Transmit Antenna Selection for Linear Dispersion
Transmit Antenna Selection
for Linear Dispersion
Codes Based on Linear Receiver Dan Deng, Ming Zhao, Jinkang Zhu PCN&SS Lab, EEIS,USTC Hefei, P. R. China, 230027 dengdan(mail.ustc.edu.cn, zhaoming(ustc.edu.cn, [email protected]
Abstract---In an effort to utilize the advantage of using multiple antennas, transmit antenna selection for linear dispersion codes (LDC-TAS) is proposed in this paper. We present three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal transmit antennas when linear, coherent receiver is used over a slowly varying channel. Simulation results suggest that the max-min post-SNR criterion outperforms the other selection methods in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS.
Keywords- antenna selection; linear dispersion codes; linear receiver; STBC; V-BLAST;
I.
INTRODUCTION
Multiple-input multiple-output (MIMO) systems can offer significant capacity gains over traditional single-input single-output (SISO) systems [1]. Practical modulation schemes for MIMO systems with receive-only channel knowledge fall principally into two areas known as diversity and multiplexing [2]. Space-time coding, such as the space-time trellis codes [3] and space-time block codes [4], uses specially designed codeword that maximize the diversity advantage or reliability of the transmitted information. Recognizing that orthogonal space-time block codes [3], designed to maximize diversity advantage do not achieve full channel capacity in MIMO channels, Hassibi and Hochwald [5] proposed the revolutionary linear dispersion codes (LDCs). And Robert W. Heath [6] introduced a family of LDCs designs based on frame theory, which guarantees good performance in terms of diversity advantage. Although Multiple-antenna system can improve the
capacity and reliability of radio communication, the multiple RF chains are costly in terms of size, power, and hardware. Antenna selection is a low-cost low-complexity alternative to capture many of the advantages of MIMO systems [7]. In an effort to utilize the advantage of using multiple antennas, several papers have appeared recently in the literature in which the notion of antenna selection was introduced for space-time coding. In [8], the authors present a comprehensive performance analysis of orthogonal STBCs with receive antenna selection. And quasi-orthogonal STBCs are investigated in [9]. Three selection criteria are proposed and compared for spatial multiplexing systems based on zero-force (ZF) receiver in [10], and they have almost the same performance. Although STBCs focused on the diversity advantage, it can't reach the full MIMO capacity; and its performance drops quickly when high order modulation is deployed. STTC is difficult to be implemented because of its complexity. While V-BLAST can't obtain diversity advantage. Transmit antenna selection for linear dispersion codes (LDC-TAS) is proposed in this paper. We present three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal transmit antennas when linear, coherent receiver is used over a slowly varying channel. Simulation results suggest that the max-min post-SNR criterion outperforms the other selection methods in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS. This paper is organized as follows. First, we review the linear dispersion codes in section II. System model of LDC-TAS is presented and some assumptions are given in section III. In section IV, we give the detail descriptions of antenna selection algorithms for LDCs based on ZF receiver. Section V contains the numerical simulation results and some analysis. Finally, Section VI presents our conclusions.
This work is sponsored by NSFC (60572066) and 863 program of China
(2005AA123920)
0-7803-9392-9/06/$20.00 (c) 2006 IEEE 2927
determine the selected M transmit antennas, given H and {Mn }n= . The selected MrXM channel matrix Hs is chosen by a function
LINEAR DISPERSION CODES
II.
