Sphere Decoding for Generalized Space-Time Shift Keying

Jianping Zheng1, Juan Tao1, Jinfang Dou2, Baoming Bai1,2 1State Key Laboratory of ISN, Xidian University, Xi’an, 710071, P.R.China

2National Key Laboratory of Science and Technology on Space Microwave, Xi’an, 710000, P.R.China Email: jpzheng@xidian.edu.cn, t377902391@126.com, jinfang_dou@163.com, bmbai@mail.xidian.edu.cn

Abstract—In this paper, the sphere decoding (SD) detector for the generalized space-time shift keying (G-STSK) scheme is developed to reduce the computational complexity of the optimal detector. The proposed SD detector consists of a series of SD algorithms with each one associated with an activated dispersion matrix pattern. Specifically, first, a lower bound of the distance measure for each possible activated dispersion matrix pattern is proposed. Then, SD algorithms are performed successively after rearranging these patterns in the order of non-decreasing lower bound. Furthermore, the list SD detector is also presented which can be employed as the soft detector in the turbo receiver for the coded G-STSK scheme. Simulation results are provided to corroborate the proposed detectors.

Keywords—sphere decoding; complexity; space-time shift keying; soft detector

I. INTRODUCTION Recently, a universal multiple-antenna architecture, termed

as generalized space-time shift keying (G-STSK) scheme, has been proposed in [1][2]. The G-STSK scheme is based on the activation of P (P�1) out of Q appropriately indexed space-time dispersion matrices. More specifically, G-STSK maps the information to both the activated dispersion matrix pattern and the constellation points which are used to modulate the corresponding activated dispersion matrices. By optimizing the number and size of the dispersion matrices, the number of activated dispersion matrices, as well as the numbers of the transmit antenna and the receive antenna, the G-STSK modulation can provide flexible diversity-multiplexing tradeoffs.

In the G-STSK scheme, to enhance the spectral efficiency, it is desirable to increase the value of P. However, upon increasing the value of P, the G-STSK receiver has to cope with the equivalent number of inter-channel interference (ICI) contributions [2]. Thus, a higher P leads to a higher computational complexity at the receiver. In fact, the optimal maximum likelihood (ML) or maximum a posteriori probability (MAP) algorithm may be prohibitively complex for a high-P scenario. Therefore, it may be urgent to develop

more efficient near-optimal detectors [2].

Sugiura and et al. [3] proposed a matched filter (MF)-based soft-decision detector (SoD), where the soft-decision information can be calculated approximately with a reduced complexity. However, the MF-based SoD has still an exponential complexity, and a medium performance gap from the optimal MAP detector. To address this problem partly, the authors of [3] also proposed to utilize the bitwise Gibbs sampler to reduce the search space with a cost of some performance loss. In [4], we proposed to employ the randomized bit-wise Markov chain Monte Carlo (R-b-MCMC) detector in the G-STSK scheme, which approaches the optimal MAP detection performance with some reduced complexity.

The sphere decoding (SD) algorithm is a well-known ML closest lattice point search method [5]-[6]. For conventional multiple-antenna system detection, SD algorithm can achieve the ML detection performance with a polynomial expected complexity in the high signal-to-noise ratio (SNR) region [6]. However, the direct extension of the SD to the G-STSK scheme is inefficient due to the constraint resulted by the activated dispersion matrix pattern.

In this paper, a smart implementation of the SD detector for the G-STSK scheme is presented. First, a lower bound of the distance measure for each activated pattern is presented, then, the activated patterns are ordered according to their lower bounds in the non-decreasing fashion, and finally, the SD algorithm is performed for each activated pattern successively in the arrangement order. This smart implementation of the SD detector can reduce the computational complexity effectively. In addition, the list SD (LSD) detector is also presented for the coded system.

II. SYSTEM MODEL Consider the G-STSK scheme G-STSK(Nt, Nr, T, Q, P) with

Nt, Nr, T, Q and P denoting the numbers of the transmit and the receive antennas, the interval per space-time codeword, the numbers of the total and the activated dispersion matrices, respectively. The input-output relation can be represented by

� �Y HS V , (1)

where rN T��Y � is the received signal over the Nr receive antennas. The entries of the channel matrix r tN N��H � and the

This work was supported in part by the National Basic Research Program of China (973 Program, No.2012CB316100), the National Natural Science Foundation of China (No: 61201140, 61372074), and National Key Laboratory Foundation of China (No: 9140C530401120C53201). ____________________________________978-1-4799-4808-6 /14/$31.00 ©2014 IEEE

noise matrix rN T��V � are assumed to be circular symmetric complex Gaussian variable with variances 1 and N0 (i.e.,

� �0,1�� and � �00, N�� ), respectively. Further, the channel matrix H is assumed to remain constant over the interval T per space-time codeword while varies from codeword to codeword independently, and is known perfectly at the receiver.

