the BER of OSD-MLD is superior to that of MMSE-MLD, we can say that OSD-MLD is superior to MMSE-MLD. Figures 14 and 15 show the SNR versus BER for QPSK and 16QAM, respectively. In Fig. 14, the thresholds ζ of
Fig. 14
Fig. 15
SNR versus BER. (QPSK)
SNR versus BER. (16QAM)
11 and 13 dB are optimum for the target BER of 10−3 and 10−5 , respectively. Similarly, in Fig. 15 the thresholds ζ of 15 and 17 dB are optimum for the target BER of 10−2 and 10−3 , respectively. From Figs. 14 and 15, we can see that the proposed detection schemes can attain almost identical BER to that of MLD with less complexity when the SNR is smaller than the required SNR for the target BER. However, the BERs of the proposed detection schemes are worse than that of MLD when the SNR is higher than the required SNR. This is because the BER of MMSE/OSD cannot be better than that of MLD. In addition the optimum threshold becomes higher as the required SNR becomes higher. Moreover, we can set a threshold ζ arbitrarily in the proposed detection schemes. If we think that BER is more important than the computational complexity, we set a threshold ζ high. Conversely, if we think that the computational complexity is more important than BER, we set a threshold ζ low. In this way we can obtain the desired BER and the desired computational complexity easily. Note that our proposed detection schemes can achieve almost identical BER to that of MLD with less complexity. Thus, the proposed detection schemes are useful for a reduction of power consumption at the receiver.
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6.
Conclusion
In this paper, we propose the reduced complexity signal detection schemes, the minimum mean square error (MMSE)Maximum Likelihood Detection (MLD) and the ordered successive MMSE detection (OSD)-MLD, that combine MMSE/OSD and MLD according to the estimated SINR. We showed that the proposed MMSE-MLD and OSD-MLD can attain almost identical BER to that of MLD but with less complexity. We also showed that when QPSK is used, MMSE-MLD is superior to OSD-MLD, and vice versa when 16QAM is used. References [1] G.J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Tech. J., vol.1, no.2, pp.41–59, Autumn 1996. [2] G.J. Foschini and M.J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol.6, no.3, pp.315–335, 1998. [3] 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,” Proc. IEEE ISSSE 1998, pp.295–300, Sept. 1998. [4] G.J. Foschini, G.D. Golden, R.A. Valenzuela, and P.W. Wolnianski, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,” IEEE J. Sel. Areas Commun., vol.17, no.11, pp.1841–1852, Nov. 1999. [5] M. Debbah, B. Muquet, M. Courville, M, Muck, S. Simoens, and P. Loubaton, “A MMSE successive interference cancellation scheme for a new adjustable hybrid spread OFDM system,” Proc. IEEE VTC’00 Spring, vol.2, pp.745–749, Tokyo, May 2000. [6] A. Benjebbour, H. Murata, and S. Yoshida, “Comparison of ordered successive receivers for space-time transmission,” Proc. IEEE VTC’01 Fall, vol.4, pp.2053–2057, Atlantic City, USA, Sept. 2001. [7] R. Van Nee, A.V. Zelst, and G. Awater, “Maximum likelihood decoding in a space division multiplexing system,” Proc. IEEE VTC’00 Spring, vol.2, pp.6–10, Tokyo, May 2000. [8] X. Li, H.C. Huang, A. Lozano, and G.J. Foschini, “Reducedcomplexity detection algorithms for system using multi-element arrays,” Proc. IEEE GLOBECOM’00, pp.1072–1076, SF, CA, Nov. 2000. [9] H. Sung, K.B. Lee, and J.W. Kang, “A simplified maximum likelihood detection scheme for MIMO systems,” Proc. IEEE VTC’03 Fall, vol.1, pp.419–423, Orlando, USA, Oct. 2003. [10] A. Beck, E.C. Burg, A.P. Djuknic, G.M. Gvoth, T.G. Haessig, D. Manji, S. Milbrodt, M.A. Rupp, M. Samardzija, D. Siegel, A.B. Sizer, T. Sizer, II, C. Walker, S. Wilkus, S.A. Wolniansky, and W. Peter, “Prototype experience for MIMO BLAST over thirdgeneration wireless system,” IEEE J., Sel. Areas Commun., vol.21, no.3, pp.440–451, April 2003.
Katsunari Honjo was born in Chiba, Japan in 1980. He received the B.S. degree in Electrical Engineering from Tokyo University of Science, Chiba, Japan in 2004. He is currently working towards M.E. degree in Graduate School of Electrical Engineering, Tokyo University of Science, Chiba, Japan. His current research interest includes deta detection in MIMO systems. He is a student member of IEEE.
Tomoaki Ohtsuki received the B.E., M.E., and Ph.D. degrees in Electrical Engineering from Keio University, Yokohama, Japan in 1990, 1992, and 1994, respectively. From 1994 to 1995 he was a Post Doctoral Fellow and a Visiting Researcher in Electrical Engineering at Keio University. From 1993 to 1995 he was a Special Researcher of Fellowships of the Japan Society for the Promotion of Science for Japanese Junior Scientists. From 1995 to 2005 he was with Tokyo University of Science. From 1998 to 1999 he was with the department of electrical engineering and computer sciences, University of California, Berkeley. He is now an Associate Professor at Keio University. He is engaged in research on wireless communications, optical communications, signal processing, and information theory. Dr. Ohtsuki is a recipient of the 1997 Inoue Research Award for Young Scientist, the 1997 Hiroshi Ando Memorial Young Engineering Award, Erricson Young Scientist Award 2000, 2002 Funai Information and Science Award for Young Scientist, and IEEE the 1st Asia-Pacific Young Researcher Award 2001. He is a senior member of the IEEE and a member of the SITA.
