Survey on an Efficient, Low-complex Tuple Search Based Sphere

Survey on an Efficient, Low-complex Tuple Search Based Sphere Detector Esther P. Adeva and Bj¨orn Mennenga and Gerhard Fettweis Vodafone Chair Mobile Communication Systems Technische Universit¨at Dresden Dresden, Germany Email: esther.perez, mennenga, [email protected]

Abstract—Nowadays, high detection complexity is known to be one of the major challenges in MIMO communications based on spatial multiplexing. Tuple search (TS) sphere detection was recently introduced, demonstrating to represent a promising approach in this context. It provides significant complexity reduction in comparison to conventional algorithms, providing in addition close to full max-log-APP BER performance. Due to the increasing multiplicity of communication standards as well as variety of mobile applications demanded by users, tackling the lack of flexibility of common receiver realizations has become an additional key challenge in MIMO detection. Aim of this work is to demonstrate that the benefit provided by the tuple search strategy is still present in a wide range of possible transmission schemes. For this purpose, a novel efficiency indicator is introduced, based on which an exhaustive analysis is performed. The existing tuple search detector has been adapted to deal with different constellation orders and transmit/receive antenna configurations. In addition, the applied MMSE strategy has been modified to support undetermined systems. The obtained results show the superiority of the proposed sphere detector under different transmission conditions, thus demonstrating its efficiency and flexibility.

Keywords— MIMO detection, sphere decoder, tuple search, efficiency, overhead, asymmetric MIMO. I. I NTRODUCTION Future mobile communication systems will make use of multiple-input multiple-output (MIMO) techniques in combination with high constellation orders to enhance spectral efficiency. Transmission of spatially multiplexed data streams allows increasing data rates as well as diversity. However, the inherent high detection complexity and lack of flexibility of common detector realizations still represent limiting factors towards efficient implementations. Tree search strategies have been shown to represent a promising approach to overcome the complexity problem. High order systems allow high data rates, but also imply great detection search spaces. As a consequence, detection algorithms exploring all or most of the possible sent symbols become impractical due to the resulting high computational complexity. This is the case of e.g. full max-log-APP detection. To solve this problem, several complexity-reduced detection algorithms presenting close to full max-log-APP detection accuracy have been widely studied. Sphere detector [1], M-algorithm [2], LISS-algorithm [3], or variations of them [4]–[7] are some examples, still subjected to numerous BER performance-complexity trade-offs.

Throughout this work a tuple search detector (TSD) with matched candidate determination is employed [8]. The TSD approach reduces the search complexity significantly, outperforming state-of-the-art detection strategies like single tree search (STS) [9] [10] or list sphere detection (LSD) [4] [11] while providing close to full max-log-APP BER performance. Several techniques for further complexity reduction like sorted QR decomposition (SQRD) [12], MMSE preprocessing with bias reduction [13], search sequence determination (SSD) [14] and metric estimation (ME) [15] can be applied while maintaining the BER performance-complexity superiority of the tuple search detector [16]. In order to handle the variety of current and future mobile communication standards and to meet the diverse requirements of user applications, detectors should be capable of supporting various types of transmission schemes. As a consequence, not only detection complexity but also receiver flexibility has become an important challenge in MIMO systems. For this reason, adaptation of BER performance-complexity or even of the utilized transmission system to the channel characteristics will be required. The proposed low-complex tuple search detector is exhaustively analyzed through this work. The utilized communication system model (section II) and the fundamentals of the tuple search detection (section III) are firstly presented. Strategies for complexity reduction employed in this work are also briefly introduced (section IV). Results of the TSD analysis are subsequently detailed (section V). Efficiency of the search algorithm (V-A) is analyzed via a novel indicator. In addition, flexibility of the proposed detector is demonstrated (V-B) by extending the analysis to different transmission schemes. In this concern, detection accuracy, constellation order and configuration of the MIMO antenna system have been considered. Regarding the last issue, symmetric antenna configurations are initially considered. Sphere detection is generally assumed not to be suitable in asymmetric transmission where the number of receive antennas is smaller than the number of transmit antennas [17], [18]. This scenario, has been therefore also covered in our analysis (V-C), extending previously the applied MMSE strategy. Results (as summarized in section VI) demonstrate the efficiency, flexibility and thus suitability of the proposed sphere detector even for the problematic asymmetric transmission.

