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Differential Spatially-Modulated Space-Time Block Codes with Temporal Permutations Ahmed G. Helmy1 , Marco Di Renzo2 , and Naofal Al-Dhahir1 1
2
The University of Texas at Dallas, TX, USA, {ahmed.g.helmy, aldhahir}@utdallas.edu Paris-Saclay University Laboratory of Signals and Systems (UMR-8506) CNRS/CentraleSupelec/University Paris-Sud XI, France,
[email protected]
Abstract—We design a differential transmission scheme based on a novel temporally- and spatially-modulated space-time block coding (STBC) codebook. Our proposed design is the first differential transmission scheme for spatial modulation with multiple active transmit antennas. Our numerical results demonstrate the significant performance gains of our proposed differential transmission scheme over the state-of-the-art schemes.
I. I NTRODUCTION : M OTIVATION AND BACKGROUND Generalized spatial modulation (GSM) enhances the spectral efficiency by activating a subset of the available transmit antennas and using their indices as part of the information bits [1]. We consider a GSM-based multi-input multi-output (MIMO) system in which a transmitter activates only NA out of NT transmit antennas in each transmission interval when communicating with a receiver equipped with NR receive antennas. The incoming data bit stream is divided into K blocks of m bits each where each block represents the conveyed message in a transmission interval. Then, each m-bit block is divided into two portions: (1) the data bits md = NA log2 M derived from a complex signal constellation Ψ of size M and (2) the active antenna selection bits ma derived from the space constellation which represents the indices of the active antenna combinations [1]. Hence, the spectral efficiency of the GSM scheme is given by m = md + ma (in bps/Hz). Let H be the NR × NT slowly-varying flat-fading MIMO channel matrix with independent and identically-distributed (i.i.d) complex zero-mean unit-variance entries. Define n to be the NR ×1 complex additive white Gaussian noise (AWGN) vector at the receiver with covariance matrix No INR . Hence, the NR × 1 received signal vector y is given by y = Hx + n
(1)
where the NT −dimensional transmit vector x contains only NA non-zero entries. Based on Eq. (1), the optimal coherent GSM detection scheme [2] requires perfect knowledge of the channel state information (CSI) at the receiver since it is part of the transmitted information. This entails high channel estimation complexity of the whole channel matrix in Eq. (1) at high pilot transmission overhead. Consequently, differential transmission/detection becomes an attractive option for GSM which eliminates pilot overhead while preserving the channelbased information component. A Differential Spatial Modulation (DSM) scheme was first proposed in [3] with a total number of bits corresponding to the spatial domain equal to 1 NT log2 bNT !c2p where b.c2p denotes the floor operation to the nearest integer that can be represented as a power of two, Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected].
and ! denotes the factorial operation. The differential scheme proposed in [3] does not require any CSI at the transmitter or receiver, while assuming only a single active transmit antenna. Extending the scheme in [3] to the GSM case of multiple active transmit antennas in each time slot while preserving the main condition for differential signaling, which is a square unitary transmission matrix, is very challenging. Several methods have been proposed to improve the transmit diversity order of GSM [4], [5]. In [4], the authors firstly proposed a spatially-modulated STBC to achieve transmit diversity gain (denoted by STBC-SM). Assuming two active transmit antennas, STBC-SM sacrifices almost half of the spectral efficiency compared to GSM to achieve a transmit diversity of order two [4]. Furthermore, a higher spectral efficiency STBC-SM scheme with cyclic structure (denoted by STBC-CSM) was proposed in [5] with a total number of codewords equal to twice the number of codewords for the STBC-SM scheme [4]. However, both STBC-SM [4] and STBC-CSM [5] adopt an NT × 2 tall transmission