Linear dispersion codes are first introduced and designed to maximize the mutual information between transmit and receive signals [5]. The linear dispersion encoder derives space-time codewords from linear combinations of certain basis matrices [6]. Let {s } N-1 be a set of scalar symbols from some complex constellation set C. Let {Mn }N I be the set of MxT codeword basis matrices, and they are known a priori to both transmitter and receiver. M is the number of transmit antennas and T is the codeword duration. Assuming that C{Sn} = 0, and £{I Sn 12}=1, where E{.} is the expectation operator. The basis matrices should satisfy the power constraint T
tr{MnMn }-_T N m
n=O,1,
nn
,N-1
(1)
h: CMxMrt"
S(SO,SI,..,SN-1) =- n=O{MnSn}
Mt
E, HS(So,
S,
SN-1) + V
N-1
E V gHsjM,sj}+
DaLa LDC Soure odig
n=O
Where Y is a MrX T matrix constructed by the receive signal, and Hs is the MrXM MIMO channel coefficient matrix. V is a MrXT AWGN matrix whose elements are i.i.d. circular complex white Gaussian noise process with distribution CN(O, No) .Es is the total transmit power from all M antennas. The SNR on each receive antenna is defined as follows: SNR Es / No. Then the spectral efficiency can be written as C IN
R l102
T
b/slHz
(4)
Where C is defined as the number of elements in constellation set (C. III. SYSTEM MODEL AND ASSUMPTIONS
The system architecture of LDC-TAS is illustrated in Fig.1, where exists a feedback path from the receiver to the transmitter. There are Mt transmit antennas and M, receive antennas, while only M transmit antennas are selected. Let H be the MrXMt channel coefficient matrix with independent entries distributed as CN(O, 1) .The receiver knows the perfect channel state information (CSI), and attempts to
(5)
v
Fading V \/ Channel
signal
Processing Outu
Antenna
sel_ction
HE _
An\ d
l I @ Deco:ding
Da
1
.------ ---- --- -- Selection Feedback.------.--------
(2)
(3)
\
m
Fig. I Transmit Antenna Selection for LDCs
Consider MIMO system with M transmit antennas and M, receive antennas. After matched filtering and symbol-rate sampling, the receiver concatenates T observations to form Y=
H2 H...H
Where P is the total size of antenna subset H, which is constructed by selecting arbitrary M columns from H. That is P = CM, and only log2 P bits of feedback are needed. We assume that the feedback path is error-free and delay-free. And the channel keeps constant during one LDCs block and independent between different blocks.
A codeword corresponding to {sn }N- is constructed by taking the corresponding linear combination of basis matrices N-1
H = {H
IV. SELECTION CRITERIA FOR LDC-TAS
Let Hs be the effective channel coefficient matrix selected by function h (6) Hs = h(H) We can rewrite the matrix input-output relationship in (3) in another equivalent column-stacked form. Define the linear transformation matrix [6]
X:= [vec(MO), vec(MI), , vec(MN 1)]
(7)
and the stacked channel matrix
Hs:=ITO0Hs
(8) Where vec(.) is vectorization operator which stacks all the columns of a matrix, and (0 is the Kronecker tensor product. Taking vec(.) for both sides of (3) gives
y= EsHsXS+v= EsHs+v (9) , s := [sl s2 ,. SN-1 ]T Where y:= vec(Y) v := vec(V) and H := HsX is the equivalent channel
matrix with size MrTXN. We may use any decoding technique already in place for BLAST. In this paper, only linear receiver is considered. An NXMrT matrix equalizer G is applied to y to obtain an estimation of s as follows:
s=Gy= EsGHs+Gv
(10)
The receiver can recover the original information from S. For 2928
the zero-force (ZF) receiver
And for MMSE receiver G
=
G=fH
-H--
V.
(11)
N0
1-H
(H H+_I IN) H
(12)
Where (+ is the matrix pseudo-inverse, and (H is the matrix conjugate transpose. Only ZF receiver is discussed within this paper, but similar performance improvements have been observed with the MMSE receiver. Then we will derive three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal M transmit antennas when linear, coherent receiver is used over a slowly varying channel. The simplest selection criterion is choosing the M transmit antennas with the highest channel gain. Sell- maximum channel gain: For every element of H in (5), compute the total power of channel matrix, and choose the antenna subset with the largest power Hs = arg max[tr(H H H)] (13)
In this section, three selection criteria are compared via numerical simulation. Considering a MIMO system with 4 transmit antennas and 2 receive antennas. And only 3 transmit antennas are selected. The LDCs used in this paper is obtained through joint capacity and error probability optimization in [6] by computer search. We only discuss the codes with T=2 and T=3, and N is fixed as N=3. The corresponding linear transformation matrixes (LTM) are given in Appendix (18) and (19) .The modulation modes include QPSK, 8PSK and 16QAM. Following the previous development, performance is measured in terms of BER (Bit Error Rate) and FER (Frame Error Rate), where a frame contains only one LDC transmit block. T=2 10
