at full rate. In this paper we present a family of adaptive, channel dependent, rate-one Extended Alamouti space- time block codes for four and eight transmit ...
IEEE Copyright 2004, Proc of SCC 04, Erlangen Germany, January 2004
Quasi-Orthogonal Space-Time Block Codes for Data Transmission over Four and Eight Transmit Antennas with Very Low Feedback Rate Biljana Badic, Markus Rupp, Hans Weinrichter Institute for Communications and Radio Frequency Engineering Vienna University of Technology Gusshausstr.25, A-1040 Vienna, Austria (bbadic, mrupp, jweinri)@nt.tuwien.ac.at
Abstract Since the publication of Alamouti’s famous space-time block code for two transmit antennas, various suboptimal space-time block codes for multi-input/multi-output fading channels have been proposed not achieving full diversity at full rate. In this paper we present a family of adaptive, channel dependent, rate-one Extended Alamouti spacetime block codes for four and eight transmit antennas. Only partial channel state information is sent back from the receiver to the transmitter. Transmit diversity is improved up to nearly the optimum diversity value. This high diversity can be exploited with zero forcing receiver as well as with maximum-likelihood (ML) receiver.
1
Introduction
Since the pioneering work of Alamouti [1], several orthogonal and quasi-orthogonal space-time codes have been investigated. The simple Alamouti scheme using two transmit antennas increases the diversity at the receiver from d = 1 to d = 2 in a frequency flat fading channel. It has been shown in [2], that orthogonal full-rate design offering full diversity for any arbitrary complex symbol constellation is limited to the case of two transmit antennas. Either diversity, data rate or decoding simplicity must be sacrificed if the number of transmit antennas is increased. To preserve the full symbol rate, however with a small loss in performance, quasi- orthogonal designs have been proposed [3]. Most of the space-time block codes (STBC) are designed under the assumption that the transmitter has no knowledge about the channel. On the other hand, it has been shown that exploiting perfect channel state information (CSI) at the transmitter and at the receiver outage performance is improved compared to the case when only the receiver has perfect channel knowledge. In some schemes, low decoding complexity, high diversity and a higher code rate can be obtained even if only partial channel information is sent back to the transmitter [4]. For instance, the receiver may return only quantized amplitude information about the channel, as it has been proposed in certain power control schemes [5]. In the scheme proposed in [4]
the relevant CSI is sent back from the receiver to the transmitter using only one bit per code block. The idea of using partial CSI by returning one feedback bit is supported in the UMTS specifications. There a closed-loop transmit diversity scheme for two transmit antennas is described where the transmitter switches between two predefined beamforming vectors. [6] Research on adapting the block code to partial feedback is an interesting area of research. In this paper a simple receiver- controlled, high rate, extended Alamouti space-time block code, designed for four and eight transmit antennas is proposed to achieve full diversity as well as low bit error ratio. These codes are very similar to the quasi-orthogonal code proposed in [3]. First, a transmission system returning only one channel information bit b per code block from the receiver to the transmitter is presented. Depending on the value of b the transmitter switches between two predefined STBCs and chooses that code matrix which achieves higher diversity. In a second approach, sending back two information bits per code block to the transmitter, the transmitter can switch between four predefined space-time block codes achieving even lower BER. The paper is organized as follows. First, an overview of the Extended Alamouti space-time block codes (EAC) for four and eight transmit antennas is given. In Section III, the generation of the feedback information is explained, which informs the transmitter about the amount of data interference introduced by the channel given a specific EAC. This feedback information consists of one or two bits per code block. Simulation
results are presented in Section IV and conclusions are summarized in Section V.
2
Extending Alamouti codes to four and eight transmit antennas
A very simple and effective coding scheme for two transmit antennas and a single receive antenna has been introduced by Alamouti [1]. The data sequence s1 , s∗2 is sent over the first antenna and s2 , −s∗1 is sent over the second antenna, where ∗ denotes complex conjugation. Assuming a flat-fading channel with transmission coefficients h1 and h2 , the received vector r is formed by stacking two consecutive data samples r1 and r2 in time. Then, r can be calculated as: r = S(2) h + v,
(2)
r1
=
s 1 h1 + s 2 h2 + s 3 h3 + s 4 h4 + v 1
r2 r3
= =
s∗2 h1 − s∗1 h2 + s∗4 h3 − s∗3 h4 + v2 s∗3 h1 + s∗4 h2 − s∗1 h3 − s∗2 h4 + v3
r4
=
s 4 h1 − s 3 h2 − s 2 h3 + s 1 h4 + v 4 .