matrix with orthogonal columns. Therefore, developing differential transmission schemes for them is very challenging. In addition, the scheme in [6] proposed specifically for only two transmit antennas where a transmit diversity order of two is guaranteed to be achieved at the cost of a reduced spectral efficiency. Moreover, the authors in the recently accepted paper [7] proposed a systematic method for any NT to achieve a diversity order of NT by obtaining a set of dispersion matrices for the differential SM systems inspired by algebraic field extensions results; named as Field Extension DSM (FE-DSM). However, the achieved diversity order is also at the cost of a reduced spectral efficiency as they transmit only a single symbol from an M -PSK signal over the NT time slots which significantly hurt the spectral efficiency. To further enhance their spectral efficiency, the authors in [7] developed a differential SM scheme to provide a better diversity-rate trade-off; named as a differential SM with diversity-rate trade-off (FE-DSM-DR) [7]. They relax the diversity order to be less than NT while transmitting multiple independent M -PSK symbols over NT time slots, however, the spectral efficiency reduction is still significant. Moreover, the authors in [7] have mentioned that the generalization of their work to the case of multiple active transmit antennas is not straightforward due to the required unitary constraint for a differential transmission and this is what we are addressing here in this letter. To the best of the authors’ knowledge, the problem of designing a differential transmission scheme has not been addressed in the literature neither for the spatially-modulated STBC schemes nor for the GSM transmission schemes where
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multiple transmit antennas are active. In this paper, we develop a differential transmission scheme based on a novel cyclic temporally- and spatially-modulated STBC, which we refer to as STBC-TSM, codebook over four consecutive time slots. We introduce the temporal permutations as an additional information-conveying dimension which leads to a two-fold advantage by constructing a square unitary transmission which is suitable for differential transmission while increasing the spectral efficiency compared the differential SM scheme proposed in [3]. The main contributions of this paper can be summarized as follows • Proposing a design framework for constructing the STBC-TSM codebook for a four-slot transmission scheme with four transmit antennas (i.e. NT = 4) and two active transmit antennas (i.e. NA = 2) while ensuring a full transmit diversity order of two. • Designing the first differential transmission scheme for GSM based on the proposed STBC-TSM codebook which is applicable to any constant-energy phase-shift keying (PSK) signal constellation. • We show how to extend our proposed differential STBCTSM scheme for more than four transmit antennas with more than two active antennas (NT > 4 and NA > 2). • For the case of NT = 4 and NA = 2, our numerical results show that our proposed differential STBC-TSM scheme enjoys higher spectral efficiency and lower error rate than the DSM in [3] due to its temporal modulation and diversity advantages, respectively. In addition, our numerical results demonstrates the spectral efficiency advantage of our proposed scheme, due to its temporal modulation, compared to the FE-DSM-DR scheme in [7] at the same diversity order. Notation: Lower and upper case bold letters denote vectors and matrices, respectively. In addition, we denote blkdiag (A, B, C, ...) as a block diagonal matrix whose blocks L are of the matrices A, B, C, ...etc. We denote by {xi }i=1 the set of the elements xi for i = 1, · · · , L. II. T HE STBC-TSM C ODEBOOK C ONSTRUCTION FOR NT = 4 AND NA = 2 We choose the 2×2 Alamouti STBC transmission matrix as the core STBC due to its orthogonal structure. In Alamouti’s STBC, two complex information symbols (x1 and x2 ) drawn from a constellation Ψ are transmitted from two transmit antennas over two transmission time slots in an orthogonal manner [8]. In our proposed STBC-TSM codebook, we modify the 2 × 2 Alamouti STBC matrix in [8] by introducing a complex rotation ejφ to the transmitted information symbols in the first time slot over both transmit antennas as follows C (x1 , x2 , φ) =