12 10
H eH
Sel2- maximum capacity: For every element of H, compute the capacity C(Hk) from equivalent channel matrix (9)
E_
-
H
C(Hk) =- log det(IMrT + -HkHk ) T NNO0
HkeH
Sel3- max-min post-SNR: The ZF receiver post-processing SNR of each of the N multiplexed streams can be derived as follows [10]
SNRn (Hk )=
NNO [Hk Hk ]nn
n = O,J,
N-l (16)
For every element of H , compute the corresponding
minimum SNR of N streams: min(SNR] n
(Hk)) , and
choose the antenna subset with the largest minimum SNR
Hs
=argmax[min(SNRzF(Hk))] HeH
n
10 10
10--6
(14)
where Hk = HkX = (IT 0) Hk )X Choose the antenna subset with the maximum capacity (15) Hs = arg max[C(Hk)]
(17)
SIMULATION AND ANALYSIS
15 SNR (dB)
Fig.2 BER curve of LDC-TAS with T=2
The BER curves of LDC-TAS with T=2 and QPSK modulation are presented in Fig.2. The performance curve of LDCs ZF receiver without antenna selection is labeled by "ZF-BER" or "ZF-FER". The corresponding curves of three selection criteria are labeled by "sell", "sel2", and "sel3". From the results we can clearly see that sel3 gives the best performance among three criteria while sel2 is 2dB loss when BER reaches 10-3, and sell is 1dB loss than sel2. And sell is still 1.5dB better than the scheme without selection. Furthermore, only the sel3 offers the additional diversity order than the other solutions. In Fig.3, we repeat the above experiment for T=3. As expected, sel3 still outperforms the other criteria. And sel2 is 1.5dB and sell is 2.5dB loss than sel3 when BER is 10-4.The sel3 method also shows benefit if higher order modulation is used.
2929
outperforms LDC-TAS, because STBC gets the full diversity advantage.
T=3
_L
R=3 b/s/Hz
1
1 It
10
N
10o2
LL
co
w LL LL
-0
10 -3
LL
K "&
m 10
10
5
15 SNR (dB)
K5_ =LDC-TAS BER
10
LDC-TAS FER_ --O--STBC-TAS BET -6 _ ---+--STBC-TAS FEI 10 4 12 6 8 10 SNR (dB)
20
Fig.3 BER curve of LDC-TAS with T=3
The sel3 method also shows benefit if higher order modulation is used. When T=2 and 8PSK modulation is deployed, the corresponding BER curves are given in Fig.4. We can see that sel3 is better than sel2 about 2dB, and 3dB than sell when BER is 10-3.
=.,%
0
=
14
16
18
Fig. 5 Perfonnance of LDC-TAS and STBC-TAS R=4 b/s/Hz 10 -o2
10
I 43
H 7
10 10
W 10
10
5
10
15
SNR (dB)
20
25
Fig. 4 BER curve of LDC-TAS with 8PSK Note that sel3 is specially tailored for ZF receiver and directly related to the BER performance, while the other two
selection criteria are based on a general theory, so the max-min post-SNR criterion will yield the best performance for ZF receiver under consideration. We compare the performance of three systems: LDC-TAS, STBC-TAS and BLAST-TAS. STBC-TAS is deployed with 4 transmit antennas and 2 receive antennas, and 8PSK modulation is used. Transmitter selects 2 from 4 transmit antennas. While QPSK modulation is used in LDC-TAS, and T=2. Then the spectral efficiency of two systems are all R=3 b/slHz. Maximum likehood (ML) decoder is used, and the selection criterion is sell. BER and FER performance of two systems are given in Fig.5. When SNR is below 14 dB, LDC-TAS is better than STBC-TAS. When FER is 10-2, LDC-TAS has 3dB gain. While in high SNR, STBC-TAS
__
LDC-TAS BER LDC-TAS FER B --D--BLAST-TAS BEt B FE FR I -----BLAST-TAS ________IIIIIIZiIL__ 5 10 15 20 25 SNR (dB)
Fig. 6 Performance of LDC-TAS and BLAST-TAS 4 transmit antennas and 2 receive antennas are also used in BLAST-TAS. Transmitter selects 2 from 4 transmit antennas,
and QPSK modulation is deployed. While 16QAM is used for LDC-TAS, and T=3. Then the spectral efficiency is R=4 b/slHz. Under ZF linear receiver and sel3 is the selection criterion. BER and FER performance of two systems are given in Fig.6. Under the same spectral efficiency, LDC-TAS shows significant diversity advantage. When BER is 10-2, LDC-TAS has about 6dB gain than BLAST-TAS. VI. CONCLUSION
Future cellular systems require reliable communication with large capacity through a simple and cheap solution. Since LDCs satisfy an information theoretic optimality property, and transmit antenna selection offers additional diversity order, we introduced the LDC-TAS system in this paper. The LDC-TAS system is an inexpensive way to obtain diversity advantage 2930
over a multiple antenna fading channel, as well as maintaining the high transport rate. We derived three selection criteria when ZF receiver is employed, and we compared these criteria via computer simulations. The simulation results show that max-min post-SNR criterion, which is specially tailored for ZF receiver, outperforms the other criteria in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS.