With the complex conjugation of the second and the third equation cited above, the received signals can be equivalently expressed as y = H(4) ¯, v s+v where
T
where h = [h1 , h2 ] is the channel vector and v is the noise vector. Here the symbol block S(2) is defined as: s1 s2 (2) . (1) S = s∗2 −s∗1
The received vector can be written alternatively as
(4)
where is y defined as y = [r1 , r∗2 ]T and the virtual 2×2 (2) channel matrix Hv results in h1 h2 (2) (2) Hv = −h∗2 h∗1 It is of vital importance that the resulting virtual chan(2) nel matrix Hv is orthogonal, i.e. H(2)H H(2) v v
(2)H H(2) v Hv
=
2
= h I2 ,
where I2 is the 2 × 2 identity matrix and h2 = |h1 |2 + |h2 |2 . In this way, the Alamouti scheme decouples the MISO-channel into two virtually independent channels with channel gain h2 and diversity d = 2.
2.1 Extended full rate Alamouti Scheme for four transmit antennas In the case of four transmit antennas (Figure 1) the signal matrix S(2) given by (1) can be used as a building block to obtain a (4 × 4) code block given in (3) [7] [8]: h1
Input
Extendend Alamouti
s
noise
h2 Receiver
h3
Output y
h4 Channel
Fig. 1.
S
(4)
Extended Alamouti Scheme for four Antennas
" (2) S1 (2)∗ S2
(4) =S1 =
# s1 (2) S2 s∗2 (2)∗ =s∗ 3 −S1 s4
s2 −s∗1 s∗4 −s3
s3 s∗4 −s∗1 −s2
s4 −s∗3 , −s∗2 s1 (3)
(4)
y1 y2
= =
r1 , r2∗ ,
v¯1 = v1 v¯2∗ = v2
y3 y4
= =
r3∗ , r4 ,
v¯3∗ = v3 v¯4 = v4
and "
Hv1 =
¯, y = H(2) v s+v
(2)
where S1 and S2 correspond to (2 × 2) code blocks defined in (1). Defining h now as h = [h1 h2 h3 h4 ]T we obtain the four received signal values
(2) Hv1 (2)∗ −Hv2
h1 −h∗2 = ∗ −h3 h4
#
(2) Hv2 (2)∗ Hv1
h2 h∗1 −h∗4 −h3
h3 −h∗4 h∗1 −h2
h4 ∗ h3 (5) h∗2 h1
is the virtual equivalent (4 × 4) channel matrix. In this way, a virtual, structured (4 × 4) MIMO channel corresponding to four transmit and four virtual receive antennas is obtained.
2.2 Extended full rate Alamouti Scheme using eight transmit antennas Applying the extension of the Extended Alamouti signalling scheme given in (3) once more, an Extended Alamouti code for eight transmit antennas can be obtained. The Extended Alamouti code for eight transmit antennas is now defined as [9]: " # (4) (4) S1 S5 (8) S = ⇒ (6) (4)∗ (4)∗ S5 −S1
s1 s∗2 ∗ s3 s4 S(8) = s∗ 5 s6 s7 s∗8
s2 −s∗1 s∗4 −s3 s∗6 −s5 s8 −s∗7
s3 s∗4 −s∗1 −s2 s∗7 s8 −s5 −s∗6
s4 −s∗3 −s∗2 s1 s∗8 −s7 −s6 s∗5
s5 s∗6 s∗7 s8 −s∗1 −s2 −s3 −s∗4
s6 −s∗5 s∗8 −s7 −s∗2 s1 −s4 s∗3
s7 s∗8 −s∗5 −s6 −s∗3 −s4 s1 s∗2
s8 −s∗7 −s∗6 s5 , −s∗4 s3 s2 −s∗1
where we have defined a new block of four symbols (4) S5 containing the symbols {s5 , s6 , s7 , s8 }. With this code block eight information symbols and their conjugates are transmitted in various combinations over the eight transmit antennas in eight successive time slots. As explained before, the system can be equivalently
described as an (8 × 8) MIMO system with an (8 × 8) (8) virtual channel matrix Hv resulting in : " # (4) (4) Hv1 Hv5 (8) Hv = ⇒ (7) (4)∗ (4)∗ −Hv5 Hv1
h1 −h∗2 ∗ −h3 h4 (8) Hv = −h∗ 5 h6 h7 −h∗8
h2 h∗1 −h∗4 −h3 −h∗6 −h5 h8 h∗7
h3 −h∗4 h∗1 −h2 −h∗7 h8 −h5 h∗6
h4 h∗3 h∗2 h1 −h∗8 −h7 −h6 −h∗5
h5 −h∗6 −h∗7 h8 h∗1 −h2 −h3 h∗4
h6 h∗5 −h∗8 −h7 h∗2 h1 −h4 −h∗3
h7 −h∗8 h∗5 −h6 h∗3 −h4 h1 −h∗2
h8 h∗7 h∗6 h5 . h∗4 h3 h2 h∗1
(4)
Unfortunately, both virtual channel matrices Hv (8) and Hv are no more orthogonal. However, in the (8) (4) following it will be shown that Hv and Hv are ”nearly orthogonal”.
