Samuel Georges

César Nicolas Bailly

Marlu C Steil

Yann Bultel

Abdelkhader Hadjar

Jean-Paul P Viricelle

Mathilde Rieu

SOFC long term operation in pure methane by gradual internal reforming

Introduction

The operation of SOFC in different carbon-based fossil fuels (natural gas, syngas) or renewable fuels (biogas, waste fuels, bioethanol), is a huge actual challenge whose success will condition the development and large scale application of fuel cells in the future. The first step to be achieved is the demonstration of stable operation in pure hydrocarbons, with high performances and without need of addition of reforming gas (H 2 O, CO 2 , O 2 ...) or carrier gas, for simplicity, flexibility, reliability and cost considerations. The demonstration of the robustness of the system in these conditions is essential while the investigation of the other limitations (sulfur poisoning, siloxanes contamination...) is carried out in parallel. If the system is demonstrated to be stable in pure hydrocarbon, the fuel flexibility can then be investigated. SOFC with original cell configurations (anode barrier or catalysts) or new anode materials have been operated or modelled in various hydrocarbons (1)(2)(3)(4)(5)(6). In this paper, after definition (7), simulation (8)(9) and experimental demonstration of the Gradual Internal Reforming concept in methane (10)(11) and in ethanol (12), we report the design and long term operation in gradual internal reforming of an anode-supported solid oxide fuel cell including a catalytic layer in pure methane.

Experimental

The fuel cells were prepared from commercial YSZ/NiO-YSZ (Forschungszentrum Julich) half cells. Composite cathodes, deposited by screen-printing, were made of a LSM:YSZ (60:40) layer, 35 µm thick covered by a 15 µm thick pure LSM layer annealed in two steps at 1200°C in air. An additional gold mesh was screen-printed onto the cathode surface and annealed at 900°C in air to improve cathodic collect. The details of the screen-printing process will be found elsewhere (13)(14). Deposition and microstructure were optimised up to an ASR of 0.48 Ω.cm 2 at 1073 K, using symmetrical cells in air. Due to the configuration of the cell including a catalytic layer on the top of the anode, an original anodic collecting system had to be designed. The development of these original anodic collectors will be detailed in a forthcoming paper. In this study, a Pt mesh shaped anodic current collector was deposited onto the oxidized cermet surface by screen-printing (Aurel, C.890) from commercial ink (ESL 5545) using a mesh 180 mask. A gold ring was fixed and pasted on the top of the mesh using the same ink as the mesh. Two wires were connected to this ring, one for potential monitoring, one for current flow. The annealing was made first at 300°C in air for 15 minutes to remove solvants and organics (as indicated by TGA), and then at 900°C in N 2 for 15 minutes. These thermal treatments were demonstrated in a preliminary study to have no effect on the cathode properties.

The anodic side of the fuel cell was then covered by a 0.1 wt% Ir-CGO catalyst coating, 200µm thick deposited by 3D controlled spray coating. This optimal thickness was evaluated from simulation studies (8)(9). The cell, 19 mm in diameter with an electrochemically active surface given by the cathode surface of 0.79 cm 2 , was inserted in a 3 atmospheres setup using an alumina ring. Gas tightness and sealing were achieved using specific glass and gold rings. More details on the experimental setup are available elsewhere (10)(11). After glass sealing in air at 860°C, the anode was gradually reduced in Ar/H 2 mixtures at 800°C. The OCV stabilized at 1.13 V. This value, close to the theoretical Nernst potential, confirms the gas tightness of the system. Electrochemical characterisation of the cells was performed into pure or Ar-diluted H 2 and/or CH 4 with flow rates controlled by calibrated mass flowmeters, using Solartron 1250 frequency response analyser and Solartron 1287 electrochemical interface. The microstructural characterisation was made with a Zeiss Ultra 55 FEG scanning electron microscope.

In this paper, the fuel utilization is assimilated to the faradaïc efficiency as defined by the Faraday's law, considering 2 electrons for H 2 and 6 for CH 4 (CO is usually not oxidized at the anode, as demonstrated (10)(11).

Results and discussion

The operation principle of the fuel cell described in this paper, Gradual Internal Reforming with a catalytic layer in dry methane, sketched in figure 1, is based on the two fastest reactions, self sustained in the system, i.e. electrochemical oxidation of H 2 [1] and catalytic steam reforming of CH 4 [2].

H 2 + O 2-→ H 2 O + 2e - [1] CH 4 + H 2 O → CO + 3H 2 [2]
The steam produced by electrochemical oxidation of hydrogen at the triple phase boundaries is used gradually along the fuel stream to complete the catalytic steam reforming of methane in the 0,1wt% Ir-CGO catalytic superficial layer (figure 1). Under load conditions and at steady state, there is no need of external steam supply if a sufficient quantity is released from the operating anode by [1], and provided that the catalytic reaction of CH 4 conversion into H 2 and CO over Ir-CGO catalysts is fast and complete, as already demonstrated (15). Based upon these considerations, looking at [1] and [2] stoichiometry, and without considering carbon monoxide which is not oxidized electrochemically at the anode, as demonstrated by electrochemistry/GPC coupled experiments (11), the simplest model of cell operation consists in assuming that the oxidation following [1] of only one of the three H 2 molecules obtained from [2], forms one water molecule. Besides, only one water molecule is needed to ensure the conversion of methane following [2]. In other words, a fuel utilization of 1/3 (based on a global faradaïc efficiency of 1) is theoretically sufficient to release enough steam to convert the entire methane stream in the catalyst layer, complete the catalytic reforming reaction, and avoid the contact between CH 4 and Ni (and subsequent carbon cracking). Below 1/3, the system will not release enough steam to ensure itself complete conversion, and methane will diffuse into the cermet yielding fast coking. More details concerning this concept will be found in our previous work (7)(8)(9)(10)(11)(12). This simplified principle mechanism was applied here study and the faradaïc efficiency was first optimized in H 2 above 1/3 by adjusting the experimental parameters (fuel concentration and flow rates). The gradual consumption of methane following the radial anodic stream (10)(11) ensures the delocalisation of the high endothermicity of steam reforming (∆H=206 kJ.mol -1 ).

During the first polarisation of the cell, if CH 4 is used as the fuel, steam is absolutely required to initiate the catalytic reaction. Without steam (or other reforming gas), methane diffuses into the cermet and yields immediate coking. Consequently, initiation was made in H 2 , as already reported for electrolyte supported cells (10)(11). When a steady state is achieved, the continuous steam release allows the direct and abrupt switch from H 2 to dry CH 4 as the fuel, provided that current (and then steam release) is maintained constant to avoid interruption in the catalytic conversion, CH 4 diffusion into the cermet and subsequent coking. During the fuel switch, fuel flows are adjusted in order to provide to the anode a theoretical constant number of electrons considering [2], where only H 2 is oxidised (see above). According to [1] and [2], H 2 and CH 4 give theoretically 2 and 6e -respectively. This hypothesis is confirmed below.

Based upon these considerations, when the fuel was switched from H 2 to CH 4 , CH 4 partial flow rate was divided by 3 to maintain a theoretical number of electrons provided to the anode, and thus maintain the current density. These two basic principles summarized in Figure 1, were applied for the experimental demonstration and durability tests of anode supported cells operating in Gradual Internal Reforming of dry methane with electro-catalytic dissociation in two different layers. The advantages of Gradual Internal Reforming with the use of a catalytic layer (delocalisation of the endothermicity, no water or reforming gas in the fuel, no carbon deposition, performances and durability increase) were already reported (10). The catalytic layer has no electrical function, and the cermet has no catalytic role towards the hydrocarbon conversion. Accordingly, the best materials can be used for each function, leading to the so-called electrocatalytic dissociation principle. In this study, specific anodic current collectors were designed between the anode and the catalyst by screen printing (see figure 1). The development of these collectors using different metals (Au, Pt, Cu) will be reported in details in another paper. In particular, neighbouring power densities were obtained in the same conditions (diluted hydrogen) with all of those metals.

