A Thesis for the Degree of Master. An Improved LLR Computation Algorithm for QRM-MLD in Coded MIMO Systems

Transcription

1 A Thesis for the Degree of Master An Improved LLR Computation Algorithm for QRM-MLD in Coded MIMO Systems Wonjae Shin School of Engineering Information and Communications University 2007

2 An Improved LLR Computation Algorithm for QRM-MLD in Coded MIMO Systems

3 An Improved LLR Computation Algorithm for QRM-MLD in Coded MIMO Systems Advisor : Professor Hyuncheol Park by Wonjae Shin School of Engineering Information and Communications University A thesis submitted to the faculty of Information and Communications University in partial fulfillment of the requirements for the degree of Master of Science in the School of Engineering Taejon, Korea June Approved by (signed) Professor Hyuncheol Park Major Advisor

4 An Improved LLR Computation Algorithm for QRM-MLD in Coded MIMO Systems Wonjae Shin We certify that this work has passed the scholastic standards required by the Information and Communications University as a thesis for the degree of Master June Approved: Chairman of the Committee Hyuncheol Park, Associate Professor School of Engineering Committee Member Hyuckjae Lee, Professor School of Engineering Committee Member Joonhyuk Kang, Assistant Professor School of Engineering

5 M.S Wonjae Shin An Improved LLR Computation Algorithm for QRM- MLD in Coded MIMO Systems School of Engineering, 2007, 54p. Major Advisor : Prof. Hyuncheol Park. Text in English Abstract For next generation wireless communication systems, multiple-input and multiple-output (MIMO) systems have been receiving a great attention due to the fact that use of multiple transmit and receive antennas dramatically increases the capacity and diversity. To achieve optimal performance for the MIMO detection, the maximum likelihood (ML) detector which minimize joint error probability is necessary. However, the main problem with ML detector is its computational complexity. The complexity of ML detector increases exponentially according to the number of transmit antennas and the size of modulation set. It becomes prohibitive when high order modulation is employed such as 16-QAM, 64-QAM and many of antennas are used. An ordered successive interference cancellation (OSIC) has been considered for implementation aspects. Although an OSIC detection scheme requires less computational complexity than ML detector, it undergos a significant performance degradation due to the error propagation. Several efforts have been focussed on achieving near-ml performance with low computation load recently. Among them, sphere decoding and maximum likelihood detection with QR-decomposition and i

6 M-algorithm (QRM-MLD) are most promising algorithms. Both algorithms are fascinating as they achieve near-ml and ML performance with only a small amount of the computational load. From the average complexity aspect, sphere decoding method has lower computational load than QRM-MLD algorithm. On the contrary, QRM-MLD algorithm has advantage over sphere decoding in implementation because its worst case complexity is much lower than that of sphere decoding. The QRM-MLD algorithm reduces the complexity by selecting M candidates with the smallest accumulated metrics at each level of the tree search. To accomplish near-ml performance for QRM-MLD algorithm, M must be larger than the constellation size. As the number of antennas and the size of modulation set are increased, a larger value of M is required, and it still requires high computational complexity. Accordingly, several approaches adaptively control the number of survival branches according to the determined threshold values with ML performance in uncoded systems, since they prune off only the unnecessary paths to find minimum accumulated metric in tree structure. In case of the QRM-MLD algorithm used in conjunction with softinput decoder, non-existing LLR bits inevitably happen when surviving symbol vectors do not contain both 1 and 0 for every coded bits at the last stage. Consequently, additional computations or an appropriate constant is required to compute non-existing LLR, but the performance is still not acceptable. Performance penalty particulary becomes worse when low order modulation is used, or the number of survival symbol vectors at the last stage is small in the tree structure. In this thesis, we introduce a new approach which computes more accurate LLR values for coded MIMO systems. The proposed method is based on two steps. The first step consists of producing the approximate LLR values at every stage in order to reflect their reliability on the nonexisting LLR bits at the last stage. It follows that the approximate ii

7 LLR values will be updated successively as tree search progress. In the second step, the LLR values which could not be computed at the last stage inherit the recently updated approximate LLR value at the upper stage. As a result, the proposed algorithm gives more accurate LLR values than that of the conventional methods and it leads to improved performance when the low order modulation is employed, or the number of candidate symbol vectors to provide LLR values is small. iii

8 Contents Abstract Contents List of Tables List of Figures List of Abbreviations i iv vi vii ix 1 Introduction Overview of MIMO System Motivation The Outline of Thesis Conventional Detection Schemes Maximum Likelihood Detection Linear Detection Zero-Forcing (ZF) Linear Detector Minimum Mean Square Error (MMSE) Linear Detector Decision Feedback Detection Near-ML Detection Schemes Sphere Decoding QRM-MLD Algorithm iv

9 3 LLR Computation Schemes LLR computation scheme for Maximum Likelihood Detection LLR computation schemes for Reduced Complexity ML Detectors LLR Clipping Euclidian Distance Estimation Schemes for Non- Existing LLR Bits Simulation Results Proposed LLR Computation Scheme Efficient LLR Computation Algorithm for reduced-complexity ML Detector Simulation Results Summary and Conclusion 48 Appendix 50 References 52 Acknowledgement 55 Curriculum Vitae 56 v

10 List of Tables 3.1 LLR Clipping Scheme The Efficient LLR Computation Algorithm vi

11 List of Figures 1.1 Multiple input multiple output (MIMO) channel model Capacity for different (N t N r ) MIMO antenna configurations General detection block diagram Performance of the ML and linear detectors for 4 4 Rayleigh-fading channels with 16-QAM Block diagram of ZF-SIC Block diagram of MMSE-SIC Performance of the ML, linear detectors, and SIC detectors for 4 4 Rayleigh-fading channels with 16-QAM Performance of the ML, SIC detectors, and OSIC detectors for 4 4 Rayleigh-fading channels with 16-QAM Idea behind sphere decoding Tree generated to determine lattice points in four-dimensional sphere Performance of the ML and sphere decoding for 4 4 Rayleigh-fading channels with 16-QAM Tree structure of 3 3 MIMO system with QRM-MLD (M = 4) and QPSK The flowchart of QRM-MLD algorithm Performance of the ML and QRM-MLD algorithm for 4 4 Rayleigh-fading channels with 16-QAM Performance of the sorted QRM-MLD and QRM-MLD algorithm for 4 4 Rayleigh-fading channels with 16- QAM vii

