Accepted for PIMRC 2006
Low Complexity Unitary Differential Space-Time Modulation with Spherical Code Chau Yuen Institute for Infocomm Research 21 Heng Mui Keng Terrace Singapore
[email protected]
Yong Liang Guan Nanyang Technological University Block S1 Nanyang Avenue Singapore
[email protected]
Abstract We show a new design of unitary differential spacetime modulation (DSTM) with low decoding complexity, i.e. its decoding can be performed by two parallel decoders where each of the decoder has a search space of only N for a codebook with N codewords. The design is based on Orthogonal Space-Time Block Code (OSTBC) and spherical code. We separate the symbols of an O-STBC into two groups and jointly modulate the symbols within a group using a joint constellation set constructed from spherical code. The proposed unitary DSTM scheme has a lower decoding complexity than many other DSTM schemes including those based on Group Codes and Sp(2) with better or comparable decoding performance. It also achieves a better performance than existing DSTM schemes with similar decoding comlexiy.
I. INTRODUCTION Modulation schemes designed for multiple transmit antennas, called space-time modulation or transmit diversity can be used to reduce fading effects effectively. Many transmit diversity schemes were designed for coherent detection, with channel estimates assumed available at the receiver. However, the complexity and cost of channel estimation grow with the number of transmit and receive antennas. Therefore, transmit diversity schemes that do not require channel estimation are desirable. To this end, several differential space-time modulation (DSTM) schemes have been proposed [1-10]. The DSTM schemes in [1-6] generally have high decoding complexity, they have a decoding search space of N for a codebook with N codewords, this leads to exponential increase in decoding complexity as the spectral efficiency increases. On the other hand, the scheme in [7] is single-symbol decodable, hence it has the simplest decoding complexity, for example, for the case of four transmit antennas, the decoding search space could be only 3 N . However, such reduction in decoding complexity is generally obtained at the scarifying of the decoding performance. Its maximum achievable code rate is limited to ¾ for four antennas and ½ for eight antennas, as it is designed based on square Orthogonal Space-Time Block Code (O-STBC).
Tjeng Thiang Tjhung Institute for Infocomm Research 21 Heng Mui Keng Terrace Singapore
[email protected]
To achieve differential transmit diversity with high code rate, low decoding complexity and good performance, two DSTM schemes (one is unitary and another is non-unitary) based on the Quasi-Orthogonal STBC (QO-STBC), and a unitary DSTM scheme based on unitary non-linear STBC have been proposed in [8], [9] and [10] respectively. All three support full rate (code rate = 1 symbol/channel use) for four transmit antennas, and both are pair-wise decodable, hence they have lower decoding complexity than the DSTM schemes in [1-6] with a decoding search space that is less than or equal to N . In this paper, we propose a new unitary DSTM scheme, whose decoding can be performed by two parallel decoders, i.e. each decoder with a search space of N , based on the concept of joint modulation. It employs the unitary matrices derived from O-STBC with joint constellation constructed from spherical code. Though also based on O-STBC, our proposed design is different from [7] in the sense that we use a special set of constellation designed from spherical code by jointly modulating several symbols. Due to this property, our proposed DSTM is able to give a better performance than [7-10], while supporting a wide range of spectral efficiencies.
