Approximate ML Detection Based on MMSE for MIMO Systems

Transcription

1 PIERS ONLINE, VOL. 3, NO. 4, Approximate ML Detection Based on MMSE for MIMO Systems Fan Wang 1, Yong Xiong 2, and Xiumei Yang 2 1 The Electromagnetics Academy at Zhejiang University, Zhejiang University, China 2 Shanghai Research Center for Wireless Communications, China Abstract We derive two types of approximate maximum likelihood (ML) detection based on minimum mean squared error (MMSE), MMSE-CML (conditional ML) detection and MMSE- CLML (conditional local ML) detection, for MIMO communication system. A simple reliability judge rule to judge the estimate of the transmit symbols is also given. For the proposed MMSE- CML detection, received signals are first sent into MMSE detector to do linear equalization, then the estimate of transmit signals is judged in reliability judge module; If the estimate is judged to be reliable, we take the estimate as the final result; if not, the received signals are then sent into conditional ML (CML) detector to get the final result; Unlike conventional ML detector, the CML detector performs a tree search till the estimate satisfies the reliability judge rule or an entire tree search has been done. For the proposed MMSE-CLML detection, we use CLML search instead of CML search in MMSE-CML, which searches in the neighborhood of the output provided by the MMSE detector. Simulation results show that the MMSE-CML detector achieves near the same performance as optimal CML detector at reduced complexity, and MMSE-CLML detector achieves suboptimal performance at remarkably reduced complexity. DOI: /PIERS INTRODUCTION Multiple input multiple output (MIMO) systems have attracted much attention because of high spectrum efficiency [1, 2]. Many different detection techniques are developed to get the diversity gain introduced by MIMO techniques. The ML detector is able to provide optimal performance, but has a disadvantage of extremely high computational complexity. Linear detectors for BLAST systems are ZF detector and minimum mean square error (MMSE) detector, which are low in complexity and poor in performance. Ordered successive interference cancellation (OSIC) detector, proposed in [3 5], which detects the transmit symbols one by one according to the post-detection SNR and does successive interference cancellation, can achieve better performance at relatively high complexity. However, there is still a big performance gap between these detectors and ML detector. Various advanced techniques have been studied to approach the performance of the ML detector, such as sphere decoding (SD) [6], m-algorithm [7] and probabilistic data association (PDA) [8]. These techniques can approach the ML performance with higher complexity than linear detectors. There are some works on combining linear detectors and ML detector [9, 10], to achieve a tradeoff between the BER performance and computational complexity. In their works, a local ML (LML) search is performed in the neighborhood of the output of zero-forcing (ZF)/MMSE detector. In this contribution we develop a novel reliability judge rule (RJR) to judge whether the output of MMSE detector and the symbol vector of the tree search are true or not. With this RJR, we can estimate part of transmit symbols using MMSE detection or conditional ML search instead of ML detection with negligible performance loss, resulting in a reduced complexity. We also propose a MMSE-CLML method using a CLML search instead of ML search, which possesses the advantage of having further reduced complexity with some performance loss to the ML detector. The rest of the paper is organized as follows. In Section 2, we give the system model. The reliability judge rule (RJR) is presented in Section 3. Two types of MMSE-based approximate ML detectors are proposed in Section 4. In Section 5 we present some simulation results, and finally conclude the paper in Section SYSTEM MODEL Consider a MIMO system with N t transmit antennas and N r receive antennas (N r N t ). The transmit vector can be denoted as s = [s 1,, s Nt ] T, where the superscript T stands for the transpose, s i is the complex signal transmitted by antenna i, with a modulation type as BPSK,