Transmitter Binary Data Source

c′

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La,Det

Fig. 1: System model with BICM transmitter and iterative receiver.

II. S YSTEM M ODEL Throughout this paper, we consider a NT × NR MIMO system based on a BICM transmission strategy with NT transmit and NR receive antennas, as depicted in Fig. 1. A vector u of i.i.d. information bits is encoded by the outer channel code with rate R. The resulting stream of vectors c′ is bit-interleaved and portioned into blocks c of NT · L bits, where L denotes the number of bits per transmit symbol. For the transmission, the corresponding bits c ∈ C, covered in the set of permitted bit vectors, are mapped (e.g. gray mapping) onto complex constellation symbols x(c) = [x0 , ...xNT −1 ]T = map(c) ∈ X , being X the set of valid transmit symbols with cardinality #X = #C = 2L = Q. The transmit energy is normalized so that E{xxH } = ES /NT I, where Es represents the average transmit energy of the transmitter. Regarding the transmission, we consider an uncorrelated, flat fading channel and an additive noise vector n ∈ CNR ×1 at the receiver with complex components of zero mean i.i.d. gaussian random variables of variance N0 /2 per real dimension (E{nnH } = N0 I). The considered passive channel is represented by H ∈ CNR ×NT with entries of a zero mean i.i.d. gaussian random process of variance 1 and is assumed to be perfectly known at the receiver. The received signal y is therefore given by: y = Hx + n and the signal-to-noise-ratio (SN R = Es /N0 ) at the receiver applied to the energy of one information bit can be stated as Eb /N0 = Es NR /N0 NT LR. In order to ensure comparability of results, a setup equivalent to the one used in e.g. [4], [8], [13] has been used for our simulations. These have been carried out for a rate 1/2 PCCC with (7R , 5) convolutional codes, an information block size of 9216 bits (including tail bits), gray mapping, and spatial and temporal fading. The detection of the transmitted bits is carried out by complex-valued tuple search sphere detector in conjunction with a BCJR based decoder with 8 internal iterations and, for the sake of simplicity, without detector↔decoder iterations. III. T UPLE S EARCH BASED MIMO D ETECTION A. Fundamentals Task of the focused detector is the determination of bits c most likely sent as well as of reliability information for these bits. This can be accomplished by calculating log-likelihood ratios (L-values):

where (1) results from application of the max-log approximation [16]. The l-th bit of a symbol sent by the m-th antenna is represented by cm,l and λ0 represents the distance metric. For the considered case without detector↔decoder iterations (i.e. without a priori information), this metric depends only on the Euclidean distance between the set of received symbols y and the representative of the symbol x ˆ(c) likely transmitted through the channel H: 2 λ0 (y, c) = ky − Hˆ x(c)k Consequently, sent symbol

besides the most arg min{λ0 } (i.e. the

probably detection

ˆ (c)|c∈C x

hypothesis) and its corresponding metric λ0 (cML ), the detector has to determine also the counter-hypotheses arg min {λ0 } and their metrics for each bit. ˆ (c)|c∈C,cm,l 6=cML x m,l

B. Tree Search Basics Since brute force (full max-log-APP) detection of (1) is known to be of exponentially growing computational complexity with the number of transmit antennas and order of the constellation, several close to optimal detection approaches have been lately proposed to find relevant candidates for the L-value calculation. Some of the most promising are based on tree search strategies. As described in detail in [4], transforming the detection problem is permitted by QRdecomposition (QRD) of H = QR, where Q is unitary and R an upper triangular matrix. With modified received symbols y′ = QH y, determination of the Euclidean distance 2 ky′ − Rˆ x(c)k (2) can be interpreted as a tree search. Resulting from this, λ0 can be recursively calculated through the layered partial metric 2 NX T −1 ′′ λi = λi+1 + yi −rii x ˆi , yi′′ = yi′− rij xˆj , (3) |{z} |{z} metric from already estimated symbols

