Spatial Lattice Modulation for MIMO Systems

# Spatial Lattice Modulation for MIMO Systems

Jiwook Choi, Yunseo Nam, and Namyoon Lee J. Choi, Y. Nam, and N. Lee are with the department of electrical engineering, POSTECH, South Korea, emails:{jiwook,edwin624,nylee}@postech.ac.kr
###### Abstract

This paper proposes spatial lattice modulation (SLM), a spatial modulation method for multiple-input-multiple-output (MIMO) systems. The key idea of SLM is to jointly exploit spatial, in-phase, and quadrature dimensions to modulate information bits into a multi-dimensional signal set that consists of lattice points. One major finding is that SLM achieves a higher spectral efficiency than the existing spatial modulation and spatial multiplexing methods for the MIMO channel under the constraint of -ary pulse-amplitude-modulation (PAM) input signaling per dimension. In particular, it is shown that when the SLM signal set is constructed by using dense lattices, a significant signal-to-noise-ratio (SNR) gain, i.e., a nominal coding gain, is attainable compared to the existing methods. In addition, closed-form expressions for both the average mutual information and average symbol-vector-error-probability (ASVEP) of generic SLM are derived under Rayleigh-fading environments. To reduce detection complexity, a low-complexity detection method for SLM, which is referred to as lattice sphere decoding, is developed by exploiting lattice theory. Simulation results verify the accuracy of the conducted analysis and demonstrate that the proposed SLM techniques achieve higher average mutual information and lower ASVEP than do existing methods.

Multiple-input-multiple-output (MIMO), spatial modulation (SM), lattice modulation.

## I Introduction

Spatial modulation (SM) [1] is a transmission method that sends information bits using the index of an active antenna and conventional quadrature-amplitude-modulation (QAM) symbols. SM has been proposed to improve both the spectral and energy efficiency of MIMO systems [2, 3, 4, 5]. For example, when a transmitter is equipped with antennas that use a radio frequency (RF) chain, bits can be modulated into a spatial symbol vector, where denotes the QAM constellation set and represents its cardinality. A simplified version of SM, referred to as space shift keying (SSK) [6] was presented to improve energy efficiency. SSK only maps information bits into the antenna index, so it is able to achieve the spectral efficiency of bits/sec/Hz when signal-to-noise ratio (SNR) is high enough.

The concepts of SM and SSK have been generalized in numerous ways by mapping information bits into multiple indices of the transmit antennas. Generalized spatial modulation (GSM) [7] and generalized space shift keying (GSSK) [8] are representative generalizations of SM. The idea of both GSM and GSSK is to map information bits onto an antenna subset that consists of elements among . Therefore, a transmitter is able to modulate information bits when using GSM with constellation set . This modulation strategy allows sending of more information bits than both SM and SSK. Multiple active spatial modulation (MA-SM)[9] is another variation of GSM, MA-SM was introduced by harnessing multiplexing gains of the MIMO system. MA-SM sends distinct QAM symbols by choosing active elements among transmit antennas; thereby, information bits are modulated to a symbol vector with QAM constellation set .

Variable set of active antenna GSM (VA-GSM) [10] is another variation of GSM. VA-GSM allows the number of active antennas to vary from to , while sending the same transmit symbol for each active antenna. In addition, quadrature spatial modulation (QSM) [11] separately exploits in-phase and quadrature signal dimensions. For instance, when a transmitter has antennas, QSM is able to modulate information bits. Recently, another generalized method of GSM, called GSM with multiplexing (GSMM) [12], was introduced. GSMM sends data symbols using a set of precoding matrices in which and . As a result, GSMM is able to modulate information bits. The common limitation of the methods in [7, 8, 9, 10, 11, 12] is that information bits are separately modulated to the index of active antenna-subsets (or the index of precoding matrices [12]) and to the transmission of QAM symbols. This separated modulation approach generally cannot achieve a higher spectral efficiency than that attained by a joint modulation strategy for a fixed and .

Adaptive-joint-mapping GSM (AJM-GSM) [13] and jointly-mapped SM (JM-SM) [14] both jointly modulate information bits into an active antenna index and transmit symbols using the proposed joint mapping rule. This joint mapping can generate more signal points than the mapping methods that separately modulate information bits into the indices of antenna subsets and transmit symbols. The limitation of the methods in [11] and [14] is that they do not jointly take into account all possible signaling dimensions, (i.e., spatial, in-phase, quadrature) when constructing signal sets for SM.

