Non-iterative Joint Detection-Decoding Receiver for LDPC-Coded MIMO Systems Based on SDR
NNon-iterative Joint Detection-Decoding Receiver forLDPC-Coded MIMO Systems Based on SDR
Kun Wang
Qualcomm Technologies, Inc.3165 Kifer RoadSanta Clara, CA 95051, USAEmail: [email protected]
Zhi Ding
Dept. of Electrical and Computer EngineeringUniversity of California, DavisDavis, CA 95616, USAEmail: [email protected]
Abstract —Semi-definite relaxation (SDR) detector has beendemonstrated to be successful in approaching maximum like-lihood (ML) performance while the time complexity is onlypolynomial. We propose a new receiver jointly utilizing theforward error correction (FEC) code information in the SDRdetection process. Strengthened by the code constraints, thejoint SDR detector provides soft information of much improvedreliability to downstream decoder and therefore outperformsexisting receivers with substantial gain.
I. I
NTRODUCTION
Multiple-input multiple-output (MIMO) transceiver technol-ogy represents a breakthrough in the advances of wirelesscommunication systems. Modern wireless systems widelyadopt multiple antennas, for example, the 3GPP LTE andWLAN systems [1], and further massive MIMO has beenproposed for next-generation wireless systems [2]. MIMOsystems can provide manifold throughput increase, or canoffer reliable transmissions by spatial diversity [3]. In orderto fully exploit the advantages promised by MIMO, thereceiver must be able to effectively recover the transmittedinformation. Thus, detection and decoding remain to be oneof the fundamental areas in state-of-the-art MIMO research.It is well known that maximum likelihood (ML) detectionis optimal in terms of minimum error probabilities for equallylikely data sequence transmissions. However, the ML detectionis NP-hard [4] and its time complexity is exponential forMIMO detection, regardless of whether exhaustive searchor other search algorithms (e.g., sphere decoding) are used[5] in data symbol detection. Aiming to reduce the highcomputational complexity for MIMO receivers, a number ofresearch efforts have focused on designing near-optimal andhigh performance receivers. In the literature, the simplistlinear receivers, such as matched filtering (MF), zero-forcing(ZF) and minimum mean squared error (MMSE), have beenwidely investigated. Other more reliable and more sophisti-cated receivers, such as successive interference cancellation(SIC) or parallel interference cancellation (PIC) receivers havealso been studied. However, these receivers suffer substantialperformance loss.In recent years, various semi-definite relaxation techniqueshave emerged as a sub-optimum detection method that canachieve near-ML detection performance [6]. Specifically, ML detection of MIMO transmission can be formulated as leastsquares integer programming problem which can then beconverted into an equivalent quadratic constrained quadraticprogram (QCQP). The QCQP can be transformed by relaxingthe rank-1 constraint into a semi-definite program. With thename semi-definite relaxation (SDR), its substantial perfor-mance improvement over algorithms such as MMSE and SIChas stimulated broad research interests as seen in the worksof [7], [8], [9], [10]. Several earlier works [7], [8] developedSDR detection in proposing multiuser detection for CDMAtransmissions. Among them, the authors of [9] proposed anSDR-based multiuser detector for M -ary PSK signaling. An-other work in [10] presented an efficient SDR implementationof blind ML detection of signals that utilize orthogonal space-time block codes. Furthermore, multiple SDR detectors of 16-QAM signaling were compared and shown to be equivalent in[11].Although most of the aforementioned studies focused onSDR detections of uncoded transmissions, forward error cor-rection (FEC) codes in binary field have long been integratedinto data communications to effectively combat noises andco-channel interferences. Because FEC decoding takes placein the finite binary field whereas modulated symbol detectionis formulated in the Euclidean space of complex field, thejoint detection and decoding typically relies on the concept ofiterative turbo processing. In this work, however, we presenta non-iterative receiver based on SDR for joint detection anddecoding. In our design, FEC codes not only are used fordecoding, but also are integrated as constraints within thedetection optimization formulation to develop a novel jointSDR detector [12], [13], [14], [15]. Instead of using themore traditional randomization or rank-one approximation forsymbol detection, our data detection takes advantage of the lastcolumn of the optimal SDR matrix solution. When comparedwith the original SDR detector in [6], our integrated SDRreceiver demonstrates substantial performance gain.II. S YSTEM M ODEL AND