interference reduced symbol

j=i+1

by adding the metric of the corresponding parent node to the squared distance between a representative of the estimated symbol and an interference reduced symbol. The search is carried out in depth-first fashion, successively extending the selected nodes by analyzing their child nodes. As presented in [19], a regularized algorithm flow and the parallel calculation of sibling parent nodes as well as of leaf nodes permit a so called one-node-per-cycle implementation [20]. Based on this, the average number of node extensions ♯n performed in the detection is taken as complexity measure throughout this paper. C. Tuple Search and Bit-Specific Candidate Determination Computing the L-values in (1) requires determination of a detection hypothesis and all counter-hypotheses as described in section III-A. Explicit search for all needed minimums leads

to impractically high ♯n [9]. Therefore, instead of searching all possible minima, the TSD introduced in [8] searches a subset of T most likely leaves, similarly to the approach used in List Sphere Detection (LSD). The metrics λ0 of these leaves are stored in a search tuple T := {λ0 (c1 ) , λ0 (c2 ) , . . . , λ0 (cT )}, defining the sphere radius as the maximum metric in the tuple: R = max {λ0,t } . ct |ct ∈T

Metrics stored by LSD for the radius determination represent in combination with the related ct the set of candidates for the L-value calculation. Resulting from this, acceptable BER performance demands huge T values, leading to high ♯n and memory requirements. On the contrary, tuple search can be combined with separated bit-specific storage of information for the L-value calculation, leading generally to much smaller T . As described in [8], this separated candidate processing enables the application of MMSE detection without BER performance degradation (in contrast to LSD). The resulting TSD achieves much better BER performance than LSD at a significantly reduced ♯n compared to STS detection. IV. C OMPLEXITY R EDUCTION Even though TSD already represents an effective approach reducing ♯n, it is still possible to further reduce detection complexity by simplifying the enclosed computations. This has direct impact on hardware complexity of a possible detector implementation, especially concerning area, power consumption and, in some cases, memory requirements. In the following, the different techniques used in this work are briefly presented. Detailed information about these approaches can be found in [8], [14] and [15]. A. Preprocessing Approaches Application of layer ordering at the preprocessing stage represents one of the possibilities to reduce ♯n. Detection of the most reliable symbols first minimizes the probability of subsequent wrong decisions. A common approach to apply layer ordering is the inclusion of sorted QRD (SQRD) as proposed in [12], [8]. Additionally, MMSE preprocessing [21], [8] might be used to further decrease ♯n. This can be done by utilizing an extended channel matrix for the (S)QR-decomposition. Following this procedure, the MMSE preprocessing incurs a data dependent bias σ which has to be removed from the metrics previously to the L-value calculation to avoid BER performance degradations [8]. B. Clipping Since the counter-hypotheses for the L-value calculation are generated only from a subset of the leaves examined during the tree search, relevant counter-hypotheses are possibly not found for every bit. To avoid overestimation of the chosen bits′ reliability, the L-values have to be clipped: |Le | ≤ Lmax . In this work a clipping level Lmax is chosen such that the average mutual information at the detector output is maximized [8], [22]. In addition, if paths of the tree leading to leaf metrics which will be clipped are excluded from the search, resulting

♯n is reduced [8]. For a non-iterative detection the radius of the sphere can be therefore limited by the maximum aimed L
value Lmax and the metric of the current hypothesis λ0 cML : 
RClipped = min R, λ0 cML + N0 Lmax . C. Search Sequence Determination (SSD) In [14], the commonly used Schnorr-Euchner (SE) enumeration as well as all the costly metric calculations (3) and sorting operations it requires are replaced by few inexpensive basic operations through a heuristic technique. This approach is based on geometrical position analysis relative to reference nodes, reducing the node selection task to simple geometrical case differentiation.′′ Reference nodes x ˆi are determined by y x ˆi = ⌊yi′′′ ⌉ = ⌊ riii ⌉, with (⌊·⌉) representing the rounding operator. Resulting enumeration is defined by fixed sequences which are mapped to constellation symbols during the detection, based on the relative position between yi′′′ and x ˆi . Computational complexity of this strategy can be further decreased by reducing the amount and length of considered sequences. As proposed in [14], in this work only two decision regions (i.e. two sequences) have been considered, sequences of 14 partially combined elements are used and adapted leaf sequences are applied. D. Metric Estimation (ME) The computationally costly operations required for the metric calculation (3) can be considerably simplified by an estimation based on the SSD geometrical approach [15]. Euclidean distances in (3) are replaced by predefined geometrical distances d defined for each decision region: 2

