EnergyEfficient Modulation Design for Reliable Communication in Wireless Networks
Abstract
In this paper, we have considered the optimization of the ary quadrature amplitude modulation (MQAM) constellation size to minimize the bit energy consumption under average bit error rate (BER) constraints. In the computation of the energy expenditure, the circuit, transmission, and retransmission energies are taken into account. A combined lognormal shadowing and Rayleigh fading model is employed to model the wireless channel. The link reliabilities and retransmission probabilities are determined through the outage probabilities under lognormal shadowing effects. Both singlehop and multihop transmissions are considered. Through numerical results, the optimal constellation sizes are identified. Several interesting observations with practical implications are made. It is seen that while large constellations are preferred at small transmission distances, constellation size should be decreased as the distance increases. Similar trends are observed in both fixed and variable transmit power scenarios. We have noted that variable power schemes can attain higher energyefficiencies. The analysis of energyefficient modulation design is also conducted in multihop linear networks. In this case, the modulation size and routing paths are jointly optimized, and the analysis of both the bit energy and delay experienced in the linear network is provided.
I Introduction
In wireless communications, the analysis of energy efficiency and reliable packet transmission has attracted much interest recently, spurred by emerging wireless applications, such as in wireless sensor networks, where recharging and/or replacing the batteries of the wireless devices are difficult. In such scenarios, a wireless node can transmit and receive a finite number of bits before its battery runs out. Therefore, minimizing the bitenergy consumption will be a key factor in maximizing the node and, eventually, the network lifetime.
In wireless networks, operation at all levels of the communication protocol stack has an impact on the energy consumption and therefore, energy efficiency has to be addressed in device, physical, link, and network layers jointly. In early work in the networking literature, minimumenergy routing algorithms, which select the paths with the minimum total transmission power over all the nodes, are developed (see e.g., [1], [2]). As described in [3], if the links are assumed to be errorfree and hence there is no need for retransmission, energyefficient routing algorithms choose the minimumhop paths in cases in which the transmitter power is fixed. On the other hand, in variabletransmission power scenarios, the sender nodes dynamically vary their transmission power proportional to , where is the link distance and is the path loss exponent, so that the received signal power is kept fixed to achieve an acceptable signaltonoise ratio (SNR) level at the receiver. In such cases, again when errorfree links are considered, energyefficient routing protocols tend to choose routes with a large number of smallrange hops. We note that the assumption of errorfree links has led to the design of algorithms which consider the energy spent only in a single transmission over each link.
However, in practical wireless channels, transmitted signals are subject to random propagation effects such as shadowing and multipath fading, leading to random fluctuations in the strength of the received signal. As a result of these random variations, reliable reception of the transmitted message signal cannot be guaranteed all the time. In cases in which the received signal power is below a certain threshold required for acceptable performance, outage is said to have occurred and retransmission of the transmitted signal is required. Due to their significant impact on the overall energy consumption, retransmissions, which are generally handled by the link layer, must be considered in the analysis of energyefficient operation. Two types of retransmission schemes, namely endtoend retransmission (EER) and hopbyhop retransmission (HHR), are elaborated in [3] and [4], where the relationship between the link error probabilty of individual links and the overall number of retransmissions needed to ensure reliable packet delivery is established. In [3], the authors designed energyefficient routing algorithms in the network layer by also taking into account the link layer retransmission probabilities and energies.
Clearly, design in the device and physical layers has a significant impact on the energy expenditure. For instance, transmission power depends on the selected modulation type and size, required error probability levels, and assumed channel attenuation models, all of which are mainly physical layer considerations. At the device level, besides the transmission energy, the circuit energy of the microchips performing as radio frequency transceivers, A/D and D/A converters and baseband processors or other application interfaces in the batterydriving nodes is an additional source of energy consumption. The circuit energy, together with the transmission energy, has to be considered in the overall optimization modeling, especially in short range range networks in which the circuit energy is comparable to the transmission energy, and also as the number of retransmissions increases.