2 PIERS ONLINE, VOL. 3, NO. 4, QPSK, 16QAM, etc., and the constellation of modulation type is denoted by with a size of M c. And y = [y 1,, y Nr ] T denotes the received vector, y = [y 1,, y Nr ] T, where y i is the received signal of antenna i. We have: y = Hs + n (1) where H is the channel matrix of dimension N r N t with the element h ij representing the channel between transmit antenna j and receive antenna i, and n is a N r 1 noise vector with each entry is a complex Gaussian noise with zero mean and a variance of σ RELIABILITY JUDGE RULE (RJR) The aim of using RJR is to judge whether the estimate of transmit symbols is reliable or not. Denoting the estimate of transmit symbols r as s, we consider the following value: α = y H s 2 /σ 2 (2) If s is the right estimate of transmit symbols, then α has a chi square distribution with N r degrees of freedom and N r mean. The PDF of α is given by: 1 f χ2 (N r)(x) = 2 Nr 2 Γ (N r /2) e x Nr 2 x 1 2, x > 0 (3) 0, x 0 If s is the wrong estimate of transmit symbols, then α has a noncentral chi square distribution with N r degrees of freedom, noncentral parameter of γ, and N t + γ mean, where γ = H( s s) 2 /σ 2 (4) And the PDF of αis given by: ( γ ) k 2 f χ2 (N r,γ)(x) = k=0 k!γ ( x Nr +k 1 2 γ+x N r 2 + k) e 2 Nr 2, x > 0 +k 2 0, x 0 (5) Now we can present an ideal RJR, which can be denoted as f χ2 (N t)(α) > max(f χ2 (N r,γ)(α)) (6) If the probability density of chi square distribution is higher than that of all noncentral chi square distributions, we judge that the estimated symbols are reliable; If not, we judge that the detector should do further detection to get the final result. Figure 1 and Fig. 2 are the PDF curves of chi square distribution (CSD) and noncentral chi square distributions (NCSD) with different γ. Denote the abscissa of intersection point of f χ2 (N r)(x) and f χ2 (N r,γ)(x) as x(n r, γ), we have: { fχ2 (N r)(x) f χ2 (N r,γ)(x) 0 < x x(n r, γ) (7) f χ2 (N r)(x) < f χ2 (N r,γ)(x) x > x(n r, γ) And if N r is fixed, we have Combining (4), (6), (7), (8), we can get the ideal RJR as: Now we can conclude ideal RJR as 1) Compute all the values of γ using (4), and get min(γ). x(n r, γ 1 ) > x(n r, γ 2 ) if γ 1 > γ 2 (8) α < x(n r, min(γ)) (9) 2) Compute x(n r, min(γ)) subject to f χ2 (N r)(x) = f χ2 (N r,min(γ))(x)

3 PIERS ONLINE, VOL. 3, NO. 4, Figure 1: PDF of chi square distribution and noncentral chi square distribution with N r = 4 degrees of freedom. Figure 2: PDF of chi square distribution and noncentral chi square distribution with N r = 6 degrees of freedom. 3) Compute α using (2). If α < x(n r, min(γ)), output the estimate of transmit symbols as the final result; if not, do further detection to get the final result. As we can see, the progress of computing min(γ) is very complex. To get all the values of γ needs to consider all possible transmit symbols, with the same complexity as ML decoding. When the channel is slow fading, the ideal RJR is feasible because we only need to compute min(γ) for several consecutive symbol periods. But when the channel is fast fading, the ideal RJR is complexity prohibitive. From Fig. 1 and Fig. 2, we can find that Combing (2), (9), (10), we can get a simplified RJR as follows. γ > 0, N r x (N r, γ) (10) y H s 2 /σ 2 < N r (11) Since σ 2 is known at the receiver, we can rewrite the simplified RJR as follows. y H s 2 < N r σ 2 (12) As we can see, the computation cost of simplified RJR will be very low. 4. MMSE-CML AND MMSE-CLML 4.1. MMSE-ML An MMSE-based conditional ML detector with reliability judge rule (RJR) is proposed for MIMO BLAST systems. The structure of MMSE-CML detector is illustrated in Fig. 3. The decoding procedure of MMSE-CML can be divided into 3 steps: Received Signals MMSE Detector Reliability Judge Module Estimated symbols EN conditional ML Detector Figure 3: Structure of MMSE-CML detector.