2

2 2 2 d . kyi′′′ − xˆi k 7→ rii |yi′′ − rii x ˆi | = rii

(4)

2 rii

Moreover it is possible to precalculate as well as 2 2 rii d , hence reducing a metric calculation to a single (real) multiplication or even completely dissolving it (excluding the addition of the parent metric in (3)). This considerable reduction in computational complexity makes this technique especially interesting for realizable hardware implementations. V. E FFICIENCY AND F LEXIBILITY A NALYSIS In addition to evaluating BER performance and ♯n, in this work a novel efficiency analysis is applied. For a given BER performance, the tree search will determine a set of counter-hypotheses for the L-value calculation, as described in section III-A. Hence, only a subset of the searched nodes belong to tree paths representing relevant counter-hypotheses. The amount of nodes belonging to these paths, excluding leaf nodes, represents therefore the minimum number of nodes ♯nmin which have to be extended to determine the corresponding counter-hypotheses. In relation to this, ♯λmin i is defined as the amount of layered metric computations λi which have to be performed to provide the metrics λ0 of the found (counter-)hypothesis. The efficiency of the algorithm can be thus measured by means of the amount of information unnecessarily processed (or redundant information), i.e. the overhead rate ∂:

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corresponding to the desired BER performance. A. Case Study: 64-QAM, 4 × 4 MIMO Fig. 2 compares ♯n, ♯nmin and ∂ versus Eb /N0 at 10−5 BER, for the STS, LSD, TSD and the TSD-SSD-ME approach focused in this work. The STS and LSD/TSD detection accuracy is adapted by varying the Lmax and T respectively, according to table I. As depicted in Fig. 2 TSD shows a significant reduction of ♯n compared to STS and LSD. At e.g. Eb /N0 = 14dB, STS extends 3 × ♯nT SD and 2 × ♯nT SD−SSD−ME . LSD extends ≈ 17% less nodes than STS, which still represents a high ♯n compared to TSD approaches (♯nLSD = 2.5 × ♯nT SD , 1.7 × ♯nT SD−SSD−ME ). TSD suffers a SNR degradation of ∼0.5dB when SSD and ME strategies are applied (TS-SSD-ME). This results mainly from the computational error incurred by estimating λi (4). Degradation caused by SSD enumeration strategy as well as by the remaining strategies for complexity reduction (IV) was shown to be irrelevant [8], [14]. Due to the significant reduction in hardware complexity provided by these strategies, this is considered an acceptable loss on behalf of an efficient detector implementation. TSD approaches examine much fewer vain nodes than LSD and STS, as can be concluded by comparing the corresponding ♯n and ♯nmin in Fig. 2. As expected from this, TSD overhead rate is significantly lower (∂ST S = 5 while ∂T SD−SSD−ME = 2 at Eb /N0 = 14dB). As shown in Fig. 3 at SNR&14.4dB, ♯n extended by TSD-SSD-ME is even smaller than the minimum number of metric calculations TABLE I: Clipping values Lmax and tuple size values T used in the simulations for STS and LSD/TDS respectively.

∞ 512

8 256

5 64

14

14.5 b

determined from the ratio between ♯n and ♯nmin corresponding to the STS approach [8]. STS is taken as reference since it finds all arg min {λ0 } within the search sphere

Lmax(ST S) T(LSD/T SD)

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Fig. 2: ♯n, ♯nmin and ∂ corresponding to different detection strategies, required for 10−5 BER using a 64-QAM, 4 × 4 MIMO system