x1 ejφ −x∗2 ejφ
x2 x∗1
(2)
where rows and columns correspond to the active transmit antennas and the symbol transmission time slots, respectively. Using the class of 2 × 2 orthogonal matrices in Eq. (2) as a building block, we propose the following framework for constructing the cyclic STBC-TSM codebook assuming NT = 4 and NA = 2 over the four consecutive time slots to transmit four complex information symbols (x1 , x2 , x3 , and
x4 ) drawn from a complex signal constellation Ψ . Constructing the cyclic STBC-TSM codebook is described next in the spatial and temporal domains 1) Spatial Domain of the Cyclic STBC-TSM Codebook: For a MIMO transmission with four transmit antennas, we denote the indices of the active transmit antennas in the 4 × 4 STBCTSM generator matrix Sk corresponding to the 1st , 2nd time (1) (2) slots by ik , ik = (1, k + 1) where 1 ≤ k ≤ NT − 1 (i.e. 1 ≤ k ≤ 3). Furthermore, the indices of the active we restrict (1) (2) transmit antenna pair jk , jk in Sk corresponding to the 3rd , 4th time slots not to overlap with the antenna index pair (1, k + 1) used in the 1st and 2nd time slots to ensure a full rank for the For example, if k = 1, i.e. transmission matrix. (1) (2) (1) (2) ik , ik = (1, 2), then jk , jk = (3, 4). On the other (1) (2) (1) (2) hand, if k = 2, i.e. ik , ik = (1, 3), then jk , jk = (2, 4). (1) (2) Based on the previous choice of ik , ik and (1) (2) jk , jk , the 4 × 4 STBC-TSM generator matrix Sk can be expressed as follows Sk
Υk × blkdiag S(1) , S(2) , 1 ≤ k ≤ 3
=
(3)
where S(1) and S(2) are the 2 × 2 matrices which follow the form in Eq. (2) corresponding to transmitting two complex information symbols (x1 and x2 ) and (x3 and x4 ) with distinct complex rotation angles α1 and α2 , respectively. We assume that α1 and α2 are the complex rotation angles for the 1st and 3rd time slot, respectively (i.e. φ = α1 at the 1st time slot and φ = α2 at the 3rd time slot as defined in Eq. (2)). We assume that α1 6= α2 to ensure a full transmit diversity order of two. In addition, Υk in Eq. (3) is a 4 × 4 row permutation matrix that determines the indices of the active transmit antennas corresponding pairs of transmit to the non-overlapping (1) (2) (1) (2) antennas ik , ik and jk , jk for each two consecutive time slots, respectively. Define G to be the 4 × 4 right-shift matrix[5] which circularly activates the pairs non-overlapping (1) (2) (1) (2) of transmit antennas ik , ik and jk , jk for each two consecutive time on the active transmit antennas slots. Based (1) (2) indices choice ik , ik = (1, k + 1) for the 1st , 2nd time slots, the cyclic spatially-modulated 4 × 4 codeword is defined as Gl−1 Sk ejθk,l where G0 = I4 for 1 ≤ l ≤ 4. Moreover, θk,l is the rotation angle corresponding to the l-th cyclic rotation for the STBC-TSM generator matrix Sk . Hence, the pair (k, l) lies in a 2−dimensional space that forms the spatial domain of the proposed cyclic STBC-TSM codebook. 2) Temporal Domain of the Cyclic STBC-TSM Codebook: Define the following time permutation πi over four consecutive time slots πi
:
{1, · · · , 4} → {πi (1) , · · · , πi (4)}
(4)
The permutation matrix acting on 4-dimensional column vectors is the following 4 × 4 matrix Pπi Pπi = eπi (1) , eπi (2) , eπi (3) , eπi (4)
(5)
where i = 1, · · · , 4! and eπi (i) is an all-zero column vector except for the entry πi (i) which equals 1.
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For a cyclic spatially-modulated 4 × 4 codeword (i) G Sk ejθk,l , the temporally- and spatially-modulated Xk,l corresponding to the i-th time permutation is defined as l−1
(i)
Xk,l = Gl−1 Sk ejθk,l Pπi ejϕi | {z } | {z } Spatial
Domain
(6)
T emporal Domain
where ϕi is the rotation angle corresponding to the i-th time permutation. Therefore, the k-th codebook corresponding to (i) the i-th time permutation χk while assuming that the indices of the active transmit antennas corresponding to the 3rd , 4th (1) (2) time slots are jk , jk is given by (i)
χk
=
(i)
4 , 1 ≤ k ≤ 3 , 1 ≤ i ≤ 4!
