Abstract: Traditional hybrid automatic repeat request (HARQ) protocols extract time diversity in a scalar fading channel. The performance depends critically on the channel variation over retransmissions. The slow fading channel model considered allows no time diversity for the HARQ protocols to exploit. But traditional protocols can not effectively exploit spatial diversity provided by a Multiple Input Multiple Output (MIMO) system. A new Alamouti-based HARQ transmission scheme for a MIMO system in a slowly varying channel is proposed. This new scheme is a combination of the packet combining HARQ and the Alamouti Space-Time Coding (STC). This technique increases the efficiency of HARQ packet transmission by exploiting both the spatial and time diversity of the MIMO channel. It uses the full diversity of Alamouti STC and the added gain of the packet combining scheme to provide reliable communication. The Packet Error Rate (PER) analysis of space-time coded MIMO-HARQ is presented. An n-dimension Pair Wise Error Probability (PWEP) analysis of the optimal Alamouti-based HARQ protocol is derived. Simulation results show that this new scheme outperforms traditional Chase Combining (CC) and reveal the gain of ARQ feedback in space-time coded MIMO systems.
INTRODUCTION
Hybrid ARQ techniques use Forward Error Correction (FEC) with the automatic repeat request (ARQ) protocol to recover erroneous packets caused by the channel noise and interferences. The HARQ schemes are usually considered to exploit both the high coding gain of FEC and the rate flexibility of ARQ protocol such that data can be transmitted with a minimum error. In a pure ARQ protocol, a received packet containing error is discarded and a retransmission of the packet is requested. In HARQ, earlier received erroneous packets are combined in an intelligent way with the subsequent received packets to improve the decoding reliability. The MIMO systems are known to increase the spectral efficiency and the capacity of a communication system. Combined with HARQ, a MIMO system can potentially provide higher throughput packet data services with higher reliability. Furthermore, through proper arrangement of the retransmitted packets, one can improve the performance of a MIMO system. However, most of the study of hybrid ARQ techniques is focused on scalar channels. Kim and Skoglund (2007) and Chuang et al. (2008) focused on the jointly design of MIMO transmission and ARQ feedback about MIMO-HARQ design. Oh et al. (2004) studied receiver processing of ARQ retransmissions. Koike et al. (2004), Carvalho and Popovski (2008) have studied HARQ schemes jointly considering STC and packet retransmission. For the sake of exploiting the additional spatial degrees of freedom, the bits or symbols rearrangement for retransmissions is studied by Carvalho and Popovski (2008) and the linear precoder design is considered in (Zheng et al., 2007). Despite the many efforts in studying MIMO-HARQ design, there are still some unsolved problems. The DMT-based approach focuses only on the high Signal-to-Noise Ratio (SNR) asymptotics and gives the tradeoff between multiplexing gain, diversity gain and ARQ delay. It requires a some of Space-Time Codes (STC) those rates ought to be proportional to the SNR. However, in practice, one is also interested in designing a fixed-rate STC that operates well within a range of finite SNRs. For the practical HARQ retransmission protocol design, the known works provide separate and ad hoc designs.
There are mainly two types of HARQ combining scheme: the packet combining (Chase, 1985) and Incremental Redundancy (IR) (Sesia et al., 2004). In IR-type ARQ, retransmissions only carry portions of the data packet. It presents an efficient technique for increasing the system throughput while keeping the error performance acceptable. Present main objective is to reduce the number of ARQ rounds required to correctly decode a data packet while keeping the receiver affordable complexity of computational load and memory requirements. We focus on space-time Bit-Interleaved Coded Modulation (BICM) transmitter schemes with Chase-type ARQ, in which the data packet is entirely retransmitted. The choice of BICM is due to the simplicity of this coding scheme and the efficiency of its iterative decoding receiver in achieving high diversity and coding gains over blockfading MIMO channels. In this study, we restrict present work to chase-type ARQ. A conventional Log-Likelihood Ratio (LLR)-level combining are used, where extrinsic LLRs corresponding to multiple transmissions are simply added together before SISO decoding. In the packet combining, the receiver combines noisy packets to obtain a packet with a code rate which is low enough such that reliable communication is possible even for low quality channels. we provide numerical simulations for some MIMO configurations demonstrating the superior performance of the proposed the Alamouti-based HARQ protocol compared with CC protocol.
Throughout the study the following notations will be used. Matrices and vectors
are denoted with inclined bold capital and lowercase letters, respectively.
