A Circulant Based Space-Time Coding for Multiple Input Multiple ...
A Circulant Based Space-Time Coding for Multiple Input Multiple Output System in Fading Channels Mandana Norouzi ,Yuteng Wu, Pradnya Chahande, Fernando Alonso Macias and G.E. Atkin ,Senior member, IEEE Department of Electrical and Computer Engineering, Illinois Institute of Technology, IL, USA Email: {mnorouzi, ywu4 , pchahand, falonsom,atkin} @hawk.iit.edu
the orthogonality of the transmitted signal in each antenna per time slots complexity of the decoding grows linearly with the underlying constellation size for general constellation [5,6]. In space-time coding, information symbols are encoded across the time and spatial dimension, which corresponds respectively to the multiple signaling interval and multiple antennas at the transmitter. By using several separated enough antennas in the transmitter each path undergoes independent fading. Therefore, transmit diversity can be achieved in distributed antenna systems (DAS) or centralized antenna system (CASs). Several methods of diversity are proposed in [5]-[13]. In this paper more focus is implied on linear spacetime block codes, in which the information symbols are linearly combined to form two dimensional coding matrixes, the rows and columns of the matrix correspond to the transmission across multiple antennas and time intervals respectively. There are several constructions of space time block codes with full diversity such as orthogonal space-time codes, diagonal algebraic space-time codes, semi orthogonal algebraic space-time codes, threaded algebraic space-time block codes and perfect space-time codes [7, 8]. First, Orthogonal spacetime block codes achieve full diversity and their code word matrix is an orthogonal matrix as Alamouti. Second, a Diagonal Algebraic Space-Time (DAST) blocks code is a linear space-time code build by the use of rotation of constellations [14].Diagonal in DAST means that the structure of the code matrix, wherein the rotated information symbols are spread over the diagonal of the square code matrix[15,20]. The rotation matrices used in DAST codes are based on using algebraic number of field theory [15]. Performance of DAST codes is better than orthogonal codes but the cost of this probability of error improvement is increased decoding complexity in the DAST. Third, Quasiorthogonal space-time codes satisfy orthogonality condition to enable rate-one in transmission with the cost of increasing decoding complexity. Quasiorthogonal codes for four antennas are proposed by Jafakhani [13], Trikkonen-Boariu [17, 18], and Foschini [1] independently. These codes outperform orthogonal designs at all spectral efficiencies for complex rotations. Fourth, singlesymbol decodable space-time block codes are a family of nonorthogonal space-time codes that achieve rate-one transmission and full diversity for three and four transmit antennas. Since a pair of real symbols defines a single complex symbol, these codes are named as single-symbol decodable. The complexity of these codes are more than orthogonal
Abstract— In this paper a new space time coding, which is suitable for high data rate communications is proposed. In this approach data is encoded by using a decomposition of a circulant matrix and the output of encoder is splitted into N streams to be transmitted simultaneously to N transmit antennas. The performance of this multiple input multiple output system is evaluated in a Rayleigh fading channel. Its performance in terms of symbol error rate, diversity gain and complexity is obtained. Simulation results show that with a perfect channel estimator, when channel state information (CSI) is available at the receiver, the Circulant Space Time code (CSTC) has a better symbol error rate via signal to noise ratio in compared to the spatial multiplexing and near to orthogonal space time block code. Index Terms—Space time coding, multiple input multiple output, Circulant matrix, Orthogonal space time coding, Alamouti space time code.
I. INTRODUCTION The key challenges in future of wireless communication are data rate and quality of service (QoS). Deploying multiple antennas at the receiver and transmitter creates multiple-input multiple-output (MIMO) systems that not only increase the data rate, but also these systems can offers high reliability and robustness and QoS in comparison with single antenna systems. Space time coding and diversity schemes are two solutions which could improve the data rate and QoS [1]. Beside channel additive with Gaussian noise, several communication channels severe channel attenuation, fading in the channel communication, one way to overcome this effect is also using space time coding. So generally, Space time coding is one way to combat fading and also simplify the implementation of the mobile terminal [1,2]. The first method of diversity is delay diversity which is inspired by Wittenben [3] to develop space-time codes. Tarokh et al in [4] develop space-time trellis codes but, it has high decoding complexity and needs a viterbi algorithms at the receiver for decoding. Regarding to decrease the high decoding complexity for the first time space-time block codes is introduced by Alamouti et al. from AT&T in 1998 as a novel means of providing transmit diversity for the multiple-antenna in fading channel [2]. Alamouti space-time block coding is simple and elegant due to low complexity of maximumlikelihood (ML) detector in the receiver side and not requiring feedback of channel state information. In fact, the Alamouti code works with two transmit antennas diversity and enables separate decoding of each complex symbol. And also, due to 1
designs and less than DAST codes and quasiorthogonal. But from data rate point of view, this design offers higher data rate in compared with orthogonal and the same with DAST and quasiorthogonal. Next category is semi-orthogonal algebraic space-time (SAST) block codes which have rate-one transmission and full diversity; they use same real rotation matrices of DAST codes [19]. They are named as semi orthogonal because half columns of the code matrix are orthogonal to the other half of them. All of the space-time codes which introduced briefly thus far achieve a maximum rate of one symbol per signaling interval. Two more categories left which has high data rate depends on number of transmit antennas. First one is threaded algebraic space time (TAST) block codes with full diversity. TAST code threads the rate-one DAST code to obtain the higher transmission rates [7, 8]. The decoding complexity of TAST codes depends on the transmission rate and whether real or complex generator matrices are used. Second category of high data rate space time block codes is perfect space time block codes for any number of antennas [20]. These codes are named because they have full diversity, high spectral efficiency [20] and decoding complexity does not depend on the transmission rate, but on the modulation alphabet [22]. In general, the design of space-time block codes is a tradeoff between minimizing decoding complexity and maximizing the diversity gain. In this paper we propose a new space time block code which is linear and orthogonal with rate one for any number of transmitters. Simulation results shows that by increasing the number of receivers wherein number of transmitters increase the probability of symbol error rate would decrease. Since this code is orthogonal the decoding complexity is low in comparison with other codes. The organization of this paper is as follow: system model of our method, including the transmitter and receiver and channel specification, is presented in section II. Proposed method is presented in the section III. In section IV performance analysis of the CSTC is presented. In section V simulation results of the CSTC method for BPSK and QPSK modulations is presented. Finally, conclusion part of the paper is drawn in section VI.