4 PIERS ONLINE, VOL. 3, NO. 4, Step 1. Get the estimate of transmit symbols in MMSE detector. Put received signals into MMSE detector. The MMSE detector gets the estimate of transmit symbols using MMSE rule and hard decisions as follows: x MMSE = (H H H + σ 2 I Nt ) 1 H H y x = Q( x MMSE ) (13) where Q(.) stands for hard decision operation. Step 2. Judge the estimate reliable or not. Judge x in reliability judge module using (12). If (12) is satisfied, output the estimate as the final result; if not, go step 3. Step 3. Do a conditional ML search. For a possible transmit symbol vector x i X, where X is the set of all possible transmit symbol vector, calculate the squared distance d i from the received vector y as follows d i = y Hx i 2 (14) if (12) is satisfied, that is, d i < N r σ 2, output x i as the final result; else if the entire tree has been searched, choose the lowest squared distance and output the corresponding x as the final result; else, continue the tree search MMSE-LML Although the MMSE-ML detector can achieve near the ML performance, which will be further proved later by the simulation results, it has a drawback of still possibly doing an exhaustive tree search. In order to further reduce the computational complexity, we consider the LML technique based on MMSE detection, which is characterized by a fixed complexity, and enables performancecomplexity tradeoffs between the MMSE and ML detector [10]. Our MMSE-CLML detector is similar to MMSE-CML detector, with a difference in Step 3. Here we do a conditional local ML search by exploring only the neighborhood of the MMSE detector s output x, which is a subset of X. We define the neighborhood of x as the set X near ( x, P ) = {x X d H (x, x) P } (15) where d H (x, x) denotes the hamming distance between x and x, and P is a integer from 0 to N t log 2 M. As we can easily find, if we set P to N t log 2 M, CLML detector does a conditional ML search using RJR, which is similar to the MMSE-CML method; and if we set P to a lower value, CLML detector does a conditional local tree search, which will further reduce the complexity. 5. SIMULATION RESULTS In this section, we compare the BER performance of the MMSE-CML detector, MMSE-CLML detector and ML detector. We further compare the computation complexity of these detectors. In our simulations, we set N t = N r = 4 and consider a QPSK modulation, no channel coding and ergodic Rayleigh fading channels. And we suppose the channel condition is perfectly known at the receiver. Figure 4 shows the performances of MMSE detector, MMSE-CML detector, MMSE-CLML detector with P = 1, 2, 4, and ML detector. We can see that MMSE-CML detector achieves near the same performance as ML detector. The MMSE-CLML detector approaches the ML performance when P increases. In this case, The MMSE-CLML detector with P = 4 approaches the ML performance in the low SNR region. When BER = 10 2, the MMSE-CLML (P = 2) detector provides about 6 db gain with respect to the MMSE detector, and about 1 db loss to the MMSE- CLML (P = 4) detector. Figure 5 presents the percentage of running the CML detector in MMSE-CML detector at different SNR levels. We can find that we can use MMSE detector, instead of ML detector, to get reliable estimate of transmit symbols at a percent of about or larger than 50%. Figure 6 gives the average number of symbol vectors visited by various detectors. As we know, the computation cost of MMSE is much lower than ML detector. So we use the average number of symbol vectors visited to compare the computation complexity of various detectors. Since the complexity of MMSE-CML detector is about half or lower to ML detector, we think the MMSE- CML is more efficient than ML detector. We also find that the MMSE-CLML detectors with

5 PIERS ONLINE, VOL. 3, NO. 4, QPSK 4*4 MIMO ergodic rayleigh fading channel BER MMSE MMSE-CLML(P=1) MMSE-CLML(P=2) MMSE-CLML(P=4) MMSE-CML ML SNR(dB) Figure 4: BER performance of various detectors. Figure 5: Percentage of running the CML detector in MMSE-CML detector. Figure 6: Average number of symbol vectors visited. P = 1, 2, 4 work at lower complexity. Considering that the MMSE-CLML (P = 4) detector can approach the ML performance with about only 25% complexity of ML detector in low SNR region, we think our MMSE-CLML detector is more efficient in this case. 6. CONCLUSIONS We have proposed two MMSE-based approximate ML detection schemes for MIMO BLAST systems together with a simplified reliability judge rule. We have shown that using the simple RJR we can estimate part of transmit symbols using MMSE method instead of ML method, with negligible performance loss, which effectively reduces the computation complexity. In the MMSE-CML detection, we do a conditional ML search when the output of MMSE detection is judged to be not reliable, which is replaced by a conditional local ML search in the MMSE-CLML detection. Simulation results demonstrate that the MMSE-CML detector can achieve near the same performance as ML detector at reduced complexity, and MMSE-CLML detector is more efficient in low SNR region at further reduced complexity. ACKNOWLEDGMENT This work was supported in part by National Hi-Tech Research and Development Program of China (National 863 Program) under Grant 2004AA

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