The tN T� space-time codeword S is generated by

� �( )

1

Ppp

ps

�

� �S A (2)

with power constraint tr , 1,...,Hq q T P q Q � �� �A A . Here,

, 1,...,tN Tq q Q�� �A � are the Q dispersion matrices which are

assumed to be known at the transceiver in advance of transmissions, and ( )ps �� (p=1,…,P) are the P signal points from the phase-shift keying (PSK) or quadrature amplitude modulation (QAM) constellation � which are to modulate the corresponding activated dispersion matrices.

S/P

dispersion-matrix

activationA1,…,AQ

M-PSK/QAM

M-PSK/QAM

b

b1

b2

...

...

...

...

...

A(1)

A(P)

s(1)

s(P)

Ssource

Fig. 1. Transmitter architecture of the G-STSK scheme.

To be more specific, as shown in Fig.1, during each block interval, the data stream � �1 ,..., Bb b�b with

� �2 2log , logB f Q P P M� � is inputted into the G-STSK mapping block, where � � 1, 2Bf Q P � satisfies the relation of

� �1 1 12 2B BQP

� � [1]. Then, the data stream b is S/P converted

to two sub-streams � �11 1 ,..., Bb b�b and � �12 1,...,B Bb b��b with length B1 and B2 (B2=B-B1), respectively. The sub-stream b1 is first mapped to one activated dispersion matrix pattern where P out of Q dispersion matrices are activated. Denote the set of the f(Q,P) activated dispersion matrix patterns as

� �1 ( , ),..., f Q PD D�� , where , 1,..., ( , )iD i f Q P� is a Q-dimensional binary vector with kth component being 1 if the kth dispersion matrix is activated in this pattern and 0 otherwise for k=1,…,Q. The sub-stream b2 is then mapped to P signal points ( )ps �� (p=1,…,P), which are used to modulate the corresponding activated dispersion matrices.

Obviously, we have 2 2logB P M� with M being the size of the signal constellation, i.e., M � � .

Applying the vectorial stacking operation vec( ) to the received signal, Eq. (1) can be rewritten as

b� �Y HK V , (3)

where � � � � � � � �1, , ,..., Qvec vec vec � � � � � �Y Y H I H � � A A

and � �vec�V V with I and� denoting the identity matrix and the Kronecker product, respectively. The Q-dimensional vector � �1,...,

T

b Qk k�K is defined as: kq is set to be the corresponding modulation symbol s(p) if the qth (q=1,…,Q) dispersion matrix is activated as � �p

q�A A , and zero otherwise. The subscript b in Kb is to emphasize that the positions and values of the non-zero elements are determined by the data stream b. It is easy to see that the complexity of the optimal ML detection for (3) is exponential with the data length B, which is prohibitively high with large B. In the next section, an efficient implementation of the SD algorithm for the G-STSK detection is presented.

III. PROPOSED SD DETECTORS

Denote jh as the j-th column of the equivalent channel

matrix H . Define � �1

( ) ,..., r

P

N T Pkk k

�� �H h h � as the matrix consisting of the P columns of the equivalent channel matrix H according to the activated dispersion matrix pattern Dk, i.e.,

1,..., Pk k are the indices of the non-zero elements of Dk, for

k=1, …, f(Q,P). And � �� �(1), ,..., Pk s sK denotes the modulation symbols (s(1),…,s(P))T associated with Dk. Then, the optimal detection of (3) can be represented as

� �opt

1,..., ,ˆ arg min kk f Q Pk

�� � (4)

� �� � � �� �� �� �

(1)

opt(1) opt (1)

,...,

ˆˆ ˆ,..., arg min , ,...,P

P P

s ss s k s s� � (5)

with

� �� � � �� � 2(1) ( ) (1), ,..., , ,...,P Pkk s s k s s� �� Y H K (6)

� �� �� �� �

(1)

(1)

,...,min , ,...,

P

Pk

s sk s s� �� (7)

Note that (7) can be solved efficiently by the SD algorithm [11]-[13]. So, to recover the transmission data, it needs to perform SD algorithm f(Q,P) times in parallel. In this section, a smarter implementation of the SD detector is given, which begins with a lower bound of (6).