Fig.1. Block diagram of space time coding in transmitter side
1 2 = 1 ⋮ 1 1
12 22 ⋮
2
⋯ 1 ⋯ 2 ⋱ ⋮ ⋯
(1)
Where x k i is the symbol transmitted from antenna ∈ {1,2, . . , } and at time {1,2, … }. The received signal at jth receiver would be ( ). (
( ) = " ℎ$ ( ) ∗ $ ( ) + '$ ( ) $)*
(2)
'$ ( ) is complex additive white Gaussian noise at the receive antenna in time t, with stationary feature, mean of zero and variance of , - ∙ ℎ$ ( ) is the channel coefficient between the ith transmitter and jth receiver at time t. It is assumed the channel is Rayleigh fading which models a scattering environment and it is modeled as independent identically distributed (i.i.d) complex Gaussian random variables. It is assumed that the fading coefficients are constant across two consecutive symbol transmission periods. They can be expressed as follow [4]:
(3) ℎ$ ( ) = ℎ$ ( + ) = ℎ$ = |ℎ$ |0 1$234 Where T is the symbol duration,|ℎ$ | amplitude gain and 5$ is the phase shift between the ith transmitter and jth receiver. In the next section the new space time block code is presented.
II. SYSTEM MODEL In this paper we consider a MIMO system with N transmits antenna and M receive antennas. Figure 1 shows the block diagram of space time coding of the transmitter [4]. In figure1, the information source passes its output through the modulator box which in this paper we consider BPSK, QPSK, for implementing our method. Generally, it is assumed that each element of the modulation constellation is scaled by a factor Es so that average energy of the constellation elements is 1.
III. PROPOSED MODEL The general idea in designing of space-time block coding is designing of the code matrix (X) wherein it has maximum diversity gain and minimum decoding complexity. The key feature for achieving minimum decoding complexity is orthogonality between the sequences transmitted by the each pair of antenna [4,26]. The idea of designing of the code matrix in this paper is based on circulant matrices. Our design criterion is not constellation-dependent and applied equally well to BPSK and QPSK modulation. Based on this method we could have any number of orthogonal sequences in the transmitter side for each antenna. Beside of orthogonality
The transmitted codeword of a space-time block code can be expressed as a nT*L matrix (M= nT):
2
$ T Where the λ$ = ∑=1* are I)8 I 0 M . Λ and Y=(G Λ) orthogonal matrix. So the transmitter could be designed based on the Λ or ^* respectively as : JπKL
another advantage of CSTC is its coding gain; coding gain is equal to one. First, briefly it is explained about the circulant matrix and then discussed about the CSTC. An N × N circulant matrix takes the form [24, 25]: 8 1* … * * 8 1* ⋱ - * 8 ⋱ ⋮ = ⋮ (4) ⋱ ⋱ ⋱ 1 1* 1* 1… * 8
∑A I)8 I 0 Λ= 0 0
And ^- =
Each row is the cyclic shift of its above row. For a circulant matrix X, based on singular value decomposition (SVD) [24], it could be decomposed to GΛG: . Where G and Λ matrix defined based on the discrete Fourier transform (DFT) as: >8,8 >*,8 * ; = >-,8 √= ⋮ >=,8
>8,* >*,* >-,* ⋮ >=,*
Where >B, = 0
>8,>*,>-,⋮ >=,-
λ* 0 Λ = 0 ⋮ 0
2πjk E
… … … ⋱ …
>8,= >*,= >-,= ⋮ >=,=
0 λ0 ⋮ 0
0 0 λA ⋮ 0
… … … ⋮ 0
(5) 0 0 0 ⋮ λ
, B = 0,1,2,3 … E − 1, =
$ 0,1,2,3 … , E − 1e, - = −1. and λ$ = ∑=1* I)8 I 0
xR x* xxA
xA xR x* x-
xxA xR S x*
8 * e R 8
e e8
e8
e$ V
Jπ
e$ V
Vπ
e$ V
Wπ
e8
e$ V
Vπ
e$ V
e$
Xπ
YJπ V
e8
λ* 0 Wπ e$ V 0 λYJπ ∗ P $ 0 0 e V YXπ 0 0 e$ V e8 e8 e8
e8 8 e e8
-π
e$ R
Rπ e R Zπ e$ R $
Rπ
e$ R
[π e R *-π e$ R $
0
JπL M
0
0
0
∑AI)8 I 0 $ M 0
A
" I
I)8 A
i " I 0 $ I)8 A
A
-π_ =
Rπ_ − " I 0 $ = I)8 A
Zπ_ −i " I 0 $ = I)8
VπL
" I
I)8 A
− " I 0 $ I)8 A
" I 0 $
I)8 A
I)8
(7) 0 WπL ∑AI)8 I 0 $ M 0
A
-π_ =
Rπ_ =
− " I 0 $
0
Zπ_ =
-π_ $ = −i " I 0 I)8 (8) A Rπ_ − " I 0 $ = I)8 A Zπ_ i " I 0 $ = I)8 " I
I)8 A
JπKL M
(9)
Since the transmitted signals are orthogonal the decoding method which is based on the maximum likelihood (ML) [26] would be much simple. Detection procedure is based on minimum distance calculation. By assuming that the channel fading coefficients could be recovered perfectly at the receiver side the minimum distance for the case of Y2 could be modeled respectively as follow:
.
