A Cross Layer Optimization Based on Variable-power Amc and Arq for Mimo Systems

To improve spectrum efficiency (SE), the adaptive modulation and coding (AMC) and automatic repeat request (ARQ) scheme have been combined for MIMO systems. In this paper, we add variable power subject to power constraint in each AMC mode. We use KKT optimization algorithm to get the optimal transmit power and AMC mode boundaries. The numerical results show that the average SE is increased by about 0.5 bps/Hz for 2 × 2 MIMO systems with Nakagami fading with parameter m = 2 when SNR is around 15 dB and the ARQ retransmission is twice.


Introduction
The demand for high data rate and quality of service (QoS) in wireless networks requires cross layer approach [1].The adaptive modulation and coding (AMC) are already considered for implementation in many wireless system standards.In order to improve the average spectrum efficiency (SE), a combining constant-power AMC scheme in physical layer and truncated automatic repeat request (ARQ) protocol which provides a trade-off between the average coding rate and the probability of undetected error at the data link layer (DLL) in single-input single-output (SISO) system [2].
In this paper, we add power adaptation proposed in [3] to the constant-power AMC and ARQ in MIMO systems in [4].In the proposed adaptive-power AMC and ARQ in MIMO systems, the power can be changed to increase average SE.The packet error rate (PER) in [2,3] which is much smaller than the target PER, so we are motivated to increase the PER, make PER as close to the target PER as possible.In this way, the switching SNR level of each rate boundary of each rate shift left in the SNR axis and we move to higher mode earlier as using higher order modulation, or higher code rate and the average SE can be improved.However the leftmost part of one each SNR region can have PER exceed limit, so we need adaptive power to compensate it.In [4] the Lagrangian multipliers λ is only one.In the propose method, the Lagrangian multipliers λ is different according to each AMC mode.
Numerical results indicates that the proposed optimization algorithm which combine adaptive power AMC scheme at physical layer and truncated ARQ protocol at data link layer can increase the average SE.
This paper is organized as follows: Section 2 describes the system model.Section 3 presented our proposed scheme with adaptive power.Numerical results are presented in Section 4, and our conclusion is in Section 5.

System Model
We consider a SISO system which combining the AMC scheme with power control at the physical layer and the truncated ARQ module at the data link layer, as shown in Figure 1.
We assume channel gains remain invariant during a packet, but vary from packet to packet.Let n R be the rate of the mode.S denotes the average transmit power, γ denotes the pre-adaptation received SNR which the receiver feed back to the transmitter, and ( ) n S  denotes the allocated power in the AMC mode n.The AMC is performed by dividing the range of the channel SNR into N + 1 non-overlapping consecutive interval, denoted by 1 [ , ), 0,1,..., , SNR range which corresponds to the outage mode.Consider of power adaptation, we modify the PER expression in the mode n [2,3] as follows: The mode dependent parameters { , , }   where t P is the target PER (0.01 usually) in (1).If considering adaptation power, we have two unknown elements, ( ) n S  is the adaptive power inside the mode n and  is the switching SNR to mode n.So we need to solve adaptive power under each mode first, then we can look for the mode-switching SNR values.

Adaptive-Power AMC
Because the AMC algorithm is the same for all SISO sub-channel, so we only discuss the AMC scheme with truncated ARQ protocol in the ith sub-channel in this section.
Here, we propose adaptive-power AMC to find the optimal SNR switching level that maximize SE under the PER constraint.The instantaneous PER is smaller than target PER, ( ) , where t P denotes the target PER, so average PER will lower than target PER, too.We now propose that, ( )   , and add power factor to make the average SE larger.From (1), we can find the power adaptation with PER constraint in mode n is Using (2), we now have the PER constraint as follows We want to find the close form of the above equation, so we first find the close form of the following: (difference from traditional, addition 1/  factor) The constraint in ( 5) is from (3) and ( 4).Using the KKT solution to solve , we got that: ( , , , ) 1 Pr( ) ( ln( ) (Pr ( ) Pr( )) where l n ( ) P r () P r () so that he optimal SNR switching level, * * * 1 2 ( , ,..., )    will larger than the bound point constraint, pn  .The general form of the optimal mode switching levels can be written as ) ln( / ) ln( / ) , 2,..., ( ) we have the optimal mode switching levels, checking the switching levels, ( , ,..., ) satisfy the constraint in (5), and the optimal mode switching levels.Finally, the proposed algorithm is summarized in Figure 2.
Change optimal parameter check the constraint in ( 5

Numerical Results
Here, we present numerical results for average spectral efficiency, power adaptation and PER using the optimal solution.We find the constraint in (5) when Nakagami channel m = 1 is a special case that ( ) Ei x in (4) has no close-form and can't give the value of constraint in (5).So we use Nakagami channel fading parameter with m = 2 instead.We use the parameters of AMC scheme that is listed in Table 1 from [5].In Figure 3, we can see power switching in each mode and the leftmost power in each AMC mode is largest that make the average PER to conform PER constraint in each AMC mode. Figure 2 shows the flowchart of optimal search algorithm.We use a Nakagami fading with parameter m = 2 2 × 2 MIMO channel model, and assume a target PLR of loss P = 0.01.In Figure 4, depict the average spectral efficiency for TM2 AMC scheme in Table 1, and we observed that using the truncated ARQ protocol helps increase the system SE ( ARQ N = 2 vs. ARQ N = 0).Also in  , we alleviate the error bound to improve the average SE, but retransmission the same packet too many times will also degrade the average SE, so we try to balance r N with target PER, such as 1 ARQ N  in 0.01 t P  , and have the maximum average SE in our proposed system.
In Figure 5, we show the different combinations of two sub-channel for different SNRs in bar diagrams to the ARQ.We consider these 8 combinations and map them to integer numbers on x-axis [3] where c i is the i sub-channel.
In Figure 5, we can see the case 2 which have better SE than the other cases.This is consistent with the conclusions of the Figure 4.
Figure 6 compares the PER in [2] and the PER of our proposed scheme.We can see we increase the PER to the maximum (target PER = 0.01) at the left most of the SNR axis.

Conclusions
In this paper, we considered optimized rate and power adaptation at the physical layer aiming at maximizing the average SE, while satisfying a target PER constraint at the data link layer in MIMO channel.We proposed that increasing the PER makes it approach PER constraint, and the optimal mode switching level of each rate will shift to the left in the SNR axis, so we can use the higher order modulation to improve the average SE.The numerical results shows that the adaptation power within each SNR mode can improve the average spectrum efficiency by 0.4 ~ 0.7 (bits/symbol) in mid-SNR range for 2 × 2 MIMO systems, and ARQ 1 ARQ N  is optimal for target PER = 0.01 to have the maximum average SE.We also show that each transmit antenna can have different maximum retransmission number due to independence of the sub-channels.

Figure 4 ,Figure 3 .
Figure 3. Adaptive power in each mode for Table 1 AMC scheme for sub-channel, target PER = 0.01.

Figure 5 .Figure 6 .
Figure 5. Different combinations of two sub-channel for different average SNRs.

Table 1 . Transmission modes in TM2 AMC scheme with convolutional coded M-QAM modulation.
are the Lagrangian multipliers.The optimal