Link energy minimization for wireless networks

Ad Hoc Networks 10 (2012) 569–585

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Link energy minimization for wireless networks Tianqi Wang ⇑, Wendi Heinzelman, Alireza Seyedi Electrical and Computer Engineering Department, Hajim School of Engineering and Applied Sciences, University of Rochester, Rochester, NY 14627, USA

a r t i c l e

i n f o

Article history: Received 24 April 2011 Received in revised form 28 September 2011 Accepted 8 November 2011 Available online 17 November 2011 Keywords: Energy minimization Fixed average power Fixed average rate Circuit power consumption

a b s t r a c t In this paper, we formalize the problem of minimizing the energy dissipated to successfully transmit a single information bit over a link, considering circuit power consumption, packetization and retransmission overhead, bit/packet error probability, and the duty cycle of the transceiver. We optimize the packet length and transmit power as a function of distance between the transmitter and the receiver for different modulation schemes. We propose a general unconstrained energy consumption model that provides a lower bound on the energy dissipated per information bit. A practical unconstrained physical layer optimization scheme is also provided to illustrate the utilization of the model. Furthermore, minimized energy consumptions of different modulation schemes are compared over an additive white Gaussian noise (AWGN) channel. We extend this general energy consumption model by considering two particular constraints: fixed average power and fixed average rate. We explore the impact of the average power and the information rate constraints on energy consumption and determine the optimum constellation size, packet length, and duty cycle. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction and literature survey In the design of communication systems, often the goal is to minimize the transmit power [1,2]. In recent years, with the advent of battery operated wireless communication nodes operating over small distances, much more attention is being paid to the overall energy consumption. A considerable amount of research has been conducted on prolonging the life-time of wireless networks from the perspective of higher communication layers [3–5]. Literature on the optimization of physical layer parameters to maximize the lifetime is less developed. Youssef-Massaad et al. studied the influence of circuit power consumption and duty cycle on energy consumed per bit from an information-theoretical perspective [6]. Cui et al. studied the energy per information bit minimization problem considering the dependency of circuit power consumption on modulation and coding schemes and the time duration of delivering a packet for different coding schemes [7]. While ⇑ Corresponding author. E-mail address: [email protected] (T. Wang). 1570-8705/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.adhoc.2011.11.002

this was the first important step towards the analytical modeling and analysis of energy minimization at the PHY layer, there are some relevant factors that are not considered. First, the authors do not consider retransmissions due to packet loss. Second, the packetization overhead, i.e., the packet header and the synchronization preamble, is not considered. Third, the target bit error probability is assumed to be fixed. Relaxation of this constraint allows for further reduction in energy consumption. Ammer and Rabaey proposed the energy-per-useful-bit (EPUB) metric to measure the PHY efficiency of wireless networks [8]. The authors conclude that, to minimize EPUB, high data rates, low carrier frequencies, and simple modulation schemes are preferred. However, the energy minimization procedure does not consider practical transceiver power models or retransmissions. The impact of packet size on the performance of wireless networks has also been investigated and proven to be significant [9–12]. In [9], the authors consider the dynamic sizing of the MAC layer frames to improve throughput, range and energy efficiency. Similarly, Yoo et al. jointly optimize the symbol rate, the packet length and the

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constellation size of M-QAM modulation to maximize the throughput of a single user link over AWGN channels in [10]. The throughput is also the optimization objective in [11], where the authors study the general relationship between the link-layer configurations, including the packet length, and the packet-success rate function (PSRF), as well as the impact of this relationship on the achievable throughput. In [12], the authors utilize optimum packet size and error control techniques to improve the energy efficiency of wireless sensor networks. However, the energy consumptions of the retransmission procedure, adaptive power control, and the power consumption of different components in the transceivers are not considered. Reddy et al. investigated the joint optimization of the transmit power and the frame length to improve the energy efficiency of a communication link in wireless sensor networks [13]. The authors concluded that transmit power control is only beneficial within a certain distance range, while at large transmission distances, full power transmission is preferred. The investigation of this paper is experimental and thus specific to a particular device type. Besides the work mentioned above, other contributions have been made to improve the energy efficiency of wireless networks. For example, Wang et al. investigated the energy efficient modulation and MAC for sensor networks with the consideration of the power consumptions of detailed transceiver components as well as the ‘‘start-up’’ energy consumption [14]. Deng et al. studied the optimum transmission range minimizing energy efficiency in ad hoc networks based on node density and node coverage area [15]. In [16], the authors derive a simple distributed optimization scheme, which is an abstracted model without the consideration of detailed channel models, packet structure, and link/MAC layer protocols. Chien et al. designed an adaptive radio to minimize energy consumption by adjusting the frame length, error control schemes, processing gain, and equalization based on channel conditions [17]. While these literature have laid the foundation for our study, a comprehensive, universal and fully extendable energy model and minimization procedure focusing on improving the PHY-level energy efficiency is still missing. Our previous work on the energy minimization problem for reliable transmission can be found in [18], where for a given modulation scheme, path loss, and noise power, we found the optimum transmit energy per symbol by minimizing the energy per successfully received bit (ESB) and find the optimum hop distance by minimizing the energy per successfully received bit per meter (ESBM). We also demonstrated that the modeling methodology can be readily adopted in the scenario when a block Rayleigh fading channel is considered. Although an effective energy efficiency metric and a solid physical layer optimization scheme are proposed in [18], there are noticeable limitations to this work. First, in the reliable transmission scheme, the retransmission cost, both of time and energy, is not considered. Second, the optimal transmit energy per symbol and the optimal hop distance are derived for a given noise power. However, the noise power is usually fixed for a given bandwidth and noise power spectral den-

sity while the transmission distances are often unknown and variable. Therefore, to better facilitate the wireless network optimization, the optimal configurations should be provided with respect to a given transmission distance instead of a given noise power. Third, although the potential impacts of a variable packet length on the energy minimization are briefly described in, the packet length is not considered as an optimization parameter. Furthermore, the work in [18] relies only on numerical calculations and did not exploit the possibility of analytical solutions. Last but not least, only unconstrained optimizations are considered in [18], while in practice communication links are often bounded by many constraints, such as transmission rate requirements and average power consumption limitations. In this paper, we extend our previous work to address these limitations. First, we derive a detailed model of an automatic repeat request (ARQ) based retransmission scheme by including the different energy costs from different stages of retransmissions. Second, we thoroughly investigate the joint influence of transmit power, packet length, modulation and coding schemes on the energy performance of a wireless link, and we provide the optimal configurations with respect to transmission distances. This allows a direct adoption of the optimal configuration in the deployment of a practical wireless network. Furthermore, we derive closed form solutions for both the optimal packet length and the optimal target bit error probabilities for a given transmission distance and M-QAM modulation scheme. Third, we derive an analytical solution for the lower bound on the energy consumption per information bit and the optimal transmit power from an information-theoretical point of view, with the consideration of circuit power consumption and retransmission overhead. Fourth, we study the link energy minimization problem given average power and average rate constraints by further including the optimization over duty cycles. The fixed average power constraint is particularly important in wireless networks that require a predictable lifetime, while the fixed average rate constraint is useful in wireless networks that must provide a certain quality of service (QoS), such as guaranteeing the continuity of a video stream. For both constrained optimization problems, we provide in-depth analyses of the impact of the constraints on the energy cost. Moreover, we derive analytical solutions of the optimal transmit power and duty cycle from an information theoretic perspective in the fixed average power case. In summary, the contribution of this work is three fold: (i) We develop a comprehensive link-level energy consumption model that includes transmit power, circuit power, retransmission overhead, packetization and duty cycle. (ii) The energy minimization procedure is highly parameterized. That is, it is straightforward to adapt the proposed energy consumption model and the energy minimization procedure to transceivers with different circuitries and different channel models, such as Rayleigh and Ricean fading channels. (iii) In some cases we derive analytical solutions which provide us with further insight into how different factors effect the optimal solution. The remainder of this paper is organized as follows. Section 2 describes the packet structure and transceiver model. In Section 3 we propose a general unconstrained