d - eb* , ^f* ℎ* + ^f- ℎ- + ^fA ℎA + ^fR ℎR g+d - eb- , ^f* ℎ* + ^f- ℎ- − ^fA ℎA − ^fR ℎR )+d - ebA , ^f* ℎ* − ^f- ℎ- + ^fA ℎA −
(10)
^fR ℎR )+d - ebR , ^f* ℎ* − ^f- ℎ- − ^fA ℎA + ^fR ℎR g
(4)
And the maximum likelihood decoding can be represented as ^f$ = hb>i j(4|ℎ$ |- − 1)k^f$ k + d - e^l$ , ^f$ gm
X matrix is circulant matrix and could be decomposed into X = GΛ; T = e8
∑AI)8 I 0
$
The received signal for these two cases would be: b c = ∑R )* ℎ c cI + c
Based on above property, we could create the circulant spacetime code, for 4 × 4 transmit matrix. Proposed codeword for the case number of four transmitted antennas (M=4) is: x* xX = Px A xR
A
" I A I)8 -π_ $ " I 0 = I)8 A Rπ_ $ " I 0 = I)8 A Zπ_ " I 0 $ = I)8
0
0 0 λA 0
e8
Zπ e$ R *-π e$ R *[π e$ R
0 0 S* 0 λR
(11)
Where^f$ is the entire possible constellation. And ^l$ , are four decision statistics constructed by combining the received signals with channel state information. The decision statistics are given by
(6)
^l* =b* ℎ*∗ + b- ℎ*∗ + bA ℎ*∗ + bR ℎ*∗
T
^l- =b* ℎ-∗ − b- ℎ-∗ − bA ℎ-∗ + bR ℎ-∗
nA =b* ℎA∗ − b- ℎA∗ + bA ℎA∗ − bR ℎA∗ ^ 3
^lR =b* ℎR∗ + b- ℎR∗ − bA ℎR∗ − bR ℎR∗
(12)
By doing the same procedure for the case of ^* = Λ the decision statistics would be: o* =b$ ℎ$∗
-
1 exp − ()dd5 = } 2 - 5 8 8
(13)
Where i is the number of transmitter. =q
=
symbol per channel use
() = ~ 0 r ()d
(14)
8
Where Er is the total number of information symbols over NT. when N channels are used in T is duration of the transmitted signal . The code rate for general CSTC is: ℛ=
( (
= 1 symbol per channel use
1
{r 1 ) (22) 2E8 And here based on this new space time block code and since the rows of − f are orthogonal the MGF can be simplified as:
() = [1 − $)*
2
codeword C and detecting it as C by p(C → C ) [12]. There are two possible methods to compute the performance of the code. First method is to compute the code distance spectrum and apply the union bound technique to calculate the average pairwise error probability, result in this case is asymptotically tight at high SNR’s for small number of receive antenna [4,12]. Second method is calculating the exact value of probability of error instead of evaluating the bounds. Here it is tried to calculate based on second method. According to [8,12] the pairwise conditional error probability on the MIMO fading channel is: w se, f ktg = u(v q ∑zc)* | tc (c − yc )|- ) -=x
-=x
x
There is also another method to calculate the probability of error which is based on the singular value decomposition of difference matrix. Probability of the error define as probability of detection of the codeword X* when X* is transmitted. λ ′ are the eigenvalues of the matrix A(X* , X* ) = (X* − X* ): (X* , −X* ) . The γ is signal to noise ratio per symbol and M is the number of receivers [12, 4, 26].