A typical polarisation curve obtained at 800°C in H 2 (2 L.h -1 ) with a Pt screen-printed collector is given in figure 2. Current density (A.cm -2 )

The maximum power density is 0.17 W.cm -2 at 0.6V, whereas 0.3 W.cm -2 (~0.5 A.cm -2 at 0.6V) were achieved by increasing the fuel flow rate, but with a lower fuel utilization. The performances of the fuel cell are probably diffusion-limited in a certain proportion by the catalytic layer as pointed out (2), but this aspect was not considered here. Due to the limitations mentioned above, the polarisation curves cannot be recorded in CH 4 . Accordingly, the fuel utilization was optimised using H 2 as the fuel, following the principles described above. This study is summarized in figure 3a for Ar-H 2 mixtures. For a given H 2 concentration, the fuel utilization decreases when the flow rate increases, and for a given flow rate, the fuel utilization decreases when H 2 concentration increases, in spite of significant increase of the current density. It means that the fuel cell efficiency is better for concentrated fuels with a low flow rate. Here, the best compromise was obtained with a total flow rate of 0.5 L.h -1 with 30 % H 2 in Ar, i.e. a partial flow rate of 0.15 L.h -1 for hydrogen, yielding fuel utilization of about 38 % and current density of 0.16 A.cm -2 . Using these conditions, long term stable operation in pure methane was achieved. As indicated above, the cell was gradually (0.5 V.min -1 ) polarised at V = 0.6V in diluted H 2 (figure 3b). After 30 minutes, at steady state, the fuel mixture was turned to 0.05 L.h -1 of methane (i.e. 1/3 of initial H 2 flow rate, see above), maintaining a constant total flow rate of 0.5 L.h -1 (10 % CH 4 ) for 25 hours (see figure 3b).

After this fuel switch, The current density remained almost at its previous value, which confirms the choice of the 1/3 ratio related to the 6e -provided by CH 4 (see above). After a second step of 25 hours with 20 % CH 4 (with a constant CH 4 partial flow rate) Ar was suppressed. Methane flow rate was manually adjusted for another 30 minutes to maintain a constant current density. The increase of the current density when the concentration of the fuel is increased is consistent with the behaviour observed in figure 3a in hydrogen. The increase of CH 4 concentration increases the fuel utilisation. The cell was then left for 1800 hours in pure methane (figure 3b). A gradual linear decrease of the performances of 15 %/1000 h. was observed. It is to be mentioned that with this degradation, the fuel utilization was slightly below the 1/3 theoretical limit. The cell was unfortunately stopped after 1800 hours (2 ½ month) of continuous operation for plant power supply maintenance. The shut down was made following the invert procedure. The fuel was turned to H 2 , maintaining the current density, left for 24 hours and then gradually depolarized. No carbon was observed at the anode or detected by EDX, no degradation of the anodic components could be observed (see figure 4).

The catalyst layer delaminated during the cell cooling and disassembly. The screenprinted Pt collector shows a very close contact with the cermet nickel, but no degradation of the components or interfaces can be evidenced from figure 4. The main modification was observed at the surface of the grains and along the grain boundaries of the electrolyte, with the apparition of sub-micrometric cracks and pores.

The degradation of the performances was then tentatively attributed to intrinsic ageing, mainly due to gradual increase of the electrolyte ohmic losses, as already mentioned for YSZ electrolysis cells long term operation (16) with higher current density, and evidenced by SEM as the presence of nanoporosity in YSZ grains, together with connexion loss in between grains. Ageing of the test bench, especially of the sealing glass can also explain this significant degradation.

Complete cell design procedure, operation and post-mortem analysis will be reported in a forthcoming paper.

Conclusion

This study demonstrates the ability of SOFC systems with decoupled electrochemical and catalytic components and functions, to operate durably in pure methane without supply of reforming gas by Gradual Internal Reforming, provided that the following conditions are respected:

-Initiation of operation and then steam release without carbon (in H 2 for example) -Adjustment of the flow rates H 2 /CH 4 to maintain the current density -Optimization of the fuel utilization, to prevent carbon deposition conditions

Figure 1 .

 1 Figure 1. Sketch of the fuel cell architecture and schematic principle of Gradual Internal Reforming associated with Electrocatalytic Decoupling with screen-printed Internal Anodic Collecting System

Figure 2 .

 2 Figure 2. Polarisation curve of the fuel cell with Pt screen-printed anodic collector and catalytic layer obtained in H 2 at 800°C

Figure 3 .

 3 Figure 3. (a) Variation of H 2 utilization rate (for a faradaïc efficiency of 1) at 800°C in Ar-H 2 mixtures as a function of concentration and total flow rate; numbers indicate the current density; the red point shows the operation conditions of the experiment of Figure 3b; (b) potentiostatic test at 800°C and V = 0,6V in pure dry methane for 1800 hours after initiation in H 2 and gradual suppression of Ar (carrier gas)

1 T

 1 h Ar + 30% H 2 II. 25 h Ar + 10% CH 4 III. 25 h Ar + 20% CH 4 IV. 1800 h pure dry CH 4 H 2 / 0.15 L.h -1 CH 4 / 0.05 L.h -

Figure. 4 :

 4 Figure. 4: SEM micrographs after 1800 hours in pure CH 4 of the complete cell (a); Detail of the anodic current collector

  This architecture allows the use of the best anode and catalyst at the same time. These results open the perspectives of fuel flexibility for SOFC. Demonstration has already been made for H 2 /CH 4 /C 2 H 5 OH[6].

Acknowledgments

The authors acknowledge ADEME (program MACSOFC 0874C0130) and GDF SUEZ for grant and financial support


Ming Liu
email: ming.liu@insa-rennes.fr

Matthieu Crussière
email: matthieu.crussiere@insa-rennes.fr

Maryline Hélard
email: maryline.helard@insa-rennes.fr

Jean-François Hélard
email: jean-francois.helard@insa-rennes.fr

Achieving Low-Complexity Maximum-Likelihood Detection for the 3D MIMO Code

Keywords: MIMO, space-time codes, maximum likelihood decoding, computational complexity

The 3D MIMO code is a robust and efficient space-time block code (STBC) for the distributed MIMO broadcasting but suffers from high maximum-likelihood (ML) decoding complexity. In this paper, we first analyze some properties of the 3D MIMO code to show that the 3D MIMO code is fast-decodable. It is proved that the ML decoding performance can be achieved with a complexity of O(M 4.5 ) instead of O(M 8 ) in quasi static channel with M -ary square QAM modulations. Consequently, we propose a simplified ML decoder exploiting the unique properties of 3D MIMO code. Simulation results show that the proposed simplified ML decoder can achieve much lower processing time latency compared to the classical sphere decoder with Schnorr-Euchner enumeration.

Introduction

Multiple-input multiple-output (MIMO) is a promising technique that can bring significant improvements to the wireless communication systems. In combination with space-time block code (STBC), it provides higher spectrum efficiency with better communication reliability [START_REF] Tarokh | Space-time codes for high data rate wireless communication: Performance criterion and code construction[END_REF]. In the last decades, MIMO has been widely employed in the latest wireless communication standards such as IEEE 802.11n, 3GPP Long Term 1 Evolution (LTE), WiMAX and Digital Video Broadcasting-Next Generation Handheld (DVB-NGH) etc. It is also seen as the key technology for the future digital TV terrestrial broadcasting standards [2].

A so-called space-time-space (3D) MIMO code [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in singlefrequency networks[END_REF] was proposed for the future TV broadcasting systems in which the services are delivered by the MIMO transmission in a single frequency network (SFN). Specifically, it is proposed for a distributed MIMO broadcasting scenario where TV programs are transmitted by two geographically separated transmission sites, each site equipping two transmit antennas. On the other hand, each receiver has two receive antennas, forming a 4 × 2 MIMO transmission. The 3D MIMO code has been shown to be robust and efficient in the distributed MIMO broadcasting scenarios where there exists strong received signal power imbalances [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF]. Hence, it is a promising candidate for the MIMO profile of future broadcasting standards. However, the 3D MIMO code suffers from a high computational complexity when the maximum-likelihood (ML) decoding is adopted. The decoding complexity is as high as O(M 8 ) when M -QAM constellation is used. Up to now, no study on the decoding complexity reduction for the 3D MIMO code has been carried out in the literature.

