ARQ protocols for MIMO systems Fatma Zouaidi(*), Hatem Boujemaa(*), Mohamed Siala(**) Higher School of Communications of Tunis, *COSIM, ** MEDIATRON Laboratory,Tunisia
[email protected],
[email protected],
[email protected] Abstract—In this paper, we evaluate the performance of Automatic Repeat reQuest (ARQ) protocols for MIMO (Multiple In Multiple Out) Systems using fixed and adaptive Modulation based on the instantaneous Signal to Noise Ratio (SNR). Transmit Antenna Selection (TAS) or/and Space Time Block Coding (STBC) are used. At the receiver, Maximum Ration Combining (MRC), Selection Combining (SC), Generalized SC (GSC), Minimum Selection GSC (MSGSC) and Maximum Likeliood (ML) detection are used. We also propose a combination of TAS and STBC where STBC is performed over two selected best antennas. We provide an analysis of the throughput performance of ARQ for MIMO systems using different transmitting and receiving techniques. Simulation results are also provided to confirm the accuracy of the theoretical analysis.
I. I NTRODUCTION ARQ protocol is used in order to increase the throughput and to obtain reliable communications for MIMO systems, even in the presence of severe propagation conditions. In this paper, we evaluate the performance of ARQ protocols for MIMO systems using fixed and adaptive modulations. The adaptation of Modulation is based on the instantaneous SNR to increase the throughput of the system. We will focus on evaluating the throughput efficiency of ARQ using different combining techniques such as MRC, SC, GSC, MSGSC with TAS technique. We select the best antenna at the transmitter and different combining techniques are performed at the receiver : TAS/MRC, TAS/SC, TAS/GSC and TAS/MSGSC. This allows to exploit spatial diversity and thus reduces the effects of fading, and increase the throughput of the system.We also propose the combination of TAS and STBC where Alamouti coding is performed over the selected best two antennas.1 The paper is organized as follows. Sections II and III give respectively the performance of ARQ protocols with adaptive and fixed modulation using different combining strategies. Section IV compares theoretical results and simulation ones of different schemes in terms of throughput efficiency(Thr). Finally, section V draws some conclusions and perspectives. II. ARQ PROTOCOLS WITH ADAPTIVE MODULATION In this paper, we assume a frequency non selective block MIMO Rayleigh fading channel using STBC code with nr and nt receiving and transmitting antennas. We consider that the channel is constant over each packet and independent among different transmissions. Selective Repeat protocol is 1 This work was supported by Qatar National Research Fund under Grant NPRP 08-577-2-241.
978-1-4673-1008-6/12/$31.00 2012 IEEE.
used. Each packet corresponds to k information bits and np parity bits are appended. We assume a noiseless feedback MIMO channel over which ACK/NACK messages are transmitted. For a given SNR, the throughput of ARQ is given by : k(1 − P bloc(γ)) (k + np )
T hr(γ) =
(1)
where the factor k/k+np reflects the loss of throughput due to the redundancy bits added by the error detection code and Pbloc (γ) is the block error probability. To increase the throughput, we will adjust the size of the modulation according to instantaneous SNR. The instantaneous throughput of ARQ for MIMO systems with M-QAM modulation is given by[1] k log2 (M ) k + np p) 2(k+n log2 (M ) 2γ3log2 (M ) 1 1 − 2(1 − √ )Q . 2(M − 1) M T hrM −QAM (γ) =
(2)
+∞ 2 where Q(x) = √12Π x exp(− u2 )du The average throughput of ARQ for MIMO systems with adaptive modulation based on the instantaneous SNR given by[1] S1 T hrQP SK (γ)fΓ (γ)dγ T hr(γ) = 0 S2 + T hr16QAM (γ)fΓ (γ)dγ S1 +∞ + T hr64QAM (γ)fΓ (γ)dγ. (3) S2
where fΓ (γ) is the Probability Density Function (PDF)of the SNR Γ fΓ (γ) =
1 (nr nt − 1)!
Eb N0
nr nt γ nr nt −1 e
−
γ Eb N0
.