Xk,l
(7)
l=1
Therefore, the total STBC-TSM codebook χ consists of a total of 3 × 4 × (4!) codewords, and can be generated as χ=
3 [ 4! [
(i)
χk
(8)
k=1 i=1
Consequently, the total number of codewords for the STBCTSM codebook is given by c = b3 × 4 × (4!)c2p and its spectral efficiency R (bps/Hz) is R=
1 log2 c + log2 M = 2 + log2 M 4
(9)
which is 1 bit more than the DSM scheme proposed in [3] for the same number of transmit antennas NT . Since there is no advantage to emphasize a specific time permutation or cyclic rotation, we choose the rotation angles θk,l and ϕi to be uniformly spaced, to reduce the design complexity, as follows ( θk,l =
(k−1)π + (l−1)π NT −1 NT (k−1)π (l−1)π + 2(NT −1) 2NT
( ϕi =
(i−1)π 4! (i−1)π 2(4!)
M = 2(BP SK) M ≥4
M = 2(BP SK) M ≥4
(10)
(11)
Moreover, since the design of our proposed STBC-TSM codebook depends on the rotation angles α1 and α2 , these angles should be selected to ensure a full transmit diversity order, as discussed next. Lemma 1. Selecting the rotation angles α1 and α2 such that α1 6= α2 ensures the transmit diversity order of two. Proof: For notational brevity, we assume that the complex signal constellation symbols (x1 , x2 , x3 , x4 ) and (x01 , x02 , x03 , x04 ) are pairwise orthogonally transmitted over the four consecutive time slots forming the STBC-TSM or(i0 ) (i) thogonal codewords Xk,l and Xk0 ,l0 , respectively. Hence, the codeword difference matrix Λχ is 0 (i ) (i) Λχ = Xk,l − Xk0 ,l0
(12)
We assume that r1 ≤ 2 and r2 ≤ 2 are the number of nonzero eigenvalues of the codeword difference matrix Gramian st ΛH and 2nd ) and (3rd and 4th ) χ Λχ corresponding to its (1 eigenvectors, respectively, such that r = r1 + r2 and r is the transmit diversity order. Based on the STBC-TSM codebook structure, it can be verified that to achieve a diversity order of 2NR , at least one of the following conditions must be satisfied 1) r1 = 1 and r2 = 1⇒ Diversity order = 2NR 2) r1 = 0 and r2 = 2⇒ Diversity order = 2NR 3) r1 = 2 and r2 = 0⇒ Diversity order = 2NR
Cases 2 and 3 will only occur when {k, l, i} = {k 0 , l0 , i0 } and either (x1 , x2 ) = (x01 , x02 ) and (x3 , x4 ) 6= (x03 , x04 ) or (x1 , x2 ) 6= (x01 , x02 ) and (x3 , x4 ) = (x03 , x04 ). Hence, a diversity order of 2NR is guaranteed regardless of the rotation angles α1 and α2 . Regarding Case 1, it can occur for any combination of {k, l, i} and {k 0 , l0 , i0 } with any combination of the complex signal constellation symbols (x1 , x2 , x3 , x4 ) and (x01 , x02 , x03 , x04 ). Hence, the scenario when (x1 , x2 ) = (x03 , x04 ) and (x3 , x4 ) = (x01 , x02 ) with {k, l} 6= {k 0 , l0 } and i = i0 represents a set of cases that can lead to a diversity of order one. For the worst-case scenario when only a single active transmit ( i0 ) (i) antenna between the codewords Xk,l and Xk0 ,l0 is common, the codeword difference matrix Gramian ΛH χ Λχ will have two all-zero columns and at least one of the other two columns will H be a function of aejα1 − bejα2 aejα2 − bejα2 , where a and b are constants that depend on the symbols (x1 , x2 , x3 , x4 ) and (x01 , x02 , x03 , x04 ). Hence, if α1 = α2 , the rank of the codeword difference matrix Gramian ΛH χ Λχ will be less than 2 (i.e. either r1 = 1 and r2 = 0 or r1 = 0 and r2 = 1) which leads to a diversity order of NR (i.e. transmit diversity order of one). Therefore, the condition α1 6= α2 should be strictly satisfied to guarantee a transmit diversity of order two. III. D IFFERENTIAL T RANSMISSION BASED ON THE STBC-TSM C ODEBOOK To develop a decodable differential transmission scheme, it is desirable to have a unitary-square transmission matrix. Although the permutation transmission matrix used in the differential scheme of [3] satisfies the unitary square constraint, it is not applicable to the case of multiple active transmit antennas. Our proposed STBC-TSM codebook for four transmit antennas can generate a class of 4×4 unitary-square transmission matrices suitable for a differential signaling for two active antennas in each time slot for any M -PSK constellation. (i) Since the square orthogonal codeword Xk,l is not a unitary (i) matrix, we normalize the codeword Xk,l , as a preprocessing step, to satisfy the square-unitary constraint for differential transmission. Hence, the k-th unitary codebook correspond(i) ing to the i-th time permutation χ ˜k which has 4 unitary (i) ˜ and the total differential STBC-TSM unitary codewords X k,l codebook χ ˜ are generated as follows 4
(i)
χ ˜k
=
˜ (i) X k,l
, 1 ≤ k ≤ 3 , 1 ≤ i ≤ 4!