3
Channel Adaptive Extended Alamouti Codes (CAEACs)
3.1 CAEACs using four transmit antennas and partial feedback In this chapter we want to further extend the idea of the Alamouti Scheme to families of Extended Alamouti Codes, where the members of a family are selected at the transmitter due to specific instantaneous parameters of the virtual channel matrix. We will start with CAEACs for four transmit antennas and a family of two Extended Alamouti Codes (EACs). The corresponding transmission scheme is shown in Fig. 2. The transmitter (4) (4) selects one of two very similar EACs S1 and S2 to transmit the signal vector s over the channel with transfer vector h. The selection algorithm is controlled by the receiver, that tells the transmitter which one of (4) (4) S1 or S2 will achieve a more reliable transmission. At the receiver this knowledge is gained from the knowledge of the channel transfer vector h (we assume that the receiver has perfect channel knowledge). In the following we will describe this selection algorithm in more detail.
y = Gs +v
S
i = 1, 2,
(8)
where ri is the (4 × 1) vector of received signals (4) resulting from the code-block Si of length T = 4, (4) (4) (4) Si is either S1 or S2 depending on the channel parameter b, defined below, which is fed back from the receiver, and v is the (4 × 1) noise vector with independent complex Gaussian components with zero (4) (4) mean and equal variance σv2 . S1 and S2 are defined as: s1 s2 s3 s4 s∗2 −s∗1 s∗4 −s∗3 (4) (9) S1 = ∗ s∗3 s4 −s∗1 −s∗2 s4 −s3 −s2 s1 −s1 s2 s3 s4 −s∗2 −s∗1 s∗4 −s∗3 (4) (10) S2 = ∗ −s∗3 s4 −s∗1 −s∗2 −s4 −s3 −s2 s1 (4)
(4)
Obviously, S1 and S2 differ only in the sign of the symbols in the first column. In the simulations the signal elements s1 to s4 are taken from a simple QPSK signal constellation. The equation (8) can be rewritten as (4) yi = Hvi s + v i = 1, 2, with s = [s1 , s2 , s3 , s4 ]T and an effective channel matrix Hvi , that results in: h1 h2 h3 h4 −h∗2 h∗1 −h∗4 h∗3 (4) (11) Hv1 = −h∗3 −h∗4 h∗1 h∗2 h4 −h3 −h2 h1 (4)
if Si
(4)
= S1 , or (4)
Hv2
if Si
(4)
H S2
(4)
ri = S i h + v
(4)
v S1
(4)
Receiver
STBC S
(4)
S1 and S2 corresponding to four transmit antennas and T = 4 time intervals. The channel transfer vector h = [h1 , h2 , h3 , h4 ]T may fade in any arbitrary way but it is assumed that h remains constant during the code block of length T. The signal transmission can be described by
−h1 −h∗2 = −h∗3 h4
h2 −h∗1 −h∗4 −h3
h4 h∗3 h∗2 −h1
h3 −h∗4 −h∗1 −h2
(4)
= S2 .