A(i, j) is the (i, j)-th element of the matrix A and a(i)
is the i-th element of the vector a.
MIMO CHANNEL MODEL WITH ARQ FEEDBACK
In Fig. 1, we focuses on a single-user multiple-antenna slow
fading wireless system with MTxMR transmit-receive antennas.
The receiver is assumed to have perfect knowledge of H is due to the slowly
varying nature of the channel helps receiver channel estimation. The transmitter
is assumed to have no knowledge of H before the transmission. The MIMO
channel
As soon as the transmission of current message including possible retransmissions
is finished, the next message is encoded and transmitted immediately. It is
assumed that there is an error-free and delay-free ARQ feedback link and there
is a very large buffer of information messages available at the transmitter.
N denotes the maximum allowable ARQ rounds, after ARQ round N retransmissions,
if the receiver still cannot decode the message successfully, no further attempt
will be tried and a decoding failure is declared. The receiver sends messages
to inform the transmitter the successful decoding by ACK and failed decoding
by NACK at each ARQ round, respectively. For the transmission of each message,
the input information message cε
| Fig. 1: | BICM with MIMO-HARQ system model |
| (1) |
The received signal corresponding to the n-th transmission can be expressed as:
and the overall received signal after the n-th ARQ round is:
where,
where,
DEGREES OF FREEDOM IN THE MIMO-HARQ PROTOCOL
In this study, a traditional stop-and-wait Chase Combining (CC) (Chase, 1985) HARQ protocol is considered, which belongs to the category of diversity combining. Upon each retransmission request, the transmitter simply repeats the same packet. Different diversity combining schemes can be used at the receiver, among which the Chase combining, which essentially is a maximum ratio combining of all the received packets in the scalar channel, gives the best performance.
Due to the randomness of the channel matrix H and the maximum ARQ round
N, the successful communication rate R is a random variable. we define two events
for ARQ round n, n = 1...,N {Dn= successful decoding at the
end of ARQ round n} and{
| (2) |
where,C (n)(H (n)eq) is the equivalent channel capacity at ARQ round n, R(n) given by Eq. 1 is the overall communication rate at round n.
Equation 2 leads to
| (3) |
| (4) |
the average rate is:
| (5) |
| (6) |
From Eq. 1, obtain R(0) = ∞ and C(0)
(Heq) = 0 due to
| (7) |
| (8) |
clearly, R(N+1) = 0.
The equivalent channel capacity C (n)(H (n)eq) at ARQ round n is determined by the HARQ protocol and the instantaneous channel matrix H, Thus, the transmit covariance matrices could be variational over retransmissions. Thus, these degrees of freedom shall be exploited in MIMO-HARQ protocol.
DETECTION ERROR PROBABILITY ANALYSIS
Here, the Pair Wise Error Probability (PWEP) (Tarokh et al., 1998) is studied, an error probability analysis of the optimal Alamouti-based HARQ protocol is performed.
Assume that the signal vector s = [s1,...,sK]T
is chosen from a uniformly distributed set
Based upon ML metric
The probability of a decoding error after ARQ round n is:
| (9) |
| (10) |
where, sj is the transmitted vector, s(n) is the detected vector after ARQ round n.
However, the performance analysis of MIMO-HARQ is intractable, which comes
from Eq. 10. Analyzing MIMO-HARQ requires the n-th pairwise
error probability
To simplify the derivation of n-PWEP, define:
Conditioning on H and sj is transmitted, obtain:
where,
With
where, N(n) has Complex Circularly Symmetric Gaussian (CCSG) entries with unit variance. The n-PWEP becomes:
Due to
Due to
and
Hence,
where, using the circularly symmetric property of the i.i.d. complex Gaussian random matrix N, Hence,
| (11) |
where, n(k) and nl–k are independent.
and have
Without loss of generality, for MIMO-HARQ with N, assuming zero mean, the n-dimensional
Q function for a real Gaussian random vector
Hence, The union bound on Pe(n) in Eq. 10 is:
Where,
RESULTS AND DISCUSSION
Alamouti-based HARQ protocol: Consider a Alamouti scheme with ARQ feedback for MIMO i.i.d. Gaussian slow fading channel with maximum ARQ rounds N = 2, which reveals the significance of ARQ feedback in different settings. Let MT = 2, MR = 1, N = 2, the following Alamouti-based HARQ protocol is studied.