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T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585 Table 1 Notations. Notation

Meaning

B d Pb N0

Signal bandwidth Transmission distance Bit error probability Noise power spectral density Bandwidth efficiency Signal-to-noise ratio Path loss Coding gain Coding rate Received signal power Transmit signal power Circuit power consumption (including both transmitter and receiver) Retransmission overhead Duty cycle Number of information bits per packet Energy consumption per information bit

g c G Gc Rc Pr Pt Pc

K U LL Eb

energy consumption model and utilize this model to minimize the energy consumption per information bit over target bit error probability and packet length for a specific modulation and coding scheme. Sections 4 and 5 investigate the energy consumption minimization for fixed average power and fixed average rate cases, respectively. Results are presented in Sections 6 and 7 concludes the paper. The notation used in this paper is summarized in Table 1. 2. System and signal model This section introduces the packet structure, transmitter/receiver structure, and automatic repeat request (ARQ) scheme that lay the foundation for analyzing the impacts of packetization, circuit power, and retransmissions on energy consumption. 2.1. Packet structure The packet structure considered in this paper is shown in Fig. 1. It consists of four components: payload, upper layer header, PHY/MAC-header, and preamble. We assume that there are LL bits in the payload of each packet. The upper layer header contains the control information added by the upper layers, namely routing information, packet ID, etc. We assume there are LUH bits in the upper layer header. From the view of the PHY and MAC layers, the payload and the upper layer header are indistinguishable. Therefore, the payload and the upper layer header are modulated and coded similarly. Conversely, PHY and MAC headers are modulated using a predefined modulation scheme, such as BPSK for an uncoded system and coded BPSK for a coded system. This is

Payload

TL

because the PHY and MAC headers carry important control information, such as information regarding modulation and coding for the payload and the upper layer header. Therefore, the modulation scheme of the PHY/MAC-header has to be robust and known to the receiver a priori, so that the receiver can always demodulate the received PHY/ MAC-header, no matter what modulation scheme the payload and upper layer header use. Finally, the preamble is a predefined sequence that serves the purpose of synchronization, automatic gain control (AGC), etc. Moreover, we assume that the transmit power is constant during the entire packet. A summary of the length and duration parameters for these components are provided in Table 2. 2.2. Transceiver model At the transmitter, energy consumption consists of the transmitted energy and the energy consumed in the circuits. At the receiver, the only energy consumption is that of the circuitry. To facilitate the analysis of the energy consumption, we assume generic transmitter and receiver models as shown in Figs. 2 and 3. 2.2.1. Transmitter As shown in Fig. 2, the major energy consuming components at the transmitter are the digital-to-analog converter (DAC), low pass filter (LPF), bandpass filter (BPF), mixer, frequency synthesizer and power amplifier (PA). In this paper, the power consumption of the LPF, BPF, mixer, and frequency synthesizer are viewed as constants, while the power consumption of the DAC follows the model in [7]. The power consumption of the power amplifier can be expressed as

Pamp ¼ bPt ;

ð1Þ

where Pt is the transmission power and b ¼ qe  1; e is the peak-to-average ratio, and q is the drain efficiency of the power amplifier. Note that e and q are both determined by the modulation scheme. 2.2.2. Receiver As shown in Fig. 3, the major energy consuming components at the receiver are the analog-to-digital converter Table 2 Packet structure parameters. Component

Length (bits)

Duration (s)

Modulation

Payload Upper layer header PHY/MAC header Preamble

LL LUH LH –

TL TUH TH TP

Adaptive Adaptive BPSK/coded BPSK –

Upper Layer Header

TUH Fig. 1. Packet structure.

PHY/MAC Header

TH

Preamble

TP

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Channel Encoder

Power Amplifier

Fig. 2. A typical transmitter structure using linear modulation.

Fig. 3. A typical receiver structure using linear demodulation.

(ADC), low pass filter (LPF), low noise amplifier (LNA), mixer, frequency synthesizer, and decoder. In this paper, the power consumption of the LPF, LNA, mixer, and frequency synthesizer are viewed as constants. The power consumptions of the ADC and Viterbi decoder follow the models in [7]. The power consumption of the circuit components of the transmitter (excluding the power amplifier) and the receiver is defined as

Pc ¼ 2Pmixer þ 2P syn þ P filter þ PDAC þ PLNA þ P ADC þ Pv ;

ð2Þ

where Pmixer,Psyn, Pfilter and PLNA are the power consumptions of the mixers, frequency synthesizer, filters, and

Table 3 Power consumption values.

Transmitter Pct Receiver Pcr

Pfilter (mW)

Pmixer (mW)

Pamp

PLNA (mW)

Psyn (mW)

2.5 2.5

30.3 30.3

bPt –

– 20

50 50

LNA, respectively. The above power consumptions are assumed to be constant. The values for these parameters are chosen based on typical implementations, as shown in Table 3 [7]. PDAC and PADC represent the power consumption of the DAC and the ADC, respectively. Pv is the power consumption of the Viterbi decoder. These power consumptions can be determined using the expressions in [7].

2.3. Automatic repeat request sessions In this paper, ARQ is assumed as the link-layer protocol, that is retransmissions are required when any bit error is detected. Considering retransmission, the procedure for successfully transmitting/receiving one packet is shown in Fig. 4. We assume that before transmission or reception of a packet, the transmitter and receiver will spend Ttr seconds to go from the off (sleep) state to an on (active) state. Also, for a given implementation, the time period to start up the frequency synthesizer, Ttr, is assumed to be fixed. TIPS denotes the inter packet space (IPS). Ton is the time duration for the transmission of one packet. Ton = (TL + TUH + TH)/

(m-1)

Ttr

Ton

TIPS TACK TIPS

Ton

TIPS TACK TIPS Ttr

Etr

Etx

EIPS ELN EIPS

Etx

EIPS EACK EIPS Etr

Ttr

Ton

TIPS TACK TIPS

Ton

TIPS TACK TIPS Ttr

Etr

Erx

EIPS EIPS EIPS

Erx

EIPS EtxACK EIPS Etr

Fig. 4. The transmission and reception of one packet using m total transmissions.