A. Lower Bound of Distance Measure

To derive the lower bound, Eq. (6) is first rewritten as

� �� � � �� �

� �� �� �

� �� �

2(1) ( ) (1)

2(1)

( )

2(1)( )

22 2( ) ( ) ( ) ( )

,1 ,1 , ,1

, ,..., , ,...,

, ,...,,

1

, ,...,

1

... ,

k

k k k

P Pk

Pk

Pk

y

Pj k P k

y j y P P P yj

k s s k s s

k s s

k s s

s s� � � � ��

� � �

� ��� ��� �� �

� �� � �� �� � �� �� �� �� �

� � � � � ��

Y H K

KH Y

KR�

0

(8)

where ( )kR and ky� are from the following QR decomposition

� � � �( )

( ) ( ) ( ) ( ), |k

kk k k k

k y

� �� �� � � �� �

� �

RH Y Q R Q y �

0� . (9)

Here, � � � �1( ) ( ) , rN T Pk kk

� �� � �Q Q y� � is an unitary matrix,ky ��

� � 1,1 ,,..., ,

k k k

T Py y P y� � � ��� , and ( )k P P��R � is an upper

triangular matrix with entries ( ),k

u v� , u, v=1,…,P. Fig. 2 illustrates the block matrix multiplication in (8).

( )kR

,1ky�

,k Py�

ky�

�K

,1

,

( )k

k P

y

k

y

�

�

� �� �

�� �� �� �

� KR

ky�

Fig. 2. Illustration of the block matrix multiplication in (8).

Then, the minimum squared Euclidean distance k� of pattern Dk from the received signal Y can be rewritten as

( )

22( ) ( )

, ,, 1,..., 1min

j k k

P Pj k

k y t j t ys j P t j t

s� � �� � �

� � � �� � , (10)

which can be lower bounded by

2 2lb ( ) ( ), ,ˆ

k k

P kk k y P P P ys� � �� � � � �� , (11)

with � �( ), ,ˆ

k

Py P P Ps g � �� . Here, � �g � denotes the closest

constellation point from the argument. Without loss of generality, we assume that the , 1,..., ( , )k k f Q P� � have been arranged in the non-decreasing order, i.e.,

1 2 ( , )f Q P� � �� .

B. SD Detector The proposed SD detector consists of a series of SD

algorithms with one for each possible activated dispersion matrix pattern, which is summarized as Algorithm I.

Algorithm I (Input 0C , lb , 1,..., ( , ),k k f Q P� � ( ),k

ky� R .

Output *k and *( ) , 1,...,ps p P� ):

For k=1, …, f(Q,P)

If lb0k C� � , perform the complex SD [13] on the

following closest point search problem

� �� �� � � �

( )

( )

( )

2(1)

, 1,...,

2( ) ( )

, 1,...,

( ) ( ) ( ) (,1 1,1 2,1 ,1( ) ( )

,2 2,2 ,2

, 1,...,

( ),,

min , ,...,

min 1: , , 1,...,

min

p k

p k

k

k

p

k

Py

s p P

k py

s p P

k k ky P

k ky P

s p P

kP Py P

k s s

P k s p P

s

�

� � � �� � �

��

�

�

�

� �

� � �

� � � �� � � �� � � �� �� � � �� � � �� �� � � �� �

� R K

����

21)

(2)

( )P

s

s

� �� �� �� �� �� �� �

�

(12)

with initial local squared radius 2

0 0 k

kyC C �� � . If the

complex SD successes with local optimal solution ( ) ,pks p � 1,..., P and the related minimum squared radius

0kC , update the global squared radius and the output

information by 2

0 0 k

kyC C �� � and * *( ) ( ), ,p p

kk k s s� � 1,...,p P� , respectively.

End for loop.