II. REVIEW OF UNITARY DSTM
A. Unitary DSTM Signal Model Consider a MIMO communication system with NT transmit antennas and NR receive antennas. Let Ht be the NR × NT channel gain matrix at a time t. The ikth element of Ht is the channel coefficient for the signal path from the kth transmit antenna to the ith receive antenna. Let Ct be the NT × P codeword transmitted at a time t. Then, the received NR × P signal matrix Rt can be written as R t = H t Ct + N t
(1)
where Nt is the additive white Gaussian noise. At the start of the transmission, we transmit a known codeword C0, which is a unitary matrix. The codeword Ct transmitted at a time t is differentially encoded by Ct = Ct −1U t
(2)
Accepted for PIMRC 2006
where Ut is a unitary matrix of size NT × NT (such that UtUtH=I), called the code matrix, that contains information of the transmitted data. Since C0 and Ut are both unitary, it follows that Ct is unitary for all time t. Hence the requirement for the code matrix Ut to be unitary is essential to ensure that all the transmitted codewords have constant power. If we assume that the channel remains unchanged during two consecutive code periods, i.e. Ht = Ht-1, the received signal Rt at a time t can be expressed [7] as R t = H t Ct −1U t + N t = (R t −1 − N t −1 )U t + N t = R t −1U t + N t
(3)
A. Orthogonal Space-Time Block Code A 4×4 codeword of reate-3/4 O-STBC in [11] (herein denoted as CO4) is shown in (7): ⎡ c1 ⎢ 0 CO4 = ⎢ * ⎢ −c2 ⎢ * ⎢⎣ c3
0 c1
c2 c*3
−c3 −c2
c1* 0
where ci, 1 ≤ i ≤ 3, represents the complex information symbol to be transmitted and
α = ∑ i =1 ci 3
The received signal Rt can be differentially decoded as it depends only on the previous received signal block Rt-1, the code matrix Ut and an equivalent additive white = −N U + N . Since Nt and Nt-1 are Gaussian noise N t t −1 t t is white white and Ut is unitary, the equivalent noise N
−c3 ⎤ ⎡α 0 0 0 ⎤ ⎢0 α 0 0⎥ * ⎥ c2 ⎥ H ⎥ =⎢ CO4CO4 ⎢0 0 α 0⎥ 0 ⎥ ⎥ ⎢ ⎥ c1* ⎥⎦ ⎣0 0 0 α⎦ (7)
2
(8)
It is obvious from (7) that CO4 is a unitary matrix. And CO4 can also be represented as: CO4 = ∑ k =1 ( A k ckR + jB k ckI ) 3
(9)
t
too [6]. The corresponding decision metric for (3) is,
(
ˆ = arg min tr {R − R U } U t t t −1 t Ut ∈U
{
H
Ut ∈U
{R t − R t −1Ut } )
}
= arg max Re tr ( R R t −1U t ) H t
(4)
where U denotes the set of all possible code matrices.
B. Diversity and Coding Gain The decoding criteria of unitary DSTM scheme have been analyzed in [1] and found to be the same as those of coherent space-time coding. Specifically, the transmit diversity level that can be achieved is given by: Min ⎡⎣ rank ( U k − U l ) ⎤⎦
∀k ≠ l .
(
)
H 1/ NT
⎤ ⎥⎦
⎛ 3 2 ⎞ det = ⎜ ∑ ∆ i ⎟ ⎝ i =1 ⎠
4
(10)
where ∆i, 1 ≤ i ≤ 3, represents the possible error in the ith transmitted constellation symbol. That is, ∆ i = ci − ei if the receiver decides erroneously in favor of ei if ci is transmitted. So O-STBC can always achieve full diversity as its determinant can never becomes zero.
(5)
In order to achieve full transmit diversity, the minimum rank in (5) has to be equal to NT and the DSTM code is said to be of full rank. For a full-rank unitary DSTM code, its coding gain is defined in [1, 7] as Min ⎡⎢ NT × det ( U k − U l )( U k − U l ) ⎣
where Ak and Bk are called the dispersion matrices [13] with orthogonal properties [11], and the superscripts R and I represent the real and imaginary parts of a symbol respectively. The determinant of the codeword distance matrix of CO4 can be easily shown to be:
∀k ≠ l . (6)
In order to achieve optimum decoding performance, the coding gain has to be maximized.
III. NEW UNITARY DSTM SCHEME BASED ON O-STBC In this section, we shall develop a new unitary DSTM scheme using the well-known square O-STBC. For simplicity, we will use the rate-3/4 O-STBC for four transmit antennas in [11] as an example. The proposed unitary DSTM technique, however, is applicable to any square O-STBC, including those for eight transmit antennas.
