Multi-Branch Successive Interference Cancellation for MIMO Spatial ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Multi-Branch Successive Interference Cancellation for MIMO Spatial Multiplexing Systems Rui Fa and Rodrigo C. de Lamare Communications Research Group, Department of Electronics, University of York, YO10 5DD, United Kingdom. Email: {rf533, rcdl500}@ohm.york.ac.uk

Abstract—In this paper we propose a novel successive interference cancellation (SIC) strategy for multiple-input multipleoutput (MIMO) spatial multiplexing systems based on multiple interference cancellation branches. The proposed detection structure employs SICs on several parallel branches which are equipped with different ordering patterns so that each branch produces a symbol estimate vector by exploiting a certain ordering pattern. The novel detector, therefore, achieves higher detection diversity by selecting the branch which yields the estimates with the best performance according to the selection rule. We consider three selection rules for the proposed detector, namely, maximum likelihood (ML), minimum mean square error (MMSE), constant modulus (CM) criteria. The simulation results reveal that our scheme successfully mitigates the error propagation and approaches the performance of the optimal ML detector, while requiring a significantly lower complexity than the ML detector.

Index Terms – MIMO systems, spatial multiplexing, Multiple Branches, Successive Interference Cancellation. I. I NTRODUCTION Recently, it has been recognized that the deployment of multiple transmit and receive antennas significantly improves wireless link performance in communication systems. The degrees of freedom afforded by the multiple antennas can offer dramatic multiplexing [1]–[5] and diversity gains [6], [7]. The multiplexing gains enable high spectral efficiencies, whereas the diversity gains make the links more reliable and allow low error rates over wireless fading channels. In these multipleinput multiple-output (MIMO) systems, the transmitter and the receiver should be appropriately designed in order to exploit the structure of the propagation channels. In a spatial multiplexing configuration, the system can obtain substantial gains in data rate. These capacity gains grow linearly with the minimum number of transmit and receive antennas, and the transmission of individual data streams from the transmitter to the receiver. In order to separate these streams, the designer may resort to several detection techniques, which are similar to multiuser detection methods [8]. The optimal maximum likelihood (ML) detector can be implemented using the sphere decoding algorithm [9], [10]. However, the complexity of this algorithm can be polynomial

or exponential depending on the signal-to-noise ratio (SNR) and the signal constellation. This has motivated the development of various alternative low-complexity strategies. A promising transmission system, called diagonal Bell Laboratories Layered Space- Time (D-BLAST) proposed by Foschini [3], is the first BLAST architecture. Owing to the large computational complexity required for the scheme, a simplified version, called the Vertical BLAST (V-BLAST) has been proposed in [4], [5]. The V-BLAST can be seen as an ordered SIC, on the other hand, there is an equivalence between the V-BLAST receiver and the generalized decision feedback equalizer (GDFE) [11]. A number of other strategies are also investigated to achieve the capacity gain of MIMO systems including the linear and the decision feedback (DF) detector [12] [13] and the parallel interference cancellation (PIC) [14]. In this work, we propose a novel SIC strategy for MIMO spatial multiplexing systems based on multiple processing branches. The proposed detection structure is equipped with SICs on several parallel branches which employ different ordering patterns. Namely, each branch produces a symbol estimate vector by exploiting a certain ordering pattern. Thus, there is a group of symbol estimate vectors at the end of the multi-branch (MB) structure. Based on different application requirements, different criteria, such as ML, MMSE and constant modulus (CM), can be used as selection rules to select the branch with the best performance. We adopt the MMSE estimator for the design of the proposed MB-SIC structure for MIMO receiver because the MMSE estimator usually has good performance, is mathematically tractable and has relatively simple adaptive implementation. In one word, the basic principle of the proposed structure is to exploit the orderings of SIC in appropriate ways such that the detector can produce a group of estimate vectors and then higher detection diversity can be obtained by selecting the most likely one based on a certain selection rule. The simulation results reveal that our scheme successfully mitigates the error propagation and approaches the performance of the optimal ML detector. This paper is organized as follows. Section II briefly de-