So far, several studies have addressed different aspects of energyefficient operation. Optimization of the onehop transmission distance without retransmission mechanism is studied in [5] while optimal routing with retransmission mechanism considering EER and HHR models is investigated [3]. In these studies, physical and device layer considerations are not incorporated in the models in detail. Authors in [6] and [7] consider both circuit and transmission energy consumption and address the optimization of modulation size under energy constraints. However, in these studies, link layer outage events, link reliabilities, and retransmission energies are not considered in the optimization.
In this paper, we provide a holistic approach by studying the energy efficiency considering device, physical, link, and network layers jointly. In particular, we investigate the optimal size of the quadrature amplitude modulation (QAM) constellation that minimizes the energy required to send one bit of information. An accurate energy consumption model is used, in which the total energy equals to the sum of circuit energy (device layer), transmission energy (physical layer), retransmission energy (link layer). Initial results are obtained for singlehop transmissions. Subsequently, results are extended to a simple linear multihop network model for which energyefficient routing paths (network layer) are studied.
The remainder of the paper is organized as follows. Section II introduces the lognormal shadowing effects and link outage probability. Section III presents the accurate energy consumption model and discusses how to compute the average receive power over the Rayleigh fading channel given an average BER constraint in uncoded square MQAM. Section IV presents the linear network model and analyzes the bit energy consumption, delay performance, the optimal routing for a specific MQAM constellation and also for the optimal MQAM constellation.
Ii LogNormal Shadowing and Link Reliability
A combined path loss and lognormal shadowing model is considered as a model for the largescale propagation effects. This has been confirmed empirically to accurately model the variations in the received power in indoor and outdoor radio propagation environments [9][11]. For this model, the ratio of the received power to transmission power in dB is given by
(1) 
where is a Gaussiandistributed random variable with zero mean and variance , modeling random shadowing effects; is a reference distance for the antenna farfield; is the path loss exponent; is a constant that depends on the wave length of the transmitted signal and . In [8], an approximation for the packet error probability has been presented for the lognormal shadowing model. We hereby adopt the same idea to define the outage probability under path loss and shadowing as the probability that falls below a threshold power . This outage probability of a certain link is denoted by and is given by
(2)  
(3) 
where is the Gaussian function [9]. We assume that if outage occurs and hence , then receiver doesn’t successfully receive the packet and the sender needs to retransmit the packet until successful reception is achieved. In our analysis, we adopt the hopbyhop retransmission (HHR) scheme. We employ to calculate the average number of retransmissions in a statistical sense. If we denote as the energy consumption of single transmission over a link between node and node with link error probability , we can derive the average total energy consumption for a reliable packet delivery over this link in the HHR mode as
(4) 
To compute this total energy and the link error probability , we have to specify the parameters in (3). Generally, we assume and employ the empirical results from [10] for and . In the following section, we describe how the threshold power level is specified.
Iii Accurate Energy Consumption Model for BER Constrained MQAM
Usually in energyconstrained networks, the emphasis on minimizing the transmission energy is reasonable in longrange networks , where the transmission energy is dominant in the total energy consumption. On the other hand, in many recently proposed densely distributed networks such as wireless sensor networks, the average distance between nodes could be within 10 to 100 meters, for which the circuit energy consumption is comparable to the transmission energy. In such cases, the circuit energy consumption should be included in the total energy consumption for more accurate analysis.
Iiia Circuit and Transmission Energies
We adopt the accurate energy consumption formulation from [7], [12] and [13]. We assume that the source node has bits in one packet to transmit. One single transmission is composed of three distinct periods: transmission period , transient period , and sleeping period . Accordingly, the total energy required to send bits is represented by
(5)  
where  
(6) 
Above , and are power consumptions for the active mode, sleep mode and transient mode, respectively. Specifically, comprises the transmission power , the amplifier power , the circuit power at the transmitter and the circuit power at the receiver . We assume that the power of the sleep mode is very small and we approximate it as . Now, the bit energy is
(7) 
When we consider uncoded MQAM as our modulation scheme, we can write
(8) 
where is the constellation size defined as in MQAM, and is the symbol duration. Note that in the second equality in (8), the approximation is used. Therefore, in a specific MQAM, if the number of bits and channel bandwidth are given, the active period is consequently derived. Finally, we note that the amplifier efficiency for MQAM can be obtained from where is a function of the MQAM constellation size.