B. New Unitary DSTM Scheme Based on O-STBC with Joint Modulation CO4 in (7) can be used as unitary code matrix of a unitary DSTM scheme, as long as α is equal to 1. The approached taken in [7] is to select the symbols ci from PSK constellation that has a constant power. In order to achieve a better performance, we let more symbols to be jointly modulated at a slight increase in decoding complexity. We propose to separate the data symbols into two groups, i.e. three real symbols in a group. Then we jointly modulate the three real symbols in each group. By doing so we achieve a decoding search space of N for a codebook with N codewords. Using CO4 in (7) as an example, we may group symbols c1R , c1I , c2R in one group and symbols c2I , c3R , c3I in another group. Hence, this suggests that c1R , c1I , c2R should be jointly mapped to a tri-symbol {ak, bk, ck}, while c2I , c3R , c3I should be mapped to another tri-symbol {al, bl, cl}, such that |ak|2 + |bk|2 + |ck|2 = 0.5 for all values of k. In addition, to maximize the coding gain, the tri-symbols should further be designed to maximize the value of detmin
Accepted for PIMRC 2006
in (10). So the data symbol must be modulated from a special joint constellation set M , which consists of realvalued tri-symbols {ak, bk, ck} that satisfy the following criteria: 2
2
(i) Power Criterion: ak + bk + ck
2
= 0.5
(11)
(ii) Performance Criterion:
{
2
2
maximize Min ∆akl + ∆bkl + ∆ckl
2
}
where ∆akl = ak − al , ∆bkl = bk − bl , and ∆ckl = ck − cl for all k ≠ l. The spectral efficiency, Eff, of the resultant unitary DSTM scheme based on rate-3/4 O-STBC is Eff = 2(log2L)/NT bps/Hz where L is the cardinality of M , i.e. total number of tri-symbols {ak, bk, ck } in M . For example, consider a system with four transmit antennas (NT = 4) and a target spectral efficiency of Eff = 1.5 bps/Hz. From the equation of Eff, the required constellation cardinality is L = 8 = 23. In the encoder of the proposed DSTM scheme, a constellation set M with 8 tri-symbols will first have to be designed according to (11). Three information bits will be mapped to a trisymbol {ak, bk, ck} in M to constitute the code symbols { c1R , c1I , c2R } in CO4 in (7), while another three information bits will be mapped to another symbol-pair {al, bl, cl} in M to constitute the code symbols { c2I , c3R , c3I } in CO4. In the decoder of the proposed DSTM scheme, with Ut in (4) set to CO4 in (7), the decision metrics in (4) can be simplified to:
{
}
⎡ ∑ Re tr ( R tH R t −1 A i ) ciR + ⎤ i =1,2 R I R ⎥ ˆ ˆ ˆ max ⎢ {c1 , c1 , c2 } = {carg H I R I R ⎢ ⎥ 1 , c1 , c2 }∈M Re tr ( R t R t −1 jB1 ) c1 ⎣ ⎦ ⎡ Re tr ( R tH R t −1 A 3 ) c3R + ⎤ ⎥ max ⎢ {cˆ2I , cˆ3R , cˆ3I } = {carg H I I R I ⎢ ⎥ Re tr R R B j c 2 , c3 , c3 }∈M ∑ ( ) 1 t t − i i ⎣ i = 2,3 ⎦ (12)
{ {
{
}
}
}
As shown in (12), the proposed DSTM scheme can be decoded by the joint detection of three real symbols ({ c1R , c1I , c2R } or { c2I , c3R , c3I }), and the two decision metrics can be computed separately. So search space is the square root of those reported in [1- 6].
C. Design of Joint Constellation Set from Spherical Code To design joint constellation for the above proposed DSTM based on O-STBC, we need a set of constellation that consists of tri-symbols satisfying the conditions in (11). It can be easily seen that (11)(i) implies that the points must lie on the surface of a 3-dimensional sphere. In addition, (11)(ii) implies that those points must be spread as far as possible, such that the minimum distance between them is maximized. In fact, this is the wellknown mathematics problem that is known as spherical code.