(13)
l=1
χ ˜
=
3 [ 4! [
(i)
χ ˜k
(14)
k=1 i=1
˜ (i) = where X k,l
(i) √1 X 2 k,l
to ensure that the unitary constraint for
˜ (i) X ˜ (i) differential transmission i.e. X k,l k,l I4 is satisfied.
H
H ˜ (i) ˜ (i) = = X X k,l k,l
A. Differential Transmission Without loss of generality, we assume that the codeword U (t) is transmitted at time index t (corresponds to time slots, 4t to 4t + 3). The differential transmission process begins with transmitting an arbitrary 4 × 4 transmission matrix U (0)
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at time index t = 0. Then, the differential encoding process proceeds as follows 1) At time index t + 1 (corresponds to the time slots 4t + 4 to 4t + 7), we map the incoming 8 + 4log2 M bits into one of the codewords from the differential STBC-TSM ˜ (i) (t + 1). codebook χ ˜ given in Eq. (14) and get X k,l 2) Calculate the 4 × 4 transmission matrix as follows ˜ (i) (t + 1) U (t + 1) = U (t) X k,l
(15)
3) Transmit the calculated 4 × 4 transmission matrix U (t + 1) over the time slots 4t + 4 to 4t + 7. B. Optimal ML Differential Detection The NR × 4 received signal matrices at time indices t and t + 1 are given by Y (t) Y (t + 1)
= = =
H (t) U (t) + N (t) (16) H (t + 1) U (t + 1) + N (t + 1) ˜ (i) (t + 1) + N (t + 1) (17) H (t + 1) U (t) X k,l
Assuming that the channel is fixed over eight consecutive time slots (i.e. H (t + 1) = H (t)), we have Y (t + 1)
=
˜ (i) (t + 1) + N (t + 1) (Y (t) − N (t)) X k,l
=
(i) ˜ (i) (t + 1) + N (t + 1) Y (t) Xk,l (t + 1) − N (t) X k,l (18)
The ML differential detector that jointly detects the indices k, l, i and the four complex information symbols (x1 , x2 , x3 , ˜ (i) (t + 1) is given by and x4 ) for the codeword X k,l n
o ˆ ˆ k, l, ˆi =
argmin k,l,i
2
˜ (i) (t + 1)
Y (t + 1) − Y (t) X
k,l
F
{x1 ,x2 ,x3 ,x4 }∈Ψ
(19)
which requires cM 4 ML metric calculations, where kDkF is the Frobenius norm of the matrix D. The detection complexity, defined by the search space size, is determined by the spectral efficiency per time slot given the same number of transmit and active antennas. Hence, the detection complexity will not change from the differential SM scheme in [3]. IV. D ISCUSSIONS AND R EMARKS Maintaining the square-unitary transmission matrix structure, the construction of the STBC-TSM codebook presented in Section II can be extended to the cases of 1) even number of transmit antennas NT > 4 for T time slots while maintaining the same number of active transmit antennas of two (i.e. NA = 2) at each time slot over NT time slots (T = NT ). This generalization will result in an NT × NT transmission matrix by extending ¯ k to the basic STBC-TSM codebook generator matrix S a block diagonal NT × NT matrix as follows ¯k S
=
Υk × blkdiag S(1) , · · · , S(n) , · · ·, S
N
T 2
(20)
where S(n) is a 2 × 2 matrix which follows the form in Eq. (2) corresponding to transmitting two complex information symbols (x2n−1 and x2n ) from an M -PSK constellation Ψ with a distinct complex rotation angle αn , for n = 1, 2, · · · , 21 NT . We denote the indices of the active transmit antennas in the block diagonal ¯ k corresponding to the n-th, (n + 1)NT × NT matrix S (1) (2) th time slots by ik,n , ik,n . We choose the index pair corresponding to the 1st , 2nd time slots to be