The overall codeword of Alamouti code (Alamouti, 1998) after N = 2 transmissions is:
The first transmission is a spatial multiplexing, which is optimal. The traditional use of Alamouti code would transmit X in two channel uses. however, while using ARQ feedback, the first and second column can be separately transmitted. x1 is sent first to try exploiting the channel, the receiver gets y1 = h1s1+h2s2+n1 and jointly decodes (s1, s2) in a Maximum-likelihood manner. when the first decoding attempt fails, a NACK will be sent back to the transmitter asking for the transmission of the second column x2 With both columns transmitted, the receiver can perform the usual Alamouti decoding to recover s1 and s2 in the second decoding attempt.
| Fig. 2: | Packet Error Rate (PER) vs. SNR for a 2x1 MISO Gaussian channel
with ARQ round N = 2. Alamouti and CC are simulated with both uncoded and
BICM coded |
Simulation setting and results: Alamouti-based protocol and the traditional CC repetition scheme are simulated. Both uncoded and coded transmission are assembled with a Gray-coded QPSK constellation on each antenna. The ML decoding is performed for each ARQ rounds, which gives the optimal decoding performance. The encoder is a 1/2-rate binary convolutional code with generators [133, 171]8. The encoder is followed by a block interleaver. The receiver detector performs ML soft-output with bit Log-Likelihood Ratio (LLR) computation concatenated with a bit deinterleaver and convolutional decoder implements a soft-input-soft-output Viterbi algorithm.
Figure 2 shows the performance comparison of Alamouti-based and CC HARQ protocols with ARQ round N = 2 and no-ARQ, with both uncoded and BICM coded transmissions in a 2x1 MISO Gaussian channel. The Packet Error Rate (PER) is a function of the average received SNR. The Alamouti-based protocol is predominant superior to the traditional CC ARQ approach in all settings. Clearly, there is a diversity loss for the CC protocol.
Analysis of gains of HARQ protocols: The gains of Alamouti-based HARQ are not only from packet error rate but also from the receiver soft-output ML detection complexity. Detector orthogonalize the two transmitted Alamouti symbols for independent bit LLR generation, whereas, the CC protocol is of higher complexity due to generated bit LLR is consists of an equivalent 2x2 caused by repetition in MIMO system.
The gains of HARQ protocols without ARQ feedback quite different with different system settings. For the uncoded case, The packet error rate gain of the two simulated HARQ protocols in no-ARQ transmission is distinct, For the coded case, these distinct is as much as negligible.
For example, the decoding error probability of at ARQ N = 2 rund is
CONCLUSIONS AND FUTURE WORK
Hybrid automatic repeat request (HARQ) is an important protocol used in packet transmission to provide reliable data communication. The MIMO systems are also well known to increase the spectral efficiency and the capacity of a communication system. In this study, a MIMO HARQ technique is proposed for slow fading MIMO Channel. The fundamental performance of Hybrid ARQ protocols are studied in a multiple-antenna channel. The new technique exploits both the space-time coding gain of Alamouti STC and the packet combining gain. It retransmits the HARQ packet using an orthogonal Alamouti STC and combined all the received packets. Alamouti-based HARQ design method based on the error probability analysis is also presented. Simulation shows that the Alamouti-based protocol is remarkably superior to the traditional CC ARQ. Note that the technique is valid only in a slow varying channel. Extension to MIMO channel with more than two transmit and receive antennas is under investigation. Note that for the Alamouti-based protocol, it is the assumption that the channel has to keep fixed over different ARQ rounds. For time-varying channels, protocols exploiting both time and spatial diversity should be considered and how to efficiently design the resulting protocol could be challenging problem. For the subjects of next work, there are several potential problems that have not been concerned herein. For MIMO ARQ schemes, the Alamouti STC is proved to be optimal performance, However, there are some other settings in existence unknown codes are close to the optimal performance. Numerical search of optimal codes is still a research topic. Incorporating the receiver lower decoding complexity into the STC HARQ design is under consideration. The Alamouti-based protocol profits from not only wonderful error performance, but also the receiver soft-output ML detection complexity at each ARQ rounds. It is ideal target that the STCs to not only satisfy the Alamouti-based HARQ design method presented herein, but also has the fast-decodable property for decoding in each ARQ round.
ACKNOWLEDGMENT
This Research Project was fully sponsored by the Future Key Technologies R and D Program of Guangdong Province Government of China under Grant No. [2005] 377.