T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585

Rc + Tp, where Rc is the channel code rate and is set to 1 for the uncoded case. TACK is the time period when the transmitter listens for an acknowledgement. We set T ACK ¼ LH þ TP. BRc Assume that to successfully deliver one packet, the total number of transmissions is m. In the first m  1 transmissions, the energy consumption during the TACK period at the transmitter is denoted by ELN. In the last delivery, the energy consumption during the TACK period at the transmitter is EACK. EACK is the energy consumption of transmittx ting the acknowledgement after receiving the mth packet. We assume that during the inter-frame space, TIPS, only the frequency synthesizer contributes to the energy consumption denoted by EIPS. Also, we assume that in the first (m  1) TACK periods, the energy consumption at the receiver is EIPS as well, since no ACK is transmitted. Etr is the energy consumption during the transient mode, Etx is the energy consumption at the transmitter to transmit one packet, and Erx is the energy consumption at the receiver to receive one packet. The detailed expressions of the energy consumptions above will be defined in the following section.

the ARQ scheme and packet structure considered in this paper, the lower bound on Eb in (4) must be modified as

Eb P

A classic model of the energy consumption per information bit when communicating at rate R is [19]

Eb ¼

Pt ; R

ð3Þ

where Pt is the transmit power and R is the information rate. However, to obtain a more refined model, we need to at least consider the following factors:  the circuit power consumption;  the reduction in information rate by packetization, duty cycle mode, and ARQ;  the increase in power consumption caused by overhead. We now provide a revised energy consumption per information bit model considering the above factors. First, considering the impact of circuit power consumption, the lower bound on energy consumption per information bit can be modeled as

Eb P

Pt ð1 þ bÞ þ Pc  ; Pt B log 1 þ 2GBN 0

ð4Þ

where Pt/G is the received signal power after path loss G, and 2B N0 is the total noise power within bandwidth B. Since B log (1 + Pt/2GBN0) is the channel capacity, it represents the maximum possible information rate and hence (4) provides a lower bound on the energy consumption per information bit. We assume that Pt/Pc = a. Furthermore we modify the initial model in (4) according to the reduction in information rate that comes from overhead in a real system. For example, packetization overhead, retransmission overhead, and duty cycle must be included. To incorporate

ð1 þ a þ abÞPc  ; aPc KB log 1 þ 2GBN 0

ð5Þ

where K represents the overhead induced by the link-layer protocol and the frame structure. For instance, inheriting previous assumptions, K can be defined as

K¼

LL   : aPc B log 1 þ 2GBN ð2T IPS þ T ACK þ T P Þ þ LL þ LUH 0

ð6Þ

It is straightforward to extend the model in (5) to include the duty cycle of wireless transceivers, since the the duty cycling can be viewed as a direct reduction in the information rate. If the transmitter/receiver only works at U fraction of the total time (we refer to U as the duty cycle), the information rate then becomes

  aPc : R ¼ UKB log 1 þ 2GBN 0

ð7Þ

The duty cycle does not affect Eb in the unconstrained case we consider here, since

3. Unconstrained energy minimization 3.1. Lower bound on energy consumption per information bit

573

Eb P

ð1 þ a þ abÞPc  ¼  ; aPc aPc UKB log 1 þ 2GBN0 KB log 1 þ 2GBN 0

U½ð1 þ bÞPt þ P c 

ð8Þ

which is the same as the model expressed in Eq. (5). However, the duty cycle mode will have a major influence in some constrained situations, such as fixed average power transmission, as discussed in Section 4. 3.2. Minimization of energy consumption with practical modulation and coding schemes The information rate in practice is much lower than the bound provided by the capacity B log (1 + Pt/2GBN0), since the information rate is reduced by the imperfections of coding, packetization overheads, etc. Therefore, to use the general Eb model in (4) in practice, we need to obtain the realistic information rate and use this as the denominator in (4). Therefore, we modify the model of Eb rom (4) to adopt practical information rates as follows: t þP c Eb ¼ ð1þbÞP ; R

¼ N ½ð1þbÞPLtLþPc T on ;

ð9Þ

where Ton is the time duration to transmit one packet, LL is the number of information bits in one packet, and N is the total number of transmissions needed to successfully deliver one packet. That is, the energy consumption per information bit can be equivalently expressed as the energy consumption per transmitting/receiving one packet multiplied by the average number of retransmissions required to successfully deliver the packet divided by the number of information bits contained in the packet. 3.2.1. Total number of retransmissions We assume that there are no errors in the PHY/MACheader. This assumption is reasonable because the robust modulation schemes used by the PHY/MAC-header ensure

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that errors rarely occur in the PHY/MAC-header. Also, we assume that whenever there is a bit error in the received packet, a retransmission is required. For a packet containing LL information bits, the probability of a packet error is

Ppe ¼ 1  ð1  Pb ÞLL þLUH :

1 ð1  Pb ÞLL þLUH

:

ð11Þ

2  3 c e 2ðM1Þ : log2 M

ð12Þ

Then, the SNR–BER relation in M-QAM modulation is

2 2 c ¼ ð2b  1Þ ln ; 3 bPb

ð13Þ

where b = log2 M. Also, based on the signal propagation model, we have Pt = G Pr, where G represents the path loss, whose decibel value is determined by

GðdBÞ ¼ G1ðdBÞ þ 10klog10 d þ LMðdBÞ ;

¼ 2BN0 Gf ðPb Þ=Gc ;

ð15Þ

where Gc denotes the coding gain and Gc = 1 for uncoded modulation. The ARQ procedure has been shown in Fig. 4. The energy consumption during each session is summarized as follows:

EIPS ¼ Psyn T IPS ;

ð18Þ

Er ðmÞ ¼ ð3EIPS þ Erx Þm þ 2Etr þ EACK tx : Consequently, to successfully deliver a packet, the average energy consumption is

E¼

1 X

½Et ðiÞ þ Er ðiÞPrfm ¼ ig;

ð19Þ

i¼1

where m is the number of transmissions and Pr{m = i} denotes the probability that the number of transmissions equals i, which is given by Prfm ¼ ig ¼ Pi1 pe ð1  P pe Þ. After simplification, we have

E ¼ ð2EIPS þ Etx þ ELN ÞN þ 2Etr þ Pv T ACK þ ð3EIPS þ Erx ÞN þ 2Etr þ EACK tx :

ð20Þ

From previous analysis, E is a function of target bit error probability Pb and packet length LL. Thus, the minimization of Eb can be conducted over LL and Pb. 3.2.3. Minimization of energy consumption per information bit Each packet contains LL information bits. Therefore, the average energy consumption per information bit is