(18)
P(X* →X* ) ≤
1
(25) γλ + ( )] 4 So for the case of 4 × 4 the eigenvalues of the A(X* , X* ) are equal to:
By using Graing’s formula [4], and z ∑c)* | tc (c − yc )| equation (18) could be rewritten
as [26] :
(23)
* rw se, f ktg = 8J ∏$)* [1 + qJ ($ − y )- ]1 d5 R= r$ 2
the channel coefficients estimation which is assumed to be perfectly estimated, and xt ‘s are all components of codeword wq
{r ( − f )- ]1 2E8
And by substituting (23) in (20) the pairwise error probability is equal to [26]: (24)
Where {r is the energy per symbol at each transmit antenna, E8 is power of the noise, L is transmitted data frame length, H t is
X. Γ=
*
() = (det [ − ( − f )( − f )T ]
(17)
2
(21)
Where here s is equal to − . For Rayleigh fading - r$ J 2 channels, the MGF can be represented as [4, 26]:
In order to evaluate the performance of this code, it is necessary to calculate the pairwise error probability of error. The pairwise error probability is defined; transmitting 1
1 1 (− )d5 } 2 - 5
Where is MGF and defined as:
Based on definition of code rate [4]:
ℛ=
(20)
1 (19) se, f ktg = ~ exp − d5 } 8 2 - 5 The average pairwise error probability of equation (19) with respect to the distribution of Γ and based on its moment generating function (MGF) is [12, 26]:
∏¡ )*[1
λ = (x − x )- ; i = 1,2,3,4
(26) Therefore the upper bound of the probability of the error for one receiver, M=1, is equal to: s(* →* ) ≤
4
1
∏R$)*[1 + ( =
£¤$ * )] 4
1 £( − $ )- * ∏R$)*(1 + ( $ )) 4
(27)
After obtaining the average probability of error in this section, we discuss about the simulation results of proposed method in the next section.
Symobl Error Rate Probability BPSK
0
10
Y1 upper bound BPSK 4Tx 1Rx Spatial Multiplexing OSTBC 4Tx 1Rx Y1 simulation BPSK for 4Tx 1Rx
-1
10
IV. SIMULATION RESULTS Ps:Symbol Error Rate
In this section, simulation results are provided to demonstrate the analysis of CSTC block theoretically and simulation based for QPSK and BPSK modulations format and compare these results with OSTBC and also spatial multiplexing(SM). Simulations have been done in MATLAB. Channel is simulated as Rayleigh fading and channel coefficients are generated based on random number function generators. So far, most work on space time coding assumes frequency-flat fading channels; an assumption that is not well justified in future broadband wireless communication systems. In broadband wireless communications, the symbols duration becomes smaller than the channel delay spread and consequently channel frequency-selectivity arises. Therefore, targeting broadband wireless communications, it is important to investigate how to design space time codes in the presence of frequency-selectivity. Figure 2 shows simulation result for the case of two transmitters and one receiver when source information is modulated by using of BPSK modulation and then modulated symbols go through space time coding based on equation(7) in this figure also the upper bound of probility of error is plotted based on equation(27) . In figure (3) also all the procedure is repeated for the case of QPSK modulation. In Fig4,5 respectively based on BPSK and QPSK modulation upper bound of probability of error and orthogonal space time block code and CSTC based on equation (8) are simulated .In these results we can see that performance of CSTC is better than SM and near to the OSTC .
-2
10
-3
10
-4
10
-5
10
-6
10
5
10
15
20
25
30
SNR (dB)
Fig. 2. Symbol error rate comparison of BPSK based on upper bound Y1_CSTC, OSTBC, spatial multiplexing, simulated Y1_CSTC. Symobl Error Rate Probability for QPSK
0
10
Y 1 Upper Bound QPSK 4Tx 1Rx Spatial Multiplexing OSTBC 4Tx 1Rx Y 1 simulation QPSK for 4Tx 1Rx
-1
10
Ps :Symbol Error Rate
-2
10
-3
10
-4
10
-5
10
-6
10
V. CONCLUSION
5
10
15
20
25
30
SNR (dB)
In this paper a new space time block coding is proposed. It is shown that the CSTC is the similar to the orthogonal space time code. It achieves multipath diversity and full rate. There is increasing complexity in the transmitter/receiver but the intrinsic diversity of the code might prove beneficial in a fast fading environment. Based on the simulation results it is shown the proposed scheme is capable of reliable transmission at relatively lower SNRs and in the different channel conditions it is more robust. Current work is in alternatives decomposition methods and decoding algorithms to simplify receiver structure.
Fig. 3. Symbol error rate comparison of QPSK based on upper bound Y1_CSTC, OSTBC, spatial multiplexing, simulated Y1_CSTC. Symobl Error Rate Probability BPSK
0
10
Y 2 upper bound BPSK 4Tx 1Rx OSTBC 4Tx 1Rx Y 2 simulation BPSK for 4Tx 1Rx
-1
P s :Symbol Error Rate
10
-2
10
-3
10
-4
10
-5
10
-6
10
5
10
15
20
25
30
SNR (dB)
Fig. 4. Symbol error rate comparison of BPSK based on upper bound Y2_CSTC, OSTBC, spatial multiplexing, simulated Y2_CSTC.
5
Symobl Error Rate Probability for QPSK
0
10
OSTBC 4Tx 1Rx Y 2 simulation QPSK for 4Tx 1Rx
-1
P s :Symbol Error Rate
10
Y 2 Upper Bound QPSK 4Tx 1Rx
[10]
-2
10
[11] -3
10
[12]
-4
10
-5
10
[13] -6
10
5
10
15
20
25
30
[14]
SNR (dB)
Fig. 5. Symbol error rate comparison of QPSK based on upper bound Y2_CSTC, OSTBC, spatial multiplexing, simulated Y2_CSTC.
