Spatial Lattice Modulation for MIMO Systems
Abstract
This paper proposes spatial lattice modulation (SLM), a spatial modulation method for multipleinputmultipleoutput (MIMO) systems. The key idea of SLM is to jointly exploit spatial, inphase, and quadrature dimensions to modulate information bits into a multidimensional 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 pulseamplitudemodulation (PAM) input signaling per dimension. In particular, it is shown that when the SLM signal set is constructed by using dense lattices, a significant signaltonoiseratio (SNR) gain, i.e., a nominal coding gain, is attainable compared to the existing methods. In addition, closedform expressions for both the average mutual information and average symbolvectorerrorprobability (ASVEP) of generic SLM are derived under Rayleighfading environments. To reduce detection complexity, a lowcomplexity 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.
I Introduction
Spatial modulation (SM) [1] is a transmission method that sends information bits using the index of an active antenna and conventional quadratureamplitudemodulation (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 signaltonoise 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 (MASM)[9] is another variation of GSM, MASM was introduced by harnessing multiplexing gains of the MIMO system. MASM 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 (VAGSM) [10] is another variation of GSM. VAGSM 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 inphase 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 antennasubsets (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 .
Adaptivejointmapping GSM (AJMGSM) [13] and jointlymapped SM (JMSM) [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, inphase, 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 MASM [9] are rotated to a different phase for each active antennasubset; 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 twodimensional hexagonal lattice to send symbols. This lattice is the densest packing lattice in a twodimensional complex domain. All these methods [15, 16, 17] demonstrate that carefullydesigned 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 inphase (or quadrature) component of each transmit antenna from the ary pulseamplitudemodulation (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, inphase, 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 lowdimensional 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 BarnesWall lattices for SLM; these are the densest lattices below 16 dimensions (or eight transmit antennas).

We also analyze the average mutual information and average symbolvectorerrorprobability (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 closedform for the proposed SLM methods. We also derive a closedform 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 lowcomplexity 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 lineartype detection methods such as zeroforcing 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.
Iia 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
(1) 
where is a complex Gaussian noise vector with zero mean and covariance matrix . We assume a richscattering and frequencyflat 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.,
(2) 
Without loss of generality, the complexbaseband inputoutput relationship in (1) can be rewritten in realvector representation as
(3) 
where
and
For notational convenience, we will use this realvalue representation of the MIMO system in this paper, except in Section IV. We also assume that ary pulseamplitudemodulation (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 pilottransmission.
IiB Preliminaries
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.,
(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 nearestneighbor quantizer associated with is defined as
(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
(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 spherepacking 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.
Iiia 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 realrepresentation 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
(7) 
where the cardinality of is
(8) 
Because , we construct an entire signal set for the joint mapping by the union of as
(9) 
and are disjoint for all , so the cardinality of is
(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 I^{1}^{1}1In case of high bitrates transmission, lookup table method is not suitable to implement practical system. Alternatively, we can avoid this lookup table by using combinatorial method [24].. These cubic lattice vectors can be also generated by using the following generating matrix of , i.e.,
(11) 
Using this set of vectors, one can send information bits at bits/sec/Hz if SNR is high enough.


Index  Index  


1  2225  
23  2629  
45  3033  
67  3441  
89  4249  
1013  5057  
1417  5865  
1821  6681  

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
(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
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:,
(13) 
where and . Using these nested lattices, the same constellation set is generated by
(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.
IiiB SLM with LowDimensional 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 multidimensional 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 BarnesWall 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 lowdimensional signal space.
The following lemma provides a method to construct the BarnesWall lattice in [20]; this lattice is used in the proposed SLM.
Lemma 1 (BarnesWall Lattice Construction [20]).
Let denote the generating matrix of a balanced BarnesWall lattice in :
(15) 
The generating matrix of lattice in is obtained from , which is the times Kronecker products of , i.e.,
(16) 
By rescaling the irrational elements in , the generating matrix of is obtained as
(17) 
The following lemma shows some coding theoretic properties of BarnesWall lattices.
Lemma 2 (Coding Properties of BarnesWall Lattices [19, 18]).
For all integer , a dimensional BarnesWall 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 BarnesWall 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 fourdimensional BarnesWall lattice, , which is equivalent to the lattice. The generating matrix of is
(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 II^{2}^{2}2 The barnesWall lattice vectors of Table II can be indexed by using arithmetic operators without lookup table [25]. This enables to reduce encoding and decoding complexity..


Index  Index  


1  4647  
25  4849  
69  5057  
1013  5865  
1417  6673  
1821  ⋮  
2225  
2641  122129  
4243  130137  
4445  138145  

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 lowdimensional 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 nonunitary.


Antennas  Dimensions  Lattice  Coding gain 


1  2  0.62 dB  
2  4  1.51 dB  
4  8  3.01 dB  
8  16  4.52 dB  
12  24  6.02 dB  
16  32  6.28 dB  

Iv Performance Analysis
This section analyzes the average mutual information and average symbolvectorerrorprobability (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).
Iva 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
(19) 
for . The conditional probability density function (PDF) of given that is
(20) 
We use the ML criterion for the performance analysis and for the comparison with other SM schemes.
IvB 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.
IvB1 Exact Expression
For a MIMO channel, the mutual information between the discrete input vector and the continuous channel output vector is
(21) 
Because the received signal vector follows a Gaussian distribution for given and , the PDF of is
(22) 
Using and , the mutual information for given is
(23) 
Using equations (21)(23), the mutual information for SLM is obtained as
(24) 
IvB2 Approximation of Average Mutual Information
The derived mutual information expression in (IVB1) is not tractable because it involves the multidimensional 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 (IVB1). The following proposition closely approximates the average mutual information.
Proposition 3: The average mutual information, for SLM, can be approximated as
(25) 
Proof.
Applying Jensen’s inequality yields a lower bound of as in [23],
(26) 
By changing the order of summation and integration in (IVB2), is lowerbounded by
(27) 
The integration in (IVB2) can be calculated as
(28) 
Because the elements of are mutually independent, the multidimensional integrations in (IVB2) are computed using elementwise integrations as
(29) 
The integration in (29) is conducted separately for real and imaginary parts to yield
(30) 
The integrals in (IVB2) are computed using the following identity
(31) 
Thus, , where and are given by
(32) 
Using the results in (IVB2), we obtain the simple
(33) 
Using the above equations (IVB2)(IVB2), the lower bound of is
(34) 
Plugging (IVB2) into (IVB1), a lower bound of is obtained as