Ebit ¼

E : LL

ð21Þ

To minimize Ebit with respect to LL, we set which gives us

A1 L2L þ B1 LL þ C 1 ¼ 0;

@Ebit @LL

¼ 0,

ð22Þ

where

A1 ¼ PonBgPb ;   on LH þ PBongLRUH ; B1 ¼ Pb 5EIPS þ ELN þ Pon T p þ PBR c c  C 1 ¼  5EIPS þ ELN þ 4Etr þ EACK tx þ P v T ACK  P on LH P on LUH þPon T p þ BRc þ BgRc ;

ð23Þ

Solving (22) yields the optimum number of information bits per packet, LL

ELN ¼ ðP cr  Pv ÞT ACK ;

EACK tx

Et ðmÞ ¼ ð2EIPS þ Etx þ ELN Þðm  1Þ þ 2Etr

Pon ¼ 2ð1 þ bÞBN0 Gc=Gc þ Pc :

Etr ¼ Psyn T tr ;

EACK ¼ P cr T ACK ; Etx ¼ ½2ð1 þ bÞBN 0 Gc=Gc þ Pct T on ;

That is, Ton is a function of packet length. The total transmit and receive energy consumptions of m deliveries are

ð14Þ

where G1 = 30 dB is the reference path loss at 1 m, k = 3.5 is the path loss constant, and LM = 40 dB is the link margin [7]. Therefore, the transmit power can be eventually denoted as

Pt ¼ 2BN 0 Gc=Gc ;

ð17Þ

¼ ðLL þ LUH Þ=ðRc gBÞ þ LH =Rc B þ T p :

þ2EIPS þ Etx þ EACK :

3.2.2. Average energy consumption per packet Since the circuit power Pc is fixed, we only need to find the transmit power Pt. The transmit power can be determined from the SNR c at the receiver and the desired bit error probability Pb. The SNR per symbol is defined as c = Pr/(2B N0), where Pr is the received power, B is the signal bandwidth, and N0 is the spectral power density of the white Gaussian noise. The function relating c to Pb of M-QAM modulations over AWGN channel is well defined as [1]

Pb 

T on ¼ ðT L þ T UH þ T H Þ=Rc þ T p ;

ð10Þ

The expected total number of transmissions to successfully deliver one packet is

N¼

and the receiver, respectively. Pv is the power consumption of the Viterbi decoder, the value of which can be calculated from [20]. Moreover, we have

ð16Þ

¼ ½2ð1 þ bÞBN0 Gc=Gc þ P ct T ACK ;

Erx ¼ Pcr T on : In the above equations, Pct and Pcr represent the power consumption of the circuits components of the transmitter

LL

¼

B1 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B21  4A1 C 1 2A1

:

ð24Þ

Correspondingly, the optimum target Pb can be found bit by solving @E j  ¼ 0. When the M-QAM family is used, @Pb LL the corresponding approximate closed-form solution of the optimum target Pb can be derived as

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Pb 

1  ; IPS þELN 1 þ ðLL þ LUH Þ lnð2bÞ þ 10 þ Pc T2onð2þ5E b 1ÞA 3

ð25Þ

1

: 1 þ ðLL þ LUH Þ lnð2bÞ þ 10

ð27Þ

Then, Eb in (5) can be expressed as

2

0 GT on where A2 ¼ 2ð1þbÞBN . Gc When transmission distance d is large, A2 approaches infinity and Eq. (25) becomes

Pb 

U½ð1 þ bÞPt þ Pc  ¼ P:

Eb P

ð1 þ bÞPt þ P c  ; Pt KB log 1 þ 2GBN 0

¼ ð26Þ

Therefore, the target bit error probability will eventually converge to a value solely determined by the packet length and the modulation scheme. The optimum target bit error probabilities of other modulation schemes and their corresponding convergence values can be obtained similarly. Furthermore, Eq. (24) reveals a one-to-one relation between Pb and LL at any given distance. Thus, as P b converges, LL will also converge for higher transmission distances. The analysis and calculation results for this model will be shown and discussed in detail in Section 6. The closed-form equations in (24) and (25) can greatly expedite the calculation of the optimal transmission configurations, since they reduce the search dimension by one. Moreover, in certain applications, e.g., when the target bit error probability is fixed or the packet length is fixed, the corresponding LL and Pb can be immediately calculated through (24) and (25). The proposed model only considers a single-link without consideration for the Multiple Access Interference (MAI) that may be caused by multiple links transmitting simultaneously. We assume that the MAI is handled by the Medium Access Control (MAC) layer after the individual links have been optimized using our approach. Also, to obtain the closed-form solutions in (24)–(26), we consider an ARQ-based retransmission scheme at the link layer, and we only consider modulation schemes with closed-form relations relationships between the signalto-noise ratio (SNR) and the bit error probability (Pb). Furthermore, the approximations used to obtain Eqs. (25) and (26) are accurate when the signal–noise-ratio (SNR) is high. Moreover, the derived lower bound can only be found when the wireless channels of interest have known capacity-power formula, such as AWGN channels and Rayleigh fading channels. So far, we have discussed the minimization of the energy consumption per information bit in an unconstrained framework. However, in practice, different constraints may apply, among which the most common ones are an average power constraint and an average rate constraint. In the following sections, we will study how these two constraints affect the minimization of energy consumption.

4. Energy minimization with a fixed average power constraint In this section, we consider bursty transmissions with a strict average power constraint. By ‘‘bursty transmission’’ we mean that the transceiver only transmits/receives for a fraction of time (duty cycle). Assume that the average power constraint is



ð28Þ

P

P=UPc UKB log 1 þ 2ð1þbÞGBN 0

:

For a given P, the minimization of Eb is equivalent to the   P=UP c . Using the maximization of R ¼ UKB log 1 þ 2ð1þbÞGBN 0 expression of K given in (6), we have the following optimization problem

max R ¼

ULL ; L þLUH 2T IPS þ T ACK þ T P þ B logð1þðP=ULP Þ=ð1þbÞG2BN Þ c

s:t:

0

U½ð1 þ bÞPt þ Pc  ¼ P; 0 6 U 6 1; ð29Þ

where LL represents the number of information bits contained in a packet. It is obvious that to achieve the maximum information rate R, we should have LL ? 1. However, in practice, the packet length is always finite and overheads are inevitable. Eq. (29) implies that overheads of both packetization and ARQ decrease the maximum possible information rate from the ideal information rate. For a given modulation scheme with bandwidth efficiency g, the channel capacity of a wireless channel with bandwidth B is limited to Bg. Moreover, if we consider the influence of finite packet length and channel distortion, the information rate can be further specified as

max R

¼ gðU; LL ; gÞPpc ¼

s:t:

ULL ð1  Pb ÞLL þLUH ; UH 2T IPS þ T ACK þ T P þ LL þL Bg

ð30Þ

U½ð1 þ bÞPt þ Pc  ¼ P; 0 6 U 6 1;

where gðU; LL ; gÞ ¼

ULL

LL þLUH Bg

2T IPS þT ACK þT P þ

which is an increasing

function of both LL and U for a given g, and Ppc is the packet-correctly-received probability. For a given distance, P and Pc, Pb is a monotonically increasing function of U, since large U implies small Pt. This makes Ppc ¼ ð1  P b ÞLL þLUH a decreasing function of U. In addition, Ppc is also a decreasing function of LL, since the larger the packet is, the greater the possibility of an error in the packet. Therefore, there  exists an optimum combination of LL ; U that balances g(U, LL, g) and Ppc and thereby maximizes R. This idea can be further explained by the following example. Take M-QAM using fixed average power as an example. The expression of Pb in this case is

Pb 

3ðP=UPc Þ=ð1þbÞ2GBN 0 2 2ðM1Þ : e log2 M

ð31Þ

Also, we have bandwidth efficiency g = log2 M in this case. Consequently, (30) becomes

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T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585

max R ¼ s:t:

ULL ð1  Pb ÞLL þLUH LL þLUH 2T IPS þ T ACK þ T P þ Blog M 2

0 6 U 6 1; 3ðP=UPc Þ=ð1þbÞ2GBN 0  2ðM1Þ

Pb ¼ log2 M e 2

c U min Eb ¼ ½Pt ð1þbÞþP ; R

s:t: ð32Þ

where R ¼

:

4.1. The influence of fixed average power The imposed average power constraint plays an important role on the resulting minimum energy consumption per bit. That is, the given average power constraint determines how close the resulting Eb is to the unconstrained global minimum value of Eb. In this subsection, we investigate the influence of the average power constraint and find the condition under which the fixed average power constrained model could give the same global minimum Eb as in the unconstrained case. The minimization model can be rewritten as LL þLUH P 2T IPS þ T ACK þ T P þ B logð1þðP=UPc Þ=ð1þbÞ2GBN0 Þ ; U LL   LL þLUH ðPt ð1 þ bÞ þ Pc Þ 2T IPS þ T ACK þ T P þ B log ð1þP t =2GBN 0 Þ

min Eb ¼

s:t:

ULL 2T IPS þT ACK þT P þ

LL þLUH Bg

ð35Þ ð1  Pb ÞLL þLUH , as defined previ-

ously, Rconst is the desired information rate and DR is the al-

The above minimization problem is readily solvable through numerical methods. The results are presented in Section 6 and explained in detail.

¼

jR  Rconst j 6 DR ;

LL

P ¼

½P t ð1þbÞþPc U ; Rconst  ½P t ð1 þ bÞ þ

Pc U :

ð36Þ

As with the fixed average power constraint, the minimization of energy consumption per information bit with the fixed average rate constraint is not the overall minimum Eb unless the target information rate is the overall optimal rate that minimizes energy consumption in the unconstrained case. 6. Numerical results

ð33Þ This model can be used to describe any modulation scheme. The minimum possible Eb should be achieved over all possible Pt > 0. However, the constraints in (33) indicate that

0 6 U 6 1 ) ðP  Pc Þ=ð1 þ bÞ 6 Pt < 1:

Eb ¼

;

UðPt ð1 þ bÞ þ Pc Þ ¼ P; 0 6 U 6 1:

UðPt ð1 þ bÞ þ Pc Þ ¼ P ) Pt ¼ ðUP  Pc Þ=ð1 þ bÞ;

lowed information rate deviation, since the deviation is unavoidable in practice due to unpredictable circumstances (e.g., processing delays from upper layers). Moreover, since an average power constraint no longer exists, the one-to-one relationship between U,a and Pt disappears. This minimization problem thus must be conducted over all possible LL,U,M, and Pt. The resulting average power and minimum Eb will be

ð34Þ

Thus, P t 2 ½ðP  Pc Þ=ð1 þ bÞ; 1Þ. To ensure that Pt is a nonnegative value, we have that P  Pc 6 0. That is, when P 6 P c , the fixed-power transmission can achieve the same minimum Eb as that of the non-constrained transmission presented in Section 3. This is because when P 6 P c ; Pt can be any nonnegative value be between [0, 1). We have discussed the energy minimization problem with fixed average power constraint. Additionally, we have investigated the impacts of the average power constraint and duty cycle on the energy consumption. 5. Energy minimization with a fixed average rate constraint In some cases, such as a sustainable video stream, a stable node-to-node throughput is desired so that a certain quality-of-service (QoS) can be guaranteed. The above analysis can be readily adapted to fixed average rate situations. The minimization of energy consumption per information bit under the fixed average rate constraint for a given modulation scheme can be modeled as

In this section, we present numerical results that verify the previous analysis and provide insight into the performance of the different optimization frameworks. We assume a bandwidth of B = 10 KHz, LUH = 160 bits, LH = 32 bits, TP = 20 ms, coding rate Rc = 1/2, and coding gain Gc = 6.47. The transient period for the transceiver is set to Ttr = 5 ls. The inter packet space TIPS = 5 ms. The power consumption values are shown in Table 3, which gives us the total circuit power consumption at the transmitter and receiver as Pc = Pct + Pcr = 0.2884 W. The power amplifier coefficient b = 0.35. The constrained optimization problems in Sections 4 and 5 are not convex. Thus, the well-developed optimization algorithms and analysis tools in convex optimization theory, such as the interior-point algorithms, cannot be used to simplify the calculation of the optimization problem [21]. Moreover, inspection of the Karush Kuhn Tucker (KKT) conditions of the constrained problems reveals that the KKT conditions do not provide any advantage over the trivial approach of exhaustive search. For instance, consider the problem formulated in (29). The KKT approach requires us to find all solutions of @R/@/ = 0, then compare the values of R for these values as well as for the boundary points / = 0, and / = 1. Unfortunately, oR/@/ = 0 does not have an analytical solution. Thus one needs to resort to an exhaustive search on [0, 1] to find all solutions. Therefore, the KKT approach would still require an exhaustive search on [0, 1]. In our work, the optimization is implemented using an exhaustive search. This approach is sufficient in practice, as the optimization can be performed and the optimum configurations determined off-line before network deployment. Then a simple look-up table can provide the optimal parameters depending on the existing conditions.

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6.1. Unconstrained energy minimization

The size of the look-up table is fairly small for practical implementations. The look-up table, for instance, can contain the optimal modulation schemes, the optimal target bit error probability, and the optimal packet length. When there are eight candidate modulation schemes, 3 bits are required to denote the modulation schemes. If double precision binary floating point data type is used to represent the optimal target bit error probability and the optimal packet length, 128 bits are required. If the transmission distance ranges from 1 meter to 120 m with a resolution of 1 meter, then a total of (128 + 3)  120 = 15,720 bits 2 KBytes are needed for the look-up table. A typical wireless sensor has sufficient memory to implement this look-up table. For example, the TMote sky node has 48 KBytes memory onboard [22].