APPENDIX
The Linear Transformation Matrixes (LTAM) used in this paper are presented as follows: For N=3, M=3, T=2 0.3003 JO. 1273,- -0.2303 jO. 1625, 0.1390 + jO.3748, - 0.2530 - jO.0734, 0.2137 + jO. 1467, 0.3677 + jO.2220, LTM= -0.0699- jO.3776, 0.2833 -jO. 1616, -0.2410 + jO.2147, 0. 1819 + jO.3560, -0.2401 - jO. 1549, 0.2294 + jO. 1207,
0.365 1 - jO. 1 192 -0.2446- jO.2106 0.1218- jO.2585 - -0.2783 + jO.0445 - 0.2324 + jO. 1240 0.4183- j0.0977
B. Hassibi, B. M. Hochwald, "High-rate codes that are linear in space and time," IEEE Transactions on Information Theory, vol.48, no.7, July 2002. [6] Robert W. Heath, Jr, A.J Paulraj,. "Linear dispersion codes for MIMO systems based on frame theory," Signal Processing, IEEE Transactions on, vol. 50, no.10, pp. 2429-2441, Oct 2002. [7] Shahab Sanayei, Aria Nosratinia, "Antenna Selection in MIMO Systems," IEEE Communications Magazine ,vol.42, no.10, pp.68-73, Oct 2004. [8] Xiang Nian Zeng, Ali Ghrayeb, "Performance Bounds for Space-Time Block Codes With Receive Antenna Selection," IEEE Transaction on Information Theory, vol. 50, no. 9, pp. 2130-2137, Sept 2004. [9] B. Badic, P. Fuxjaeger and H. Weinrichter, "Performance of quasi-orthogonal space-time code with antenna selection", IEEE Electronics Letters, vol. 40, no. 20, pp. 1282-1284, Sept 2004. [10] Robert W. Heath, Sumeet Sandhu, and Arogyaswami Paulraj, "Antenna Selection for Spatial Multiplexing Systems with Linear Receivers," IEEE Communications Letters, vol. 5, no. 4, pp. 142-144, April 2001. [11] G. Kutz, A. Chass, "Low Complexity Implementation of a Downlink CDMA Generalized RAKE Receiver," Proc. IEEE Vehicular Technology Conference 2002, Canada, September 24-28, 2002. [5]
(18) For N=3, M=3, T=3
LTM
=
0. 1960 + jO. 1412, 0.2942 +jO.3268, 0.1579 + jO.2381 -0.2860 - jO.2102, - 0.1274 + jO. 1802, 0.3982 + jO.0062 -0.1191+ jO.3671,--0.0557- jO.2969, 0.2215 + jO.2098 0.0607 - jO.2792, - 0.1607+ jO.1804,- 0.4257 - jO. 1 103 -0.273 1 jO.2899, 0.2387 -jO.2627, 0.0069 -jO.2206 -0.0039- jO.3051,-- 0.3528 - jO. 1565, 0.2102 +jO.2170 0.4198 + jO. 1308,--0.1904+ jO.2130, 0.2057 + jO. 1267 -0.0176 + jO.2200, 0.0949 + jO.3868,- 0.3005 - jO. 1 890 -0. 1995 - jO.2269, - 0.1469 + jO.2674,- 0.0923 + jO.3748
(19) REFERENCES [1]
[2] [3]
[4]
I.E.Telatar, "Capacity of multi-antenna Gaussian channels," Euro. Trans. Telecommun, vol. 10, no.6, pp. 585-595, Nov 1999. R.W.Heath Jr, A. Paulraj, "Switching between spatial multiplexing and transmit diversity based on constellation distance," in Proc. AllertonConf Commun. Cont. Comput., Illinois, Oct. 2000. V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Performance criterion and code construction," IEEE Trans. Inform. Theory, vol. 44, no.2, pp. 744-765, Mar. 1998. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE Trans. Inform. Theory, vol. 48, no.5, pp. 1456-1467, July 1999.