SDR D
ETECTION
A. Maximum-likelihood MIMO Signal Detection
Consider an N t -input N r -output spatial multiplexing MIMOsystem with memoryless channel. The baseband equivalent a r X i v : . [ c s . I T ] A ug odel of this system at time k can be expressed as y ck = H ck s ck + n ck , k = 1 , . . . , K, (1)where y ck ∈ C N r × is the received signal, H ck ∈ C N r × N t denotes the MIMO channel matrix, s ck ∈ C N t × is thetransmitted signal, and n ck ∈ C N r × is an additive Gaussiannoise vector, each element of which is independent and fol-lows CN (0 , σ n ) . In fact, besides modeling the point-to-pointMIMO system, Eq. (1) can be also used to model frequency-selective systems [16], multi-user systems [17], among others.The only difference lies in the structure of channel matrix H ck .To simplify problem formulation, the complex-valued signalmodel can be transformed into the real field by letting y k = (cid:20) Re { y ck } Im { y ck } (cid:21) , s k = (cid:20) Re { s ck } Im { s ck } (cid:21) , n k = (cid:20) Re { n ck } Im { n ck } (cid:21) , and H k = (cid:20) Re { H ck } − Im { H ck } Im { H ck } Re { H ck } (cid:21) . Consequently, the transmission equation is given by y k = H k s k + n k , k = 1 , . . . , K. (2)In this study, we choose capacity-approaching LDPC codefor the purpose of forward error correction. Further, weassume the transmitted symbols are generated based on QPSKconstellation, i.e., s ck,i ∈ {± ± j } for k = 1 , . . . , K and i = 1 , . . . , N t . The codeword (on symbol level) is placedfirst along the spatial dimension and then along the temporaldimension.Before presenting the code anchored detector, we begin witha brief review of existing SDR detector in uncoded MIMOsystems for the convenience of subsequent integration. By theabove assumption of Gaussian noise, it can be easily shownthat the optimal ML detection is equivalent to the followingdiscrete least squares problemmin. x k ∈{± } Nt K (cid:88) k =1 (cid:107) y k − H k x k (cid:107) . (3)However, this problem is NP-hard. Brute-force solution wouldtake exponential time (exponential in N t ). Sphere decod-ing was proposed for efficient computation of ML problem.Nonetheless, it is still exponentially complex, even on averagesense [5]. B. SDR MIMO Detector
SDR can generate an approximate solution to the MLproblem in polynomial time. More specifically, the time com-plexity is O ( N . t ) when a generic interior-point algorithmis used, and it can be as low as O ( N . t ) with a customizedalgorithm [6]. The trick of using SDR is to firstly turn the MLdetection into a homogeneous QCQP by introducing auxiliary variables { t k , k = 1 , . . . , K } [6]. The ML problem can thenbe equivalently written as the following QCQPmin. { x k ,t k } K (cid:88) k =1 (cid:2) x Tk t k (cid:3) (cid:20) H Tk H k H Tk y k − y Tk H k || y k || (cid:21) (cid:20) x k t k (cid:21) s.t. t k = 1 , x k,i = 1 , k = 1 , . . . , K, i = 1 , . . . , N t . (4)This QCQP is non-convex because of its quadratic equalityconstraints. To solve it approximately via SDR, define therank-1 semi-definite matrix X k = (cid:20) x k t k (cid:21) (cid:2) x Tk t k (cid:3) = (cid:20) x k x Tk t k x k t k x Tk t k (cid:21) , (5)and for notational convenience, denote the cost matrix by C k = (cid:20) H Tk H k H Tk y k − y Tk H k || y k || (cid:21) . (6)Using the property of trace v T Qv = tr ( v T Qv ) = tr ( Qvv T ) ,the QCQP in Eq. (4) can be relaxed to SDR by removing therank-1 constraint on X k . Therefore, the SDR formulation ismin. { X k } K (cid:88) k =1 tr ( C k X k ) s.t. tr ( A i X k ) = 1 , k = 1 , . . . , K, i = 1 , . . . , N t + 1 , X k (cid:23) , k = 1 , . . . , K, (7)where A i is a zero matrix except that the i -th position on thediagonal is 1, so A i is used for extracting the i -th elementon the diagonal of X k . It is noted that A i ≡ A i,k , ∀ k ; thus,the index k is omitted for A i,k in Eq. (7). Finally, we wouldlike to point out that the SDR problems formulated in mostpapers are targeted at a single time snapshot, since their systemof interest is uncoded. Here, for subsequent integration ofcode information, we consider a total of K snapshots thatcan accommodate an FEC codeword.III. FEC C ODES IN J OINT
SDR R
ECEIVER F ORMULATION
If MIMO detector can provide more accurate information todownstream decoder, an improved decoding performance canbe expected. With this goal in mind, we propose to use FECcode information when performing detection.