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~z [i ] 1

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MIMO spatial multiplexing system.

r[i ] NR×1

scribes a MIMO spatial multiplexing system model. Section III is dedicated to the presentation of the novel multi-branch SIC detector. Section IV presents and discusses the numerical simulation results, while Section V gives the conclusions. II. S YSTEM M ODEL

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Let us consider a spatial multiplexing MIMO system, as depicted in Fig. 1, with QW transmit antennas and QU receive antennas, where QU QW . At each time instant [l], the system transmits QW symbols which are organized into a £ ¤W QW ×1 vector s[l] = v1 [l]> v2 [l]> = = = > vQW [l] taken from a modulation constellation D = {d1 > d2 > = = = > dQ }, where (·)W denotes transpose. The symbol vector s[l] is then transmitted over flat fading channels and the signals are demodulated and sampled at the receiver, which is equipped with QU antennas. into a QU ×1 vector ¤W £ The received signal is collected r[l] = u1 [l]> u2 [l]> = = = > uQU [l] given by r[l] = Hs[l] + v[l]>

~ z l ,1 [i ]

ù l ,1[i ]

(1)

where the QU × 1 vector v[l] is a zero mean complex circular symmetric Gaussian noise with covariance matrix ¤ £ H v[l]vK [l] = y2 I, where H[·] stands for expected value, (·)K denotes the Hermitian operator, y2 is the noise variance and I is the identity matrix. The £symbol vector s[l] has zero ¤ mean and a covariance matrix H s[l]sK [l] = v2 I, where v2 is the signal power. The elements kqU >qW of the QU × QW channel matrix H correspond to the complex channel response from the qW th transmit antenna to the qU th receive antenna. III. M ULTI -B RANCH SIC D ETECTION This section is devoted to the description of the proposed multi-branch successive interference cancellation (MB-SIC) detector for MIMO systems. We present the overall principles and structures of the proposed scheme in the first place, and then we introduce the selection rules and ordering schemes which are employed in our proposed detector in the next. Based on different application requirements and system structures, better performance and lower complexity can be achieved by choosing a proper selection rule and ordering scheme.

rl , NT [i ]

~z [i ] l , NT

ù l , NT [i ]

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SICl

Fig. 2. (a) The global block diagram of the proposed multi-branch SIC detector. (b) The schematic structure of the o-th SIC branch.

A. Proposed Scheme In this subsection, we detail the principles and structures of the proposed MB-SIC detector for MIMO systems. The proposed detection structure employs SICs on several parallel branches which are equipped with different ordering patterns. Namely, each branch produces a symbol estimate vector by exploiting a certain ordering pattern. Thus, there is a group of symbol estimate vectors at the end of the multi-branch (MB) structure. We present MMSE-SIC for the design of the proposed multi-branch MIMO receiver because the MMSE estimator usually has good performance, is mathematically tractable and has relatively simple adaptive implementation. The novel structure for detection exploits different patterns and orderings for the modification of the original V-BLAST architecture and achieves higher detection diversity by selecting the branch which yields the estimates with the best performance. Fig. 2(a) depicts the global block diagram of the proposed detector. In order to detect the transmitted signals using the proposed MB-SIC structure, the detection process for each branch uses linear MMSE nulling and symbol successive cancellation to compute ˜ zo [l] = [˜ }o>1 [l]> }õ>2 [l]> ===> }õ>QW [l]]W , which ˜ zo [l] denotes the QW ×1 symbol estimate vector detected for the o-th branch by using the ordering transformation matrix To . Here, we introduce the ordering transformation matrix To > o = 1> ===> O, which corresponds to the ordering pattern employed in the o-th branch. The process in the o-th SIC


branch, as shown in Fig. 2(b), is mathematically given as follow }õ>q [l] = $K o>q [l]ro>q [l]>

(2)

where ro>q [l] = r[l]> H0o

q = 1>

= To H>

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(3)