Since the minimum Euclidean distance between adjacent symbols in a constellation set affects the detection accuracy, numerous methods for the signal set design of SM have been proposed to maximize the minimum distance [15, 16, 17]. For instance, the QAM symbols of MA-SM [9] are rotated to a different phase for each active antenna-subset; as a result, the minimum distance increases. Enhanced spatial modulation (ESM) [15, 16] uses a similar approach. The idea of ESM is to construct a primary and a secondary signal constellation set. Then the secondary set is interpolated into the primary set to increase the minimum distance between symbol vectors. Another constellation design method for SM exploits Eisenstein integer [17]; in this method, a transmitter uses a two-dimensional hexagonal lattice to send symbols. This lattice is the densest packing lattice in a two-dimensional complex domain. All these methods [15, 16, 17] demonstrate that carefully-designed constellation sets is able to achieve higher SNR gains than SM methods that use the conventional QAM constellation set.

In this paper, we consider a MIMO system in which a transmitter is equipped with transmit antennas and a receiver is equipped with receive antennas. We assume that the transmitter can send one symbol per in-phase (or quadrature) component of each transmit antenna from the -ary pulse-amplitude-modulation (PAM) signal set . The contributions of this paper are summarized as follows:

• Our major contribution is to propose a novel multidimensional spatial modulation method called spatial lattice modulation (SLM). Unlike the existing SM techniques in which information bits are separately mapped to a set of antenna indices and modulation symbols [7, 8, 9, 10, 12], the key idea of the proposed SLM is to modulate information bits into a set of lattice points in by jointly exploiting spatial, in-phase, and quadrature signal dimensions. An element of each lattice point is chosen from the set with , where the null element indicates that the input signal for a chosen signal dimension is deactivated for SM. In particular, we present two SLM methods: SLM using a simple cubic lattice and SLM that uses a dense packing lattice that has a large nominal coding gain in a low-dimensional vector space [18, 19]. We show that the proposed SLM methods achieve the spectral efficiency of bits/sec/Hz when SNR is high enough. This result is interesting because it enables the transmission of additional information bits per channel use compared to the conventional spatial multiplexing; this gain is unbounded as increases. In particular, to attain a nominal coding gain for a given target spectral efficiency, we propose a signal set design algorithm that uses Barnes-Wall lattices for SLM; these are the densest lattices below 16 dimensions (or eight transmit antennas).

• We also analyze the average mutual information and average symbol-vector-error-probability (ASVEP) of the proposed SLM methods under a Rayleigh MIMO channel. Although the mutual information expressions have been characterized for SM and GSM in [21, 22, 23] for a fixed MIMO channel, the average mutual information expression is unknown for general SM methods. We derive a tight approximation of the average mutual information in a closed-form for the proposed SLM methods. We also derive a closed-form upper bound of ASVEP for SLM to complete our analysis. Simulation results verify the effectiveness of our analysis. One major observation is that SLM using dense lattices provides SNR gains in both the average mutual information and the ASVEP.

• Lastly, we present a low-complexity SLM detection method, which is called lattice sphere decoding (LSD). The key idea of LSD is to exploit the property that a lattice is closed under addition. Using this lattice property, the proposed LSD algorithm reduces the effective search space of SLM by calculating ML metrics only in the closest lattice vectors from an initially estimated lattice vector. We show that the complexity order of the proposed LSD algorithm is , which is the same detection complexity order with linear-type detection methods such as zero-forcing MIMO detection. Simulation results show that the error performance of LSD closely matches that of ML detection in practical MIMO systems, while significantly diminishing the detection complexity.

## Ii System Model and Preliminaryies

This section presents the system model considered in this paper and provides some useful mathematical definitions that will be used subsequently.