12 3.1 Block diagram of a N t N r Coded MIMO system model Euclidian distance estimation schemes for non-existing LLR bits Squared Euclidian distance-based LLR calculation and Euclidian distance-based calculation Performance of the MLD and QRM-MLD with several LLR computation schemes for 4 4 MIMO channels with 16-QAM An example of QRM-MLD(M = 2) with the proposed algorithm in a 3 3 MIMO system with QPSK The distribution of the LLR values according to different computation methods for QRM-MLD(M = 2) in a 4 4 MIMO system with QPSK when E b /N 0 = 5 [db] The BER performance of QRM-MLD (M = 16, 4) for 4 4 MIMO system with 16-QAM The BER performance of QRM-MLD for 4 4 MIMO system according to the modulation schemes viii

13 List of Abbreviations BER Bit Error Rate CSI Channel State Information DF Decision Feedback ED Euclidian Distance LLR Log Likelihood Ratio MGS Modified Gram Schmidt MIMO Multiple Input Multiple Output ML Maximum Likelihood MMSE Minimum Mean Square Error MSE Mean Squared Error OSIC Ordered Successive Interference Cancellation PDF Probability Density Function QAM Quadrature Amplitude Modulation QPSK Quadrature Phase Shift Keying QRD QR Decomposition QRM-MLD Maximum Likelihood Detection employing QR Decomposition and M-algorithm ix

14 SIC Successive Interference Cancellation SISO Single Input Single Output SNR Signal to Noise Ratio ZF Zero Forcing x

15 1. Introduction 1.1 Overview of MIMO System In this thesis, we consider the MIMO channel model with N t transmitter and N r receiver antennas as shown in Fig In the MIMO channel, an complex input vector s = [ s 1, s 2,, s Nt ] T is sent and an complex output vector r = [ r 1, r 2,, r Nr ] T is received. Input-output relationship of the form can be formulated as follows: r = Hs + n, (1.1) where H is a N r N t channel matrix and n = [ n 1, n 2,, n Nr ] T is the noise vector. Furthermore, we assume that H is a random matrix with independent complex Gaussian elements {h i,j } with zero mean r 1 s 1 H N r N t r 2 s Nt r Nr Figure 1.1: Multiple input multiple output (MIMO) channel model 1

16 and unit variance, denoted h i,j CN (0, 1). We also assume that n is a complex Gaussian random vector with independent and identically distributed (i.i.d.) elements n i CN (0, σ 2 ). It is assumed that H and n are independent of each other and of the input vector s. Further, we assume that the average power of each antennas is normalized to one. Lastly, we assume that the receiver has perfect knowledge of the channel realization H, while the transmitter has no such channel state information (CSI). The shannon capacity of a communication channel is the maximum asymptotically error-free transmission rate supported by the channel. In the followings, we will examine the capacity benefits of MIMO channels with mathematical analysis. The capacity over the Rayleigh flat fading MIMO channel can be expressed as [5] [ ( C = log 2 det I Nr + ρ ) ] HH H, (b/s/hz) (1.2) N t where I Nr is the N r N r identity matrix and ρ is the average signal-tonoise ratio (SNR) at each receive antenna. By the law of large numbers, the term 1 N t HH H I Nr as N t becomes large and N r is fixed. It follows that the capacity in the condition of large N t approximate to C = N r log 2 (1 + ρ). (1.3) Further analysis of the MIMO channel capacity given in (1.2) is possible by diagonalizing the product matrix HH H by eigenvalue decomposition. By using eigenvalue decomposition, the matrix product is written as HH H = EΛE H, (1.4) 2

17 x 4 3 x 3 2 x 2 1 x 1 Channel capacity [b/s/hz] SNR [db] Figure 1.2: Capacity for different (N t N r ) MIMO antenna configurations. where E is the eigenvector matrix with orthogonal columns and Λ is a diagonal matrix with the eigenvalues on the main diagonal. Consequently, the channel capacity over MIMO system given in(1.2) can be expressed as [ ( C = log 2 det I Nr + ρ ) ] EΛE H, (b/s/hz) (1.5) N t which may be converted into C = k i=1 log 2 ( 1 + ρ N t λ i ), (1.6) where k = rank{h} min(n t, N r ) and λ i are the eigenvalues of the 3

18 matrix Λ. It is obvious that the total capacity over a MIMO channel is composed of the summation of parallel single-input single-output (SISO) subchannels with corresponding channel gains λ i (i = 1, 2,, k). The number of parallel subchannels is determined by the rank of the channel matrix. It follows that multiple scalar spatial data pipes (also known as spatial modes) open up between transmitter and receiver resulting in significant performance gains over the SISO case. Fig. 1.2 describes the capacity of several MIMO configurations as a function of SNR. As expected, the capacity increases with enlarging ρ and with N t and N r as well. 1.2 Motivation To support high data rates in the wireless channel environments, latest research on wireless communication systems has focussed on the usage of the multiple antennas at both transmitter and receiver. The information-theoretic capacity of these multiple-input multiple-output (MIMO) channels was shown to grow linearly with the smaller value of the numbers of transmit and receive antennas in rich scattering environments, and at sufficiently high signal-to-noise ratio [1]. On the other hand, the complexity to recover the transmitted signals at the receiver is increased as the multiple antennas at the transmitter and receiver are employed. As the capacity increases linearly with the number of antennas, the complexity of signal detection increases exponentially with the number of transmit antennas and the size of modulation set. As a result, the maximum likelihood (ML) detector requires prohibitively high complexity even for small numbers of antennas. Several detection algorithms have been introduced in order to achieve near-ml performance [2], [3], [4]. Among them, tree search based max- 4