Recently a lot of efforts have been made in the STBC design to obtain both high code rate and low decoding complexity [START_REF] Sharma | Full-rate full-diversity linear quasi-orthogonal space-time codes for any number of transmit antennas[END_REF][START_REF] Dao | Four-Group Decodable Space-Time Block Codes[END_REF][START_REF] Biglieri | On fast-decodable space-time block codes[END_REF][START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF][START_REF] Sinnokrot | Embedded Alamouti space-time codes for high rate and low decoding complexity[END_REF][START_REF] Ren | Fast-group-decodable space-time block code[END_REF][START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF]. The decoding complexity reduction is commonly achieved by exploiting the orthogonality embedded in the STBC codeword. When there exists group-wise orthogonality in the codeword, the joint detection of many information symbols is converted into independent, group-wise detections [START_REF] Dao | Four-Group Decodable Space-Time Block Codes[END_REF][START_REF] Ren | Fast-group-decodable space-time block code[END_REF], yielding low decoding complexity. For other cases such as DjABBA code [START_REF] Hottinen | Multi-antenna Transceiver Techniques for 3G and Beyond[END_REF], Biglieri-Hong-Viterbo (BHV) code [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], Srinath-Rajan code [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF] and Ismail-Fiorina-Sari (IFS) code [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF] in which the orthogonality only exists in a part of information symbols, some symbols can be detected in a group-wise manner once we condition them with respect to other symbols. The ML solutions can be obtained with a lower complexity compared with the ML detector. In other words, their decoding complexity is less than O(M κ ) where κ is the number of information symbols in a codeword. Such kind of STBCs are referred to as fast decodable STBCs [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF].

However, most of fast-decodable STBCs are not optimized for distributed MIMO broadcasting scenarios and they are not robust under the received signal power imbalance conditions [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF].

A partial interference cancellation (PIC) group decoding scheme has been presented aiming at reducing the decoding complexity of the STBCs containing group-wise orthogonalities in the codewords [START_REF] Dai | An efficient detector for combined space-time coding and layered processing[END_REF][START_REF] Guo | On full diversity space-time block codes with partial interference cancellation group decoding[END_REF]. A number of STBCs that are optimized for this decoding scheme have also been proposed [START_REF] Guo | On full diversity space-time block codes with partial interference cancellation group decoding[END_REF][START_REF] Zhang | Two designs of space-time block codes achieving full diversity with partial interference cancellation group decoding[END_REF]. This scheme actually uses a linear equalization to convert the joint detection of a large number of symbols to several groups of ML decodings for few symbols. However, the overall performance of this decoding scheme cannot achieve the ML optimality. Some alternatives with reduced decoding complexity have been presented for the distributed MIMO broadcasting. Polonen et al. described a STBC with less decoding complexity based on orthogonal basis [START_REF] Polonen | Reduced complexity space-time coding in single-frequency networks[END_REF].

However, such a code does not achieve full-diversity or full-rate for 4 × 2 MIMO transmissions and therefore performs worse than 3D MIMO code. A "punctured version" of 3D MIMO code that is full-rate for 4 × 2 MIMO transmissions with low decoding complexity has also been proposed [START_REF] Liu | Distributed MIMO coding scheme with low decoding complexity for future mobile TV broadcasting[END_REF]. However, it does not achieve full-diversity and is hence less robust in harsh channel conditions.

In this paper, we propose a reduced-complexity ML decoder for the 3D MIMO code which exploits the embedded orthogonality in the codeword. The main contributions are:

• We propose to modify the original 3D MIMO codeword through some permutations of information symbols which leads to an ML decoding algorithm with reduced complexity without affecting all desirable properties of the 3D MIMO code.

• We prove that the 3D MIMO code is fast decodable. Moreover, we show that the worst case decoding complexity is O(M 4.5 ) for M -ary square QAM modulations which is the least among all square full-rate STBCs for 4 × 2 MIMO transmission.

• Based on the unique properties of the new form of 3D MIMO codeword, we propose a novel implementation of the simplified decoder that achieves a lower average complexity in terms of time latency without losing the ML optimality. The proposed implementation is also applicable for other fast decodable STBCs.

The remainder of the paper is organized as follows. Some fundamentals of the MIMO detection are presented in Section 2. In Section 3, the 3D MIMO code is first recalled. Consequently, a modification of the codeword is proposed to facilitate the decoding process. Three important properties of the new codeword are also revealed. With this knowledge, in Section 4, the ML decoder with a worst case decoding complexity of O(M 4.5 ) is derived. Then in Section 5, a new implementation of the reduced-complexity ML decoder is described. Section 6 presents the symbol error and complexity performance of the new decoder. Conclusions are drawn in Section 7.

Notations: Vectors and matrices are written in boldface letters. Superscript X T represents transposition of matrix X. x R and x I denote the real and imaginary parts of a complex number x, respectively. The operator (•) performs the complex-real conversion from C to R 2×2 :

x x R -x I x I x R . (1)
When (•) operator is applied to a matrix X ∈ C m×n , the operation in ( 1) is performed for all elements x j,k 's in the matrix, i.e. the (j, k)th 2 × 2 submatrix of X is xj,k . For a complex vector x = [x 1 , x 2 , . . . , x n ] T ∈ C n , the operator (•) separates the real and imaginary parts of the given vector, i.e. x [x R 1 , x I 1 , . . . , x R n , x I n ] T . For a matrix X = [x 1 , x 2 , . . . , x n ] where x j is the jth column of X, the operator vec(X) stacks the columns of X to form one column vector, i.e. vec(X) [x T 1 , x T 2 , . . . , x T n ] T . vec(X) denotes vectorizing matrix X followed by the real/imaginary part separation. The inner product of two real-valued vectors x and y is denoted by x, y = x T y. The n × n identity matrix is denoted by I n . The operator ⊗ denotes the Kronecker product.

Finally, i represents √ -1.
2 System Model

MIMO system model

We consider a MIMO transmission with N t transmit and N r receive antennas over flat-fading channel. The received signal Y ∈ C Nr×T is:

Y = HX + W, (2)
where X ∈ C Nt×T is the STBC codeword matrix which is transmitted over T channel uses; W ∈ C Nr×T is a complex-valued additive white Gaussian noise (AWGN) component; H ∈ C Nr×Nt is the channel matrix whose (j, k)th element h j,k denotes the channel coefficient of the link between the kth transmit antenna and the jth receive antenna. The channel is assumed to be quasi-static. That is, the channel coefficients keep constant over the duration of one STBC codeword, but change from one codeword to another. Moreover, h j,k 's are assumed to be independent from each other.

For linear STBCs, the codeword matrix X can be obtained through a linear operation [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF]:

vec(X) = G s, (3)
where s = [s 1 , s 2 , . . . , s κ ] T is the vector containing κ independent information symbols. The code rate of STBC is κ/T information symbols per channel use. The generator matrix G ∈ R 2NtT ×2κ is obtained:

G [ vec(A 1 ), vec(B 1 ), . . . , vec(B κ )], (4)
where A j ∈ C Nt×T and B j ∈ C Nt×T are the complex weight matrices representing the contribution of the real and imaginary parts of the jth information symbol s j in the final codeword matrix.

Separating the real and imaginary parts of the transmitted and received signals, and stacking the columns of the codeword, the received MIMO signal (2) can be expressed in an equivalent real-valued form:

y = H eq s + w, (5)
where y = vec(Y), w = vec(W) and H eq ∈ R 2NrT ×2κ is the equivalent channel matrix and is obtained by:

H eq = (I T ⊗ Ȟ)G. ( 6
)
Note that the real-valued expression of the signal can be obtained from the complex-valued form via a linear transform. Hence, we will jointly use both real-and complex-valued forms in the sequel.