(12)
In both cases we obtain a quasi orthogonal Grammian matrix [8]
(4)
Xmin
(4)
Gi Feedback Information: quant |Xmin|
Fig. 2. Scheme with One Feedback Bit per Code Block and Four Transmit Antennas
The family of EACc consists of two (4×4) code blocks
(4)H
= Hvi
(4)
(4)
(4)H
Hvi = Hvi Hvi
(i = 1, 2) with 1 I2 = 0
0 1
,
= h2
J2 =
Xi J 2 , I2
I2 −Xi J2
0 −1
1 0
h2 = |h1 |2 + |h2 |2 + |h3 |2 + |h4 |2
,
(13)
and Xi equal to 2Re(h1 h∗4 − h2 h∗3 ) (4) , if S1 transmitted (14) h2 2Re(−h1 h∗4 − h2 h∗3 ) (4) X2 = , if S2 transmitted h2
X1 =
(4)
It is well known that Gi should approximate a scaled identity-matrix as far as possible to get full diversity and optimum BER performance. This means, (4) |Xi | should be as small as possible. As Gi indicates, our scheme inherently supports full diversity d = 4 if Xi = 0. Therefore, the strategy is to transmit that (4) (4) code S1 or S2 that minimizes |Xi |. In any case, a performance loss due to the non vanishing value of Xi is expected caused by some interference between the signal elements y1 to y4 . As it is assumed that the receiver has full information of the channel state knowing h1 to h4 , the receiver can compute X1 and X2 . With this information the receiver returns the feedback bit b informing the transmitter to select that code block (4) Si (i = 1, 2) which leads to the smaller value of |Xi |. With this information the transmitter switches between (4) (4) the EACs S1 and S2 such that the resulting |Xi | will correspond to min(|X1 |, |X2 |). Obviously the control information sent back to the transmitter consists only of one feedback bit per code block. In the simulation it has been assumed that the channel varies slowly such that the delay of the feedback information can be neglected.
3.2 Statistics of the interference parameter
3.3 CAEACs using two feedback bits per code block In a quite similar way as discussed in section A we can switch between four different EACs. In this case, the receiver must send back two control bits per code block to support the transmitter in choosing the best code matrix for the actual channel realization. (4) (4) The family of EACs consist of S1 and S2 already (4) (4) defined in (9) and (10), and additionally of S3 , S4 defined as: js1 −js2 s3 s4 js∗2 js∗1 s∗4 −s∗3 (4) S3 = (19) ∗ ∗ js3 −js4 −s∗1 −s∗2 js4 js3 −s2 s1 (4)
S4
js1 js∗2 = js∗3 js4
js2 −js∗1 js∗4 −js3 (4)
s3 s∗4 −s∗1 −s2
s4 −s∗3 . −s∗2 s1
(20)
(4)
In case of transmitting S3 and S4 the corresponding (4) virtual channel matrices Hv are: jh1 −jh2 h3 h4 −jh∗2 −jh∗1 −h∗4 h∗3 (4) (21) Hv3 = ∗ ∗ −h∗3 −h4 −jh1 jh∗2 h4 −h3 jh2 jh1
If the channel coefficients hi are i.i.d. complex Gaussian random variables, then the probability density function of Xi is calculated in [8] as: 3 2 4 (1 − x ) ; |x| < 1, (15) fXi (x) = 0 ; else
and
To derive the probability density function of min(|X1 |, |X2 |) a new random variable is defined as: X1 ;if |X1 | < |X2 | . (16) W = X2 ;else
respectively. (4) The resulting Grammian matrices Gi and the channel gain h2 can be written as in (13). The channel depen(4) dent interference parameters Xi in case of S3 and (4) S4 result now in:
Due to its symmetry only the PDF of W ≥ 0 must be considered as fW (w) = fX1 (w) + fX2 (w) − fX1 (w)FX2 (w) − FX1 (w)fX2 (w) =
2fX1 (w)(1 − FX1 (w)),
(17)
assuming that X1 , X2 are two independent random variables. The cumulative probability distribution function of Xi is denoted by Fxi (w). The final solution for fW (w) is : ( h i 2 3 (1 − w2 ) 1 − 32 |w|(1 − w3 ) ; |w| ≤ 1 2 fW (w) = 0 ; else (18) Simulation results verifying this result are shown in Fig. 3 further ahead.