19 imum likelihood detection with QR decomposition and M-algorithm (QRM-MLD) has been receiving a special attention as a favorable detection in that it requires low average complexity as well as low worst-case complexity. Although the complexity of QRM-MLD algorithm is lower than that of ML detector, it still requires high complexity for practical application. Accordingly, several approaches in [13], [14] adaptively control the number of survival branches according to the determined threshold values with ML performance in uncoded systems, since they prune off only the unnecessary paths to find minimum accumulated metric in tree structure. In case of the QRM-MLD algorithm used in conjunction with softinput decoder, non-existing LLR bits inevitably happen when surviving symbol vectors do not contain both 1 and 0 for every coded bits at the last stage. Consequently, some extra computation methods [15], [16], [17] are introduced to compute appropriate LLR values as non-existing LLR bits. The degradation in the soft input decoding performance becomes significant when the LLR values is incorrectly overestimated such a extremely large absolute value. Since it commonly occurs in the reduced-complexity ML detectors, an efficient LLR computation scheme is essential to mitigate the effect of the overestimation of the LLR values. In this thesis, we offer the new LLR computation method to rectify the problem. 1.3 The Outline of Thesis Firstly, we describe several conventional MIMO detection scheme in chapter 2. Next, chapter 3 will deal with the conventional LLR computation methods for reduced complexity ML detectors. In chapter 4, we propose novel LLR computation algorithm which computes more accurate LLR values through updating approximate LLR values at ev- 5

20 ery stage in the tree structure. It also contain the simulation results to demonstrate the superiority of the proposed method compared with conventional scheme. Finally, we provide summary and conclusions in chapter 5. 6

21 2. Conventional Detection Schemes In this chapter, we present the methods to estimate the transmitted vector s from received vector r = Hs + n over MIMO channels. The procedure is illustrated as the general block diagram in Fig In the followings, we assume that the channel knowledge H is perfectly known to the receiver in the absence of channel state information at the transmitter. The problem faced by a receiver is the presence of interference, since the signals launched from the transmit antennas interfere with each other. n s H r Detector ŝ Figure 2.1: General detection block diagram 2.1 Maximum Likelihood Detection To achieve optimal performance, the maximum likelihood (ML) detector is necessary. It is defined from likelihood of observing r given the transmit vector s. If the channel is known to the receiver and the noise is zero-mean Gaussian, the conditional distribution of r given s is defined as 7

22 p(r s) = 1 ( 2π exp ) r Hs 2. (2.1) σ 2 To maximize of the conditional probability in (2.1), the search for the ML decision vector can be formalized succinctly as ŝ = arg min r Hs 2, (2.2) s Ω N t where Ω denotes the modulation set. The ML decision rule needs an exhaustive search over Ω N t candidate symbol combinations, where Ω is the cardinality of Ω. Therefore, the complexity of ML detection is high and even prohibitive when the number of transmit antennas and modulation order are large. 2.2 Linear Detection We can reduce the detection complexity of the ML detection significantly by employing following linear receiver: zero-forcing (ZF) and maximum-mean-square-error (MMSE). Further, we show the similarity of two linear detectors by introducing an extended system model Zero-Forcing (ZF) Linear Detector Among the detection schemes, zero-forcing linear detector is the simplest detection, where the received vector r is pass through a filter matrix W. It follows that the interference between the signals launched from different antennas is eliminated completely regardless of noise enhancement. The filter matrix simply inverts the channel matrix. For the case when the inverse of the channel does not exist, the pseudoinverse of the channel is employed by the Moore-Penrose-inverse as follows: 8

23 W ZF = ( H H H ) 1 H H, (2.3) We assume that H has full column rank. The decision step consists of mapping each element of the filter output vector ŝ ZF = W ZF r = s + ( H H H ) 1 H H n (2.4) onto an element of the symbol alphabet by a slicer. The estimation errors of the different layers correspond to the main diagonal elements of the error covariance matrix Φ ZF = E { (ŝ ZF s)(ŝ ZF s) H } = σ 2( H H H ) 1. (2.5) which equals the covariance matrix of the noise after the receive filter. When the eigenvalues of H H H are small, it would lead to noise enhancement due to the fact that its insistence on forcing the interference to zero, regardless of the interference strength Minimum Mean Square Error (MMSE) Linear Detector To rectify the noise enhancement problem in ZF linear detector, MMSE linear detector minimizes the mean squared error (MSE) between the transmit vector and the output of the linear detector. It provides the following filter matrix [6] W MMSE = ( H H H + σ 2 I Nt ) 1H H. (2.6) The corresponding output of the filter is given by 9

24 ŝ MMSE = W MMSE r = ( H H H + σ 2 I Nt ) 1H H r. (2.7) The main diagonal elements of the error covariance matrix imply the estimation errors of the different layers. Φ MMSE = E { (ŝ MMSE s)(ŝ MMSE s) H } = σ 2( H H H + σ 2 I Nt ) 1. (2.8) We take a (N t + N r ) N t extended channel matrix, and a (N t + N r ) 1 extended receive vector r to demonstrate similarity two linear detectors as follows. H = [ H σi Nt ] and r = [ r 0 Nt,1 ], (2.9) The output of the MMSE filter computed by (2.7) becomes ŝ MMSE = ( H H H ) 1 H H r = H + r. (2.10) Furthermore, the error covariance matrix (2.8) can be reformulated as Φ MMSE = σ 2( H H H ) 1. (2.11) There is only difference that channel matrix H has been replaced by H between (2.10), (2.11) for ZF linear detector and (2.4), (2.5) for MMSE linear detector. ZF detector perfectly separates the co-channel 10