In this subsection, we compare the unconstrained energy consumption per information bit lower bound in (4) with the practically minimized energy consumption per information bit (21). In the practical scheme, the modulations considered are confined to coded and uncoded BPSK, QPSK, 16-QAM and 64-QAM. Fig. 5 shows a comparison of the practical minimum Eb of different modulation schemes and the theoretical lower bound of Eb, which is given by (4). As transmission distance increases, the total energy consumption per information bit increases. This is mainly caused by the increasing transmitted energy. As shown in Fig. 5, uncoded 64-QAM, uncoded 16-QAM, uncoded QPSK, and coded QPSK are

1010

15 BPSK QPSK 16QAM 64QAM Coded BPSK Coded QPSK Coded 16QAM Coded 64QAM Theoretical lower bound

5

b

E* (dBmJ)

0

BPSK QPSK 16QAM 64QAM Coded BPSK Coded QPSK Coded 16QAM Coded 64QAM

109 * LL (in bits), with LUH = 160 bits

10

−5 −10 −15 −20

108

107

106

105

104 −25 −30

103 1

10

20

30

40

50

60

70

80

90

1

100 110 120

10

20

30

40

50

70

80

90

100 110 120

 Fig. 7. Optimized packet length versus transmission distance P b ¼ P b .

Fig. 5. The minimum Eb of different modulation schemes versus distance compared with the theoretical lower bound of Eb.

10−4

10−6

10−8

*

Pb

10−10

10−12 BPSK QPSK 16QAM 64QAM Coded BPSK Coded QPSK Coded 16QAM Coded 64QAM

10−14

10−16

10−18

60

Distance (m)

Distance (m)

1

10

20

30

40

50

60

70

80

90

100 110 120

Distance (m)  Fig. 6. Optimized target bit error probability versus transmission distance LL ¼ LL .

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preferred for very short, short, medium and long distances, respectively. This observation is justified by noting the fact that at short distances, the energy consumption is dominated by that of the circuitry. Consequently, bandwidth efficient modulation schemes that lead to shorter on time will have an advantage. On the other hand, at longer distances, the energy consumption is dominated by the transmitted energy. Hence, modulation and coding schemes that require lower SNR will have an advantage. The curve labeled ‘‘Theoretical lower bound’’ in Fig. 5 is obtained by numerically finding the minimum Eb from (4). Fig. 6 presents Pb at different transmission distances. As the transmission distance increases, Pb will increase as well. This is because, as transmission distance increases, a higher target Pb is preferred lest the transmission energy increase dramatically to mitigate the path loss. Moreover, as transmission distance increases, a flattening of Pb can be observed, which is consistent with (26). Fig. 7 depicts LL at different transmission distances. LL decreases as transmission distance increases and con-

verges to a certain value at large transmission distances. Recall that P b increases as d increases, which gives rise to a higher retransmission probability. Therefore, to reduce the retransmission cost, a shorter packet length is preferred. Also, the convergence of LL occurs at large transmission distances as P b flattens. Fig. 8 shows the optimum transmit power, P t , that minimizes Eb. The theoretical and practical P t s have the same trend. However, the theoretical P t curve is smooth, while the practical Pt curve exhibits a saw-toothed shape. This irregularity is caused by the limitations of using a discrete modulation and the packetization parameters used in the calculations for the practical model. Fig. 9 compares the optimum rate, R⁄, obtained through (32) and the optimum information rate from (7). We can tell that the framework from (32) provides an upper bound for the optimum information rate. Also, the staircase type curve of the calculated optimum information rate is caused by the discrete nature of modulation used in the numerical calculations. 5

2

0.7 Calculation results Theoretical results

x 10

R (eq. (9)) R’ (Packetized, eq. (26)) Maximum R’ (MQAM) M=2 M=4 M=8 M =16 M = 32 M = 64 M =128 M = 256

1.8

0.6

0.5

Information rate (bps)

Optimized transmit power

1.6

0.4

0.3

0.2

1.4 1.2

R

R’

Maximum R’

1 M = 256 M = 128 M = 64 M = 32 M = 16 M=8 M=4 M=2

0.8 0.6 0.4

0.1 0.2 0

0 1

10

20

30

40

50

60

70

80

90

0

100 110 120

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1

Fig. 10. The information rate at distance d = 1 m.

Fig. 8. The optimized transmit powers versus transmission distance (theoretical and practical). 4

250 Calculation results Theoretical results

x 104 R (eq. (9)) R’ (Packetized, eq. (26)) Maximum R’ (MQAM) M=2 M=4 M=8 M =16 M = 32 M = 64 M =128 M = 256

3.5

200

3

Information rate (bps)

Optimized information rate (kbps)

0.5

Φ

Distance (m)

150

100

2.5

R

R’

2 Maximum R’ 1.5 M = 256

M=2

1 50 0.5 0 1

10

20

30

40

50

60

70

80

90

100 110 120

Distance (m) Fig. 9. The optimized information rate versus transmission distance.

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Φ Fig. 11. The information rate at distance d = 40 m.

0.9

1

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T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585 4

2.5

x 10

Information rate (bps)

2

1.5

R (eq. (9)) R’ (Packetized, eq. (26)) Maximum R’ (MQAM) M=2 M=4 M=8 M =16 M = 32 M = 64 M =128 M = 256

R

R’ M = 256 1 M=2 Maximum R’ 0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Φ Fig. 12. The information rate at distance d = 70 m.

6.2. Energy minimization with fixed average power constraint

Information rate (bps)

This subsection presents the optimization results of the energy minimization with fixed average power constraint. The average power constraint P ¼ 0:2894 W, and the circuit power consumption Pc = 0.2884 W. Figs. 10–12 show the achievable information rates at distances 1 m, 40 m and 70 m, respectively. Note that the

packet length LL and target bit error rate Pb have been optimized for any specific U. The bound R is achieved using R ¼ /B logð1 þ ðP=/  Pc Þ=2ð1 þ bÞGBN0 Þ. The packetized/ ARQ information bound is obtained using (29), and the information rates are achieved using (32). The maximum R is achieved through (32) by searching the constellation size M up to 1012. Here we have allowed non-integer M. As shown in Fig. 10, at very short distances, a large M and a large U will maximize the information rate and thereby minimize the energy per bit under fixed average power constraint. Moreover, Figs. 11 and 12 show that,  as distance increases, the parameters M ; U ; LL will de    crease. For example, the set M ; U ; LL is (8, 0.58,3  104) at d = 40 m, while the set M  ; U ; LL is (4, 0.32,1  104) at d = 70 m. This trend is caused by the fact that the reliable (energy efficient) modulation and coding schemes gradually outweighs the high-speed (bandwidth efficient) modulation and coding schemes as the communication environment deteriorates. The cost we pay to save energy is the information rate. As shown in Fig. 13, to achieve energy efficiency, the information rate will drop rapidly. At d = 120 m, the R⁄ is only about 1 Kbps over a bandwidth of B = 10 kHz. For a given distance and a given family of modulation schemes, it is straightforward to find the optimum U⁄ with respect to a fixed constellation size M, or the optimum constellation size M⁄ with respect to a fixed U by taking partial derivatives of (32) and setting them to zero. For instance, the U⁄ and M⁄ of M-QAM at different distances are shown in Fig. 13. From Fig. 13, we can see that, under a strict aver-

106

104

102

1

20

45

70

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95

120

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Constellatoin size

Distance (m) 1010

105

100

1

20

45

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102

100

10−2

1

20

45

Duty cycle (%)

Distance (m) 100

50

0

1

20

45

Distance (m) Fig. 13. The information rate, M ; P t , and U⁄ of M-QAM at different transmission distances.