ML decoding of MIMO signals

Once the channel H eq is known by the receiver1 , the information symbols can be retrieved from the received signal y in [START_REF] Sharma | Full-rate full-diversity linear quasi-orthogonal space-time codes for any number of transmit antennas[END_REF]. The maximum-likelihood (ML) solution of the transmitted signal is the combination of information symbols s = [s 1 , s 2 , . . . , s κ ] that minimizes the Euclidian distance between the channel distorted information signal H eq s and received signal y, namely:

ŝML = arg min s∈Θ κ y -H eq s 2 , ( 7
)
where Θ is the set of the constellation symbols. [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] indicates that the ML solution is found by jointly determining κ independent information symbols. In other words, when the modulation of these symbols is M -QAM, the ML decoding should exhaustively check all M κ combinations. The search complexity grows dramatically with higher modulation order or larger number of information symbols in one codeword. Hence, the ML decoding is computationally demanding.

Fast ML decoding of MIMO signals

More efficient STBC decoding is achieved with the help of orthogonal-triangular (QR) decomposition [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF][START_REF] Agrell | Closest point search in lattices[END_REF].

The QR decomposition of the equivalent channel matrix H eq yields H eq = QR, where Q ∈ R 2NrT ×2κ is a unitary matrix, and R ∈ R 2κ×2κ is an upper triangular matrix. The definitions of the "classical Gram-Schmidt algorithm" based QR decomposition can be found in the Appendices. Note that other numerically stable QR decomposition algorithms can also be used without affecting the properties of the 3D MIMO code as well as the resulting low-complexity decoding methods. Instead of solving [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF], the ML solution can be alternatively found by: ŝML = arg min

s∈Θ κ ∩S z -R s 2 , (8)
where z = Q T y ∈ R 2κ is a linear transformation of received signal; S is a hypersphere centered on the received signal. Only the codewords inside the hypersphere are checked during the search in order to reduce the search complexity. The size of the hypersphere is represented by its radius. The decoding process is turned into a bounded search over a κ-level tree with complex-valued nodes. Hence, the worst case decoding

complexity is O(M κ ).
Moreover, according to the property of QR decomposition, some information symbols can be decoded independently from the others if some elements of R are equal to zero. It suggests that the joint search in a high dimension is converted into a bunch of parallel, independent searches in low dimensions. This results in a significant reduction of the worst-case decoding complexity [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF][START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF][START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF].

3D MIMO Code

The 3D MIMO code [START_REF] Nasser | 3D MIMO scheme for broadcasting future digital TV in singlefrequency networks[END_REF] possesses a better robustness against receive signal power imbalances in the distributed MIMO broadcasting scenarios but suffers from high decoding complexity [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF]. In this section, we

propose a new 3D MIMO codeword that enables low sphere decoding complexity via exchanging the positions of information symbols in the original 3D MIMO codeword. The basic idea behind this modification comes from the facts that the orthogonality embedded in the information symbols essentially enables independent detections and the sphere decoding complexity is mainly determined by the orthogonality among the first several symbols. Hence, exploiting the underlying orthogonality in the codeword and carefully choosing the sequence of information symbols can bring benefits in terms of decoding complexity.

A new proposal of the 3D MIMO codeword

The initially proposed codeword matrix of the 3D MIMO code is explicitly written as:

X 3D = X Golden,1 -X * Golden,2 X Golden,2 X * Golden,1 = 1 √ 5     α(s 1 + θs 2 ) α(s 3 + θs 4 ) -α * (s * 5 + θs * 6 ) -α * (s * 7 + θs * 8 ) i ᾱ(s 3 + θs 4 ) ᾱ(s 1 + θs 2 ) i ᾱ * (s * 7 + θs * 8 ) -ᾱ * (s * 5 + θs * 6 ) α(s 5 + θs 6 ) α(s 7 + θs 8 ) α * (s * 1 + θs * 2 ) α * (s * 3 + θs * 4 ) i ᾱ(s 7 + θs 8 ) ᾱ(s 5 + θs 6 ) -i ᾱ * (s * 3 + θs * 4 ) ᾱ * (s * 1 + θs * 2 )     , (9)
where

θ = 1+ √ 5 2 , θ = 1 -θ, α = 1 + i(1 -θ) and ᾱ = 1 + i(1 -θ).
It is constructed in a hierarchical manner: eight information symbols (κ = 8) are first encoded to two Golden codewords [START_REF] Belfiore | The golden code: a 2×2 full-rate space-time code with nonvanishing determinants[END_REF], i.e. X Golden,1 and X Golden,2 , which are consequently arranged in an Alamouti manner [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] over four channel uses (T = 4)2 .

This results in a code rate of 2 which is full-rate for the 4 × 2 MIMO transmission. Previous study shows that the 3D MIMO code achieves efficient and robust performance. However, since eight information symbols are stacked in one codeword, the ML decoding complexity is up to O(M 8 ).

It was shown that it is possible to achieve lower sphere decoding complexity through permuting the sequence of information symbols [START_REF] Jithamithra | Minimizing the complexity of fast sphere decoding of STBCs[END_REF]. We propose to slightly modify the codeword by exchanging the positions of information symbols (s 3 , s 4 ) and (s 5 , s 6 ), yielding a new form of codeword:

X 3D,new = 1 √ 5     α(s 1 + θs 2 ) α(s 5 + θs 6 ) -α * (s * 3 + θs * 4 ) -α * (s * 7 + θs * 8 ) i ᾱ(s 5 + θs 6 ) ᾱ(s 1 + θs 2 ) i ᾱ * (s * 7 + θs * 8 ) -ᾱ * (s * 3 + θs * 4 ) α(s 3 + θs 4 ) α(s 7 + θs 8 ) α * (s * 1 + θs * 2 ) α * (s * 5 + θs * 6 ) i ᾱ(s 7 + θs 8 ) ᾱ(s 3 + θs 4 ) -i ᾱ * (s * 5 + θs * 6 ) ᾱ * (s * 1 + θs * 2 )     . (10)
Since we only change the sequence of the information symbols in the codeword (the third and fourth information symbols become the fifth and sixth, and vice versa) and the information symbols are independent from each other, the new codeword preserves all the good attributes of the original 3D MIMO code in distributed MIMO scenarios illustrated in [START_REF] Liu | Distributed MIMO schemes for the future digital video broadcasting[END_REF]. More importantly, this modification is based on the embedded orthogonalities in the 3D MIMO codeword and yields an interesting codeword structure which will be exploited to achieve lower decoding complexity. The advantages brought by the new codeword structure will be highlighted in the following sections.

Key properties of the proposed 3D MIMO codeword

Due to the underlying Alamouti and Golden structures, the 3D MIMO code has some unique properties which lead to simplified decoding algorithms.

For the modified 3D MIMO code (10) over a 4 × 2 MIMO channel, the R matrix in ( 8) is a 16 × 16 real-valued matrix. Rewrite R in a block-wise form:

R =     R 11 R 12 R 13 R 14 0 R 22 R 23 R 24 0 0 R 33 R 34 0 0 0 R 44     , (11)
where R jk 's are 4 × 4 submatrices containing q m , h n 's with m = 4(j -1) + 1, . . . , 4j and n = 4(k -1) + 1, . . . , 4k.

Based on the new codeword in [START_REF] Ren | Fast-group-decodable space-time block code[END_REF] and taking into account ( 6), ( 3) and ( 4), we can obtain a few interesting properties of R that can be made use of to achieve a low decoding complexity.

Theorem 1. R 11 is an upper triangular matrix with q 1 , h 2 = q 1 , h 4 = q 2 , h 3 = q 3 , h 4 = 0.

Theorem 2. R 12 is a null matrix when the channel is quasi-static, i.e. q j , h k = 0, ∀j = 1, 2, 3, 4 and k = 5, 6, 7, 8.