2931
for Linear Dispersion
Codes Based on Linear Receiver Dan Deng, Ming Zhao, Jinkang Zhu PCN&SS Lab, EEIS,USTC Hefei, P. R. China, 230027 dengdan(mail.ustc.edu.cn, zhaoming(ustc.edu.cn, [email protected]
Abstract---In an effort to utilize the advantage of using multiple antennas, transmit antenna selection for linear dispersion codes (LDC-TAS) is proposed in this paper. We present three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal transmit antennas when linear, coherent receiver is used over a slowly varying channel. Simulation results suggest that the max-min post-SNR criterion outperforms the other selection methods in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS.
Keywords- antenna selection; linear dispersion codes; linear receiver; STBC; V-BLAST;
I.
INTRODUCTION
Multiple-input multiple-output (MIMO) systems can offer significant capacity gains over traditional single-input single-output (SISO) systems [1]. Practical modulation schemes for MIMO systems with receive-only channel knowledge fall principally into two areas known as diversity and multiplexing [2]. Space-time coding, such as the space-time trellis codes [3] and space-time block codes [4], uses specially designed codeword that maximize the diversity advantage or reliability of the transmitted information. Recognizing that orthogonal space-time block codes [3], designed to maximize diversity advantage do not achieve full channel capacity in MIMO channels, Hassibi and Hochwald [5] proposed the revolutionary linear dispersion codes (LDCs). And Robert W. Heath [6] introduced a family of LDCs designs based on frame theory, which guarantees good performance in terms of diversity advantage. Although Multiple-antenna system can improve the
capacity and reliability of radio communication, the multiple RF chains are costly in terms of size, power, and hardware. Antenna selection is a low-cost low-complexity alternative to capture many of the advantages of MIMO systems [7]. In an effort to utilize the advantage of using multiple antennas, several papers have appeared recently in the literature in which the notion of antenna selection was introduced for space-time coding. In [8], the authors present a comprehensive performance analysis of orthogonal STBCs with receive antenna selection. And quasi-orthogonal STBCs are investigated in [9]. Three selection criteria are proposed and compared for spatial multiplexing systems based on zero-force (ZF) receiver in [10], and they have almost the same performance. Although STBCs focused on the diversity advantage, it can't reach the full MIMO capacity; and its performance drops quickly when high order modulation is deployed. STTC is difficult to be implemented because of its complexity. While V-BLAST can't obtain diversity advantage. Transmit antenna selection for linear dispersion codes (LDC-TAS) is proposed in this paper. We present three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal transmit antennas when linear, coherent receiver is used over a slowly varying channel. Simulation results suggest that the max-min post-SNR criterion outperforms the other selection methods in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS. This paper is organized as follows. First, we review the linear dispersion codes in section II. System model of LDC-TAS is presented and some assumptions are given in section III. In section IV, we give the detail descriptions of antenna selection algorithms for LDCs based on ZF receiver. Section V contains the numerical simulation results and some analysis. Finally, Section VI presents our conclusions.
This work is sponsored by NSFC (60572066) and 863 program of China
(2005AA123920)
0-7803-9392-9/06/$20.00 (c) 2006 IEEE 2927
determine the selected M transmit antennas, given H and {Mn }n= . The selected MrXM channel matrix Hs is chosen by a function
LINEAR DISPERSION CODES
II.