(4)
where Eb is the transmitted energy per bit and N0 is the Power Spectral Density (PSD) of the channel noise. The instantaneous SNR is compared to some thresholds S1, S2 and S3 as follows : - if γ < S1, then QPSK modulation is used - if S1 < γ < S2, 16-QAM modulation is used - if γ > S3, 64-QAM modulation is used
III. ARQ PROTOCOLS USING FIXED MODULATION AND DIFFERENT COMBINING STRATEGIES
This section presents the MIMO system using TAS at the transmitter and different combining techniques at the receiver : MRC, SC, GSC and MSGSC. In the following, perfect channel knowlege is assumed at the transmitter and receiver.
A. TAS/SC We are interested in a MIMO system using TAS at the transmitter and SC at the receiver. It was originally designed to reduce the receiver complexity while having full spatial diversity. The transmission is made from the best antenna and the reception is done at the best receiving antenna. The throughput of ARQ for TAS/SC using a BPSK modulation is expressed as follows T hT AS−SC =
k T AS−SC (1 − Pbloc ) k + np
(5)
T AS−SC where Pbloc is the average block error probability of ARQ protocol for TAS/SC is +∞ √ k+np T AS−SC 1 − Q( 2x) Pbloc =1− pΓTmax AS−SC (x)dx. 0
(6) is the PDF of Γmax = maxi,j (γij ), γij is The the SNR between the i − th receiving and j − th transmitting antenna. nt nr −1 x x nt nr AS−SC (x) = 1 − e− Γ e− Γ . pΓTmax Γ AS−SC (x) pTΓmax
This expression is found by a simple differentiation of the CDF : nt nr x PΓTmax AS−SC (x) = 1 − e− Γ . (7)
This expression is found by differentiation of CDF nt AS−M RC (x) = PΓT AS−M RC (x) = PΓT AS−M RC (x) . PΓTmax i
1≤i≤nt
(11) r γji is the best SNR given by Γmax = maxi Γi = maxi nj=1 the selected antenna, pΓT AS−M RC (x) and PΓT AS−M RC (x) are i i the PDF and CDF of Γi respectively. Γi follows a chi-square distribution. C. TAS/GSC We are interested in a MIMO system using TAS at the transmitter and GSC at the receiver. The transmission is from the best antenna and the receiver combines the signals received over the best Lc antennas. This technique is characterized by the use of a maximum number of combining antennas Lc < nr . After ranking the SNR for each transmitting antenna, we combine the Lc strongest receiving antennas. Then, the antenna with the strongest SNR is chosen for transmission. In order to simplify the illustration, we assume that the received SNRs are ranked as follows : Γ1i ≥ Γ2i ≥ · · · ≥ Γji ≥ · · · ≥ ΓLc i ≥ · · · ≥ Γnr i . The receiver combines all the strongest Lc channels. When the i-th antenna is activated, the instantaneous SNR can be written as ΓTi AS−GSC =
T hT AS−MRC =
k T AS−MRC (1 − Pbloc ) k + np
(8)
Lc
Γji .
(12)
j=1
As the selected antenna provides the best SNR, we have AS−GSC = max ΓTi AS−GSC ΓTmax 1≤i≤nt
AS−GSC the CDF of ΓTmax is
PΓTmax AS−GSC (x)
=
1≤i≤nt
B. TAS/MRC We are interested in a MIMO system using TAS at the transmitter and MRC at the receiver. The transmission is made from the best antenna and the receiver uses an MRC strategy. The throughput of ARQ for TAS/MRC using a BPSK modulation is expressed as follows
i
=
PΓT AS−GSC (x) i
nt PΓT AS−GSC (x) . i
(13)
(14)
AS−GSC By differentiation, we deduce the expression of ΓTmax PDF: nt −1 pΓTmax AS−GSC (x) = nt p T AS−GSC (x) P T AS−GSC (x) . Γi Γi (15) pΓT AS−GSC and PΓT AS−GSC are the PDF and CDF of Γi i i respectively,they are provided in [3]. Similarly to the previous section, we deduce the block error probability and the throughput efficiency (see (8)-(9)).