Here, the initial radius is set as 20 1,ZF-DFEC d� with 2

1,ZF-DFEd being the squared distance between the zero-forcing decision- feedback equalizer (ZF-DFE) solution for pattern D1 and the received signal Y . The ZF-DFE solution for pattern D1 is

1

( ) ( ) (1) (1), , ,

1

ˆ ˆ , 1,...,P

t jy t j t t t

j ts g s t P� � �

� �

� �� �� � �� �� �� �� �� �

� . (13)

And its squared distance from the received signal is

1

222 ( ) ( )

1,ZF-DFE , ,1

ˆk

P Pj k

y t j t yt j t

d s� � �� �

� � �� � . (14)

It is easy to see that, with the above SD detector, the total search space can be reduced enormously compared with the paralleled SD detector in (4)-(7). The two key components of the proposed SD detector are i) the successive SD in the order of non-decreasing lower bound, and ii) the initial radius update in the successive SD. The first component can reduce the time of the SD algorithm implementation, while the second can reduce the search space in the next SD algorithm implementation if it is necessary.

C. LSD Detector Here, the extension of the SD detector to the turbo receiver,

called list SD (LSD) detector, is presented. For the LSD

detector, a candidate set � of the most likely vectors with size NCAND should be generated, which can be acquired with a simple modification on the SD detector. First, the initial radius is set to be positive infinite, i.e., 0C � �� . Then, every time the SD algorithm finds a valid point inside the current radius C0, it: 1) adds this point to � if the set � is not already full; or 2) if � is full, it replaces the point with the largest radius in � and decrease C0 to the largest radius in the new � . Given the candidate set � and the prior information � �a kL b provided by the outer channel decoder, the max-log version of the LSD detector calculates the extrinsic information by

� �� �� � � �

� �� � � �

1

1

(1)

0

(1)

0

, ,..., 1max2

, ,..., 1 max ,2

kb

kb

P

e i j a jj k

P

j a jj k

k s sL b b L b

N

k s sb L b

N

�

�

� �

� �

� ��� �� � �� �� �� ��� �� � �� �� �

�

�

K

K

�

�

(15)

where 1k�� and 1

k�� denote the sets of bit vectors in � whose

k-th bit is +1 and -1, respectively.

D. Complexity Analysis In this section, the complexity per bit of the proposed

detectors in a rapidly block-fading environment is analyzed. The complexity is quantized by the number of real-valued multiplications. First, the complexity to compute

� �� �H I H � is 4 t rN N TQ [3]. Second, the complexity of the f(Q,P) QR decompositions of (9), through Gram-Schmidt orthogonalization, can be shown to be � �QRComp ,f Q P�

� � � � � �� �2 1 2 2 1rN T P P P P� � � � . Further, the complexity to

compute the initial value 21,ZF-DFEd from (13)-(14) can be found

to be � �� �2 2 1 1P P� � . Denote Alg. IComp as the complexity of Algorithm I, which can be acquired by Monte Carlo simulation. Then, the total complexity of the SD detector is

� �� �QR

SDAlg. I

4 Comp1 Comp2 2 1 1 Comp

t rN N TQ

B P P

�� �� � �� �� � � �� �

(16)

For the soft-decision detector, the complexity in the first turbo iteration is analyzed. With this assumption, the complexity of the MAP detector is the same as that of the ML detector. Note that the value of � �� �(1), ,..., Pk s s� in (15) has been computed and stored during the procedure to generate the candidate set � . Hence, the complexity of the LSD detector can be presented as

� �LSD QR Cand1Comp 4 Comp Compt rN N TQB

� � � � � , (17)

where CandComp denotes the complexity to generate � and can be acquired from Monte Carlo simulation.

For comparison, from [4], the complexities of the ML detector, the MAP detector and the R-b-MCMC detector are also listed by

� �� �MAP ML Comp Comp

1 4 4 2 2 2B Bt r rN N TQ N T P

B

�

� � � �, (18)

� �� �R-b-MCMC1Comp 4 4 2 e e

t r rN N TQ N T PB

� � � �� � (19)

with e� being the bit expanding set of � [10].

IV. SIMULATION RESULTS In this section, we provide computer simulation results to

demonstrate the performance of the proposed detectors. The 8-PSK modulated G-STSK(3,2,3,5,P) schemes with P=2 and 3 are considered, which correspond to the data length B=9 and B=12, respectively. Both the dispersion matrices for the two schemes are acquired from Monte Carlo simulations based on the rank-determinant criterion [1].

The un-coded bit-error-rate (BER) performance of the proposed SD detector together with the ML detector is shown in Fig. 2. The abscissa in logarithmic SNR measure is defined as

� �100

SNR 10log dBsEN

� , (20)

where Es is the average signal energy per receive antenna. And the complexities of the detectors for the G-STSK(3,2,3,5,2) scheme are shown in Fig. 3. From Figs. 2 and 3, it is obvious that the proposed SD can achieve the same performance as the ML detector with an enormous complexity reduction.