25 signals at the cost of noise enhancement. On the other hand, the MMSE detector can minimize the overall error caused by noise and mutual interference between the co-channel signals Linear ZF Linear MMSE ML Bit Error Rate Eb/No[dB] Figure 2.2: Performance of the ML and linear detectors for 4 4 Rayleigh-fading channels with 16-QAM. The ML detector and two linear detector are evaluated in terms of bit error performance as shown in Fig Note that linear MMSE detector is approximate to ML detector in low SNR, but both linear ZF and MMSE detector show a diversity order of d = 1. On the other hand, ML curve has a diversity order of d = 4 in given MIMO channel environment. 11

26 2.3 Decision Feedback Detection In this section, we describe the zero-forcing decision-feedback (ZF-DF) detector, and MMSE decision-feedback (MMSE-DF) detector. It is also called as successive interference cancellation (SIC). For this decision process, ZF-SIC begins from the QR decomposition of the channel matrix H = QR, which is based on the modified Gram-Schmidt (MGS) method [18]. The N r N t matrix Q is unitary matrix which has orthogonal columns with unit norm. And the N t N t matrix R is upper triangular. By multiplying the receive vector r with Q H, we have y = Rs + n, (2.12) where y is the N t 1 vector and the statistics of the N t 1 noise vector n = Q H n remains unchanged since the mean and variance of n are perfectly equal to these of n as follows. E{Q H n} = E{Q H } E{n} = 0 var(q H n) = {Q H n n H Q} E{Q H n} 2 = E{Q H } (σ 2 I) E{Q} (2.13) = σ 2 E{Q H Q} (2.14) = σ 2 I (2.15) Due to the upper triangular structure of R, the k th element of y is given by y k = R k,k s k + N t i=k+1 R k,i s i + n k (2.16) 12

27 s H n r H 1 Q y Γ sˆ B Figure 2.3: Block diagram of ZF-SIC. and is free of interference from layers 1,, k 1. Therefore, y Nt can be used to estimate ŝ Nt after dividing with R Nt,Nt, since it is totally free from the interference. With assuming correct decisions in previous steps and proceeding successively with ŝ Nt 1,, ŝ 1, the interference can be perfectly cancelled in each step. However, it could lead to serious problem called error propagation when incorrect decision is performed in previous symbol due to the imperfect interference cancellation. We can develop a model of the ZF-SIC in matrix terms as shown in Fig. 2.3, and its forward filter Γ 1 and feedback filter B is given by Γ = diag( R) B = R Γ. (2.17) The purpose of the forward filter is to suppress interference due to the undetected symbols yet, and that of the feedback filter is to suppress interference already detected symbols. In case of MMSE-SIC, QR decomposition of the extended channel matrix H in (2.9) is performed instead of original channel matrix H. There is no difference between ZF-SIC and MMSE-SIC except for uti- 13

28 lizing the extended channel matrix H in MMSE-SIC. ŝ = ( ) 1H H H H + σ 2 I H Nt r. [ ] H [ ] H H = 1 H H r = σi Nt [ H σi Nt σi Nt ] H [ H ] 1[ H σi Nt σi Nt ] H r, r = [ r 0 Nt,1 ] (2.18) = H + r Fig. 2.4 illustrates the block diagram of MMSE-SIC receiver. s H n r y H 1 Q Γ sˆ 0 B Figure 2.4: Block diagram of MMSE-SIC. 14

29 [ Q, R ] = QR decomposition (H) Γ = diag( R) B = R Γ y = Q H r = R s + n (2.19) = Γ s + B s + n Linear ZF Linear MMSE ZF-SIC MMSE-SIC ML Bit Error Rate Eb/No[dB] Figure 2.5: Performance of the ML, linear detectors, and SIC detectors for 4 4 Rayleigh-fading channels with 16-QAM. Fig. 2.5 shows the BER performance with the DF detectors and linear detectors versus the E b /N 0. DF detectors outperform linear detectors 15

30 due to the SNR gain not diversity gain. Up to this point we have assumed that the DF detector detects the symbols in the natural order such as ŝ Nt 1,, ŝ 1. To minimize the error propagation from the incorrect interference cancellation in DF detector, the receiver may detect the symbols in optimal order. Inspired by this motivation, ordered successive interference cancellation (OSIC) method is proposed [8]. For the case of ZF-OSIC method, it is optimal to select the symbol with the largest post-detection SNR. The post-detection SNR for the k th detected component of s is expressed as follows W = ( H H H ) 1 H H SNR k = 1 σ 2 W k 2. (2.20) The following procedure implies the ZF-OSIC detection algorithm from the initialization to the recursive process. 16

31 Initialization : i = 1 W 1 = H + (2.21) k 1 = arg min (W1 ) j 2 j Recursion : w ki = (W i ) ki y ki = w ki r i ŝ ki = Q(y ki ) r i+1 = r i ŝ ki (H) ki (2.22) W i+1 = H + k i k i+1 = arg min i = i + 1 j / {k 1,,k i } (Wi+1 ) j 2 where (W i ) j is the j th row of W i, and Q( ) is a slicer to the nearest symbol alphabet. Also, (H) ki denotes the k i th column of H, and H ki corresponds to the matrix obtained by zeroing the columns k 1, k 2,, k i of H. MMSE-OSIC receiver suppresses not only the interference but also noise components, whereas ZF-OSIC receiver removes only the interference components. In other words, minimizing the mean square error (MSE) between the transmit vector and estimate of the receiver is main criterion to determine the detection order for MMSE-OSIC. The detection algorithm is summarized as follows 17