(1) (2) ik,1 , ik,1 = (1, k + 1). Hence, Υk is an NT ×NT row permutation matrix that determines the active transmit antennas indices corresponding to each two consecutive time slots where 1 ≤ k ≤ NT −1. We ensure a full diversity order of two by restricting the indices of the active transmit antennas corresponding to the n-th, (n + 1)-th (for n 6= 1) time slots not to overlap with the antenna ¯ index pair (1, k + 1) used in the 1st and 2nd . Define G to be the NT ×NT right-shift matrix [5] which circularly activates the non-overlapping pairs of transmit antennas for each two consecutive time slots. Based on the active transmit antennas indices choice, the temporally- and ¯ (i) spatially-modulated modified NT × NT codeword X k,l corresponding to the i-th time permutation is defined as ¯ ejϕi ¯ (i) = G ¯ l−1 S ¯ k ejθk,l P X k,l | {z } | πi{z } Spatial
Domain
(21)
T emporal Domain
¯ 0 = INT for 1 ≤ l ≤ NT and P ¯ πi is the where G NT ×NT permutation matrix acting on NT -dimensional column vectors. In addition, the rotation angles θk,l and ϕi have the same definitions presented earlier. It can be verified that this generalization leads to STBC-TSM codebook with an NT −1 c = NT × 21 2 (NT − 1)! × (NT )! 2p
codewords and an overall spectral efficiency R = N1T log2 c + log2 M . 2) three active transmit antennas (i.e. NA = 3) over T = 8 time slots (i.e. NT = T = 8) while maintaining the relation of NT = T = 4ρ using the rate-3/4 orthogonal STBC (OSTBC-TSM) designs in [9], where ρ is any integer. For the case of ρ = 2 (i.e. NT = T = 8), We modify the rate-3/4 OSTBC transmission matrix defined in [9] as follows jφ ∗ ∗ x1 e x2 ejφ C (x1 , x2 , x3 , φ) = x3 ejφ 0
−x2 x∗1 0 x3
x3 0 −x∗1 x2
0 x∗3 (22) −x∗2 −x1
The non-overlapping indices of the three active transmit antennas in the modified 8 × 8 OSTBC-TSM gener¯ k corresponding to the 1st to 4th time ator matrix S slots and the 5th to 8th time slots are denoted by (1) (2) (3) (1) (2) (3) ik , ik , ik and jk , jk , jk , respectively. By (1) (2) (3) (2) (3) setting ik , ik , ik = 1, ik , ik , it can be easily (2) (3) verified that ik , ik can take any of the total remain(NT −1)! NA !(NT −NA −1)! (2) (3) such that ik 6= ik .
ing
=
(NT −1)! 3!(NT −4)!
= 35 index choices
Consequently, the remaining steps to generate the OSTBC-TSM codebook will still be the same as presented in Section II with 1 ≤ k ≤ 35, i = 1, · · · , 8!, and l = 1, · · · , 7 to transmit six complex information symbols over eight consecutive time slots (the detailed structure of the the 8 × 8 OSTBC-TSM ¯ k is omitted due to space limitations). generator matrix S It can be verified that this extension leads to an OSTBCTSM codebook with an overall spectral efficiency R = 23 3 8 + 4 log2 M . Moreover, for the case of ρ > 2, the extension presented in Point 1 above will be applicable for this case as well.