T AS−MRC is the block error probability of ARQ where Pbloc protocol for TAS/MRC +∞ √ k+np T AS−MRC T AS−MRC = 1− pΓmax (x)dx.D. TAS/MS-GSC 1 − Q( 2x) Pbloc 0 (9) For each transmitting antenna the MS-GSC receiver begins AS−MRC Where pTΓmax (x) is given by to estimate the SNR of all available antennas and rank from nt −1 best to worst. Then the strongest must be with SNR Γ1i . If pΓTmax AS−M RC (x) = nt p T AS−M RC (x) P T AS−M RC (x) Γ1i > γT , only the strongest antenna will be used, so that Γi Γi (10) the output SNR Γci = Γ1i . If Γ1i < γT , the receiver activate
another receiving antenna to have an output SNR equal to Γ1i + Γ2i . If Γ1i + Γ2i γT , only two branches are combined. j−1 Similarly, if the SNR output of j-1 branch MRC l=1 Γli < j j γT and l=1 Γli > γT , then Γci = l=1 Γli This operation is repeated until the combined SNR is greater than or equal to γT or Lc antennas are activated. Finally, the transmission is made from the best transmitting antenna that has the strongest SNR Γci . When the i-th antenna is activated, the instantaneous SNR can be written as follows
- TAS/STBC with MRC: TAS/STBC is used at the transmitter and MRC at the receiver. We select the best two antennas over which Alamouti coding is performed and the receiver uses an MRC strategy.
ΓTi AS−MSGSC = Γci
- TAS/STBC with MSGSC: TAS/STBC is used at the transmitter and MSGSC is used at the receiver. We select the best two antennas over which Alamouti coding is performed and the receiver uses an MSGSC strategy.
where
⎧ Γ ⎪ ⎪ 1ii ⎪ ⎪ ⎪ ⎨ j=1 Γji
(16)
- TAS/STBC with GSC: TAS/STBC is used at the transmitter and GSC is used at the receiver. We select the best two antennas over which Alamouti coding is performed and the receiver uses an GSC strategy.
if Γ1i ≥ γT i−1 if j=1 Γji < γT i−1 IV. N UMERICAL AND SIMULATION RESULTS Γci = and j=1 Γji ≥ γT ⎪ L L −1 ⎪ c c ⎪ In this section, we provide some theoretical and simulation if ⎪ j=1 Γji j=1 Γji < γT ⎪ ⎩ and Γi1 ≥ Γ2i ≥ · · · ≥ ΓLc i ≥ · · · ≥ Γnr i . results of ARQ protocols for MIMO Rayleigh channels in terms of throughput efficiency and block error probability. Since the selected antenna provides the best SNR, we have For the sake of illustration, we consider the following values for the parameters: AS−MSGSC = max ΓTi AS−MSGSC , (17) Each packet consists of k = 168 bits and np = 20 parity bits ΓTmax 1≤i≤nt are appended for error detection. MIMO nr = 4, nt = 2. T AS−MSGSC is given by The CDF of Γmax figure 1 shows the throughput of ARQ for MIMO (2 × 4), SIM0 (1 × 2) and SISO systems using STBC and a fixed AS−M SGSC (x) = PΓTmax PΓT AS−M SGSC (x) BPSK modualtion. i 1≤i≤nt Since the average number of transmissions tends to 1 when nt = PΓT AS−M SGSC (x) . (18) the SNR goes to infinity, the throughput will increase and i 1, we also see that the goes to k/(k + np ).In figure By differentiation, we deduce the expression of the probability throughput increases as the diversity order nt × nr increases. AS−MSGSC : density function of ΓTmax Finally, we notice that the simulation results agree with the theoretical ones for all values of the SNR. nt −1 AS−M SGSC (x) = nt p T AS−M SGSC (x) P T AS−M SGSC (x) pΓTmax Γi Γi In (figure 2), the instantaneous throughput of ARQ protocol (19) for QPSK, 16-QAM and 64-QAM modulations is plotted where the expressions of PΓT AS−M SGSC (x) and versus the instantaneous SNR. We notice the presence of i pΓT AS−M SGSC (x) are provided in [2] two thresholds in figure 2. The first threshold (S1) is i Similarly to section III.A, we deduce the throughut of the intersection between the curve of throughput of QPSK ARQ using TAS/MS-GSC. modulation and that of 16 QAM, and the second (S2) is the intersection of 16 QAM and 64 QAM. For the system to be adaptive, we must combine the three types E. TAS/STBC of modulation while enjoying the best throughput efficiency In this techniques, we are interested in a MIMO system for each instantaneous SNR, thus we obtain: using TAS/STBC at the transmitter and different combining - If instantaneous SNR < S1: the mobile uses a QPSK techniques at the receiver. We select the best two antennas modulation. over which Alamouti coding is performed and the receiver - If S1