32 Initialization : i = 1 R ee,1 = σ 2( H H H + σ 2 I Nt ) 1 W 1 = R ee,1 H H (2.23) k 1 = arg min j {diag ( ) } R ee,1 j Recursion : w ki = (W i ) ki y ki = w ki r i ŝ ki = Q(y ki ) r i+1 = r i ŝ ki (H) ki R ee,i+1 = σ 2( H H k i H ki + σ 2 I Nt ) 1 (2.24) W i+1 = R ee,i+1 H H k i { k i+1 = arg min diag ( ) } R ee,i+1 j / {k 1,,k i } j i = i + 1 where (H) ki denotes the k i th column of H, and H ki corresponds to the matrix obtained by zeroing the columns k 1, k 2,, k i of H. In addition, R ee is the error covariance matrix related to MSE. We consider a 4 4 Rayleigh fading channels with 16-QAM, and shows the performance achieved by SIC detectors with optimal ordering compared to SIC detectors. ZF-OSIC receiver outperforms ZF-SIC receiver by about 3.7 db. Beside, MMSE-OSIC receiver has performance gain about 4.1 db compared with MMSE-SIC receiver. Notes that the performance improvement from OSIC detectors comes from diminishing the error propagation by optimal ordering. 18

33 ZF-SIC MMSE-SIC ZF-OSIC MMSE-OSIC ML Bit Error Rate Eb/No[dB] Figure 2.6: Performance of the ML, SIC detectors, and OSIC detectors for 4 4 Rayleigh-fading channels with 16-QAM. 2.4 Near-ML Detection Schemes In this section, we review the signal detection schemes which obtain the near-ml performance while requiring relatively low computational load, which is generally called as near-ml detection. First, we briefly introduce the sphere decoding method. After that, QRM-MLD algorithm is described Sphere Decoding From now on, we observe the sphere decoding method, a depth-first tree search algorithm. We attempt to search over only lattice points s Z m that lie in a certain sphere of radius d around the given vector r, thereby 19

34 d r Figure 2.7: Idea behind sphere decoding. reducing the search space as shown in Fig Obviously, the closest lattice point inside the sphere will also be the closest lattice point for the whole lattice. Although it is difficult to determine the lattice points inside a general m-dimensional sphere, it is trivial to do so in the case of m = 1. The reason is that a one-dimensional sphere reduces to the endpoints of an interval. So, the desired lattice points will be the integer values that lie in this interval. We can use this observation to go from dimension k to dimension k + 1. Suppose that we have determined all k-dimensional lattice points that lie in a sphere of radius d. Then, for any such k-dimensional point, the set of possible values of the (k + 1)th dimensional coordinate that lie in the higher dimensional sphere of the same radius d forms an interval. With this brief introduction, we can now be more specific about the problem. We shall assume that n m. Note that the lattice point Hs lies inside a sphere of radius d centered at r if and only if r Hs 2 d 2. (2.25) It is useful to consider the QR decomposition of the matrix H 20

35 k = 1 k = 2 m = 4 k = 3 k = 4 Figure 2.8: Tree generated to determine lattice points in fourdimensional sphere. H = Q [ R 0 (n m) m ] (2.26) where R is an m m upper triangular matrix and Q = [Q 1 Q 2 ] is an n n orthogonal matrix. The matrices Q 1 and Q 2 represent the first m and last n m orthogonal columns of Q, respectively. Therefore, (2.25) can be written as [ r Q 1 ] [ ] R Q 2 s 0 [ ] [ ] 2 Q T = 1 R r s = Q T 0 1 r Rs 2 + Q T 2 r 2 d 2. 2 Q T 2 In other words, Q T 1 r Rs 2 d 2 Q T 2 r 2. (2.27) 21

36 By defining y = Q T 1 r and d 2 = d 2 Q T 2 r 2, we can rewrite (2.27) as ( ) 2 m m y i R i,j s j d 2. (2.28) i= 1 j= i The right-hand side of the above inequality can be expanded as (y m R m,m s m ) 2 +(y m 1 R m 1,m s m R m 1,m 1 s m 1 ) 2 + d 2 (2.29) where the first term depends only on s m, the second term on {s m, s m 1 }, and so on. Therefore, considering the first term only, a necessary condition for Hs to lie inside the sphere is that (y m R m,m s m ) 2 d 2. (2.30) This condition is equivalent to s m belonging to the interval d + y m d + y m s m R m,m R m,m (2.31) where denotes rounding to the nearest larger elements in the set of numbers that spans the lattice. Similarly, denotes rounding to the nearest smaller elements. For every s m satisfying (2.31), defining d 2 m 1 = d 2 (y m R m,m s m ) 2, (2.32) and y m 1 m = y m 1 R m 1,m s m, (2.33) 22

37 a stronger necessary condition can be found by looking at the first two terms in (2.29), which generates s m 1 belonging to the interval d m 1 + y m 1 m d s m 1 R m 1,m 1 m 1 + y m 1 m R m 1,m 1. (2.34) We can repeat in a similar process for s m 2, and so on until s 1, thereby obtaining all lattice points satisfying (2.25). The squared Euclidean distances between such lattice points and r are given by 10 0 ML Sphere decoding 10-1 Bit Error Rate Eb/No[dB] Figure 2.9: Performance of the ML and sphere decoding for 4 4 Rayleigh-fading channels with 16-QAM. 23

38 d 2 (r, Hs) = m y i i= 1 m 2 R i,j s j j= i (2.35) The algorithm outputs the point ŝ whose distance is minimum. If no point in the sphere is found, the sphere is declared empty and the search fails. In this case, radius d must be increased and the search is restarted with the new radius. Fig. 2.9 illustrates the bit error rate (BER) performance of ML and sphere decoding algorithm. The BER curve of sphere decoding coincides with ML curve in that the closest lattice point inside the sphere will also be the closest lattice point for the whole lattice QRM-MLD Algorithm In this subsection, we introduce the QRM-MLD signal detection algorithm. QRM-MLD signal detection method is breadth-first tree search algorithm whereas sphere decoding method is a depth-first algorithm. It is very promising algorithm in practical applications since it achieves near-ml performance, while requiring relatively low computational load. Sphere decoding has lower complexity than QRM-MLD algorithm with respect to the average complexity. However, QRM-MLD algorithm has advantage over sphere decoding from the hardware implementation perspective because its worst-case complexity is much lower than that of sphere decoding [9]. The channel matrix H is decomposed as H = QR by using the QR decomposition based on the modified Gram-Schmidt (MGS) method [18], where Q is an N r N t unitary matrix and R is an N t N t upper 24