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6
5
4
3
4
6
8
10
12
DSM [3],
14
16
N T (Number of Transmit Antennas) (a)
18
20
10
R
DSM [3],
R
FE-DSM-DR [7],
= 2, BPSK (3 bps/Hz)
R
N = 4, N T
Differential STBC-TSM, N = 4, N T
NR = 2, 8PSK (4 bps/Hz)
Differential STBC-TSM , N
10-1
FE-DSM-DR [7] - N T = 8 Proposed Differential STBC-TSM - NT = 8
100
N = 2, QPSK (3 bps/Hz)
Differential STBC-TSM, N
FE-DSM-DR [7] - N T = 6 Proposed Differential STBC-TSM - NT = 6
FE-DSM-DR [7],
= 2, QPSK (4 bps/Hz)
T
R
R
N = 4, N
R
= 2, BPSK (1.25 bps/Hz) = 2, BPSK (3 bps/Hz) = 3, BPSK (1.25 bps/Hz)
Differential STBC-TSM, N T = 4, N R = 3, BPSK (3 bps/Hz)
10-1
8
10-2 6
10-2
BER
7
2
100
FE-DSM-DR [7] - N T = 4 Proposed Differential STBC-TSM - N = 4 T
Spectral Efficiecny per Time Slot (Bps/Hz)
8
Spectral Efficiecny per Time Slot (Bps/Hz)
12
Proposed Differential STBC-TSM - BPSK DSM [3] - BPSK Proposed Differential STBC-TSM - QPSK DSM [3] - QPSK Proposed Differential STBC-TSM - 8PSK DSM [3] - 8PSK
BER
9
10-3
10-3
10-4
10-4
4
2
0
10-5 1
2
3
4
5
6
7
8
0
3
6
Fig. 1. Comparison of the spectral efficiency per time slot from the spatial domain between our proposed differential STBC-TSM scheme, the DSM scheme [3], and the FE-DSM-DR scheme [7] assuming FBE for a diversity order of two. V. N UMERICAL R ESULTS
9
12
15
18
21
10-5
Eb / N o (dB)
log2 M (b)
0
3
6
9
12
15
Eb / N o (dB)
(a)
(b)
Fig. 2. (a) BER performance of the DSM scheme [3] versus Eb /No compared to our proposed Differential STBC-TSM scheme with NT = 4 for NR = 2 at 3 bps/Hz and 4 bps/Hz. (b) BER performance of the FEDSM-DR scheme in [7] versus Eb /No compared to our proposed Differential STBC-TSM scheme with NT = 4 and BPSK for NR = 2 and NR = 3 . 10
0
Coherent STBC-TSM, BPSK (3bps/Hz) Differential STBC-TSM, BPSK (3bps/Hz)
10
Coherent STBC-TSM, QPSK (4bps/Hz)
-1
Differential STBC-TSM, QPSK (4bps/Hz)
10-2
BER
In Fig. 1, we compare the spectral efficiencies due to the spatial domain between our proposed differential STBC-TSM scheme (assuming only two active transmit antennas), the DSM scheme in [3], and the FE-DSM-DR scheme in [7]. We compare the spectral efficiency assuming a fractional bit encoding (FBE) can be applied, i.e. the spectral efficiency can be a real number [3]. Fig. 1-a shows the significant spectral efficiency increase obtained by introducing the temporal permutations to our proposed STBC-TSM codebook structure compared to the DSM in [3] at the same number of transmit antennas in addition to the diversity advantage of our scheme over the one in [3]. Assuming the diversity order of two, Fig. 1-b demonstrates the spectral efficiency enhancement obtained by introducing the temporal permutations compared to the FEDSM-DR scheme in [7] at the same M -PSK constellation size. In addition, it is clear that increasing the number of transmit antennas doesn’t enhance the spectral efficiency of the FE-DSM-DR scheme in [7] compared to the enhancement obtained by our proposed scheme. Assuming a slowly-varying Rayleigh flat-fading channel, in Fig. 2-a, we compare the BER performance of our proposed differential scheme with the DSM scheme in [3] versus Eb /No where Eb is the bit energy. We set α1 = π4 and α2 = π2 and leave their optimization for future work. Assuming a total of four transmit antennas, we consider the case of two active antennas for our proposed differential scheme and a single active antenna for the DSM scheme in [3]. For