μ ¶1 y2 K 0 0 ¯ ¯ $ o>q = H o>q H o>q + 2 I (H0o )q > v }o>q [l])> vˆ õ>q [l] = Q(˜ where H0o is the transformed channel matrix for the o-th branch, ¯ 0 o>q denotes the (H0o )q denotes the q-th column of H0o , H matrix obtained by taking columns q> q + 1> ===> QW of H0 . ro>q [l] denotes the received vector after the cancellation of the previously detected (q 1) symbols, so that ro>1 [l]=r[l]. Without loss of generality, we assume that the signal stream q = 1 is detected first. The interference due to the first stream is then regenerated and subtracted before q = 2 is detected. This procedure is repeated successively until all streams are detected. The ordered symbol estimate vector ˜ zo [l] can be arranged in original order by zo [l] = TWo ˜ zo [l]=

B. Selection Rules The proposed MB-SIC detector selects the branch that optimizes the corresponding cost function J according to 1oO

(5)

The final detected symbol is ˆ si [l] = Q(zorsw [l])=

The ML criterion can provide the best performance among these candidate criteria while channel information is available. Although the channel estimation would cost extra complexity, the performance improvement by employing the ML criterion is considerable. 2) Minimum Mean Square Error Criterion: While channel information is not available, MMSE criterion can be used to select the branch which minimizes the mean square error of transmitted symbols. The cost function is given by J PPVH = kˆs[l] z[l]k2 >

(8)

where ˆs[l] is symbol estimation in the decision directed mode, thus, the MMSE criterion would be greatly impaired by the error propagation. 3) Constant Modulus Criterion: The CM algorithm originally proposed by Godard [15], has widely been applied to the blind detection because of its robustness and easy implementation. In this context, CM criterion attempts to minimize the cost function h¯ ¯2 i J FP = H ¯(}q [l])2 1¯ = (9)

(4)

In summary, the basic principle of the proposed structure is to exploit the orderings of SIC in appropriate ways such that the detector can produce a group of estimate vectors and then higher detection diversity can be obtained by selecting the most likely one based on a certain selection rule, which will be introduced in the following section. The simulation results reveal that our scheme successfully mitigates the error propagation and approaches the performance of the optimal ML detector.

orsw = arg min J(zo [l])>

1) Maximum Likelihood (or Minimum Euclidean Distance) Criterion: The cost function for the ML criterion, which is equivalent to the minimum Euclidean distance criterion, is written as 2 (7) J PO = kr[l] Hˆs[l]k =

(6)

Based on different application requirements, different criteria, such as ML, MMSE and CM, can be used as selection rules to select the branch with the best performance.

We will show how these selection rules perform later in the simulation section. Note that for non constant modulus constellations such as QAM, one can replace the cost function in (9) with a square contour. C. Ordering Schemes Here, we propose the optimal ordering scheme and three alternative ordering schemes for designing the proposed receiver, where the common framework is the use of parallel branches with ordering patterns that yield a group of symbol estimate vectors. The number of parallel branches O is a parameter that must be chosen by the designer. In this context, the optimal ordering scheme conducts an exhaustive search O = QW ! where ! is the factorial operator. It is clearly very complex for practical systems, especially when QW is large. Therefore, an ordering scheme with low complexity, which renders itself to practical implementation, is of great interest. In order to design the transformation matrices To , we propose three sub-optimal schemes to constrain them to be appropriate structures such that they can be used for low-complexity implementation of the detector. These three schemes are developed based on an assumption that the original detection ordering is the optimal ordering instead of an arbitrary ordering.