Linear dispersion codes are first introduced and designed to maximize the mutual information between transmit and receive signals [5]. The linear dispersion encoder derives space-time codewords from linear combinations of certain basis matrices [6]. Let {s } N-1 be a set of scalar symbols from some complex constellation set C. Let {Mn }N I be the set of MxT codeword basis matrices, and they are known a priori to both transmitter and receiver. M is the number of transmit antennas and T is the codeword duration. Assuming that C{Sn} = 0, and £{I Sn 12}=1, where E{.} is the expectation operator. The basis matrices should satisfy the power constraint T
tr{MnMn }-_T N m
n=O,1,
nn
,N-1
(1)
h: CMxMrt"
S(SO,SI,..,SN-1) =- n=O{MnSn}
Mt
E, HS(So,
S,
SN-1) + V
N-1
E V gHsjM,sj}+
DaLa LDC Soure odig
n=O
Where Y is a MrX T matrix constructed by the receive signal, and Hs is the MrXM MIMO channel coefficient matrix. V is a MrXT AWGN matrix whose elements are i.i.d. circular complex white Gaussian noise process with distribution CN(O, No) .Es is the total transmit power from all M antennas. The SNR on each receive antenna is defined as follows: SNR Es / No. Then the spectral efficiency can be written as C IN
R l102
T
b/slHz
(4)
Where C is defined as the number of elements in constellation set (C. III. SYSTEM MODEL AND ASSUMPTIONS
The system architecture of LDC-TAS is illustrated in Fig.1, where exists a feedback path from the receiver to the transmitter. There are Mt transmit antennas and M, receive antennas, while only M transmit antennas are selected. Let H be the MrXMt channel coefficient matrix with independent entries distributed as CN(O, 1) .The receiver knows the perfect channel state information (CSI), and attempts to
(5)
v
Fading V \/ Channel
signal
Processing Outu
Antenna
sel_ction
HE _
An\ d
l I @ Deco:ding
Da
1
.------ ---- --- -- Selection Feedback.------.--------
(2)
(3)
\
m
Fig. I Transmit Antenna Selection for LDCs
Consider MIMO system with M transmit antennas and M, receive antennas. After matched filtering and symbol-rate sampling, the receiver concatenates T observations to form Y=
H2 H...H
Where P is the total size of antenna subset H, which is constructed by selecting arbitrary M columns from H. That is P = CM, and only log2 P bits of feedback are needed. We assume that the feedback path is error-free and delay-free. And the channel keeps constant during one LDCs block and independent between different blocks.
A codeword corresponding to {sn }N- is constructed by taking the corresponding linear combination of basis matrices N-1
H = {H
IV. SELECTION CRITERIA FOR LDC-TAS
Let Hs be the effective channel coefficient matrix selected by function h (6) Hs = h(H) We can rewrite the matrix input-output relationship in (3) in another equivalent column-stacked form. Define the linear transformation matrix [6]
X:= [vec(MO), vec(MI), , vec(MN 1)]
(7)
and the stacked channel matrix
Hs:=ITO0Hs
(8) Where vec(.) is vectorization operator which stacks all the columns of a matrix, and (0 is the Kronecker tensor product. Taking vec(.) for both sides of (3) gives
y= EsHsXS+v= EsHs+v (9) , s := [sl s2 ,. SN-1 ]T Where y:= vec(Y) v := vec(V) and H := HsX is the equivalent channel
matrix with size MrTXN. We may use any decoding technique already in place for BLAST. In this paper, only linear receiver is considered. An NXMrT matrix equalizer G is applied to y to obtain an estimation of s as follows:
s=Gy= EsGHs+Gv
(10)
The receiver can recover the original information from S. For 2928
the zero-force (ZF) receiver
And for MMSE receiver G
=
G=fH
-H--
V.
(11)
N0
1-H
(H H+_I IN) H
(12)
Where (+ is the matrix pseudo-inverse, and (H is the matrix conjugate transpose. Only ZF receiver is discussed within this paper, but similar performance improvements have been observed with the MMSE receiver. Then we will derive three criteria: maximum channel gain, maximum capacity and max-min post-SNR, for selecting the optimal M transmit antennas when linear, coherent receiver is used over a slowly varying channel. The simplest selection criterion is choosing the M transmit antennas with the highest channel gain. Sell- maximum channel gain: For every element of H in (5), compute the total power of channel matrix, and choose the antenna subset with the largest power Hs = arg max[tr(H H H)] (13)
In this section, three selection criteria are compared via numerical simulation. Considering a MIMO system with 4 transmit antennas and 2 receive antennas. And only 3 transmit antennas are selected. The LDCs used in this paper is obtained through joint capacity and error probability optimization in [6] by computer search. We only discuss the codes with T=2 and T=3, and N is fixed as N=3. The corresponding linear transformation matrixes (LTM) are given in Appendix (18) and (19) .The modulation modes include QPSK, 8PSK and 16QAM. Following the previous development, performance is measured in terms of BER (Bit Error Rate) and FER (Frame Error Rate), where a frame contains only one LDC transmit block. T=2 10
12 10
H eH
Sel2- maximum capacity: For every element of H, compute the capacity C(Hk) from equivalent channel matrix (9)
E_
-
H
C(Hk) =- log det(IMrT + -HkHk ) T NNO0
HkeH
Sel3- max-min post-SNR: The ZF receiver post-processing SNR of each of the N multiplexed streams can be derived as follows [10]
SNRn (Hk )=
NNO [Hk Hk ]nn
n = O,J,
N-l (16)
For every element of H , compute the corresponding
minimum SNR of N streams: min(SNR] n
(Hk)) , and
choose the antenna subset with the largest minimum SNR
Hs
=argmax[min(SNRzF(Hk))] HeH
n
10 10
10--6
(14)
where Hk = HkX = (IT 0) Hk )X Choose the antenna subset with the maximum capacity (15) Hs = arg max[C(Hk)]
(17)
SIMULATION AND ANALYSIS
15 SNR (dB)
Fig.2 BER curve of LDC-TAS with T=2
The BER curves of LDC-TAS with T=2 and QPSK modulation are presented in Fig.2. The performance curve of LDCs ZF receiver without antenna selection is labeled by "ZF-BER" or "ZF-FER". The corresponding curves of three selection criteria are labeled by "sell", "sel2", and "sel3". From the results we can clearly see that sel3 gives the best performance among three criteria while sel2 is 2dB loss when BER reaches 10-3, and sell is 1dB loss than sel2. And sell is still 1.5dB better than the scheme without selection. Furthermore, only the sel3 offers the additional diversity order than the other solutions. In Fig.3, we repeat the above experiment for T=3. As expected, sel3 still outperforms the other criteria. And sel2 is 1.5dB and sell is 2.5dB loss than sel3 when BER is 10-4.The sel3 method also shows benefit if higher order modulation is used.