### Ii-a System Model

We consider a MIMO channel in which a transmitter equipped with transmit antennas sends information symbols to a receiver equipped with receive antennas. We denote the complex baseband transmit vector by , and the MIMO channel matrix by . Then the complex baseband received signal at the receiver is

 ¯y=¯H¯x+¯v, (1)

where is a complex Gaussian noise vector with zero mean and covariance matrix . We assume a rich-scattering and frequency-flat channel model, in which all elements of are chosen from complex Gaussian random variables with zero mean and unit variance. The transmit vector is normalized to satisfy the power constraint , i.e.,

 Tr(E[¯x¯xH])=Es. (2)

Without loss of generality, the complex-baseband input-output relationship in (1) can be rewritten in real-vector representation as

 y=Hx+v, (3)

where

 x=[Re{¯x}Im{¯x}]∈R2Nt,y=[Re{¯y}Im{¯y}]∈R2Nr,v=[Re{¯v}Im{¯v}]∈R2Nr,

and

 H=[Re{¯H}−Im{¯H}Im{¯H}Re{¯H}]∈R2Nr×2Nt.

For notational convenience, we will use this real-value representation of the MIMO system in this paper, except in Section IV. We also assume that -ary pulse-amplitude-modulation (PAM), , is used for the input constellation points per dimension; therefore the input signal set is finite. We also assume that perfect channel state information is known to the receiver, i.e., perfect CSIR, which can be reliably estimated using conventional pilot-transmission.

### Ii-B Preliminaries

We provide some useful definitions which also can be found in [19, 18].

Definition 1 (Lattice): Let be a set of linearly independent vectors in , in which each vector constitutes a basis for the lattice. A real lattice is the countably infinite set defined by integer combinations of basis vectors, i.e.,

 Λ={x∈Rn:x=c1g1+c2g2+⋯+cngn}, (4)

where . Thus, a lattice, , is a discrete additive subgroup of , and is closed under addition and reflection. Matrix is referred to as a generating matrix of the lattice.

Definition 2 (Nearest neighbor quantizer): The nearest-neighbor quantizer associated with is defined as

 Q(v)=xi∈Λ if v∈{v∈Rn:∥v−xi∥2≤∥v−xj∥2}, (5)

for any other point . The Voronoi cell associated with is the set of points in closest to , i.e.,.

Definition 3 (Nested Lattice): A pair of -dimensional lattices is called nested if , i.e., corresponding generator matrices and exist such that , where is an integer matrix that has a determinant . The volumes of the Voronoi cells of and satisfy .

Definition 4 (Normalized Coding Gain): Let be the minimum squared distance of the lattice points in . The Hermite parameter of is the normalized density parameter or the normalized coding gain, which is defined as

 γc(Λ)=d2min(Λ)V0(Λ)2n. (6)

Asymptotically, for very high SNR, determining the maximum possible nominal coding gain of an -dimensional lattice code is equivalent to finding the densest lattice in a sphere-packing sense.

## Iii Spatial Lattice Modulation

In this section, we present the idea of SLM. Unlike the existing SM techniques in which information bits are separately mapped to a set of antenna indices and modulation symbols, the key idea of SLM is to jointly map information bits to one of lattice vectors in . This joint mapping strategy using lattices makes it possible to obtain the maximum entropy of input symbol vectors for an given -ary PAM condition per dimension. Also, by using a dense lattice, we are able to achieve the largest nominal coding gain for a given . Depending on different lattice structures, we propose two SLM methods: SLM using a cubic lattice and SLM using a dense lattice.

### Iii-a SLM using Cubic Lattices

This proposed SLM method uses a joint mapping strategy to map information bits into a set of information symbol vectors, each in a cubic lattice. Let be a set of transmit symbol vectors where is the real-representation of . We denote the number of activated dimensions in an -dimensional space by , where . Under the premise that dimensions are active, a set of all possible transmit vectors using generalized spatial modulation method under the constraint of the -ary PAM input signal per dimension is

 SNa={sNa1,…,sNaLNa}, (7)

where the cardinality of is

 LNa=(2NtNa)MNa. (8)

Because , we construct an entire signal set for the joint mapping by the union of as

 SCB(Nt,M)=∪2NtNa=0SNa. (9)

and are disjoint for all , so the cardinality of is

 |SCB(Nt,M)|=2Nt∑i=0(2Nti)Mi=(M+1)2Nt, (10)

where the last equality follows from the binomial expansion.