1) Pre-Stored Patterns (PSP): The transformation matrix T1 for the first branch is chosen as the identity matrix IQW to keep the optimal ordering as described by T1 = IQW . The remaining ordering patterns can be described mathematically by ¸ Iv 0v>QW v > 2 o QW > (10) To = 0QW v>v ![Iv ] where 0p>q denotes an p×q-dimensional matrix full of zeros, the operator ![·] rotates the elements of the argument matrix column-wise such that an identity matrix becomes a matrix with ones in the reverse diagonal. The proposed ordering algorithm shifts the ordering of the antennas according to shifts given by v = b(o 2)QW @Oc > 2 o QW >

TABLE I F REQUENTLY S ELECTED B RANCHES (FSB) O RDERING S CHEME 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

dH { NULL> Llg{ { NULL> LI VE { NULL Orsw { QW !> o { 1 Lr { PERMS(QW ) for qh = 1 to Qh do for o = 1 to Orsw do To { Lr (o) ˆ so [l] { SIC(To H) so [l]k dH [o] { kr[l] 3 Hˆ end for Llg{ (qh ) { MIN Index(dH ) end for LI VE { SELECT(HIST(Llg{ )) TABLE II L ISTING PATTERNS A PPROACH (LPA) O RDERING S CHEME

(11)

where O is the number of parallel branches and b·c rounds the argument to the lowest integer according to the o-th branch. 2) Frequently Selected Branches (FSB): The basic principle of the proposed FSB algorithm is to build a codebook which contains the ordering patterns for the most likely selected branches. In order to build such codebook, we resort to a simulation approach, where we identify the statistics of each selected branch and construct the codebook with the O most likely selected branches to be encountered. The algorithm is summarised in Table I, where dH denotes the vector of Euclidean distance for all possible branches, Qh denotes the total number of experiments we did, Llg{ is defined for the storage of the selected branches for every experiment and Lr is the codebook for optimal ordering patterns computed by PERMS(QW ), which provides the list containing all possible permutations of the QW elements. We highlight that in each run, after we measure the Euclidean distances for all branches, the branch which brings the minimum Euclidean distance is stored in Llg{ at step 10. Finally, the FSB codebook LI VE is created by selecting the most frequently selected O branches according to the histogram of Llg{ . We discover an interesting fact that the FSB codebooks with 10 elements for 4 × 4, 6 × 6 and 8 × 8 systems, are almost the same, [1> 2> 3> 5> 7> 8> 13> 17> 19> 21]. The only difference is that for 6×6 and 8×8 systems, reverse ordering should be included in the codebooks. 3) Listing Patterns Approach (LPA): Motivated by the fact that we have to do a lot of prior work before the FSB algorithm can be employed, we propose an online codebook updating algorithm, which is called listing patterns approach (LPA). However, this approach is restricted by the number of antennas. We suppose that the channel is block-fading in which a block of symbols are affected by the same fading value. Thus, once the channel changes, we would re-select a list of ordering patterns to update the codebook. In this case, the LPA algorithm is proposed to fulfil the online updating

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

dH { NULL> LOS D { NULL Orsw { QW !> o { 1 Lr { PERMS(QW ) for l = 1 to Oe do if l == 1 then for o = 1 to Orsw do To { Lr (o) ˆ so [l] { SIC(To H) so [l]k dH [o] { kr[l] 3 Hˆ end for LOS D { SELECT(dH ) OOS D { LENGTH(LOS D ) else for o = 1 to OOS D do To { LOS D (o) ˆ so [l] { SIC(To H) so [l]k dH [o] { kr[l] 3 Hˆ end for end if end for

of the codebook. We formalize the algorithm in Table II. In each block which is supposed to contain Oe frames, we use the optimal ordering scheme which exhaustively searches all possible orderings in the first frame, then online update the codebook LOS D by listing the first O ordering patterns which minimize the cost function. Thereafter, we detect the remaining frames by using the updated codebook LOS D . IV. S IMULATIONS In this section, we assess the bit error rate (BER) performance of the proposed scheme and the existing MIMO detection schemes, namely, the ML detector, the linear MMSE detector, the V-BLAST, the parallel interference cancellation (PIC) and the proposed MB-SIC algorithm. Here, we consider two channel models in the simulation: the first one is independent and identically-distributed (i.i.d.) random fading, whose coefficients are taken from complex Gaussian random variables with zero mean and unit variance, and the second one is the 3GPP spatial model (SCM) [16], which was developed to be a common reference for evaluating different MIMO


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Fig. 5. BER vs. Number of Elements performance comparison between proposed algorithm and existing algorithms for MIMO spatial multiplexing. QW = QU , SNR = 12dB.