the same spectral efficiency, Fig. 2-a shows that our proposed differential STBC-TSM scheme achieves about 4 dB and 6 dB SNR gains compared to the DSM scheme in [3] for NR = 2 at BER of 10−3 for a spectral efficiency of 3 bps/Hz and 4 bps/Hz, respectively. These results demonstrate the diversity advantage of our proposed differential scheme over the DSM scheme [3]. For two active antennas (i.e. two transmit RF chains), Fig. 2-b shows that an SNR gain of 1.5 dB and 1 dB is achieved by our differential STBC-TSM scheme compared to the FE-DSM-DR scheme [7] at BER of 10−3 for NR = 2 and BER of 1.5 × 10−4 for NR = 3, respectively, while achieving a 1.75 bps/Hz higher spectral efficiency compared to the FEDSM-DR scheme [7]. Finally, Fig. 3 confirms that the performance gap is no more than 3 dB between the differential and coherent transmission schemes based on our proposed STBC-TSM codebook.
10-3
10-4
NR = 1
NR = 2 10-5
0
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15
18
21
24
E b / No (dB)
Fig. 3. BER performance of the coherent and differential transmission based on STBC-TSM codebook versus Eb /No with NT = 4 for NR = 1, 2 at 3 bps/Hz and 4 bps/Hz. VI. C ONCLUSIONS
We designed a differential transmission scheme based on a novel STBC-TSM codebook. Our design addresses the challenge of designing a differential transmission scheme for spatial modulation schemes with multiple active transmit antennas and can be applied to any constant-energy PSK constellation. Our numerical results demonstrate that our proposed scheme enjoys a significant BER gain at higher spectral efficiency compared to the DSM scheme and suffers no more than 3 dB from its coherent counterpart. R EFERENCES [1] M. D. Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatial Modulation for Generalized MIMO: Challenges, Opportunities, and Implementation,” Proceedings of IEEE, vol. 102, pp. 56 – 103, Jan. 2014. [2] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Spatial Modulation: Optimal Detection and Performance Analysis,” IEEE Communications Letters, vol. 12, pp. 545 – 547, Aug. 2008. [3] Y. Bian, X. Cheng, M. Wen, L. Yang, H. V. Poor, and B. Jiao, “Differential Spatial Modulation,” IEEE Transactions of Vehicular Technology, vol. 64, no. 7, pp. 3262–3268, Jul. 2015. [4] E. Basar, U. Ayg, E. Panayirci, and H. V. Poor, “Space-time Block Coded Spatial Modulation,” IEEE Transactions on Communications, vol. 59, no. 3, pp. 823–832, Mar. 2011. [5] X. Li and L. Wang, “High Rate Space-Time Block Coded Spatial Modulation with Cyclic Structure,” IEEE Communications Letters, vol. 18, no. 4, pp. 532–535, Apr. 2014. [6] W. Zhang, Q. Yin, and H. Deng, “Differential Full Diversity Spatial Modulation and Its Performance Analysis With Two Transmit Antennas,” IEEE Communications Letters, vol. 19, no. 4, pp. 677–680, Apr. 2015. [7] R. Rajashekar, N. Ishikawa, S. Sugiura, K. V. S. Hari, and L. Hanzo, “Full-Diversity Dispersion Matrices from Algebraic Field Extensions for Differential Spatial Modulation,” to appear IEEE Transactions on Vehicular Technology, Mar. 2016. [8] S. Alamouti, “ A Simple Transmit Diversity Technique for Wireless Communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451 – 1458, Oct. 1998. [9] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–Time Block Codes from Orthogonal Designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999.
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