(4)
Hv4
jh1 jh∗2 = −h∗3 h4
X3 X4
jh2 −jh∗1 −h∗4 −h3
h3 −h∗4 −jh∗1 −jh2
h4 h∗3 −jh∗2 jh1
2Im(h1 h∗4 + h2 h∗3 ) h2 2Im(h1 h∗4 − h2 h∗3 ) . = − h2
= −
(22)
(23)
Now, the receiver has to calculate X1 to X4 and to (4) inform the transmitter which code Si leads to the minimal interference. Using these two feed back bits the transmitter switches between the four space-time (4) block codes Si to obtain still higher diversity gain and a smaller bit error rate than in case of relying only (4) on one bit feedback information and the two EACs S1 (4) and S2 . In fact, the transmitter chooses that block(4) code Si which leads to an interference parameter Z with minimum absolute value of X1 to X4 .
3.4 Statistics of the resulting interference parameter Z As explained in the last section the main idea of our adaptive coding is to reduce the relevant interference parameter in order to improve the ”quasi orthogonality” of the equivalent channel matrix. Therefore, we calculate now the corresponding probability density of the random variable Z. Starting from the four variables Xi , the variable Z is defined as: X1 ;if |X1 | = min(|X1 |, |X2 |, |X3 |, |X4 |) X2 ;if |X2 | = min(|X1 |, |X2 |, |X3 |, |X4 |) . Z= X3 ;if |X3 | = min(|X1 |, |X2 |, |X3 |, |X4 |) X4 ;else (24) According to[10], [p.186, n = 4], the probability density of Z results in: fZ (z)
=
4[1 − FX (z)]3 fx (z).
(
3 (1 2
h − z 2 ) 1 − 23 |z|(1 − 0
(8)
For Gi (8)
i3
; |z| ≤ 1 ; else (26)
The PDFs of all three interference parameters X, W and Z are shown in Fig. 3. A comparison with MonteCarlo simulations, also shown in Fig. 3, shows excellent agreement between simulation results and analytical formulas given in (18) and (26). Obviously, the mean
(8)
Hvi
I2 2−Ui J2 =h Vi J2 Yi I 2
U i J2 I2 −Yi I2 Vi J2
−Vi J2 −Yi I2 I2 −Ui J2
Yi I 2 −Vi J2 , Ui J2 I2
where the channel dependent interference parameters Ui , Yi and Vi can be calculated as: U1 = 2Re(h1 h∗4 − h2 h∗3 + h5 h∗8 − h6 h∗7 )/h2 Y1 = 2Re(h1 h∗7 − h3 h∗5 + h2 h∗8 − h4 h∗6 )/h2 V1 = 2Re(−h1 h∗6 + h2 h∗5 + h4 h∗7 − h3 h∗8 )/h2 (8)
if S1
is sent and
U2 = 2Re(−h1 h∗4 − h2 h∗3 + h5 h∗8 − h6 h∗7 )/h2 Y2 = 2Re(−h1 h∗7 − h3 h∗5 + h2 h∗8 − h4 h∗6 )/h2 V2 = 2Re(h1 h∗6 + h2 h∗5 + h4 h∗7 − h3 h∗8 )/h2
6 |X| analytic |W| analytic |W| simulated |Z| analytic |Z| simulated
5
we get in both cases:
(8)H
Gi =Hvi z2 ) 3
(4)
The corresponding virtual channel matrices # are: " (4) (4) Hv5 Hv1 (8) (29) Hv1 = (4)∗ (4)∗ Hv1 −Hv5 # " and (4) (4) Hv5 Hv2 (8) (30) Hv2 = (4)∗ (4)∗ Hv2 −Hv5
(25)
With (15) we find the PDF of Z as: fZ (z) =
(4)
back to the transmitter the codes S1 given in (9), S2 (4) in (10) and S5 given in (6) are used as building blocks (8) (8) for S1 and S2 . " # (4) (4) S5 S1 (8) S1 = (27) (4)∗ (4)∗ S5 −S1 # " and (4) (4) S5 S2 (8) , (28) S2 = (4)∗ (4)∗ S5 −S2
(8)
PDF
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x, w, z
Fig. 3.