[15]
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the orthogonality of the transmitted signal in each antenna per time slots complexity of the decoding grows linearly with the underlying constellation size for general constellation [5,6]. In space-time coding, information symbols are encoded across the time and spatial dimension, which corresponds respectively to the multiple signaling interval and multiple antennas at the transmitter. By using several separated enough antennas in the transmitter each path undergoes independent fading. Therefore, transmit diversity can be achieved in distributed antenna systems (DAS) or centralized antenna system (CASs). Several methods of diversity are proposed in [5]-[13]. In this paper more focus is implied on linear spacetime block codes, in which the information symbols are linearly combined to form two dimensional coding matrixes, the rows and columns of the matrix correspond to the transmission across multiple antennas and time intervals respectively. There are several constructions of space time block codes with full diversity such as orthogonal space-time codes, diagonal algebraic space-time codes, semi orthogonal algebraic space-time codes, threaded algebraic space-time block codes and perfect space-time codes [7, 8]. First, Orthogonal spacetime block codes achieve full diversity and their code word matrix is an orthogonal matrix as Alamouti. Second, a Diagonal Algebraic Space-Time (DAST) blocks code is a linear space-time code build by the use of rotation of constellations [14].Diagonal in DAST means that the structure of the code matrix, wherein the rotated information symbols are spread over the diagonal of the square code matrix[15,20]. The rotation matrices used in DAST codes are based on using algebraic number of field theory [15]. Performance of DAST codes is better than orthogonal codes but the cost of this probability of error improvement is increased decoding complexity in the DAST. Third, Quasiorthogonal space-time codes satisfy orthogonality condition to enable rate-one in transmission with the cost of increasing decoding complexity. Quasiorthogonal codes for four antennas are proposed by Jafakhani [13], Trikkonen-Boariu [17, 18], and Foschini [1] independently. These codes outperform orthogonal designs at all spectral efficiencies for complex rotations. Fourth, singlesymbol decodable space-time block codes are a family of nonorthogonal space-time codes that achieve rate-one transmission and full diversity for three and four transmit antennas. Since a pair of real symbols defines a single complex symbol, these codes are named as single-symbol decodable. The complexity of these codes are more than orthogonal
Abstract— In this paper a new space time coding, which is suitable for high data rate communications is proposed. In this approach data is encoded by using a decomposition of a circulant matrix and the output of encoder is splitted into N streams to be transmitted simultaneously to N transmit antennas. The performance of this multiple input multiple output system is evaluated in a Rayleigh fading channel. Its performance in terms of symbol error rate, diversity gain and complexity is obtained. Simulation results show that with a perfect channel estimator, when channel state information (CSI) is available at the receiver, the Circulant Space Time code (CSTC) has a better symbol error rate via signal to noise ratio in compared to the spatial multiplexing and near to orthogonal space time block code. Index Terms—Space time coding, multiple input multiple output, Circulant matrix, Orthogonal space time coding, Alamouti space time code.
I. INTRODUCTION The key challenges in future of wireless communication are data rate and quality of service (QoS). Deploying multiple antennas at the receiver and transmitter creates multiple-input multiple-output (MIMO) systems that not only increase the data rate, but also these systems can offers high reliability and robustness and QoS in comparison with single antenna systems. Space time coding and diversity schemes are two solutions which could improve the data rate and QoS [1]. Beside channel additive with Gaussian noise, several communication channels severe channel attenuation, fading in the channel communication, one way to overcome this effect is also using space time coding. So generally, Space time coding is one way to combat fading and also simplify the implementation of the mobile terminal [1,2]. The first method of diversity is delay diversity which is inspired by Wittenben [3] to develop space-time codes. Tarokh et al in [4] develop space-time trellis codes but, it has high decoding complexity and needs a viterbi algorithms at the receiver for decoding. Regarding to decrease the high decoding complexity for the first time space-time block codes is introduced by Alamouti et al. from AT&T in 1998 as a novel means of providing transmit diversity for the multiple-antenna in fading channel [2]. Alamouti space-time block coding is simple and elegant due to low complexity of maximumlikelihood (ML) detector in the receiver side and not requiring feedback of channel state information. In fact, the Alamouti code works with two transmit antennas diversity and enables separate decoding of each complex symbol. And also, due to 1
designs and less than DAST codes and quasiorthogonal. But from data rate point of view, this design offers higher data rate in compared with orthogonal and the same with DAST and quasiorthogonal. Next category is semi-orthogonal algebraic space-time (SAST) block codes which have rate-one transmission and full diversity; they use same real rotation matrices of DAST codes [19]. They are named as semi orthogonal because half columns of the code matrix are orthogonal to the other half of them. All of the space-time codes which introduced briefly thus far achieve a maximum rate of one symbol per signaling interval. Two more categories left which has high data rate depends on number of transmit antennas. First one is threaded algebraic space time (TAST) block codes with full diversity. TAST code threads the rate-one DAST code to obtain the higher transmission rates [7, 8]. The decoding complexity of TAST codes depends on the transmission rate and whether real or complex generator matrices are used. Second category of high data rate space time block codes is perfect space time block codes for any number of antennas [20]. These codes are named because they have full diversity, high spectral efficiency [20] and decoding complexity does not depend on the transmission rate, but on the modulation alphabet [22]. In general, the design of space-time block codes is a tradeoff between minimizing decoding complexity and maximizing the diversity gain. In this paper we propose a new space time block code which is linear and orthogonal with rate one for any number of transmitters. Simulation results shows that by increasing the number of receivers wherein number of transmitters increase the probability of symbol error rate would decrease. Since this code is orthogonal the decoding complexity is low in comparison with other codes. The organization of this paper is as follow: system model of our method, including the transmitter and receiver and channel specification, is presented in section II. Proposed method is presented in the section III. In section IV performance analysis of the CSTC is presented. In section V simulation results of the CSTC method for BPSK and QPSK modulations is presented. Finally, conclusion part of the paper is drawn in section VI.