Corollary 1. R 22 is an upper triangular matrix with a similar structure as R 11 , i.e. q 5 , h 6 = q 5 , h 8 = q 6 , h 7 = q 7 , h 8 = 0. The proofs of Theorem 1, Theorem 2 and Corollary 1 are presented in Appendices. The above properties are visualized in Figure 1.

Remark 1: Theorem 1 and Corollary 1 actually suggest the independency between real and imaginary parts of the information symbols. For instance, q 1 , h 2 = q 1 , h 4 = q 3 , h 4 = 0 means that the real parts of the first and second received symbols, namely z(1) and z(3), do not contain any contribution from s I 1 and s I 2 . Similarly, q 2 , h 3 = 0 means that their imaginary parts, namely z(2) and z(4), do not contain any contribution from s R 1 and s R 2 , either. As we will show later, this real/imaginary independency leads to independent and parallel detections for real part and imaginary part, respectively.

The real/imaginary part independency comes from the underlying Golden and Alamouti structures. It has been revealed that the complex-valued R matrix of the Golden code has a real upper left submatrix [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF],

which coincides with the structure as presented in Theorem 1. It shows the real/imaginary part independency of the Golden code in its 2×2 codeword matrix. The Alamouti-like arrangement of the two Golden codewords, on the other hand, helps creating this independency in the 4 × 4 codeword matrix of the 3D MIMO code.

Remark 2: Theorem 2 indicates that the information symbols s 1 and s 2 are uncorrelated with s 3 and s 4 in the received symbols. It means that these two symbol groups can be determined independently. We only have to jointly determine six information symbols to obtain the ML solutions. In other words, the ML decoding complexity is expected to be O(M 6 ) instead of O(M 8 ). Therefore, the 3D MIMO code is fast decodable. Details on the simplified ML decoding will be discussed in Section 4.

It should be noted that Theorem 2 is partially enabled by the embedded Alamouti structure in the codeword. The channel coefficients should be constant within the duration of one codeword to validate the orthogonalities in Alamouti structure. Hence Theorem 2 is only valid in the quasi-static channels3 .

Comparison with the original 3D MIMO codeword

Figure 1 illustrates the R matrix of the new 3D MIMO codeword. Compared with the original one as shown in Figure 2, the new structure is actually more favorable for the MIMO decoding. In the new codeword, the contributions of information symbol groups (s 1 , s 2 ) and (s 3 , s 4 ) are totally uncorrelated in the received signal, which means that the ML detection of eight information symbols can be achieved by two independent and less complex detections of six information symbols. Moreover, the structure of R 11 and R 22 enables the independent detections of real and imaginary parts of (s 1 , s 2 ) and (s 3 , s 4 ), which leads to further complexity reduction. Yet, this real/imaginary parts separation is not straightforward in the original codeword.

It should be emphasized that the new codeword only changes the sequence of the information symbols in the codeword to facilitate the decoding process. It does not affect all the good properties of the 3D MIMO code.

Proposed ML Decoder with Low Complexity

In this section, a low-complexity ML decoding algorithm exploiting the unique properties highlighted in previous section is proposed for the 3D MIMO code. Generally speaking, the complexity reduction is achieved in two steps. Based on Theorem 2 the joint detection of eight information symbols is converted into two partially independent detections of six information symbols. This step reduces the worst-case decoding complexity from O(M 8 ) to O(M 6 ). Consequently, using Theorem 1 and Corollary 1, the detections of complex information symbols are converted into independent detections of real and imaginary parts, which further reduces the worst-case complexity to O(M 4.5 ).

Group-wise parallel detections

We divide the information symbols and received symbols into four groups, i.e.

a = [s 1 , s 2 ] T , b = [s 3 , s 4 ] T , c = [s 5 , s 6 ] T , d = [s 7 , s 8 ] T , z 12 = [z 1 , z 2 ] T , z 34 = [z 3 , z 4 ] T , z 56 = [z 5 , z 6 ]
T and z 78 = [z 7 , z 8 ] T . Taking into account the structure of R and Theorem 2, the decoding metric in (8) can be rewritten as:

z -R s 2 = z 12 -R 11 a -R 13 c -R 14 d 2 (12)
+ z 34 -R 22 b -R 23 c -R 24 d 2 (13)
+ z 56 -R 33 c -R 34 d 2 + z 78 -R 44 d 2 .
From ( 12) and ( 13), it can be seen that the contributions from the information symbol groups a and b are uncorrelated in the received symbol. For instance, z 12 does not contain any information from b, and z 34 is irrelevant to a, either. This enables us to use group-wise conditional detections to retrieve the ML solutions [START_REF] Sirianunpiboon | Fast optimal decoding of multiplexed orthogonal designs by conditional optimization[END_REF].

In particular, the ML solution ŝML = [â, b, ĉ, d] T is achieved in two search steps, namely a joint "outer"

search for [ĉ, d]:

[ĉ, d] = arg min [c,d]∈Θ 4 v 12 -R 11 a ( * ) (c, d) 2 + v 34 -R 22 b ( * ) (c, d) 2 + z 56 -R 33 c -R 34 d 2 + z 78 -R 44 d 2 , (14)
and two independent "inner" searches for â and b, respectively:

â = a ( * ) (ĉ, d), b = b ( * ) (ĉ, d) (15)
where ). We note that this complexity reduction does not rely on the constellation that is adopted by the information symbols. In other words, the 3D MIMO code requires a worst decoding complexity of O(M 6 ) for arbitrary modulation.

a ( * ) (c, d) = arg min a∈Θ 2 v 12 -R 11 a 2 , (16) b
( * ) (c, d) = arg min b∈Θ 2 v 34 -R 22 b 2 , (17)

Independent detections of real and imaginary parts

If square shape QAM modulations are considered, the decoding complexity can be further improved. The square M -QAM symbol can be separated into two independent √ M -PAM symbols on the real and imaginary axes, respectively. Using Theorem 1 and Corollary 1, the real and imaginary parts can be decoded separately.

Take the detection of a as an example. Denote its real and imaginary parts as

a R = [s R 1 , s R 2 ] T and a I = [s I 1 , s I 2 ]
T , respectively. Given [c, d] and using Theorem 1, the detection of a in ( 16) is rewritten as [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF]:

âR = arg min a R ∈Ψ 2 v R 12 -R R 11 a R 2 , âI = arg min a I ∈Ψ 2 v I 12 -R I 11 a I 2 , ( 18
)
where

Ψ is the set of √ M -PAM constellation symbols; v R 12 = [v R 1 , v R 2 ] T , v I 12 = [v I 1 , v I 2 ] T ; R R
11 and R I 11 are tailored upper-triangular matrices associated with real and imaginary parts, respectively:

R R 11 = R 11 (1, 1) R 11 (1, 3) 0 R 11 (3, 3) , R I 11 = R 11 (2, 2) R 11 (2, 4) 0 R 11 (4, 4) . (19)
(18) means that the detections of real and imaginary parts are similar and can be performed separately.