IiiB Average BER Constraint for MQAM
We consider a wireless channel model in which, in addition to the largescale lognormal shadowing effects discussed in Section II, smallscale Rayleigh fading is experienced. First, we note that for squareconstellation MQAM, the symbol error probability as a function of the instantaneous received SNR is given by [9]:
(9) 
where the Gaussian function and its square can be expressed, respectively, as
(10)  
(11) 
The average symbol error probability is computed by averaging the instantaneous error probability over the specific fading distribution as follows
(12) 
Substituting (9)(11) into (12), we can express the average BER for Rayleigh fading approximately as
(13) 
Above, denotes the moment generating function
(14) 
of the fading distribution in the Rayleigh fading channel where
(15) 
For a given BER constraint , we can use (13) to compute the required bit SNR numerically. Then, the minimum average received power that guarantees an average BER of is computed by [9]:
(16) 
where is the noise power spectrum density. Below this power level, the received information packets might undergo unacceptable distortion and the transmission is considered unreliable, which is quantitatively modeled by the outage probability in the lognormal shadowing model in Section II.
Iv Optimal Modulation for SingleHop and MultiHop Transmissions
In this section, we numerically evaluate the optimal QAM constellation size that minimizes the bit energy consumption under different BER constraints, and establish the tradeoffs associated with this analysis. The bit energy is computed by taking into account circuit, transmission, and retransmission energies. Initially, we consider singlehop transmissions and then investigate multihop links.
Iva Optimal MQAM Constellation Size in SingleHop Transmissions
In this section, we study the bit energy consumption in singlehop transmissions where retransmission, circuit energy, uncoded square MQAM and constraint are considered based on equations (3), (4), (7), (13), and (16). We consider two transmit power policies: fixed and variable .

=0.35 

=3.12  
=20000 bits  
=100mW  =98.2mW 
=3.8dB  =112.5mW 
=10KHz  =5 

The values of the set of parameters used in numerical results are provided in Table 1. For instance, the maximum allowable average BER is . In fixed transmit power scheme, we assume =100mW and vary the constellation size at link distance . The bit energy consumption vs. constellation size curves for different transmission distances are given in Figure 1.
We immediately observe that the bit energy increases as the constellation size gets either large or very small, and there exists an optimal value at which the bit energy is minimized. This tradeoff is due to the following. We note that an MQAM with large constellation sends the information at a faster rate decreasing and hence the transmit energy. However, a large constellation requires a larger , leading to a higher outage probability and consequently more retransmissions. On the other hand, a small constellation size requires less retransmissions but increases and hence the transmit energy. Hence, the optimal constellation size should provide a balance between these effects. In Fig. 1, we observe that the optimal is 8 at and the optimal is 6 at . We note that the energyminimizing constellation size gets smaller with increasing distance. In Fig. 2., the bit energy is plotted as a function of transmission distance for different constellation sizes. We see that while large constellation sizes are performing well at small distances, small constellations should be preferred when the distance gets large.
Now, we consider a variable transmit power scheme, where is dynamically adjusted so that the average received power is
(17) 
In this case, it can be easily seen that the link error probability in (3) is always equal to 0.5 independent of the transmission distance. We apply the same configuration to compute the bit energy for different MQAM constellation sizes and transmission distance. Figures 3 and 4 provide the numerical results. Conclusions similar to those for the fixedtransmit power case can be drawn in the variablepower case as well. However, we note that smaller bit energies and hence higher energy efficiency can be attained in the variablepower case as evidenced in the numerical results.