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1

0.6

Φ

*

0.8

Average power = 0.28 W Average power = 0.4 W Average power = 0.7 W Average power = 1 W

0.4

0.2 0

1

20

45

70

95

120

70

95

120

Distance (m) 2

No average power constraint Average power = 0.28 W Average power = 0.4 W Average power = 0.7 W Average power = 1 W

*

Pt (W)

1.5 1 0.5 0 1

20

45

Distance (m) −5

*

Eb (dbmJ)

−10 −15

No average power constraint Average power = 0.28 W Average power = 0.4 W Average power = 0.7 W Average power = 1 W

−20 −25 −30

1

20

45

70

95

120

Distance (m)  Fig. 14. The influence of the average power on U ; P t ; Eb from (33).

age power constraint, the transceiver has to sacrifice its maximum possible information rate by slashing its duty cycle to satisfy this constraint. For example, as shown in Fig. 13, at d = 100 m, the duty cycle should be as low as 24.6% to provide an appropriate amount of transmit power Pt to guarantee a reasonably low Pb even when the constel-

lation size M = 2. This limits the maximum possible information rate R0 6 BU log2 M = 2.46 Kbps. 6.2.1. The influence of average power Fig. 14 illustrates the impact of average power on the choice of duty cycle, minimum energy consumption per 2

106 1.8 1.6

Average power (W)

Information rate (bps)

No average power constraint Average power = 0.28 W Average power = 0.4 W Average power = 0.7 W Average power = 1 W

105

1.4

Inefficient zone 1.2 1 0.8

104 0.6

Efficient zone 0.4 0.2 103 5

10

15

20

25

1

20

45

70

95

120

Distance (m)

Distance (m) Fig. 15. The influence of the average power on R⁄ from (33).

Fig. 16. Range of efficient versus inefficient zones for average power versus distance from (33).

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2.5

x 10

5

d=5m d = 30 m d = 95 m

Information rate (bps)

2

1.5

1

0.5

0 0.1

1

1

2

3

4

5

6

7

8

9

Average power (W) Fig. 17. Information rate as a function of average power from (33).

Information rate (bps)

information bit, and transmit power, where the circuit power consumption is Pc = 0.288 W. The average power configurations are P ¼ 0:28; 0:4; 0:7, and 1 W. Fig. 14 shows that, for a given distance, U⁄ is larger for a higher P. That is, when Pt is fixed as Pt ðdÞ (the unconstrained optimal transmit power), the transmitter has to be on for a longer period of time to meet a higher P. However, when P increases beyond a point when U = 1, the transmit power

4

x 10

Pt has to start increasing to maintain the average power consumption requirement P. In this case, Pt > P t ðdÞ and the resulting minimum Eb becomes suboptimal. The sufficient average power constraint at any given distance is P ðdÞ ¼ Pt ðdÞð1 þ bÞ þ Pc . That is, for a given transmission distance d, any P above P is unnecessary and not energy efficient. In fact, Fig. 14 shows that, under an average power constraint where U½P t ð1 þ bÞ þ P c  ¼ P, the transmit power sometimes has to be higher than the unconstrained optimum value just to maintain an unnecessarily high average power constraint. Although a low average power constraint P < P t ð1þ bÞ þ P c benefits energy efficiency, it lowers the information rate correspondingly. An example is shown in Fig. 15 where a higher average power obviously provides higher information rate. This is because a low P requires the transmitter to sleep for a larger portion of the duty cycle (small U). On the other hand, the information rate R is proportional to U. Therefore, the information rate will drop with decreasing P. Thus, the average power P reflects the tradeoff between the energy-efficiency (low P) and the information rate (high P). However, the tradeoff of energy-efficiency and the information rate caused by P should always be evaluated in the range ð0; Pt ð1 þ bÞ þ P c ; since beyond this range the transmitter will operate in a classic inefficient trade-off between Eb (linear increase with respect to P) and R (logarithmic increase with respect to P). The efficient versus inefficient zone of P is shown in Fig. 16 where the boundary is P t ð1 þ bÞ þ P c . Fig. 17 gives

4

3 2 1

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Constellatoin size

Distance (m) 1010

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1

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45

Distance (m) Transmit power (W)

102

100

10−2

1

20

45

Duty cycle (%)

Distance (m) 100

50

0

1

20

45

Distance (m) Fig. 18. M ; P t , and U⁄ for M-QAM at different transmission distances (jR0  20 Kbpsj < 100 bps).

T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585

6.3. Energy minimization with fixed average rate constraint In the case of energy minimization with fixed average rate constraint, some illustrative results are presented in Figs. 18–21, where the information rate constraint Rconst = 20 Kbps with the rate deviation DR = 100 bps.  Fig. 18 shows the optimum M  ; P t ; U of M-QAM at different transmission distances under this average rate constraint. As the transmission distance increases, M⁄ decreases due to the fact that a robust modulation scheme becomes energy efficient as the communication environment worsens. In contrast to the fixed average power case, U⁄ also increases in the fixed average rate scenario. This is because, U⁄ is only used to control the information rate in the fixed average rate transmission. Therefore, at short distances, when large M⁄ is energy efficient, a small U⁄ needs to be chosen to meet the information rate requirement. As M⁄ decreases, U⁄ has to increase to maintain information rate requirement. Therefore, for fixed average rate transmissions, at short distances where Pc is comparable with Pt, a time sharing use of the channel is energy efficient; at large distances, where Pt  Pc, a continuous use of the channel is energy efficient.

0 Unconstrained Fixed−average−rate constrained (|R − 20Kbps| < 100bps ) Fixed−average−power constrained (0.5 Watt)

−10

b

some examples to show the trend of information rate versus average power. There clearly exists a point of P where the slope of the curve changes. For instance, at d = 95 m, this point is about P ¼ 1 W, below which the R curve increases linearly (operating in efficient zone) and above which the R curve increases logarithmically (operating in inefficient zone).

E* (dBmJ)

582

−20

−30 1

20

45

70

95

Fig. 20. The Eb of M-QAM at different transmission distances.