Fig.1. Block diagram of space time coding in transmitter side
1 2 = 1 ⋮ 1 1
12 22 ⋮
2
⋯ 1 ⋯ 2 ⋱ ⋮ ⋯
(1)
Where x k i is the symbol transmitted from antenna ∈ {1,2, . . , } and at time {1,2, … }. The received signal at jth receiver would be ( ). (
( ) = " ℎ$ ( ) ∗ $ ( ) + '$ ( ) $)*
(2)
'$ ( ) is complex additive white Gaussian noise at the receive antenna in time t, with stationary feature, mean of zero and variance of , - ∙ ℎ$ ( ) is the channel coefficient between the ith transmitter and jth receiver at time t. It is assumed the channel is Rayleigh fading which models a scattering environment and it is modeled as independent identically distributed (i.i.d) complex Gaussian random variables. It is assumed that the fading coefficients are constant across two consecutive symbol transmission periods. They can be expressed as follow [4]:
(3) ℎ$ ( ) = ℎ$ ( + ) = ℎ$ = |ℎ$ |0 1$234 Where T is the symbol duration,|ℎ$ | amplitude gain and 5$ is the phase shift between the ith transmitter and jth receiver. In the next section the new space time block code is presented.
II. SYSTEM MODEL In this paper we consider a MIMO system with N transmits antenna and M receive antennas. Figure 1 shows the block diagram of space time coding of the transmitter [4]. In figure1, the information source passes its output through the modulator box which in this paper we consider BPSK, QPSK, for implementing our method. Generally, it is assumed that each element of the modulation constellation is scaled by a factor Es so that average energy of the constellation elements is 1.
III. PROPOSED MODEL The general idea in designing of space-time block coding is designing of the code matrix (X) wherein it has maximum diversity gain and minimum decoding complexity. The key feature for achieving minimum decoding complexity is orthogonality between the sequences transmitted by the each pair of antenna [4,26]. The idea of designing of the code matrix in this paper is based on circulant matrices. Our design criterion is not constellation-dependent and applied equally well to BPSK and QPSK modulation. Based on this method we could have any number of orthogonal sequences in the transmitter side for each antenna. Beside of orthogonality
The transmitted codeword of a space-time block code can be expressed as a nT*L matrix (M= nT):
2
$ T Where the λ$ = ∑=1* are I)8 I 0 M . Λ and Y=(G Λ) orthogonal matrix. So the transmitter could be designed based on the Λ or ^* respectively as : JπKL
another advantage of CSTC is its coding gain; coding gain is equal to one. First, briefly it is explained about the circulant matrix and then discussed about the CSTC. An N × N circulant matrix takes the form [24, 25]: 8 1* … * * 8 1* ⋱ - * 8 ⋱ ⋮ = ⋮ (4) ⋱ ⋱ ⋱ 1 1* 1* 1… * 8
∑A I)8 I 0 Λ= 0 0
And ^- =
Each row is the cyclic shift of its above row. For a circulant matrix X, based on singular value decomposition (SVD) [24], it could be decomposed to GΛG: . Where G and Λ matrix defined based on the discrete Fourier transform (DFT) as: >8,8 >*,8 * ; = >-,8 √= ⋮ >=,8
>8,* >*,* >-,* ⋮ >=,*
Where >B, = 0
>8,>*,>-,⋮ >=,-
λ* 0 Λ = 0 ⋮ 0
2πjk E
… … … ⋱ …
>8,= >*,= >-,= ⋮ >=,=
0 λ0 ⋮ 0
0 0 λA ⋮ 0
… … … ⋮ 0
(5) 0 0 0 ⋮ λ
, B = 0,1,2,3 … E − 1, =
$ 0,1,2,3 … , E − 1e, - = −1. and λ$ = ∑=1* I)8 I 0
xR x* xxA
xA xR x* x-
xxA xR S x*
8 * e R 8
e e8
e8
e$ V
Jπ
e$ V
Vπ
e$ V
Wπ
e8
e$ V
Vπ
e$ V
e$
Xπ
YJπ V
e8
λ* 0 Wπ e$ V 0 λYJπ ∗ P $ 0 0 e V YXπ 0 0 e$ V e8 e8 e8
e8 8 e e8
-π
e$ R
Rπ e R Zπ e$ R $
Rπ
e$ R
[π e R *-π e$ R $
0
JπL M
0
0
0
∑AI)8 I 0 $ M 0
A
" I
I)8 A
i " I 0 $ I)8 A
A
-π_ =
Rπ_ − " I 0 $ = I)8 A
Zπ_ −i " I 0 $ = I)8
VπL
" I
I)8 A
− " I 0 $ I)8 A
" I 0 $
I)8 A
I)8
(7) 0 WπL ∑AI)8 I 0 $ M 0
A
-π_ =
Rπ_ =
− " I 0 $
0
Zπ_ =
-π_ $ = −i " I 0 I)8 (8) A Rπ_ − " I 0 $ = I)8 A Zπ_ i " I 0 $ = I)8 " I
I)8 A
JπKL M
(9)
Since the transmitted signals are orthogonal the decoding method which is based on the maximum likelihood (ML) [26] would be much simple. Detection procedure is based on minimum distance calculation. By assuming that the channel fading coefficients could be recovered perfectly at the receiver side the minimum distance for the case of Y2 could be modeled respectively as follow:
.