Example 1: Suppose and . In this case, we can create 81 lattice vectors by using the joint mapping strategy. The corresponding 81 vectors are listed in Table I111In case of high bit-rates transmission, look-up table method is not suitable to implement practical system. Alternatively, we can avoid this look-up table by using combinatorial method [24].. These cubic lattice vectors can be also generated by using the following generating matrix of , i.e.,

 GCB4=⎡⎢ ⎢ ⎢⎣1000010000100001⎤⎥ ⎥ ⎥⎦. (11)

Using this set of vectors, one can send information bits at bits/sec/Hz if SNR is high enough.

Proposition 1: Let be an input symbol vector that is chosen uniformly from the proposed signal set . Then the maximum input entropy of the proposed signal set is

 H(x)=2Ntlog2(M+1). (12)
###### Proof.

The proof follows from the definition of the entropy and the uniform input distribution, i.e., for . ∎

We offer some remarks and examples on our joint mapping strategy to clarify understanding.

Remark 1: The proposed joint mapping method has an effective modulation size of per dimension. Because we assume that -ary PAM is used for the input constellation set per dimension, the modulation size of is trivial. However, because we use spatial modulation, we can add one hidden constellation point of ‘0‘ as an input vector by physically deactivating the input per dimension; this process provides the effective modulation size of . This increment provides a considerable gain in the input entropy. For example, suppose a spatial multiplexing transmission method with uniform -ary PAM input signaling. The maximum input entropy of this method is . Therefore, one can achieve the gain of

 2Ntlog2(M+1)−2Ntlog2(M)=2Ntlog2(1+1M).

This entropy gain implies that one can send more information bits per transmission using the proposed method when SNR is high enough. The gain increases linearly with and is therefore unbounded.

Remark 2: The proposed joint mapping technique generalizes the existing SM methods in [1, 6, 7, 9, 11, 14] and spatial multiplexing method. For example, supposing that and , the signal sets generated by the conventional SM and QSM method in [1, 11] are contained in , which is a subset of . Similarly, the signal set of the conventional spatial multiplexing method is , which is same with of .

Remark 3: When the effective modulation size is a prime number, the proposed constellation set generated by the joint spatial mapping can also be constructed by a nested lattice modulation method. Letting , we consider two nested -dimensional cubic lattices with the generating matrices and respectively:,

 Λc ={x∈R2Nt:x=c⊤Gc}  and Λs ={x∈R2Nt:x=c⊤Gs}, (13)

where and . Using these nested lattices, the same constellation set is generated by

 SCB(Nt,M)=Λc∩V0(Λs), (14)

where is the Voronoi region associated with , i.e., the volume . Therefore, the joint spatial mapping method can be interpreted as the nested lattice modulation using the corresponding cubic generating matrices.

### Iii-B SLM with Low-Dimensional Dense Lattices

The cubic lattice used for the previous SLM is a baseline lattice, because by the definition it offers no nominal coding gain, i.e., . Therefore, a natural extension is to use a dense lattice to construct a set of multi-dimensional lattice vectors that yield a higher nominal coding gain than in a given number of dimensions, i.e., .

Finding the densest lattice packing in an arbitrary number of dimensions is a difficult mathematical problem. The lattices that have the largest nominal coding gain are well characterized up to 24 dimensions [19, 18, 26]. The Barnes-Wall lattice is a good one due to its simplicity of lattice construction and its tractability to analyze. It also provides high nominal coding gains in a low-dimensional signal space.

The following lemma provides a method to construct the Barnes-Wall lattice in [20]; this lattice is used in the proposed SLM.

###### Lemma 1 (Barnes-Wall Lattice Construction [20]).

Let denote the generating matrix of a balanced Barnes-Wall lattice in :

 M1=[√2011]. (15)

The generating matrix of lattice in is obtained from , which is the times Kronecker products of , i.e.,

 Mm+1=M1⊗M1⊗⋯⊗M1m+1. (16)

By rescaling the irrational elements in , the generating matrix of is obtained as

 GBW2m+1(i,j) ={Mm+1(i,j),if Mm+1(i,j) is rational,1√2Mm+1(i,j),if Mm+1(i,j) is% irrational. (17)

The following lemma shows some coding theoretic properties of Barnes-Wall lattices.