One sided PDF of X, W , and Z
values of the resulting interference parameters W and Z are substantially reduced compared to the mean value of the original parameter X (E[|X|] = 0.3, E[|W |] = 0.2, E[|Z|] = 0.1). These mean values of the interference parameter can be interpreted as a quantitative measure of the ”non-orthogonality” of the CAEACs.
3.5 CAEACs for eight transmit antennas In a similar way as described above, the feedback scheme can be applied to a family of extended Alamouti codes for eight transmit antennas. In case if one bit sent
if S2 is sent. Matrices I2 and J2 are defined in (13). If two bits are returned to the transmitter, the transmit(8) (8) ter can switch between four EACs, namely, S1 , S2 (8) (8) given in (27) and (28) and S3 and S4 defined as: " # (4) (4) S3 S5 (8) S3 = (31) (4)∗ (4)∗ S5 −S3 " # (4) (4) S4 S5 (8) S4 = . (32) (4)∗ (4)∗ S5 −S4 The corresponding virtual channels can be calculated as: " # (4) (4) Hv3 Hv5 (8) Hv3 = (33) (4)∗ (4)∗ −Hv5 Hv3 and (8) Hv4
=
"
(4)
Hv4 (4)∗ −Hv5
(4)
Hv5 (4)∗ Hv4
#
(34)
with U3 = (−2Im(h1 h∗4 + h2 h∗3 ) + 2Re(h5 h∗8 − h6 h∗7 ))/h2 Y3 = (−2Im(h1 h∗7 − h2 h∗8 ) + 2Re(−h3 h∗5 − h4 h∗6 ))/h2 V3 = (−2Im(−h1 h∗6 − h2 h∗5 ) + 2Re(h4 h∗7 − h3 h∗8 ))/h2
(8)
if S3
is sent and
U4 = (−2Im(h1 h∗4 − h2 h∗3 ) + 2Re(h5 h∗8 − h6 h∗7 ))/h2 Y4 = (−2Im(h1 h∗7 + h2 h∗8 ) + 2Re(−h3 h∗5 − h4 h∗6 ))/h2 V4 = (−2Im(−h1 h∗6 + h2 h∗5 ) + 2Re(h4 h∗7 − h3 h∗8 ))/h2 (8)
if S4
is sent.
In case of eight transmit antennas we now have three channel parameters Ui , Yi and Vi that lead to cross signal interference. To obtain full diversity and optimum BER performance, all three parameters Ui , Yi and Vi should be as small as possible. The (8) best method to select the best transmit code Si we have found, relies on minimizing the condition number of the Grammian matrix G defined as the ratio of the largest to the smallest eigenvalue of G G) ). In [9] it is shown that the (κ = cond(G) = max(λ min(λG ) eigenvalues of G can easily be computed as:
4
λ1
=
λ2 = [(1 − Ui ) + (Yi − Vi )]/h2 ,
λ3 λ5 λ7
= = =
λ4 = [(1 + Ui ) − (Yi + Vi )]/h2 , λ6 = [(1 + Ui ) + (Yi + Vi )]/h2 , λ8 = [(1 − Ui ) − (Yi − Vi )]/h2 .
BER Simulation Results
In our simulations, we have used a QPSK signal constellation. The Rayleigh fading channel has been kept constant during the transmission of a each code block but has faded independently between successive blocks (fast fading). At the receiver side, we have implemented a zero forcing (ZF) receiver in case of the four transmit antennas and in case of the eight transmit antenna scheme. Additionally, a maximum likelihood (ML) receiver has been tested in case of the four antenna scheme. Fig. 4 shows the resulting BER as a function of Eb /N0 using the ZF receiver. Fig. 5 shows the results for the ML receivers. The BER results have been averaged over 2,048 QPSK information symbols and 10,000 realizations of the i.i.d. channel matrix. The resulting curves are compared with an ideal two and four path diversity transmission. Obviously a substantial improvement of the BER can be achieved by sending back only a small amount of partial feedback information about the channel state to the transmitter enabling this device to switch between two or four predefined code matrices. Note, that there is only a small difference between ZF receiver and ML receiver performance due the reduced interference parameter of G or the small amount of ”nonorthogonality” respectively. Fig. 6 shows the simulated BER for QPSK modulation using eight transmit antennas and one receive
antenna. Again, a ZF receiver has been used. The BER curves are compared with the results obtained with ideal eight path diversity. In this case only a small improvement can be achieved by using on or two feedback bits per code block. The ideal two, four and eight- path diversity curves are simulated using (2.1) in [9]: ! r 2h 1 . BER = erfc 2 4σ 2 for 106 realizations of the i.i.d. channel matrix.