∂=

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λmin required, on average. This behavior is enabled by the i parallel processing of sibling nodes and additionally of leaf nodes, as mentioned in section III-B, and further detailed in [19]. At higher Eb /N0 , LSD and STS still extend ∼3-4 times more nodes than the minimum required, while TSDSSD-ME ∂ tends to ∼1. A considerably higher efficiency than common state-of-the-art algorithms is thus achieved by TSD(SSD-ME), reaching near 100% efficiency at high SNR. With regard to the given scenario and for a reasonable work scope (SNR ≤14dB), further significant reduction (≥ factor 1/2) of ♯n can not be therefore expected by refining depth-first search strategies. B. Flexible Detector: Q-QAM, NT × NT MIMO Fig. 4 depicts ♯n ⇔ Eb /N0 relationship required for 10−5 BER and corresponding to different transmission schemes with TSD-SSD-ME detection. As shown in previous section, this relationship can be adapted by varying T and, for a sufficiently high value (T = 512), TSD BER performance limit is reached. Based on this, the ♯n ⇔ Eb /N0 relationship has been individually analyzed for each transmission scheme and T has been selected to ensure a SNR performance loss ≤0.2dB with respect to the TSD BER limit. Results corresponding to unclipped (Lmax = ∞) STS detection are also depicted, as reference of full max-log-APP detection BER performance bound. As shown in Fig. 4, significant reduction of ♯n is achieved by TSD-SSD-ME compared to STS for different Q (4×4 MIMO transmission). TSD extends ∼1 order of magnitude less nodes than STS detector in the case of a 4QAM constellation, ∼2 orders of magnitude for the 16-QAM case and >3 orders of magnitude for a 64-QAM scenario. In addition, an extension rate ♯n/bit ≈ 1.6 nodes/bit was shown to be nearly constant for all Q, thus becoming almost independent of the constellation size. The enormous reduction of ♯n is achieved at the cost of ∼0.7-1dB SNR performance loss relative to the STS approach, mainly caused by the metric estimation strategy. Similar results can be observed in Fig. 4 for different NT = NR antenna configurations. The proposed TSD extends ∼2 orders of magnitude less nodes than STS for

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Fig. 5: Eb /N0 ⇔ ♯n relationship of TSD-SSD-ME for different transmission schemes using asymmetric NT > NR MIMO.

NT = 2, >3 orders for NT = 4 and ∼6 orders for NT = 8. SNR performance loss of ∼0.5-0.75dB is observed (due to estimation of the metrics) with respect to STS. As depicted in Fig. 4 for different antenna configurations and for the case of 4 and 16-QAM constellations, the focused TSD extends even less nodes than the minimum amount required by STS (♯nmin ST S ). Through this analysis it has been consequently shown that the benefit in ♯n reduction provided by the proposed TSD grows exponentially with the size of the constellation and of the MIMO antenna system, while the SNR performance loss keeps approximately constant. This fact makes TSD strategy especially interesting for high data rate scenarios, where it provides greater benefit. The proposed TSD-SSD-ME approach has been shown to achieve ♯n reduction between 1 and more than 5 orders of magnitude in comparison to STS for the considered transmission schemes, while maintaining an acceptable BER performance. Resulting growth of ♯n is shown to be linear with Q and approximately linear with NT , in contrast to the exponential trends typical of STS. In addition to this, TSD achieves near 100% efficiency in different transmission situations, assuming T conveniently adjusted. Consequently, the benefit provided by TSD in the case of 64-QAM, 4×4 MIMO (V-A) is also present in a wide range of possible transmission configurations.

spatially multiplexed MIMO systems is found at the preprocessing stage. The channel matrix is typically rotated onto a space such that the spatially multiplexed signals are only sequentially dependent. Application of successive interference cancelation is then possible. In this work (S)QR decomposition is used. Same difficulties affect nevertheless other widely used transformations like Cholesky factorization [17]. H corresponding to a NT > NR system is not square, thus impeding a direct matrix inversion. This problem can be avoided by applying a convenient unbiased MMSE strategy and adapting the utilized submatrixes compared to [21]. As described in section IV-A, tuple search together with separate candidate processing enable the application of MMSE without loss of BER performance. As a result, by extending the channel matrix and adapting QRD and y ¯ ′ = QH y ¯ accordingly (5), (S)QR decomposition of the originally undetermined system is enabled.