A. FEC Code Anchoring
Consider an ( N c , K c ) LDPC code. Let M and N be theindex set of check nodes and variable nodes of the parity checkmatrix, respectively, i.e., M = { , . . . , N c − K c } and N = { , . . . , N c } . Denote the neighbor set of the m -th check nodeas N m and let S (cid:44) {F | F ⊆ N m with |F| odd } . Then onecharacterization of fundamental polytope is captured by thefollowing forbidden set (FS) constraints [18] (cid:88) n ∈F f n − (cid:88) n ∈N m \F f n ≤ |F| − , ∀ m ∈ M , ∀F ∈ S (8)plus the box constraints for bit variables ≤ f n ≤ , ∀ n ∈ N . (9)in. { X k ,f n } K (cid:88) k =1 tr ( C k X k ) s.t. tr ( A i X k ) = 1 , X k (cid:23) , k = 1 , . . . , K, i = 1 , . . . , N t + 1 , tr ( B i X k ) = 1 − f N t ( k − i − , k = 1 , . . . , K, i = 1 , . . . , N t , tr ( B i + N t X k ) = 1 − f N t ( k − i , k = 1 , . . . , K, i = 1 , . . . , N t , (cid:88) n ∈F f n − (cid:88) n ∈N m \F f n ≤ |F| − , ∀ m ∈ M , ∀F ∈ S ;0 ≤ f n ≤ , ∀ n ∈ N . (12)Recall that the bits { f n } are mapped by modulators intotransmitted data symbols in x k . It is important to note that theparity check inequalities (8) can help to tighten our detectionsolution of x k by explicitly forbidding the bad configurationsof x k that are inconsistent with FEC codewords. Thus, a jointdetection and decoding algorithm can take advantage of theselinear constraints by integrating them within the SDR problemformualtion.Notice that coded bits { f n } are in fact binary. Hence, thebox constraint of (9) is a relaxation of the binary constraints. Infact, if variables f n ’s are forced to be only 0’s and 1’s (binary),then the constraints (8) will be equivalent to the original binaryparity-check constraints. To see this, if parity check node m fails to hold, there must be a subset of variable nodes F ⊆ N m of odd cardinality such that all nodes in F have the value 1 andall those in N m \F have value 0. Clearly, the correspondingparity inequality in (8) would forbid such outcome. B. Symbol-to-Bit Mapping
To anchor the FS constraints into the SDR formulation inEq. (7), we need to connect the bit variables f n ’s with thedata vectors x k ’s or the matrix variables X k ’s.As stated in [6], if ( x ∗ k , t ∗ k ) is an optimal solution to (7), thenthe final solution should be t ∗ k x ∗ k , where t ∗ k controls the sign ofthe symbol. In fact, Eq. (5) shows that the first N t elements oflast column or last row are exactly t k x k . We also note that thefirst N t elements correspond to the real parts of the transmittedsymbols and the next N t elements correspond to the imaginaryparts. Hence, for QPSK modulation, the mapping constraintsfor time instant k = 1 , . . . , K are simply as followstr ( B i X k ) = 1 − f N t ( k − i − , i = 1 , . . . , N t , tr ( B i + N t X k ) = 1 − f N t ( k − i , i = 1 , . . . , N t , (10)where B i is a selection matrix designed to extract the i -thelement on the last column/row of X k (except last element): B i = . . . . . . . . . / ... . . . ...... / ... . . . ... / . . . / . . . , ≤ i ≤ N t . (11) The non-zero entry of B i is the i -th element on the lastcolumn. For the same reason as that of A i , the index k isomitted in B i . Moreover, note the subtle difference that A i is defined for ≤ i ≤ N t + 1 while B i is defined for ≤ i ≤ N t . C. Joint ML-SDR Receiver
Having defined the necessary notations and constraints, ajoint ML-SDR detector can be formulated as the optimizationproblem in Eq. (12) for QPSK modulation. For higher orderQAM beyond QPSK, the necessary changes for our jointSDR receiver include the relaxed box constraints for diagonalelements [11] and the symbol-to-bit mapping constraints. Werefer interested readers to the works [17], [19], [20], [15] forthe details of mapping higher order QAM constraints.Recall that the matrix X k (cid:23) is a relaxation of the rankone matrix X k = (cid:20) x k t k (cid:21) (cid:2) x Tk t k (cid:3) After obtaining the optimal solution { X k } of the SDR,one must determine the final detected symbol values in x k .Traditionally, one “standard” approach to retrieve the finalsolution is via Gaussian randomization that views X k as thecovariance matrix of x k , and another method is to apply rank-one approximation of X k [6].However, a more convenient way is to directly use the first N t elements in the last column of X k . If hard-input hard-output decoding algorithm (such as bit flipping) is used, wecan first quantize t ∗ k x ∗ k into binary values before feeding themto the FEC decoder for error correction. On the other hand, forsoft-input soft-output decoder such as sum-product algorithm(SPA), log-likelihood ratio (LLR) can be generated from theunquantized t ∗ k x ∗ k .IV. S IMULATION R ESULTS