d - eb* , ^f* ℎ* + ^f- ℎ- + ^fA ℎA + ^fR ℎR g+d - eb- , ^f* ℎ* + ^f- ℎ- − ^fA ℎA − ^fR ℎR )+d - ebA , ^f* ℎ* − ^f- ℎ- + ^fA ℎA −
(10)
^fR ℎR )+d - ebR , ^f* ℎ* − ^f- ℎ- − ^fA ℎA + ^fR ℎR g
(4)
And the maximum likelihood decoding can be represented as ^f$ = hb>i j(4|ℎ$ |- − 1)k^f$ k + d - e^l$ , ^f$ gm
X matrix is circulant matrix and could be decomposed into X = GΛ; T = e8
∑AI)8 I 0
$
The received signal for these two cases would be: b c = ∑R )* ℎ c cI + c
Based on above property, we could create the circulant spacetime code, for 4 × 4 transmit matrix. Proposed codeword for the case number of four transmitted antennas (M=4) is: x* xX = Px A xR
A
" I A I)8 -π_ $ " I 0 = I)8 A Rπ_ $ " I 0 = I)8 A Zπ_ " I 0 $ = I)8
0
0 0 λA 0
e8
Zπ e$ R *-π e$ R *[π e$ R
0 0 S* 0 λR
(11)
Where^f$ is the entire possible constellation. And ^l$ , are four decision statistics constructed by combining the received signals with channel state information. The decision statistics are given by
(6)
^l* =b* ℎ*∗ + b- ℎ*∗ + bA ℎ*∗ + bR ℎ*∗
T
^l- =b* ℎ-∗ − b- ℎ-∗ − bA ℎ-∗ + bR ℎ-∗
nA =b* ℎA∗ − b- ℎA∗ + bA ℎA∗ − bR ℎA∗ ^ 3
^lR =b* ℎR∗ + b- ℎR∗ − bA ℎR∗ − bR ℎR∗
(12)
By doing the same procedure for the case of ^* = Λ the decision statistics would be: o* =b$ ℎ$∗
-
1 exp − ()dd5 = } 2 - 5 8 8
(13)
Where i is the number of transmitter. =q
=
symbol per channel use
() = ~ 0 r ()d
(14)
8
Where Er is the total number of information symbols over NT. when N channels are used in T is duration of the transmitted signal . The code rate for general CSTC is: ℛ=
( (
= 1 symbol per channel use
1
{r 1 ) (22) 2E8 And here based on this new space time block code and since the rows of − f are orthogonal the MGF can be simplified as:
() = [1 − $)*
2
codeword C and detecting it as C by p(C → C ) [12]. There are two possible methods to compute the performance of the code. First method is to compute the code distance spectrum and apply the union bound technique to calculate the average pairwise error probability, result in this case is asymptotically tight at high SNR’s for small number of receive antenna [4,12]. Second method is calculating the exact value of probability of error instead of evaluating the bounds. Here it is tried to calculate based on second method. According to [8,12] the pairwise conditional error probability on the MIMO fading channel is: w se, f ktg = u(v q ∑zc)* | tc (c − yc )|- ) -=x
-=x
x
There is also another method to calculate the probability of error which is based on the singular value decomposition of difference matrix. Probability of the error define as probability of detection of the codeword X* when X* is transmitted. λ ′ are the eigenvalues of the matrix A(X* , X* ) = (X* − X* ): (X* , −X* ) . The γ is signal to noise ratio per symbol and M is the number of receivers [12, 4, 26].