5
Conclusion
In this paper, a family of EACs for four and eight transmit antennas is combined with limited feedback information sent back to the transmitter. We have shown that this simple transmission scheme with only one or two feedback bits per code block used to select the most appropriate member of the code family improves diversity and bit error ratio over the whole SNR range compared to the case of an open loop transmission system. In case of four transmit antennas only one bit per code block fed back can improve BER performance substantially. Sending two bits back to the transmitter using four transmit antennas we in fact achieve full diversity order with a low complexity ZF receiver. In case of eight transmit antennas our CAEACs are not as useful as in the four transmit antenna case. The improvements obtained by feedback are not so high as in a case of four transmit antennas due to the increased spatial interference.
References [1] S.M Alamouti ”A Simple Diversity Technique for Wireless Communications,” IEEE J. Sel. Ar. Comm., vol.16, no.8, pp. 1451-1458, Oct. 1998. [2] V.Tarokh, H. Jafarkhani and A.R. Calderbank, ”SpaceTime block codes from orthogonal designs,” IEEE Trans.Inf. Theory, vol.45, pp. 1456-1467, July 1999. [3] H. Jafarkhani, ”A quasi orthogonal space-time block code,” IEEE Trans. Comm., vol. 49, pp. 1-4, Jan. 2001. [4] J.Akhtar, D.Gesbert ”Partial Feedback Based Orthogonal Block Coding,” Proceedings IEEE Vehicular Technology Conference, VTC 2003-Spring, Vol.1, pp.287-291, April 22-25, 2003. [5] S. Bhashyam, A. Sabharwal and B. Aazhang, ”Feedback gain in multiple antenna systems, ” IEEE Trans, Comm., vol.48, pp. 83-94, May 2002. [6] 3GPP TR25.869 v1.0.0.(2001-06) 3rd Generation Partnership Project; Technical Specification Group Access Network; Tx diversity solutions for multiple antennas (Release 5). [7] Papadias C. and Foschini G. ”A Space-Time coding approach for systems employing four transmit antennas”, Acoustics, Speech and Signal Processing, 2001 Proceedings, vol.4, pp. 2481-2484, 7-11 May 2001.
0
10
−1
10
−2
BER
10
−3
10
−4
10
−5
10 −15 Fig. 4.
ideal 2−path diversity ZF without feedback ZF with one bit feedback ZF with two bits feedback ideal 4−path diversity −10
−5
0
5 Eb/N0
10
15
20
25
BER of (4 × 1) extended Alamouti scheme with feedback and ZF Receiver 0
10
ideal 2−path diversity ML without feedback ML with one bit feedback ML with two bits feedback ideal 4−path diversity
−1
10
−2
BER
10
−3
10
−4
10
−5
10 −15 Fig. 5.
−10
−5
0
5 Eb/N0
10
BER of (4 × 1) extended Alamouti scheme with feedback and ML Receiver
[8] C.F.Mecklenbr¨auker, M.Rupp ”On Extended Alamouti Schemes for Space-Time Coding, ” Proc. WPMC’02, Honolulu, pp. 115-119, Oct. 2002. [9] C.F.Mecklenbr¨auker, M.Rupp ”Flexible Space-Time Block Codes for Trading Quality of Service against Data Rate in MIMO UMTS”, EURASIP Journal on Applied Signal Processing, Special Issue on MIMO Communication and Signal Processing, to appear 2004. [10] A.Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 3.Ed., 1991.
15
20
25
0
10
−1
10
−2
BER
10
−3
10
−4
10
ZF without feedback ZF with one bit feedback ZF with two bits feedback ideal 8−path diversity
−5
10 −15 Fig. 6.
−10
−5
0
Eb/No
5
BER of (8 × 1) extended Alamouti scheme with feedback and ZF Receiver
10
15
20