Take the real part as an example. We apply again the conditional detection here. For a given s R 2 , the metric for the real part detection becomes:

v R 12 -R R 11 a R 2 = w R 1 -R 11 (1, 1)s R 1 2 + (w R 2 ) 2 , (20)
where

w R 1 = v R 1 -R 11 (1, 3)s R 2 and w R 2 = v R 2 -R 11 (3, 3)s R 2 .
For a given s R 2 , the best s R 1 that minimizes the decoding metric can alternatively be found by minimizing a quadratic function of s R 1 given on the right hand side of [START_REF] Belfiore | The golden code: a 2×2 full-rate space-time code with nonvanishing determinants[END_REF]. The best solution of s R 1 is easily found by:

s R 1 = Q v R 1 -R 11 (1, 3)s R 2 R 11 (1, 1) , (21)
where Q(•) is the slicing operation providing the PAM symbol that is closest to the given value. The best

combination of [ŝ R 1 , ŝR 2 ] T given [c, d] is obtained after testing (21) with all ( √ M ) possible values of s R 2 : ŝR 2 = arg min s R 2 ∈Ψ |v R 1 -R 11 (1, 1)s R 1 -R 11 (1, 3)s R 2 | 2 + |v R 2 -R 11 (3, 3)s R 2 | 2 . ( 22
)
Consequently, ŝR 1 is obtained by using the solution ŝR 2 in [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]. Similar process can be applied to solve the imaginary parts. The best solution of [ŝ I 1 , ŝI 2 ] T given [c, d] can be found by:

ŝI 2 = arg min s I 2 ∈Ψ |v I 1 -R 11 (2, 2)s I 1 -R 11 (2, 4)s I 2 | 2 + |v I 2 -R 11 (4, 4)s I 2 | 2 , (23)
where ) O(M 4.5 ) DjABBA [START_REF] Hottinen | Multi-antenna Transceiver Techniques for 3G and Beyond[END_REF] O(M 7) O(M 6 ) Perfect code (2-layer) [START_REF] Oggier | Perfect space-time block codes[END_REF] O(M 6 ) O(M 5.5 ) BHV [START_REF] Biglieri | On fast-decodable space-time block codes[END_REF] O(M 6 ) O(M 4.5 ) EAST [START_REF] Sinnokrot | Embedded Alamouti space-time codes for high rate and low decoding complexity[END_REF] O(M 5) O(M 4.5 ) Srinath-Rajan [START_REF] Srinath | Low ML-decoding complexity, large coding gain, full-rate, full-diversity STBCs for 2×2 and 4×2 MIMO systems[END_REF] O(M 5) O(M 4.5 ) IFS [START_REF] Ismail | A new family of low-complexity STBCs for four transmit antennas[END_REF] O(M 5) O(M 4.5 ) ŝI 1 is computed by applying the solution ŝR 2 in [START_REF] Oggier | Perfect space-time block codes[END_REF]. Using the same technique, the best solutions of b in [START_REF] Liu | Distributed MIMO coding scheme with low decoding complexity for future mobile TV broadcasting[END_REF] can also be converted into independent detections of b R and b I . Substituting R 11 , v 1 , v 2 , s 1 and s 2 in ( 21), ( 22), ( 23) and ( 24 STBCs are presented in Table 1. It can be seen that the 3D MIMO code is among the simplest full-rate STBCs when the square QAM modulations are considered.

s I 1 = Q v I 1 -R 11 (2, 4)s I 2 R 11 (2, 2) . ( 24
)

Proposed Implementation of the Simplified ML Decoder

In the previous sections, we have illustrated the fast decodability of the 3D MIMO code in theory. With this knowledge, we propose an implementation of the simplified ML decoder that can be used in practice. Using the two-stage tree-search structure and leveraging the symmetry structure in the codeword, the proposed implementation requires a low average complexity in practice. Moreover, various performance-complexity trade-offs can be easily achieved by replacing the sphere decoder by other suboptimal tree search algorithms such as K-best algorithm [START_REF] Wong | A VLSI architecture of a K-best lattice decoding algorithm for MIMO channels[END_REF], fixed-complexity sphere decoder [START_REF] Barbero | Fixing the complexity of the sphere decoder for MIMO detection[END_REF] etc.

Two-stage decoding structure

Recall that the fast decodability is achieved by concatenating the joint search of four complex symbols and several detections in parallel. Figure 3 presents the general structure of the proposed simplified ML decoder. Detailed pseudo code is presented in Algorithms 1, 2, 3 and 4 so that the proposed decoder can be implemented without major effort.

4-level tree search phase

The joint detection of [c, d] is realized by a complex sphere decoder with Schnorr-Euchner enumeration, which is visualized by the search over a 4-level tree as shown in Figure 3. The nodes of the same level represent all the solutions of a complex information symbol. Each path from the root to a leaf node represents a possible combination of [c, d].

The details of the tree search is explicitly presented in Algorithm 2. The search starts from the root node and traverses the nodes of lower levels in a depth-first manner. An adaptive search radius is used to speed up the convergence of the algorithm by limiting the search within a hypersphere S. For the node under checking, the partial distance resulted by the current path is compared with the radius. If the partial distance is smaller than the radius, the search moves on to the children nodes on the next level. Otherwise, the search jumps to another sibling node on the current level. When all the nodes of the level have already been checked, the search goes back to the upper level. The radius is initially set to infinity and is adaptively decreased according to the best solution already found in the search. Specifically, the radius is updated taking into account the best combination of [c, d] and [a, b] (line 16 of Algorithm 2). The latter is obtained from the parallel decisions phase. The tree search is terminated when all nodes within the hypersphere have minimum branch distance (τ 1 > p 1 ); or 2) if the overall distance is beyond the current radius of the sphere decoder in the tree search phase ((

τ 1 + d 2 + d 3 + d 4 + d) > radius).
Once the searches on all branches are terminated, the solution [a, b] and the resulting distance d p are returned to the tree search phase. The tree search process is resumed. The overall distance is compared with the current radius (line 14 in Algorithm 2) to determine whether the current solution is a better one.

If a better solution is found, the radius is updated accordingly (line 16 in Algorithm 2). The tree search process is moved on to the next unchecked node.

Column switch based on ZF estimation

In the proposed algorithm, the search of eight symbols is divided into a tree search for four symbols and parallel detections for the other four symbols. Due to the symmetric structure of the codeword matrix [START_REF] Ren | Fast-group-decodable space-time block code[END_REF], some parts of the codewords can be exchanged without changing the properties of the 3D MIMO code. For instance, we have the same properties as illustrated in Section 3 after exchanging the positions of [s 1 , s 2 , s 3 , s 4 ] with [s 5 , s 6 , s 7 , s 8 ]. Similarly, if we exchange [s 1 , s 2 ] with [s 3 , s 4 ] and exchange [s 5 , s 6 ] with [s 7 , s 8 ] simultaneously, the structure of R matrix maintains, as well. That is to say, besides the original symbol sequence, the proposed low-complexity decoding algorithm is also valid with other three permuted symbol sequences, i.e. [s 5 , s 6 , s 7 , s 8 , s 1 , s 2 , s 3 , s 4 ], [s 3 , s 4 , s 1 , s 2 , s 7 , s 8 , s 5 , s 6 ] and [s 7 , s 8 , s 5 , s 6 , s 3 , s 4 , s 1 , s 2 ].

The exchanging of the symbol sequences can be achieved by permuting the corresponding columns in the equivalent channel matrix H eq . Note that, the aforementioned column permutations do not affect the decoding performance. This permits us to choose the symbols that will be determined by the tree search and the ones that will be decoded in the parallel detections.

The proposed column switch method is presented in Algorithm 4. The basic idea is to use the tree search to determine the more difficult half part and use the parallel detections to find the easier half part. The reason behind this idea is that the parallel decoding is more efficient to decode the reliable symbols separately.

The more accurate the linear estimation, the faster convergence speed for each individual detection branch.

On the other hand, the tree search phase is a joint serial detection in nature which is more suitable to decode those unreliable symbols.

The next question is how to properly choose the unreliable symbols. In the literature, Barbero et al. proposed to sort the decoding sequence based on the norm of subchannels in the fixed-complexity sphere decoder [START_REF] Barbero | Fixing the complexity of the sphere decoder for MIMO detection[END_REF]. However, it is not applicable here because the 3D MIMO code achieves full-diversity and the equivalent subchannels have similar norm values. In addition, as we have to maintain the structure of R matrix, the unconstrained subchannel sorting proposed in [START_REF] Wiesel | Efficient implementation of sphere demodulation[END_REF] is not applicable, either.

Alternatively, we propose to sort the information symbols according to the aggregate error of the linear estimation:

ǫ jk = k l=j |ŝ ZF (l) -s ZF (l)| 2 , (25)
where s ZF = H † eq y is the unconstrained estimation of the information symbols in which H † eq represents the inverse of equivalent channel matrix; ŝZF = Q(s ZF ) is the constellation point that is closest to s ZF . The metric is the distance between the estimated information symbols and the nearest constellation points, i.e. an indicator of the estimation accuracy.