(18)
P(X* →X* ) ≤
1
(25) γλ + ( )] 4 So for the case of 4 × 4 the eigenvalues of the A(X* , X* ) are equal to:
By using Graing’s formula [4], and z ∑c)* | tc (c − yc )| equation (18) could be rewritten
as [26] :
(23)
* rw se, f ktg = 8J ∏$)* [1 + qJ ($ − y )- ]1 d5 R= r$ 2
the channel coefficients estimation which is assumed to be perfectly estimated, and xt ‘s are all components of codeword wq
{r ( − f )- ]1 2E8
And by substituting (23) in (20) the pairwise error probability is equal to [26]: (24)
Where {r is the energy per symbol at each transmit antenna, E8 is power of the noise, L is transmitted data frame length, H t is
X. Γ=
*
() = (det [ − ( − f )( − f )T ]
(17)
2
(21)
Where here s is equal to − . For Rayleigh fading - r$ J 2 channels, the MGF can be represented as [4, 26]:
In order to evaluate the performance of this code, it is necessary to calculate the pairwise error probability of error. The pairwise error probability is defined; transmitting 1
1 1 (− )d5 } 2 - 5
Where is MGF and defined as:
Based on definition of code rate [4]:
ℛ=
(20)
1 (19) se, f ktg = ~ exp − d5 } 8 2 - 5 The average pairwise error probability of equation (19) with respect to the distribution of Γ and based on its moment generating function (MGF) is [12, 26]:
∏¡ )*[1
λ = (x − x )- ; i = 1,2,3,4
(26) Therefore the upper bound of the probability of the error for one receiver, M=1, is equal to: s(* →* ) ≤
4
1
∏R$)*[1 + ( =
£¤$ * )] 4
1 £( − $ )- * ∏R$)*(1 + ( $ )) 4
(27)
After obtaining the average probability of error in this section, we discuss about the simulation results of proposed method in the next section.
Symobl Error Rate Probability BPSK
0
10
Y1 upper bound BPSK 4Tx 1Rx Spatial Multiplexing OSTBC 4Tx 1Rx Y1 simulation BPSK for 4Tx 1Rx
-1
10
IV. SIMULATION RESULTS Ps:Symbol Error Rate
In this section, simulation results are provided to demonstrate the analysis of CSTC block theoretically and simulation based for QPSK and BPSK modulations format and compare these results with OSTBC and also spatial multiplexing(SM). Simulations have been done in MATLAB. Channel is simulated as Rayleigh fading and channel coefficients are generated based on random number function generators. So far, most work on space time coding assumes frequency-flat fading channels; an assumption that is not well justified in future broadband wireless communication systems. In broadband wireless communications, the symbols duration becomes smaller than the channel delay spread and consequently channel frequency-selectivity arises. Therefore, targeting broadband wireless communications, it is important to investigate how to design space time codes in the presence of frequency-selectivity. Figure 2 shows simulation result for the case of two transmitters and one receiver when source information is modulated by using of BPSK modulation and then modulated symbols go through space time coding based on equation(7) in this figure also the upper bound of probility of error is plotted based on equation(27) . In figure (3) also all the procedure is repeated for the case of QPSK modulation. In Fig4,5 respectively based on BPSK and QPSK modulation upper bound of probability of error and orthogonal space time block code and CSTC based on equation (8) are simulated .In these results we can see that performance of CSTC is better than SM and near to the OSTC .
-2
10
-3
10
-4
10
-5
10
-6
10
5
10
15
20
25
30
SNR (dB)
Fig. 2. Symbol error rate comparison of BPSK based on upper bound Y1_CSTC, OSTBC, spatial multiplexing, simulated Y1_CSTC. Symobl Error Rate Probability for QPSK
0
10
Y 1 Upper Bound QPSK 4Tx 1Rx Spatial Multiplexing OSTBC 4Tx 1Rx Y 1 simulation QPSK for 4Tx 1Rx
-1
10
Ps :Symbol Error Rate
-2
10
-3
10
-4
10
-5
10
-6
10
V. CONCLUSION
5
10
15
20
25
30
SNR (dB)
In this paper a new space time block coding is proposed. It is shown that the CSTC is the similar to the orthogonal space time code. It achieves multipath diversity and full rate. There is increasing complexity in the transmitter/receiver but the intrinsic diversity of the code might prove beneficial in a fast fading environment. Based on the simulation results it is shown the proposed scheme is capable of reliable transmission at relatively lower SNRs and in the different channel conditions it is more robust. Current work is in alternatives decomposition methods and decoding algorithms to simplify receiver structure.
Fig. 3. Symbol error rate comparison of QPSK based on upper bound Y1_CSTC, OSTBC, spatial multiplexing, simulated Y1_CSTC. Symobl Error Rate Probability BPSK
0
10
Y 2 upper bound BPSK 4Tx 1Rx OSTBC 4Tx 1Rx Y 2 simulation BPSK for 4Tx 1Rx
-1
P s :Symbol Error Rate
10
-2
10
-3
10
-4
10
-5
10
-6
10
5
10
15
20
25
30
SNR (dB)
Fig. 4. Symbol error rate comparison of BPSK based on upper bound Y2_CSTC, OSTBC, spatial multiplexing, simulated Y2_CSTC.
5
Symobl Error Rate Probability for QPSK
0
10
OSTBC 4Tx 1Rx Y 2 simulation QPSK for 4Tx 1Rx
-1
P s :Symbol Error Rate
10
Y 2 Upper Bound QPSK 4Tx 1Rx
[10]
-2
10
[11] -3
10
[12]
-4
10
-5
10
[13] -6
10
5
10
15
20
25
30
[14]
SNR (dB)
Fig. 5. Symbol error rate comparison of QPSK based on upper bound Y2_CSTC, OSTBC, spatial multiplexing, simulated Y2_CSTC.
[15]
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