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concepts in outdoor environments at a centre frequency of 2GHz and a system bandwidth of 5MHz. We define the SNR Q 2 as 10 log10 W2 v , where v2 is the variance of the transmitted q symbols and q2 is the noise variance, respectively. We average the experiments over 1000 runs and use packets with 100 symbols employing the QPSK modulation. Let us first compare the BER performance against SNR for our proposed detector by employing the three candidate selection rules. As shown in Fig. 3, the detector with the ML criterion outperforms the other criteria while the channel information is known. The selection rule can be chosen according to the different application requirements. In our following simulations, the ML criterion is the selection rule because we assume that the channel information is perfectly known. In Fig. 4, we evaluate the BER performance against SNR

for MIMO systems with QW = QU = 4 antennas. We compare the proposed ordering algorithms with the optimal ordering scheme described in the previous section. We also compare the proposed MB-SIC detectors with different ordering schemes against the existing linear MMSE detector, VBLAST, MMSE-PIC and ML detector. For our proposed ordering schemes, we have to configure the number of branches O. In this context, the maximum O is set to QW for PSP scheme due to the algorithm limitation. For the FSB and the LPA schemes, we set O = 10 considering the tradeoff between computational complexity and the performance. The indexes of branches which are selected into the codebook are [1> 2> 3> 5> 7> 8> 13> 17> 19> 21] for the FSB scheme. It is presented that the performance of the proposed MB-SIC detectors outperforms the linear MMSE detector, V-BLAST and MMSE-PIC detector. The plots also show that the performance of the proposed detector with optimal ordering scheme, which tests all QW ! possible branches and selects the most likely estimate, approaches the optimal ML detector closely and the proposed detector with the FSB and the LPA schemes perform as good as that with optimal ordering. As depicted in Fig. 6, the BER performance against SNR is investigated when the MIMO system with QW = QU = 4 antennas working in the 3GPP SCM enviroment. We use the MATLAB implementation of SCM developed by Salo etc. [17]. The plots show a similar result as in Fig. 4. The performance of the proposed detector with optimal ordering scheme approaches the optimal ML detector and the FSB scheme is slight better than the LPA scheme. In the following experiment, shown in Fig. 5, we investigate the BER vs. the number of antennas performance of the proposed MB-SIC detectors and the V-BLAST when SNR =


The proposed MMSE-MB-SIC detector, which achieves higher detection diversity, was compared with several existing detectors in the literature via computer simulations and was shown to approach the optimal maximum likelihood detector while reducing the complexity significantly.

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Fig. 6. BER vs. SNR performance comparison between proposed algorithm and existing algorithms for MIMO spatial multiplexing using SCM. QW = QU = 4.

12dB. Due to the computational complexity of the optimal ordering increasing with the number of antennas, the optimal ordering scheme and the LPA scheme are not available in this case. We compare the BER performance of the V-BLAST, the proposed detector with PSP scheme when O = QW and the proposed detector with the FSB scheme when O = 4 and O = 10 respectively. It is obvious that even though the number of branches in the FSB scheme is less than that in PSP scheme when the number of antennas is greater than or equal to 6 (QW = QU 6), The performance of the FSB scheme is better than the PSP scheme. V. C ONCLUSIONS We presented a novel MMSE SIC detector based on multiple parallel branches for MIMO spatial multiplexing system. The proposed detection structure is equipped with SICs on several parallel branches which employ different ordering patterns. Namely, each branch produces a symbol estimate vector by exploiting a certain ordering pattern. Thus, there are a group of symbol estimate vectors at the end of multi-branch (MB) structure. Based on different application requirements, different criteria, such as ML, MMSE and CM, can be used as selection rules to select the branch with the best performance. We also proposed three sub-optimal ordering schemes together with the optimal ordering scheme.

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