Using [START_REF] Wong | A VLSI architecture of a K-best lattice decoding algorithm for MIMO channels[END_REF], the decoding sequence can be determined in two levels. We first compare the aggregate errors of the first half and second half parts of symbols (line 2 in Algorithm 4). The half with worse accuracy is assigned to the tree search (put in the latter part of the decoding sequence). Consequently, within this half part, the errors of the first two symbols and the second two are compared. The two symbols with worse accuracy are put closer to the root of the tree. If this two-symbol by two-symbol exchange takes place in the second half of the symbols which are to be decoded using tree search, the same two-symbol by two-symbol exchange should be done accordingly in the other half in order to maintain the structure of R matrix. If only the symbol exchange between two halves of the symbols is carried out, it is referred to as "4-by-4 column switch". Otherwise, if the exchange within each half is also performed, it is called "2-by-2 column switch".

The advantage of the column switch will be shown in the next section. 6 Simulation Results

BER performance

Figure 5 presents the uncoded symbol error rate of the proposed simplified decoders in quasi-static independent Rayleigh flat fading channel. The performances of the ML decoder and the sphere decoder with Schnorr-Euchner (S-E) enumeration proposed by Guo and Nilson [START_REF] Guo | Reduced complexity Schnorr-Euchner decoding algorithms for MIMO systems[END_REF] are also given as references. The Guo-Nilson's sphere decoder is a low-complexity implementation of sphere decoder with S-E but is sub-optimal in terms of symbol error rate. It can be seen that the proposed decoders achieve the same performance as ML decoder with both QPSK and 16-QAM modulations. In addition, the proposed decoders outperform the Guo-Nilson's sphere decoder with 16-QAM modulation. A gain of around 0.7 dB can be observed at symbol error rate level of 1 × 10 -4 .

Computational complexity

Figure 6 and Figure 7 present the complexity in terms of number of visited nodes for decoding each codeword with QPSK and 16-QAM, respectively. For the proposed decoders, this number is calculated as the number of visited nodes in the tree search phase plus the maximum visited nodes among the four search branches.

The Guo-Nilson's sphere decoder is also given as a reference. Since processing each node requires roughly the same operations for both decoders, these experiments actually give the comparison of the processing time latency [START_REF] Sinnokrot | Fast maximum-likelihood decoding of the golden code[END_REF][START_REF] Guo | Reduced complexity Schnorr-Euchner decoding algorithms for MIMO systems[END_REF]. It can be seen from the results that the proposed decoders require much less processing time than the ML decoder which needs to traverse all M 8 possibilities. In QPSK case, the proposed decoders always yield less latency than Guo-Nilson's sphere decoder. For instance, the proposed decoder with 2-by-2 column switch visits only 254.6 nodes on an average at SNR of 0 dB. Compared with the Guo-Nilson's decoder which visits 1276.6 nodes at SNR of 0 dB, the proposed one achieves a processing time reduction of 80%. The reductions are 50% and 49% at 10 dB and 20 dB, respectively. In addition, the improvements brought by the proposed column switch technique can also be seen in the results. For instance, the 2-by-2 column switch yields a processing time reduction of 38% at 0 dB compared with the decoder without column switch. This improvement is less significant in high SNR region (e.g. greater than 15 dB). Moreover, the 2-by-2 column switch offers better performance than the 4-by-4 counterpart because it is more likely to allocate the four most unreliable complex symbols to the tree-search stage, which helps improving the global convergence speed. In 16-QAM case (see Figure 7), the proposed decoder with 2-by-2 column switch also brings processing time reduction in low SNR region. At 8 dB, it visits 1301.1 nodes on an average, yielding a time reduction of 84% compared with Guo-Nilson's decoder. The proposed decoder needs similar time latency as Guo-Nilson's decoder in higher SNR region (e.g. greater than 15 dB). Taking into account the symbol error rate performance given in Figure 5, 8 ∼ 15 dB is the SNR region where the error correction ability of the channel coding will be carried out significantly. This region also represents the minimum SNR level at which the receiver can still work properly. Figure 8 and Figure 9 give the overall required multiplications to decode each codeword. For the proposed decoders, the multiplications spent by the tree search and by all four search branches are taken into account.

The computation overheads such as QR decomposition, linear estimation, are also included in the results to give the overall complexity of the decoders. It can be seen from the results that, in QPSK case, the proposed decoder with 2-by-2 column switch spends 11%, 21% and 3% more multiplications than Guo-Nilson's decoder at SNR of 0 dB, 6 dB and 20 dB, respectively. In 16-QAM case, it needs 5% less multiplication at 8 dB but spends 85% and 9% more multiplications at 14 dB and 28 dB, respectively. However, it is worth noting that with the cost of increased multiplications the proposed decoders provide less processing latencies. For instance, the proposed decoder with 2-by-2 column switch achieves 62% processing time reduction at SNR of 6 dB with QPSK and 42% reduction at 14 dB with 16-QAM, respectively. Finally, Figure 10 and Figure 11 present the overall divisions spent by the decoders. The proposed decoders require less divisions than the Guo-Nilson's. For instance, the proposed decoder with 2-by-2 column switch requires 47%, 9% and 5% less divisions at SNR of 0 dB, 10 dB and 20 dB, respectively, with QPSK.

In 16-QAM case, it achieves 79% reduction of divisions at 8 dB. In the meantime, the two decoders spend roughly the same number of divisions in higher SNR region, e.g. greater than 18 dB.

In general, we can see the different trade-offs achieved by different decoders. Proposed decoders achieves ML performance with less time latency and less divisions than Guo-Nilson's. On the other hand, the Guo-Nilson's decoder needs less multiplications with some performance loss with 16-QAM.

Figure 1 :

 1 Figure 1: Illustration of the R matrix of the new 3D MIMO codeword (10) in quasi-static channel.

Figure 2 :

 2 Figure 2: Illustration of the R matrix of the original 3D MIMO codeword given in (9) in quasi-static channel.

with v 12 =

 12 z 12 -R 13 c-R 14 d, v 34 = z 34 -R 23 c-R 24 d. The outer search is carried out over the combinations of [c, d]. For a given [c, d], the search of a and the search of b are performed in parallel. The concatenation of outer and inner searches (either a or b) results in a joint search of six information symbols. Therefore, the worst-case decoding complexity is reduced from O(M 8 ) to O(M 6

  scheme and O(M4.5 ) for square M -QAM modulations. Recall that the 3D MIMO code achieves a coding rate of 2 which is full-rate for 4 × 2 MIMO transmissions. The comparisons with other state-of-the-art full-rate

Figure 3 :

 3 Figure 3: Illustration of the two-stage sphere decoding.

Figure 5 :

 5 Figure 5: Uncoded symbol error rate obtained with Guo-Nilson's sphere decoder with S-E for 3D MIMO code, ML decoder and proposed simplified ML decoders in quasi-static Rayleigh channel with QPSK and 16QAM.

  's SD with S-E proposal, no column switch proposal, 4-by-4 column switch proposal, 2-by-2 column switch

Figure 6 :

 6 Figure 6: Computational complexity in terms of visited nodes required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with QPSK modulation.

Figure 7 :

 7 Figure 7: Computational complexity in terms of visited nodes required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with 16-QAM modulation.

  's SD with S-E proposal, no column switch proposal, 4-by-4 column switch proposal, 2-by-2 column switch

Figure 8 :

 8 Figure 8: Computational complexity in terms of multiplications required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with QPSK modulation.

  's SD with S-E proposal, no column switch proposal, 4-by-4 column switch proposal, 2-by-2 column switch

Figure 9 :

 9 Figure 9: Computational complexity in terms of multiplications required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with 16-QAM modulation.

  's SD with S-E proposal, no column switch proposal, 4-by-4 column switch proposal, 2-by-2 column switch

Figure 10 :

 10 Figure 10: Computational complexity in terms of divisions required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with QPSK modulation.