The coded BER performance of the LSD detector is given in Fig. 4 for the G-STSK(3,2,3,5,2) scheme. Here, a code-rate Rc=1/2 regular LDPC code (column degree 3 and row degree 6) of length 10800 is employed. In the turbo receiver, the maximum number of iterations between the detector and the LDPC decoder is set to 8, and the maximum number of message propagation iterations in LDPC decoder is set to 50. In this figure, LSD-NCAND denotes the LSD detector with a candidate set of size NCAND. Define the signal energy per transmitted information bit at the receiver as

� �b s r cE E N T BR� . The abscissa is

� �100 0

10 log dBb s r

c

E E N TN BR N

��

. (21)

From Figs. 3 and 4, first, the performance of the proposed LSD detector improves with the increasing NCAND. The performance gain of the LSD detector is about 0.7dB from NCAND=16 to NCAND=32, while only about 0.1dB from NCAND=32 to NCAND=64 at BER=3×10-4. The LSD detector with NCAND=64 has only a slight performance loss and large complexity reduction compared with the MAP detector. Second, the proposed LSD detector can achieve similar performance while lower complexity than the R-b-MCMC detector. For example, the LSD detector with NCAND=32 has

nearly the same performance as the R-b-MCMC detector, while the complexity of the former is only about 0.16 times of that of the latter.

0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

SD, P=2ML, P=2SD, P=3ML, P=3

Fig. 2. BER curves for the un-coded G-STSK(3,2,3,5,P) scheme with P=2, 3.

0

500

1000

1500

2000

2500

3000

3500

4000

Com

plex

ity p

er b

it

ML/Max-Log-MAPMax-Log-R-b-MCMCLSD-64LSD-32LSD-16SD

Fig. 3. Complexities of different detectors for the G-STSK(3,2,3,5,2) scheme.

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.010-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BE

R

Max-Log-MAPMax-Log-R-b-MCMCLSD-16LSD-32LSD-64

Fig. 4. BER curves of different detectors for the coded G-STSK(3,2,3,5,2) scheme.

7.0 7.5 8.0 8.5 9.010-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BE

R

LSD-16LSD-32LSD-64LSD-128

Fig. 5. BER curves of the LSD detector for the coded G-STSK(3,2,3,5,3) scheme.

The BER results for P=3 are also given in Fig. 5, where the curves of both the Max-Log-MAP detector and the Max-Log-R-b-MCMC detector are not shown due to the very high implementation complexity. Similarly, the performance of the LSD detector improves with the increasing NCAND, and the gain become smaller with larger NCAND.

V. CONCLUSION In this paper, the SD detector for the G-STSK scheme is

developed. The novelty lies in the proposed lower bound of the distance measure for each activated dispersion matrix pattern, based on which the SD detector can be realized in an efficient fashion. Both the proposed SD detector and its soft-decision variant, the LSD detector, can approach the optimal detection performance with reduced complexity.

REFERENCES [1] S. Sugiura, S. Chen, and L. Hanzo, “Generalized space-time shift keying

designed for flexible diversity-, multiplexing- and complexity tradeoffs,” IEEE Trans. Wirel. Commun., vol. 10, no. 4, pp. 1605-1615, Apr. 2011.

[2] S. Sugiura, S. Chen, and L. Hanzo, “A universal space-time architecture for multiple-antenna aided systems,” IEEE Commun. Surveys & Tutorials, vol. 14, no. 2, pp. 401-420, Second Quarter 2012.

[3] S. Sugiura, C. Xu, S. Ng, and L. Hanzo, “Reduced-complexity iterative-detection aided generalized space-time shift keying,” IEEE Trans. Veh. Technol., vol. 61, no. 8, pp. 3656-3664, Oct. 2012.

[4] J. Zheng, Y. Sun, “Generalized space-time shift keying: Randomized bitwise Markov chain Monte Carlo detection and code design,” in Proc. of 2013 IEEE ICSPCC, Kunming, China, pp.1-5, August 5-8, 2013.

[5] M. O. Damen, H. El Gamal, and G. Caire, “On maximum-likelihood detection and the search for the closest lattice point,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2389-2402, Oct. 2003.

[6] B. M. Hochward, and S. ten Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389-399, Mar. 2003.