IvB Linear Network Model
In this section, we study multihop scenarios and consider a simple linear network with nodes: source, destination and intermediate nodes equally distributed in between. Each intermediate node can behave as either an active relay or just a sleeping node. The working condition of intermediate nodes could be visualized by a binary code, where the active nodes are denoted as 1 and the sleeping nodes as 0. In this model, totally routing paths are available in the HHR relay transmission scenario. Accordingly, we can either choose a routing path with less intermediate relay nodes and longer hops, or a routing path with more intermediate relay nodes and shorter hops. Note that longer hops tend to have higher , resulting in more retransmissions while more intermediate relay nodes imply a potentially higher circuit energy consumption. Therefore, the selection of the optimal route should strike a balance between these tradeoffs.
In the linear network, we set the distance between the source and destination to 100m, consider 9 intermediate nodes, and choose from the following set: {0.0001, 0.0003, 0.0005, 0.0008, 0.001}. We apply the parameters in Table 1 and employ the method used in Section IV.A for each possible relay link. Given the constellation size and a specific , we search the routing paths to find the optimal route that requires the minimum bit energy consumption defined as:
(18) 
At fixed =100mW scheme, among all possible routings, the optimal routing is found in terms of the minimum bit energy consumption when the packet containing 20000 bits is relayed in HHR mode successfully from the source to the destination via intermediate nodes. Fig. 6 shows the minimum bit energy consumption corresponding to the optimal routing for each square uncoded MQAM under different BER constraints when is fixed. Hence, in the figure, the bit energies for different constellation sizes are the ones required by the optimal route. Again, there exists a certain constellation size that minimizes the bit energy consumption. Moreover, we observe that lower error probabilities expectedly require more energy. Fig. 7 shows the corresponding delay experienced in optimal routing. The delay is formulated as the sum of time intervals consumed over the relay links, whose individual delay is computed as:
(19) 
Fig. 7 is showing that on points which achieve optimal bit energy consumption, the corresponding delay is also minimal. This is due to the fact that the optimal route and modulation size that minimize the energy requirements are favorable in terms of delay as well, because they try to diminish both and the number of retransmissions by finding a balance between competing factors in the fixedtransmitpower case.
Finally we consider the variable scheme, and adopt all previous parameters to repeat the bit energy as well as delay analysis. Figure 8 shows the minimum bit energy consumption corresponding to the optimal routing for each square uncoded MQAM when is variable. Compared to Fig. 6, the variable scheme shows better performance, requiring about 11.5 dBmJoule less than that of the fixed scheme at optimal MQAM points. Fig. 9 plots the optimal bit energydelay curves. Here, we note that it no longer holds that optimal bit energy and optimal delay are achieved at the same point for all BER constraints .
Intuitively, in the fixed scheme, the transmit power would strongly affect . Accordingly, we can also vary as well as constellation size to find the global minimum point under a given constraint. Fig. 10 illustrates optimal bit energy consumption corresponding to the optimal routing as a function of and . The numerical result shows the global minimum is 19.71 for the error rate constraint =0.0001, found when =4 and =25mW.
V Conclusion
In this paper, we have considered the optimization of the MQAM constellation size to minimize the bit energy consumption under average BER constraints. In the computation of the energy expenditure, the circuit, transmission, and retransmission energies are taken into account. A combined lognormal shadowing and Rayleigh fading model is employed to model the wireless channel. The link reliabilities and retransmission probabilities are determined through the outage probabilities under lognormal shadowing effects. Both singlehop and multihop transmissions are considered. Through numerical results, the optimal constellation sizes are identified. Several interesting observations with practical implications are made. It is seen that while large constellations are preferred at small transmission distances, constellation size should be decreased as the distance increases. Similar trends are observed in both fixed and variable transmit power scenarios. We have noted that variable power schemes can attain higher energyefficiencies. The analysis of energyefficient modulation design is also conducted in multihop linear networks. In this case, the modulation size and routing paths are jointly optimized, and the analysis of both the bit energy and delay experienced in the linear network is provided.
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