Spherical code (or spherical packing) is the problem to distribute n points on a sphere in d dimension such the minimum distance (or equivalently the minimal angle) between any pair of points is maximized, and the maximum distance is called the covering radius. Using the examples of four transmit antennas with spectral efficiency of 1.5bps/Hz, we need a spherical code with 3dimension and eight points (i.e. eight sets of tri-symbols). An analytical solution to this problem is difficult to find, fortunately, a list of optimal spherical code has been found in [12]. For example, in Table 1 we list a few set of spherical codes with their dimension, number of points, and the minimum separation in terms of angle that is obtained from [12]. The exact configuration of spherical code with 3-dimension and 16-points is shown in Appendix A as an example. Table 1 Minimal separation for some optimal spherical codes Dimension Minimum separation Points n d θmin (degree) 3 8 74.8585 3 16 52.2444 4 64 42.3062
IV. PERFORMANCE RESULTS In Table 2, we compare the coding gain and decoding complexity of our proposed unitary DSTM scheme, i.e. OSTBC with spherical code, against the unitary DSTM schemes based on group codes [1, 2], square O-STBC [7], and QO-STBC in [8] for four transmit antennas. Table 2 Comparison of coding gains and decoding complexity for unitary DSTM for four transmit antennas Eff
1.5
2
Unitary DSTM scheme [1, 2]
Constellation
64PSK Spherical code Proposed 3 dimension – 8 points [8] “Two QPSK” [7] QPSK [1, 2] 256PSK Spherical code Proposed 3 dimension – 16 points [8] “Two 8PSK” [7] 16-PSK
No. of Coding Search parallel gain space decoder 1.85 64 1 2.95
8
2
2.83 2.70 0.78
8 4 256
2 3 1
1.55
16
2
1.17 0.31
16 16
2 2
Table 2 shows that our proposed unitary DSTM provides the highest coding gain at both spectral efficiency values of 1.5bps/Hz and 2bps/Hz. Our proposed unitary DSTM scheme also has lower decoding complexity than those in [1, 2], and can be decoded with two parallel decoder, each has a decoding search space dimension of 8 at 1.5bps/Hz and 16 at 2 bps/Hz. The unitary DSTM based on rate-3/4 square O-STBC [7] has a
Accepted for PIMRC 2006
lower decoding complexity than ours at a spectral efficiency 1.5bps/Hz, but at a lower coding gain. At a spectral efficiency 2 bps/Hz, our proposed unitary DSTM has higher coding gain and equal decoding search space dimension as the unitary DSTM based on rate-1/2 square O-STBC with 16PSK [7]. It is also worthwhile to mention that the use of joint modulation allows our scheme, which is based on a rate-3/4 QO-STBC, to support both spectral efficiency of 1.5bps/Hz and 2bps/Hz. Such flexibility is not known for the scheme proposed in [7]. 0
10
its decoding can be implemented by two parallel decoders; each has a decoding search space of 64. The second is DSTM based on rate-3/4 QO-STBC from [8]. A rate-3/4 QO-STBC for six transmit antennas transmit six complex symbols over eight period of time. In this setting, we jointly modulate two complex symbols. The decoding of this DSTM can be performed by three parallel decoders; each has a search space of 16. Finally is the DSTM based on rate-1/2 O-STBC from [7] that employs 8-PSK, its decoding can be performed by four parallel decoders; each with a search space of 8. It can be seen that our proposed schemes have the highest coding gain. Table 3 Comparison of coding gains and decoding complexity (decoding search space dimension) for unitary DSTM for eight transmit antennas
-1
10
Unitary DSTM scheme
Eff
-2
BLER
10
1.5
-4
10
-5
10
10
Rate-1/2 O-STBC 2bps/Hz [7] SP(2) 1.94bps/Hz [6] Rate-1 QO-STBC 2bps/Hz [8] Rate-3/4 O-STBC SP 2bps/Hz [proposed] 12
14
16
18
20
No. of Coding Search parallel gain space decoder
Spherical code Proposed 4 dimension – 2.08 64 points [8] “Two 8PSK” 1.56 [7] 8-PSK 1.17
-3
10
Constellation
64
2
16 8
3 4
0
10
22
24
SNR -1
Figure 1 Block error rates of different DSTM schemes for four tx and one rx antennas
10
In Figure 1, we compare the simulated block error rate (BLER) performance of our proposed unitary DSTM scheme and the DSTM schemes reported in [6, 7, 8], assuming four transmit and one receive antennas. Compared with the 2 bps/Hz unitary DSTM scheme based on rate-1/2 square O-STBC [7], our proposed unitary DSTM has better BLER performance with the same spectral efficiency and decoding search space dimension. Compared with the DSTM based on rate-1 QO-STBC in [8], we achieve a better decoding performance at the same decoding complexity. Finally, compared with the Sp(2) DSTM scheme of spectral efficiency 1.94 bps/Hz and decoding search space dimension of 225 (obtained from [6] with M=5, N=3), our proposed unitary DSTM scheme performs