###### Lemma 2 (Coding Properties of Barnes-Wall Lattices [19, 18]).

For all integer , a -dimensional Barnes-Wall lattice exists that has the minimum squared Euclidean distance and the normalized volume ; therefore its nominal coding gain is . In addition, the kissing number of is .

By exploiting the Barnes-Wall lattice that is constructed by in (17), we propose Algorithm 1, which creates a signal set for SLM using dense lattices. The algorithm finds a signal set such that each lattice vector is created by generating matrix and satisfies the maximum power constraint . The first step is to exhaustively search all possible integer vectors that satisfy the given power . For a vector satisfying the power constraint, the symmetric group is selected as a candidate. For example, with lattice at power condition , 1,1,0,0, 1,0,1,0, 0,0, , 0,1,1,0, 0,1,0,1, 0,0,,1 will be the possible candidates for constellation vectors. After collecting the candidates, we check the condition whether they consist of an integer linear combination of the basis . Then, vectors that satisfy this condition are included in our SLM signal set. Algorithm 1 iterates this procedure by increasing until the power condition reaches the maximum power constraint .

Example 2: Suppose a four-dimensional Barnes-Wall lattice, , which is equivalent to the lattice. The generating matrix of is

 GBW4=⎡⎢ ⎢ ⎢⎣2000110010101111⎤⎥ ⎥ ⎥⎦. (18)

The minimum squared Euclidean distance , the volume , and therefore its nominal coding gain is dB. Using this generating matrix and Algorithm 1, we can generate 145 transmit signal vectors in with . The corresponding 145 vectors are listed in Table II222 The barnes-Wall lattice vectors of Table II can be indexed by using arithmetic operators without look-up table [25]. This enables to reduce encoding and decoding complexity..

Remark 4: The proposed algorithm is valid when the generating matrix has integer elements. Therefore, except , we are able to construct a signal set for the SLM with any low-dimensional dense lattices as and Leech lattice . As an extreme case, when the channel matrix is chosen from the set of unitary matrices, the maximum nominal coding gains achieved by the known lattices are known (Table III) [26]. For a general MIMO channel, however, these gains are not obtainable because the minimum distance of the signal set seen by the receiver diminishes when the channel matrix is non-unitary.

## Iv Performance Analysis

This section analyzes the average mutual information and average symbol-vector-error-probability (ASVEP) for the MIMO systems when input symbol vectors are uniformly drawn from an arbitrary finite input constellation . For ease of exposition, we use the complex baseband MIMO model in (1).

### Iv-a Maximum Likelihood (ML) Detection

We assume that the receiver and transmitter share the joint mapping rule between information bits and the SLM signal set. The active dimension and constellation points of a transmit symbol vector are jointly decoded using ML principle

 [l]=argmaxmf¯y(¯y|¯xm)=argminm∥¯y−¯H¯xm∥2, (19)

for . The conditional probability density function (PDF) of given that is

 f(¯y|¯xm)=1(πσ2)Nrexp(−∥¯y−¯H¯xm∥2σ2). (20)

We use the ML criterion for the performance analysis and for the comparison with other SM schemes.

### Iv-B Average Mutual Information Analysis for SLM

When input signal vectors are uniformly drawn from a finite constellation set (e.g., -ary PAM, -ary QAM), an exact expression of the mutual information with perfect CSIR was derived in [23] when the channel is fixed. For the completeness of our analysis, we first present the exact expression of the mutual information. Then, to improve intuition, we also provide a tight approximation expression for the average mutual information.

#### Iv-B1 Exact Expression

For a MIMO channel, the mutual information between the discrete input vector and the continuous channel output vector is

 I(¯x;¯y|¯H) =H(¯y|¯H)−H(¯y|¯x,¯H) =H(¯y|¯H)−H(¯v) =H(¯y|¯H)−Nrlog2(πeσ2). (21)

Because the received signal vector follows a Gaussian distribution for given and , the PDF of is

 f(¯y|¯H)=1|S|∑¯xi∈S1(πσ2)Nrexp(−∥¯y−¯H¯xi∥2σ2). (22)

Using and , the mutual information for given is

 I(¯x;¯y|¯H)=∑¯xi∈S1|S|∫¯yf(¯y|¯x=¯xi,¯H)log2f(¯y|¯x=¯xi,¯H)f(¯y|¯H)d¯y. (23)