2929
outperforms LDC-TAS, because STBC gets the full diversity advantage.
T=3
_L
R=3 b/s/Hz
1
1 It
10
N
10o2
LL
co
w LL LL
-0
10 -3
LL
K "&
m 10
10
5
15 SNR (dB)
K5_ =LDC-TAS BER
10
LDC-TAS FER_ --O--STBC-TAS BET -6 _ ---+--STBC-TAS FEI 10 4 12 6 8 10 SNR (dB)
20
Fig.3 BER curve of LDC-TAS with T=3
The sel3 method also shows benefit if higher order modulation is used. When T=2 and 8PSK modulation is deployed, the corresponding BER curves are given in Fig.4. We can see that sel3 is better than sel2 about 2dB, and 3dB than sell when BER is 10-3.
=.,%
0
=
14
16
18
Fig. 5 Perfonnance of LDC-TAS and STBC-TAS R=4 b/s/Hz 10 -o2
10
I 43
H 7
10 10
W 10
10
5
10
15
SNR (dB)
20
25
Fig. 4 BER curve of LDC-TAS with 8PSK Note that sel3 is specially tailored for ZF receiver and directly related to the BER performance, while the other two
selection criteria are based on a general theory, so the max-min post-SNR criterion will yield the best performance for ZF receiver under consideration. We compare the performance of three systems: LDC-TAS, STBC-TAS and BLAST-TAS. STBC-TAS is deployed with 4 transmit antennas and 2 receive antennas, and 8PSK modulation is used. Transmitter selects 2 from 4 transmit antennas. While QPSK modulation is used in LDC-TAS, and T=2. Then the spectral efficiency of two systems are all R=3 b/slHz. Maximum likehood (ML) decoder is used, and the selection criterion is sell. BER and FER performance of two systems are given in Fig.5. When SNR is below 14 dB, LDC-TAS is better than STBC-TAS. When FER is 10-2, LDC-TAS has 3dB gain. While in high SNR, STBC-TAS
__
LDC-TAS BER LDC-TAS FER B --D--BLAST-TAS BEt B FE FR I -----BLAST-TAS ________IIIIIIZiIL__ 5 10 15 20 25 SNR (dB)
Fig. 6 Performance of LDC-TAS and BLAST-TAS 4 transmit antennas and 2 receive antennas are also used in BLAST-TAS. Transmitter selects 2 from 4 transmit antennas,
and QPSK modulation is deployed. While 16QAM is used for LDC-TAS, and T=3. Then the spectral efficiency is R=4 b/slHz. Under ZF linear receiver and sel3 is the selection criterion. BER and FER performance of two systems are given in Fig.6. Under the same spectral efficiency, LDC-TAS shows significant diversity advantage. When BER is 10-2, LDC-TAS has about 6dB gain than BLAST-TAS. VI. CONCLUSION
Future cellular systems require reliable communication with large capacity through a simple and cheap solution. Since LDCs satisfy an information theoretic optimality property, and transmit antenna selection offers additional diversity order, we introduced the LDC-TAS system in this paper. The LDC-TAS system is an inexpensive way to obtain diversity advantage 2930
over a multiple antenna fading channel, as well as maintaining the high transport rate. We derived three selection criteria when ZF receiver is employed, and we compared these criteria via computer simulations. The simulation results show that max-min post-SNR criterion, which is specially tailored for ZF receiver, outperforms the other criteria in a variety of modulation modes, such as QPSK, 8PSK and 16QAM. Compared with BLAST-TAS, under the same spectral efficiency, LDC-TAS shows significant diversity advantage. In low SNR environment, LDC-TAS is still better than STBC-TAS.