39 triangular matrix. The matrix R is represented as: R Nt,Nt R Nt,Nt 1 R Nt,1 0 R Nt 1,N t 1 R Nt 1,1 R = R 1,1. (2.36) By pre-multiplying the receive vector r by Q H, we have y = Q H r = Rs + n, (2.37) where the statistics of the noise vector n = Q H n remains unchanged. The eq. (2.37) is converted into the tree structure by using the property of the matrix R. Therefore, the ML metric can be expressed as follows. r Hs 2 = y Rs 2 N t i 2 = y i R i,j s j = i=1 N t i=1 j=1 y i 1 2 i R i,i s i R i,j s j. j=1 (2.38) The QRM-MLD algorithm [4], [10] starts from calculating the first branch metrics for all possible s 1. The branch metrics are calculated as: y 1 R 1,1 s 1 2. (2.39) At the first stage, constant M branches with the smallest accumulated metrics are selected as survival paths. At the second stage, each survival path is extended to Ω branches, where Ω is the cardinality of modulation set Ω. Therefore, there are M Ω combinations of s 1 and s 2. Only M paths with the smallest accumulated metrics out of M Ω are selected. This process is iterated until N t tree depth. At the last stage, a path with the minimum accumulated metric is detected as ŝ. 25

40 1st stage 2nd stage 3rd stage ŝ Figure 2.10: Tree structure of 3 3 MIMO system with QRM-MLD (M = 4) and QPSK. As shown in Fig. 2.10, the original QRM-MLD algorithm has fixed complexity regardless of the channel environment due to the constant selection of survival branches at each stage. The complexity is defined by the total number of branch metric calculations. The parameters which determine the complexity of the QRM-MLD algorithm are the number of transmit antennas N t, the number of survival candidates vector at each stage M, and modulation scheme. The flowchart for the QRM-MLD algorithm is illustrated in Fig While the QRM-MLD algorithm maintains constant branches every stage, the adaptive QRM-MLD algorithm [13], [14] selectively controls the number of survival branches according to threshold value calculated at each stage. As a result, it result in reducing the complexity of tree searching in the sense of average complexity. In Fig. 2.12, we present the BER curves of using ML and QRM- MLD algorithm for a 4 transmit, 4 receive antenna MIMO system with 26

41 QRD of channel matrix, H = QR i = N t Premultiply receive vector r H with Q Extend all branches to all constellation points Calculate branch metrics for the extended branches Order the list and retain M branches, discarding the rest i = i 1 No i = 0 Yes Output Figure 2.11: The flowchart of QRM-MLD algorithm. 16-QAM. This implies the effects on the error performance for QRM- MLD as M values are set to 4, 8, 12, and 16 respectively. When M is set to the number of constellation size, 16, QRM-MLD algorithm achieves ML performance approximately. However, the error performance are degraded gradually as M value is decreased due to the diversity loss. To mitigate the performance penalty, we can change the tree structure using the sorted QR decomposition instead of QR decomposition. It make better accuracy of the tree search with small M numbers since the error propagation effect could be diminished as similar to chase 27

42 ML QRD-M (M=16) QRD-M (M=12) QRD-M (M=8) QRD-M (M=4) Bit Error Rate Eb/No[dB] Figure 2.12: Performance of the ML and QRM-MLD algorithm for 4 4 Rayleigh-fading channels with 16-QAM. #! " $ % & ' ( )+*-,./ $ % & ' ( )+*-,./ $ % & ' ( )+*-,./ $ % & ' ( )+*-,./ *-,./ 01 0: *-,./ 01 0:24 6 *-,./ 01 0:27 6 *-,./ 01 0:298 6 Figure 2.13: Performance of the sorted QRM-MLD and QRM-MLD algorithm for 4 4 Rayleigh-fading channels with 16-QAM. 28

43 detector [12]. Consequently, it significantly outperforms the original QRM-MLD algorithm with small M such as 4, 8 by 2 db approximately as shown in Fig In case of QRM-MLD with large M number, the performance advantage is decreased because of sufficient surviving number of candidate vectors in tree structure already. 29

44 3. LLR Computation Schemes In this chapter, we focus on the tree-based MIMO detectors used in conjunction with soft-input decoders. Fig. 3.1 shows the transmitter and receiver structure of the coded MIMO systems considered in this paper. When soft input decoders are employed, soft decision is performed based on the candidate symbol vectors to compute the log likelihood ratio (LLR) values of the coded bits as an input to the decoder. Since the ML detector searches all possible symbol vectors to compute the LLR values, an ideal LLR sequence for the subsequent soft input decoder can be obtained. On the other hand, since the reduced-complexity ML detectors compute LLR values only from a handful of candidate symbol vectors, the resultant LLR sequence is less accurate, degrading the performance of the subsequent decoding. Since this problem occurs in any of the detectors except ML detector, an effective solution is great necessary. Nt H Nr n Nr sn t r Nr Data... Encoder S/P s Detection 2 r 2... n 2! " Estimated Data n 1 s 1 MIMO Channel r 1 Figure 3.1: Block diagram of a N t N r Coded MIMO system model 30