  's SD with S-E proposal, no column switch proposal, 4-by-4 column switch proposal, 2-by-2 column switch

Figure 11 :

 11 Figure 11: Computational complexity in terms of divisions required by Guo-Nilson's sphere decoder with S-E for 3D MIMO code and proposed simplified ML decoders, in quasi-static Rayleigh channel with 16-QAM modulation.

FiguresFigure 1 -

 1 Figures Figure 1 -R matrix of the new 3D MIMO codewordIllustration of the R matrix of the new 3D MIMO codeword given in[START_REF] Ren | Fast-group-decodable space-time block code[END_REF] in quasi-static channel.

Figure 2 -

 2 Figure 2 -R matrix of the original 3D MIMO codeword Illustration of the R matrix of the original 3D MIMO codeword given in (9) in quasi-static channel.

Figure 3 -

 3 Figure 3 -Two-stage sphere decoding Illustration of the two-stage sphere decoding.

Figure 4 -

 4 Figure 4 -Parallel decisionsParallel decisions of [s R 1 , s R 2 ], [s I 1 , s I 2 ], [s R 3 , s R 4 ] and [s I 3 , s I 4 ].

Table 1 :

 1 Comparison of ML decoding complexities of full-rate STBCs for 4 × 2 MIMO transmission

	STBC	ML decoding complexity any QAM square QAM
	new 3D MIMO	O(M 6

Table 1 -

 1 Comparison of ML decoding complexities of STBCs for 4 × 2 MIMO transmissionThe table presents the ML decoding complexities of several STBCs that are suitable for 4 × 2 MIMO transmission.

	Tables

We assume that the receiver has perfect knowledge of the channel in our work. In practice, the channel coefficients should be estimated using some channel estimation techniques.

Note that this construction is different from that of the quasi-orthogonal code[START_REF] Sharma | Full-rate full-diversity linear quasi-orthogonal space-time codes for any number of transmit antennas[END_REF] and the EAST code[START_REF] Sinnokrot | Embedded Alamouti space-time codes for high rate and low decoding complexity[END_REF].

The fast decodability of other STBCs such as BHV, Srinath-Rajan, IFS, also requires quasi-static channel assumption.

Acknowledgements

This work is supported by French ANR "Mobile Multi-Media (M3)" project and "Pôle Images & Réseaux".

S-E enumeration

and [s I 3 , s I 4 ].

been checked. The best solution is the ML solution.

The sequence in which the sibling nodes are visited is determined according to the their partial distances in an ascending order. This is to guarantee that the promising candidates are visited first in order to reduce the search complexity. This ordering process is referred to as Schnorr-Euchner enumeration [START_REF] Agrell | Closest point search in lattices[END_REF][START_REF] Schnorr | Lattice basis reduction: Improved practical algorithms and solving subset sum problems[END_REF][START_REF] Guo | Reduced complexity Schnorr-Euchner decoding algorithms for MIMO systems[END_REF]. It can simply be implemented by a lookup table [START_REF] Wiesel | Efficient implementation of sphere demodulation[END_REF][START_REF] Tsai | A 4×4 64-QAM reduced-complexity K-Best MIMO detector up to 1.5 Gbps[END_REF] (line 4 in Algorithm 1) and its complexity is merely the computation of the linear estimation ŝZF .

Parallel decision phase

Once a leaf node is achieved in the tree search, a better solution of [c, d] is found. Consequently, the tree search process is suspended and the new [c, d] is used to trigger the parallel detections of rest symbols.

The parallel detection is depicted in Figure 4. The implementation details are presented in Algorithm 3.

As shown in Figure 4, the detections of

and [s I 3 , s I 4 ] are carried out in parallel. For each branch, a one-level sphere decoder is used to traverse all possible PAM symbols as given in [START_REF] Jithamithra | Minimizing the complexity of fast sphere decoding of STBCs[END_REF].

The visiting sequence is also determined by the Schnorr-Euchner enumeration. The detections in different branches are synchronized by a common clock signal because the operations are exactly the same for all branches. All branches simultaneously check the first candidate PAM symbol and then move on to the second one, and so on.

Moreover, we propose a mechanism that terminates the search in each branch not only based on its own results, but also taking into account the results from other branches. In particular, once the best solution of the jth branch is found ahead of others, the resulting branch distance d j is recorded and shared with other branches to speed up the overall search process.

Take the search of the first branch as an example. The most promising PAM symbol in the unchecked symbol list is assigned to s R 2 (line 8 in Algorithm 3). The partial distance τ 1 is calculated (line 9 in Algorithm 3). The search is terminated in two cases: 1) if this partial distance is greater than the current

Conclusion

The 3D MIMO code has been shown to be efficient and robust in distributed MIMO scenarios. Yet, it suffers from high ML decoding complexity. In this paper, we first proposed a new form of the 3D MIMO codeword and investigated some important properties of the new codeword. With these properties, the 3D MIMO code is proved to be fast decodable. Consequently, we proposed a reduced-complexity ML decoder for the 3D MIMO code which offers the same performance as ML decoder. Simulation results demonstrate that the novel low-complexity decoder yields much less processing time latency than the classical Guo-Nilson's sphere decoder with Schnorr-Euchner enumeration. Moreover, the proposed 2-by-2 column switch technique can significantly reduce the average decoding complexity, especially with 16-QAM modulation.

Algorithm 1: Simple ML decoder for 3D MIMO code. Input: y, H eq Output:

% start from root node 8 run SimpML ( z, R, ŝ, s, radius, l, d, Ω);

Appendices Definition of QR decomposition

If we write H eq = [h 1 , . . . , h 2κ ], the H eq 's QR decomposition H eq = QR is achieved by Gram-Schmidt procedure such that: Q [q 1 , . . . , q 2κ ], where columns q j 's are orthogonal, and

where

k=1 q k , h j q k , q j = r j / r j , j = 1, . . . , 2κ.

Proof of Theorem 1

Based on H eq , after some straightforward computation, it yields

According to the definition of QR decomposition, In addition,

Moreover, r 3 = h 3 -2 j=1 q j , h 3 q j = h 3q 1 , h 3 q 1 and q 3 = r 3 / r 3 = (h 3q 1 , h 3 q 1 )/ r 3 . Therefore, q 3 , h 4 = ( h 3 , h 4q 1 , h 3 q 1 , h 4 )/ r 3 = 0. This completes the proof of Theorem 1.

Proof of Theorem 2

Based on H eq , after some straightforward computation, it yields h j , h k = 0, ∀j = 1, 2, 3, 4 and k = 5, 6, 7, 8.

Using q 1 = h 1 / h 1 and q 2 = h 2 / h 2 which have been proven in the proof of Theorem 1, it yields q j , h k = 0, ∀j = 1, 2 and k = 5, 6, 7, 8.

Using q 3 = (h 3q 1 , h 3 q 1 )/ r 3 which has been proven in the proof of Theorem 1, it yields

This completes the proof of Theorem 2. % decision flags

repeat lines 16 to 19 for other branches;

% overall distance of [ŝ1, . . . ŝ4] T

Proof of Corollary 1

Using the similar method as in the proof of Theorem 1, it can be computed from the definition of H eq that h 5 , h 6 = h 5 , h 8 = h 6 , h 7 = h 7 , h 8 = 0.

In addition, using Theorem 2, it can be obtained that q 5 = h 5 / h 5 . Hence, q 5 , h 6 = q 5 , h 8 = 0.

Using q 5 , h 6 = 0 and Theorem 2, it yields q 6 = h 6 / h 6 . Hence, q 6 , h 7 = 0.

Finally, using q 6 , h 7 = 0 and Theorem 2, it yields r 7 = h 7q 5 , h 7 q 5 and q 7 = r 7 / r 7 = (h 7q 5 , h 7 q 5 )/ r 7 . Therefore, q 7 , h 8 = ( h 7 , h 8q 5 , h 7 q 5 , h 8 )/ r 7 = 0.

This completes the proof of Corollary