APPENDIX
The Linear Transformation Matrixes (LTAM) used in this paper are presented as follows: For N=3, M=3, T=2 0.3003 JO. 1273,- -0.2303 jO. 1625, 0.1390 + jO.3748, - 0.2530 - jO.0734, 0.2137 + jO. 1467, 0.3677 + jO.2220, LTM= -0.0699- jO.3776, 0.2833 -jO. 1616, -0.2410 + jO.2147, 0. 1819 + jO.3560, -0.2401 - jO. 1549, 0.2294 + jO. 1207,
0.365 1 - jO. 1 192 -0.2446- jO.2106 0.1218- jO.2585 - -0.2783 + jO.0445 - 0.2324 + jO. 1240 0.4183- j0.0977
B. Hassibi, B. M. Hochwald, "High-rate codes that are linear in space and time," IEEE Transactions on Information Theory, vol.48, no.7, July 2002. [6] Robert W. Heath, Jr, A.J Paulraj,. "Linear dispersion codes for MIMO systems based on frame theory," Signal Processing, IEEE Transactions on, vol. 50, no.10, pp. 2429-2441, Oct 2002. [7] Shahab Sanayei, Aria Nosratinia, "Antenna Selection in MIMO Systems," IEEE Communications Magazine ,vol.42, no.10, pp.68-73, Oct 2004. [8] Xiang Nian Zeng, Ali Ghrayeb, "Performance Bounds for Space-Time Block Codes With Receive Antenna Selection," IEEE Transaction on Information Theory, vol. 50, no. 9, pp. 2130-2137, Sept 2004. [9] B. Badic, P. Fuxjaeger and H. Weinrichter, "Performance of quasi-orthogonal space-time code with antenna selection", IEEE Electronics Letters, vol. 40, no. 20, pp. 1282-1284, Sept 2004. [10] Robert W. Heath, Sumeet Sandhu, and Arogyaswami Paulraj, "Antenna Selection for Spatial Multiplexing Systems with Linear Receivers," IEEE Communications Letters, vol. 5, no. 4, pp. 142-144, April 2001. [11] G. Kutz, A. Chass, "Low Complexity Implementation of a Downlink CDMA Generalized RAKE Receiver," Proc. IEEE Vehicular Technology Conference 2002, Canada, September 24-28, 2002. [5]
(18) For N=3, M=3, T=3
LTM
=
0. 1960 + jO. 1412, 0.2942 +jO.3268, 0.1579 + jO.2381 -0.2860 - jO.2102, - 0.1274 + jO. 1802, 0.3982 + jO.0062 -0.1191+ jO.3671,--0.0557- jO.2969, 0.2215 + jO.2098 0.0607 - jO.2792, - 0.1607+ jO.1804,- 0.4257 - jO. 1 103 -0.273 1 jO.2899, 0.2387 -jO.2627, 0.0069 -jO.2206 -0.0039- jO.3051,-- 0.3528 - jO. 1565, 0.2102 +jO.2170 0.4198 + jO. 1308,--0.1904+ jO.2130, 0.2057 + jO. 1267 -0.0176 + jO.2200, 0.0949 + jO.3868,- 0.3005 - jO. 1 890 -0. 1995 - jO.2269, - 0.1469 + jO.2674,- 0.0923 + jO.3748
(19) REFERENCES [1]
[2] [3]
[4]
I.E.Telatar, "Capacity of multi-antenna Gaussian channels," Euro. Trans. Telecommun, vol. 10, no.6, pp. 585-595, Nov 1999. R.W.Heath Jr, A. Paulraj, "Switching between spatial multiplexing and transmit diversity based on constellation distance," in Proc. AllertonConf Commun. Cont. Comput., Illinois, Oct. 2000. V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless communication: Performance criterion and code construction," IEEE Trans. Inform. Theory, vol. 44, no.2, pp. 744-765, Mar. 1998. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE Trans. Inform. Theory, vol. 48, no.5, pp. 1456-1467, July 1999.
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