45 3.1 LLR computation scheme for Maximum Likelihood Detection We consider a MIMO system with N t transmitter antennas and N r receiver antennas. For each symbol period, let s = [s Nt, s Nt 1, s 1 ] T denote the N t 1 vector of transmit symbols, then the corresponding N r 1 received signal vector r = [r Nr, r Nr 1, r 1 ] T can be expressed as r = Hs + n, (3.1) where the N r 1 noise vector n represents the complex additive white Gaussian noise with variance N 0 I. The N r N t channel matrix H contains uncorrelated complex Gaussian fading gains. Using maximum likelihood detection(mld), the optimum LLR values can be computed by searching all possible candidate vectors. Due to the definition of the log likelihood ratio(llr), we can express LLR value for kth interleaved coded symbol under given received signal vector r as follows: L(c k r) = log P r(c k = 0 r) P r(c k = 1 r) = log P r(r c k = 0) P r(c k = 0) P r(r c k = 1) P r(c k = 1) s Ω N t exp ( 1 r Hs 2) P r(s) c = log k =0 2σ 2 exp ( 1 r Hs 2) (3.2) P r(s) 2σ 2 s Ω N t c k =1 where Ω denotes the modulation set, and Ω N t c k =i is the set of possible candidate symbol vectors on the condition that kth interleaved coded bit is i (i {0, 1}) to be transmitted. Assuming all the transmitted vector symbols are equally likely it can be simplified by applying max-log approximation, log( e P i ) 31

46 max {P i }, as follow: L(c k r) min s S k,0 r Hs 2 min s S k,1 r Hs 2 (3.3) where S k,p {s = Ξ(c 1, c 2,, c Nt log 2 Ω ) c k = p}, k = 1, 2,, N t log 2 Ω, p {1, 0}, and c k is kth interleaved coded symbol, and Ξ indicates modulation function. Computing the LLR values from (3.3) requires a search for all possible symbol vectors, s. This exhaustive search becomes computationally prohibitive for a larger number of transmit antennas and a higher modulation order such as 16-QAM and 64-QAM. Therefore, LLR computation schemes for reduced complexity ML detectors are essential from hardware implementation perspective. 3.2 LLR computation schemes for Reduced Complexity ML Detectors From now on, we will focus on the LLR calculation method for the reduced complexity ML detector. In this paper, we used QRM-MLD as a reduced-complexity ML detector. The LLR values through multiplying (3.3) by Q H can be obtained as where L(c k r) min s S k,0 y Rs 2 min s S k,1 y Rs 2 (3.4) S k,p {s = Ξ(c 1, c 2,, c Ntlog 2 Ω ) s S Nt, c k = p}, k = 1, 2,, N t log 2 Ω, p {1, 0}, y = Q H r 32

47 and c k is kth interleaved coded symbol, S Nt denotes the set of candidate symbol vectors survived at the last stage, Ξ indicates modulation function. In case of the QRM-MLD algorithm, non-existing LLR bits inevitably happen when surviving symbol vectors do not contain both 1 and 0 for every coded bits at the last stage. In other words, S k,p = ø causes a serious problem that LLR value cannot be computed without additional process. In this case, some extra computation methods are required to compute appropriate LLR values as non-existing LLR bits. The degradation in the soft input decoding performance becomes significant when L(c k ) is incorrectly overestimated such a extremely large absolute value. To illustrate this, if no candidate symbol vector includes c k = 0 for the kth coded bit at the last stage, the event of c k = 1 is supposed to be extremely likely. Then it assigns L(c k ) as very large value through choosing the larger likelihood functions. Consequently, L(c k ) is overestimated and it accounts for severely poor performance if c k = 0 was transmitted. Since it commonly occurs in the reducedcomplexity detectors, an efficient LLR computation scheme is essential to mitigate the effect of the overestimation of L(c k ). In this chapter, we will deal with several conventional method to overcome this problem LLR Clipping A LLR clipping method is briefly introduced in [15]. This method computes the LLR values as described below in Table 3.1. This LLR clipping scheme can obviously impose an upper limit on the resulting LLR values, and can thus mitigate the detrimental effect caused by the overestimated L(c k ) slightly. In practice, ±8 values are employed as the threshold value of the LLR value, which results in good results [15]. However, it can fail to produce the M best candidate symbol vectors. 33

48 Table 3.1: LLR Clipping Scheme Step 1: Compute LLR values of the coded bits by using survived symbol vectors in (3.4). Step 2: In case with no candidate symbol vectors for c k = 1 (or c k = 0), assign minimum distance of it a number with a very large absolute value Step 3: Clip the amplitude of the computed LLR values obtained in Step 1), and 2) by using threshold value T h = A c where A c is a predetermined constant value. This can happen when the signal to noise ratio (SNR) is low due to the deep fading, or when the number of candidate symbol vectors M is small. In this case, it is very difficult to find appropriate threshold by such a heuristic approach in a variety of environments. Therefore, improved method which considers the reliability more is necessary to rectify overestimation of the LLR values for non-existing LLR bits Euclidian Distance Estimation Schemes for Non-Existing LLR Bits Euclidian distance estimation scheme for the bits when the surviving symbol does not exist at the last stage in the QRM-MLD algorithm was proposed via likelihood function generation in [16]. Furthermore, Euclidian distance instead of squared Euclidian distance is used to mitigate LLR estimation errors in the generated LLR values for non-existing LLR 34

49 1.5 Figure 3.2: Euclidian distance estimation schemes for non-existing LLR bits bits. In this scheme, the likelihood function is generated in the following process. First, The candidates among the survived symbol vectors in the last stage, in which bit 1, and 0 remains, are selected independently. Then, for the candidate vectors selected for each bit 1, and 0 the likelihood function of each bit for each symbol candidate vector is obtained using the squared Euclidian distance for the selected candidate vector. For each bit in the data symbols, the likelihood function of bit 1 is derived as the minimum squared Euclidian distance obtained in previous step. The likelihood function for bit 0 is obtained in the same manner. After that, for a bit which has both bits 1 and 0 in the remaining candidate symbol vectors in the last stage, the larger likelihood function of the two bits(1 and 0) is selected. Among the bits, which have both bits 1 and 0 in the remaining candidate symbol vectors in the last stage, the selected larger likelihood function of each bit in the data symbols are averaged. Finally, The likelihood function calculated in previous work is multiplied by 1.5, which is come from near optimization based on simulation. Fig. 3.2 shows the likelihood function generation process for non-existing LLR bits in QRM-MLD. In general, the squared Euclidian distance is used for computing the LLR values. However, the Euclidian distance might be near optimum 35