Using equations (21)-(23), the mutual information for SLM is obtained as

 I(¯x;¯y|¯H)=log|S| −1|S|∑¯xi∈SE¯v⎡⎣log2∑¯xj∈Sexp(−∥¯H(¯xi−¯xj)+¯v∥2−∥¯v∥2σ2)⎤⎦. (24)

#### Iv-B2 Approximation of Average Mutual Information

The derived mutual information expression in (IV-B1) is not tractable because it involves the multi-dimensional integral with respect to and it does not capture the randomness in channels. We derive a closed form tight approximation of the average mutual information by exploiting the lower bound of (IV-B1). The following proposition closely approximates the average mutual information.

Proposition 3: The average mutual information, for SLM, can be approximated as

 E¯H [I(¯x;¯y|¯H)] ≈2log2|S|−log2∑xi∈S∑xj∈S(1+∥¯xi−¯xj∥22σ2)−Nr. (25)
###### Proof.

Applying Jensen’s inequality yields a lower bound of as in [23],

 H(¯y|¯H)≥−log2E¯y[f(¯y|¯H)]= −log2∫¯y⎛⎝1|S|∑¯xi∈S1(πσ2)Nrexp(−∥¯y−¯H¯xi∥2σ2)⎞⎠2d¯y. (26)

By changing the order of summation and integration in (IV-B2), is lower-bounded by

 H(¯y|¯H) ≥2log2|S|(πσ2)Nr −log2∑¯xi∈S∑¯xj∈S∫¯ye−∥¯y−¯H¯xi∥2σ2e−∥¯y−¯H¯xj∥2σ2d¯y. (27)

The integration in (IV-B2) can be calculated as

 ∫¯ye−∥¯y−¯H¯xi∥2σ2e−∥¯y−¯H¯xj∥2σ2d¯y=e−∥¯H¯xi∥2+∥¯H¯xj∥2σ2× ∫¯yexp(−2∥¯y∥2−2Re{¯yH(¯H¯xi+¯H¯xj)}σ2)d¯yΘ. (28)

Because the elements of are mutually independent, the multi-dimensional integrations in (IV-B2) are computed using element-wise integrations as

 Θ=Nr∏k=1∫ykexp⎛⎜ ⎜⎝−2∥yk∥2−2Re{y∗k{¯H(¯xi+¯xj)}k}σ2⎞⎟ ⎟⎠dyk. (29)

The integration in (29) is conducted separately for real and imaginary parts to yield

 ∫yk,Reexp⎛⎝−2y2k,Re−2yk,Re{¯H(¯xi+¯xj)}k,Reσ2⎞⎠dyk,Re ×∫yk,Imexp⎛⎝−2y2k,Im−2yk,Im{¯H(¯xi+¯xj)}k,Imσ2⎞⎠dyk,Im. (30)

The integrals in (IV-B2) are computed using the following identity

 ∫∞−∞exp(−2y2−2xyσ2)dy=√πσ22exp(x22σ2). (31)

Thus, , where and are given by

 ΘRe=Nr∏k=1√πσ22exp⎛⎜⎝{¯H(¯xi+¯xj)}2k,Reσ2⎞⎟⎠, ΘIm=Nr∏k=1√πσ22exp⎛⎜⎝{¯H(¯xi+¯xj)}2k,Imσ2⎞⎟⎠. (32)

Using the results in (IV-B2), we obtain the simple

 Θ =Nr∏k=1(πσ22)exp⎛⎜⎝∥∥{¯H(¯xi+¯xj)}k∥∥22σ2⎞⎟⎠ =(πσ22)Nrexp(∥¯H(¯xi+¯xj)∥22σ2). (33)

Using the above equations (IV-B2)-(IV-B2), the lower bound of is

 H(¯y|¯H) ≥2log2|S|+log2(πσ2)Nr+Nrlog22 −log2∑¯xi∈S∑¯xj∈Sexp(−∥¯H(¯xi−¯xj)∥22σ2). (34)

Plugging (IV-B2) into (IV-B1), a lower bound of is obtained as

 ILow(¯x;¯y|¯H) =2log2|S|+Nrlog22e −log2∑¯xi∈S∑¯xj∈S