In the simulation tests, a MIMO system with N t = 4 and N r = 4 is assumed. The MIMO channel coefficients areassumed to be ergodic Rayleigh fading. QPSK modulation isused and a regular (256,128) LDPC code with column weight3 is employed.In this section, we will demonstrate the power of codeanchoring. We term the formulation in Eq. (7) as disjoint ML-SDR , while that in Eq. (12) as joint ML-SDR . With the optimalDR solution { X ∗ k } , there are several approaches to retrievethe final solution ˆs k .- Rank-1 approximation : Perform eigen-decomposition on X ∗ k to obtain the largest eigenvalue e k and its cor-responding eigenvector v k . The final solution ˆs k = √ e k v k [1 : 2 N t ] × v k [2 N t + 1] .- Direct approach : The final solution is retrieved from thelast column of X k , i.e., ˆs k = X k [1 : 2 N t , N t + 1] .- Randomization : Generate v k ∼ CN ( , X k ) for a certainnumber of trials, and pick the one that results in smallestcost value. Note that when evaluating the cost value, theelements of v k are quantized to {− , +1 } .We caution that, among the methods mentioned above,randomization is not suitable for soft decoding, because themagnitudes of the randomized symbols do not reflect theactual reliability level. Therefore, in the following, we willonly consider rank-1 approximation and direct method, theBER curves of which are shown in Fig. 1 and Fig. 2,respectively. In the performance evaluation, we consider 1)hard decision on symbols, 2) bit flipping (BF) decoding and3) SPA decoding. In some sense, hard decision shows the“pure” gain by incorporating code constraints. BF is a harddecoding algorithm that performs moderately and SPA usingLLR is the best. If we compare the SPA curves within eachfigure, the SNR gain is around 2 dB at BER = 1e-4. For othercurves, the gains are even larger. On the other hand, if wecompare the curves across the two figures, their performancesare quite similar. Therefore, we do not need an extra eigen-decomposition; the direct approach is just as good.Moreover, we compare ZF and MMSE against the SDRreceivers in Fig. 3. All BER curves are shown after SPAdecoding. It is clear that ZF and MMSE receivers are farworse than the disjoint SDR, let alone joint SDR. Given thatZF and MMSE are O ( N t ) complexity and SDR receiveris O ( N . t ) complexity, the performance gap is quite largegiven the relatively small difference in complexity. In addition,the performance of the exponential-complex ML receiver isplotted. Here we use a soft-output ML detector [21] andthen feed the LLRs to SPA decoder. It is seen that the BERperformances of ML and joint SDR are very close, eventhough joint SDR is polynomial-complex.V. C ONCLUSION
This work introduces joint SDR detectors integrated withcode constraints for MIMO systems. The joint ML-SDR detec-tor takes advantage of FEC code information in the detectionprocedure, and it demonstrates significant performance gaincompared to the SDR receiver without code constraints. Incurrent stage, this joint receiver works well with short-to-medium length FEC code. However, since the computationcapability is ever increasing, this design should be able toaccommodate longer codes. In the meantime, we would liketo conduct complexity reduction of the joint receiver in futureworks [22]. It is also interesting to investigate the robustreceiver’s performance against RF imperfections, such as I/Q
SNR (dB) -5 -4 -3 -2 -1 BE R Disjoint HardDisjoint BFDisjoint SPAJoint HardJoint BFJoint SPA
Fig. 1:
BER comparisons of disjoint and joint SDR receivers: Rank1 approximation.
SNR (dB) -5 -4 -3 -2 -1 BE R Disjoint HardDisjoint BFDisjoint SPAJoint HardJoint BFJoint SPA
Fig. 2:
BER comparisons of disjoint and joint SDR receivers: Directapproach using the final column of X k . SNR (dB) -5 -4 -3 -2 -1 BE R ZFMMSEDisjoint SDRJoint SDRML
Fig. 3:
BER comparisons after SPA: ZF, MMSE, Disjoint SDR, JointSDR and ML. mbalance and phase noise [23]. Moreover, joint design ofprecoder [24] and receiver would be a good topic to pursue.R
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