C. Flexible Detector: Q-QAM, NT × NR (NT > NR ) MIMO The asymmetric transmission scheme where NT > NR has recently received considerable attention, especially within the context of multiuser transmission. E.g. base stations require sufficient number of antennas to fulfill the high data rates demanded by users while mobile terminals call for small number of antennas to reduce size, power consumption and hence cost [17]. Many of the already performed analyses claim sphere detection is not suitable in such systems, at least in its current form [17], [18]. In order to contrast this fact, we have extended the analysis of the proposed low-complex TSD, covering this problematic case. One of the main difficulties with regard to NT > NR ,

" # " #" # # " yNR ×1 HNR ×NT RNT QNT Q′′ ¯ H= σI = Q′ Q′′′ 0 , y ¯= 0 NT NR ×NT (NT −NR )×1

(5)

An additional difficulty is observed in those cases where NT is significantly greater than NR (e.g. 8×4). Zero entries in the diagonal of the NR × NR -submatrixes contained in H lead to infinite values when inversion of these entries is applied for (S)QRD. To solve this problem, these zero entries have been replaced by a very small value γ = 10−6 , as illustrated by eq. (6) for a sample 4×2 system. Similar approach was applied in [17].   h1,1  h2,1   σ H′ =   0  γ 0

h1,2 h2,2 0 σ 0 γ

... ... γ 0 σ 0

h1,4 h2,4 0 γ 0 σ

     

(6)

(NT +NR )×NT

Figure 5 shows ♯n ⇔ Eb /N0 relationship to achieve 10−5 BER using TSD-SSD-ME with the proposed modified MMSE approach. 8×4 and 4×2 MIMO configurations have been considered since they represent the worst case scenario (NR = 12 NT ). Results are compared to the 8×8 and 4×4

systems respectively, representing the better BER performance achievable in each case. T has been fixed to 8 for simplicity. By halving NR respect to NT and considering Q = 4, ∼50% SNR degradation is observed for the 8×4 system. Regarding the 4×2 scheme a marginally inferior loss is observed. Using Q = 16, slightly greater (55%) SNR degradation is shown. Considering the poor BER performance achieved in the case of 16-QAM constellations, it is not necessary to further analyze higher data rate systems (64-QAM), since unacceptable BER performance is directly expected. For low order constellations (Q = 4) halving NR respect to NT leads to a ∼10% increase of ♯n. For higher order (Q = 16), ♯n increases considerably, especially for high order systems (NT = 8). It should be noticed that the BER performance provided by a NT ×NT , 16QAM system is almost equivalent to achieved by a NT ×NT /2 4-QAM configuration. ♯n of the latter is ∼15% and ∼35% smaller for NT = 4 and NT = 8, respectively. Based on the proposed adaptations, TSD has therefore demonstrated to be applicable to asymmetric MIMO systems with NT > NR . Moreover, for a worst case scenario where NR = 12 NT , acceptable Eb /N0 at low ♯n has been shown for low order systems. VI. C ONCLUSION In this work a low-complex TSD is presented and exhaustively analyzed. A novel overhead indicator is introduced, representing the amount of redundant information processed. In this concern, TSD has shown considerably higher efficiency than state-of-the-art algorithms, reaching near 100% efficiency in high SNR scenarios. As a consequence, further dramatic reduction of ♯n can not be expected via refined search algorithms. Flexibility of the presented detector has been also demonstrated and analyzed, showing ♯n to grow linearly with Q and approximately linearly with NT . In addition, the benefit in reduction of ♯n provided by TSD has been shown to increase with NT and Q. In order to handle asymmetric NT > NR MIMO transmission, a modified MMSE strategy has been introduced. Based on this, TSD has shown low ♯n in worst case scenarios where NR = 12 NT . Through the presented analysis, superiority of the proposed TSD(SSD-ME) has been shown in terms of ♯n ⇔ Eb /N0 . Flexibility and high efficiency of the detector have been also demonstrated. As a result, the proposed detection approach represents a promising candidate for MIMO detection in a wide range of applications. ACKNOWLEDGMENT This work has been supported by the German Federal Ministry of Education and Research (BMBF) under grant 01BU1001. The authors would like to thank the project partner Blue Wonder Communications (Dresden, Germany) for an excellent cooperation. R EFERENCES [1] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Transactions on Information Theory, vol. 45, pp. 1639– 1642, Jul. 1999.

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