50 Ideal squared Euclidian distance Large Estimation error Small estimation error Estimated squared Euclidian distance for non-existing bits Ideal Euclidian distance Estimated Euclidian distance for non-existing bits Minimum squared Euclidian distance Figure 3.3: Squared Euclidian distance-based LLR calculation and Euclidian distance-based calculation rather than the squared Euclidian distance in QRM-MLD since estimation errors in the likelihood function of non-existing LLR bits can be mitigated. Therefore, [16] proposed using the Euclidian distance instead of the squared Euclidian distance and investigate the effect of the employing the root of the squared value for the Euclidian distance. This is because the Euclidian distance can reduce the fluctuation in the signal level compared to that for the squared Euclidian distance as shown in Fig. 3.3, leading to a decrease in the error in the generated LLR values for the bits when the surviving symbol does not exist at the last stage in QRM-MLD. To illustrate this, let us assume that estimated squared minimum Euclidian distance is obtained as 4, if ideal squared Euclidian distance is 8. The amount of error through inaccurate 36

51 Euclidian distance estimation diminish when Euclidian distance-based LLR calculation is employed instead of squared Euclidian distance by approximately 3.0. This is because the squared root function, y = x is less than y = x for all positive number x except for x 1. Beside, it will have an great effect on diminishing estimation error for overestimated LLR values through any detectors. 3.3 Simulation Results In this section, we compare the performance of the conventional schemes in 4 4 MIMO channels. The simulation parameters are summarized as follows. We assume that channel state information (CSI) is perfectly known at the receiver and channel response is static during a frame period. Number of transmit antenna (N t ): 4 Number of receive antenna (N r ) : 4 Modulation scheme : BPSK, QPSK, 16QAM Channel coding: Convolutional coding (rate=1/2, Soft viterbi decoding, G=[133, 171]) Interleaver : Random interleaver Channel estimation : Perfect (Known CSI) In Fig. 3.4, we present the BER curves of using MLD, and several conventional scheme for reduced-complexity ML detector, with 16- QAM. It shows that LLR clipping method suffer from a significant performance degradation since the LLR values for non-existing LLR bits are still overestimated with 16-QAM. Clearly, the degradation of the 37

52 "$# % & %'& ( ) * +-, ) *.0/ # # * 66 * /708 9:0;< = # # * 66 * /708 9:0;> =! Figure 3.4: Performance of the MLD and QRM-MLD with several LLR computation schemes for 4 4 MIMO channels with 16-QAM LLR clipping scheme over MLD method becomes more significant for lager threshold value. This is because when the threshold value is set to a lager value, the LLR values for non-existing LLR bits are overestimated more, which then makes error performance worse. It shows that when the LLR clipping scheme is employed, the performance for T h = 3 outperforms these for T h = 8 by approximately 3 db at the BER of 10 4, and its degradation to the MLD is approximately 3 db. In addition, the Euclidian distance estimation scheme outperforms the LLR clipping scheme for both T h = 3 by approximately 1 db at the BER of However, it still have performance loss due to the non accurate LLR values through Euclidian distance estimation for non-existing LLR bits. 38

53 4. Proposed LLR Computation Scheme In this chapter, we explain the proposed LLR calculation algorithm in order to mitigate degradation due to the overestimated LLR values for non-existing LLR bits. Moreover, we demonstrate how accurate the LLR values via proposed method are, compared to previous method by using statistical property of the computed LLR values. After that, improved BER performance through proposed method is shown according to the modulation order and the number of selected candidate vector at the every stage in QRM-MLD algorithm. 4.1 Efficient LLR Computation Algorithm for reduced-complexity ML Detector The main feature of the proposed scheme is that we consider both signs and reliability for non-existing LLR bits by computing approximate LLR values at every stage. The magnitudes of LLR values indicate the degree of reliability as input of the soft-decision decoder. The approximate LLR produced at the middle stage can sufficiently reflect reliability of the LLR value calculated at the last stage. If M candidate symbol vectors at the middle stage have low reliability, surviving candidate symbol vectors at the last stage also can be regarded as inaccurate information. From this observation, proposed algorithm calculates approximate LLR values in the middle stage in order to estimate the LLR values precisely when non-existing LLR bits occur at the final stage. We also employ Euclidian distance [16] instead of squared Euclidian distance. 39

54 Therefore, the approximate LLR value produced by the proposed algorithm at each stage can be written as follows: where L(c k y) min s l χ k,0 y l R l s l min s l χ k,1 y l R l s l (4.1) l = k/log 2 Ω, and k = 1, 2,, N t log 2 Ω, χ k,p {s l = Ξ(c 1,, c l log2 Ω ) c k = p}, p {0, 1} y l = [y l,, y 2, y 1 ] T, R l = [r T l,, r T 2, r T 1 ] T, s l = [s l,, s 2, s 1 ] T, and r l is denoted as the lth row from the bottom of the matrix R. An example of the proposed algorithm is shown in Fig. 4.1, where N t = 3 and QPSK with M = 2. All possible metrics are calculated in the first stage with 1, 6, 9, and 2. Using these values, approximate LLR values are achieved according to (4.1). It results in the approximate LLR value of the first and the second bit as -1 and -5 at the first stage, respectively. After that, only two paths are chosen as survival paths for next stage. At the second stage, two survival paths at the first stage branch out with the size of QSPK signal constellation and accumulated metric are produced. The approximate LLR at the second stage is employed by (4.1) again. The approximate LLR values then are updated by latest approximate LLR value if it can be produced at the second stage successfully. However, the problem occurs when we produce the approximate LLR value of the second bit due to the fact that there is no the candidate symbol vector which contains 1 as the second bit at the second stage in Fig In the proposed algorithm, the approximate LLR value of the second bit inherits from the approximate LLR value of the second bit computed at the first stage. By doing this process, 40