Fig. 19 shows the information rate and P  of M-QAM at different transmission distances, under average power constrained and average rate constrained cases, respectively. It is easy to see why P does not change with distance in the average-power-constrained transmission case (here P ¼ P ¼ 0:2894 W). In the fixed average rate transmission case, P will increase with distance. This is directly caused by the increase in the duty cycle and the transmit power as shown in Fig. 18. In this particular example, average rate constrained transmission achieves

5

Information rate (bps)

2

x 10

fixed−average−rate constrained (20 Kbps) fixed−average−power constrained (0.2894 W) 1.5

1

0.5

0 1

20

40

60

80

120

60

80

120

Distance (m)

6

fixed−average−rate constrained (20 Kbps) fixed−average−power constrained (0.2894 W)

Average power (W)

5 4 3 2 1 0

1

20

40

120

Distance (m)

Distance (m) Fig. 19. The information rate and P  of M-QAM at different transmission distances (fixed average power compared with fixed average rate).

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Theoretical results (M ∈ [2 2 ], uncoded, avereage power = 0.3 Watt) Calculated results (M ∈ [2, 4, 8, 16, 64], coded/uncoded) 20

Theoretical results (M ∈ [2 2 ], uncoded, R ∈ [19.9Kbps 20.1Kbps])

109

*

L L (in bits)

108

107

106

105

104

103

1

20

45

70

95

120

Distance (m) Fig. 21. The LL of M-QAM at different transmission distances for the unconstrained, fixed-rate-constrained and fixed-power-constrained cases.

a lower P than average power constrained transmission when d 6 35 m. That is, for fixed average rate transmission, a high average power budget is not necessarily beneficial in terms of energy efficiency when transmission distance is below a certain value dth, where dth is a relative value determined by Rconst and P. For example, when Rconst ? 0, the fixed average rate transmission always achieves a lower P  than the fixed average power transmission with P ¼ 0:2894 W for any d > dth = 0. On the other hand, although the fixed average power constrained transmission provides a low power consumption at longer distances, the provided information rate drops dramatically. Fig. 20 shows the Eb of M-QAM at different transmission distances for the unconstrained, fixed average rate transmission, and fixed average power transmission, respectively. The Eb of fixed average rate constrained transmissions is always higher than or equal to the unconstrained optimal Eb . From Fig. 20, Rconst = 20 Kbps approaches the global energy efficient information rate in the transmission range d 6 55 m. Compare this with Fig. 18, this is the range where U⁄ increases linearly. In this distance range, the transceiver can achieve the desired information rate yet obtain the overall minimum energy consumption by increasing U⁄. Beyond this range, since U⁄ cannot be further increased, extra energy has to be wasted to maintain this information rate. Therefore, a reasonable target information rate should be set according to the transmission distance, or in general, the communication environment. In this case, the target information rate Rconst = 20 Kbps is suitable in terms of energy efficiency  6 55 m. for a wireless network with average distance d  is larger than this range, the target information When d rate should be reduced to save energy. On the other hand, for a given target information rate, this algorithm gives us a target node density that is energy efficient. A fixed average power constrained case is also shown in Fig. 20 with P ¼ 0:5 W. At short distances, the minimized Eb is suboptimal due to the superfluous average power.

However, as d increases, the minimized Eb becomes the global minimum value since there is no more wasted P. Fig. 20 indicates that, in general, fixed average rate transmissions and duty cycle are energy efficient at short transmission distances, while fixed average power transmissions and duty cycle are energy efficient at large transmission distances. Fig. 21 presents the optimum LL of M-QAM at different transmission distances for unconstrained and fixed average rate constrained transmissions, respectively. The calculated results of the optimal packet length in Section 3 (unconstrained) are also provided as a comparison. Note that the calculated results are obtained when the modulation schemes are restricted within coded/uncoded BPSK, QPSK, 16-QAM, and 64-QAM. We can see that the theoretical LL in the unconstrained transmissions is almost the same as the optimum packet lengths given by the calculations. The differences between the calculated and theoretical results are mainly caused by the limited resolutions of constellation sizes in the calculation. However, in the case of the fixed average rate (Rconst = 20 Kbps) transmission, the optimum packet length (denoted by b L L ) is smaller than the overall optimum packet  length LL at short distances, while larger than LL at large distances. This is caused by the requirement of maintaining throughput. At short distances (d < 15 m in this case), nodes should use larger constellation sizes to save energy. However, a larger constellation size will increase the throughput beyond the acceptable range. Thus, other parameters must be adjusted to shrink the information rate. These parameters include: adopting lower duty cycle and using shorter packet. Therefore, the transmitter adopts a low duty cycle at very short distances as shown in Fig. 18 and a packet length b L L < LL . On the other hand, as distance increases, the effect of increasing U⁄ to maintain the information rate starts to fade (U⁄  1), the b L L then needs to be  larger than the LL to increase the effective information rate by amortizing the overhead over a larger number of information bits.

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7. Conclusions

References

We investigated the energy consumption minimization problem for a single link in a wireless network. Specifically, we proposed a generic model for energy consumption per information bit, considering circuit power, packetization, overhead and duty cycle. We have considered the unconstrained, fixed average power, and fixed average rate cases. For the unconstrained case, the results reveal that when transmission distance is short, a system adopting large packet length, small target bit error probability, and high bandwidth-efficient modulation schemes (e.g., high-order uncoded QAM) is more energy efficient. On the other hand, when transmission distance is large, a system using small packet length, large target bit error probability, and high energy efficient modulation schemes (e.g., coded BPSK) is energy efficient. Moreover, as transmission distance increases, a flattening of the optimum values of packet length and target bit error probability is observed. In the fixed average power case, we conclude that the minimization of energy consumption per information bit is equivalent to the maximization of information rate. At short distances, large constellation sizes and large duty cycle are energy efficient, while the optimum constellation size and duty cycle both decrease with distance. This indicates that, within the limits of average power constraint, bandwidth efficient modulations and continuous use of channels are energy efficient at short distances, while robust modulations and duty cycling are energy efficient at large distances. The cost associated with maintaining a fixed average power is the decrease of information rate with distance. In the fixed average rate case, at short distances, large constellation sizes and small duty cycle are energy efficient. As transmission distance increases, the optimum constellation size decreases and optimum duty cycle increases to get data through while minimizing the energy consumption. That is, under a strict average rate constraint, bandwidth efficient modulations and duty cycling are energy efficient at short distances, while robust modulations and continuous use of the channels are energy efficient at long distances. The proposed optimization scheme features a low memory requirement and is highly parameterized. The resulting optimal modulation schemes, the optimal target bit error probability, and the optimal packet length can be stored in a small look-up table, for example, when there are eight candidate modulation schemes, only about 2 KBytes of memory is needed. Moreover, the algorithm is highly parameterized, so interested researchers can easily plug the power consumption values of their own platform into our model to determine the optimal transmission parameters for their platform.1

[1] J.G. Proakis, Digital Communications, fourth ed., Addison-Wesley, MA, 1972. [2] T.S. Rappaport, Wireless Communications: Principles and Practice, second ed., Prentice-Hall Inc., Upper Saddle River, NJ 07458, 2002. [3] W. Heinzelman, A. Chandrakasan, H. Balakrishnan, An applicationspecific protocol architecture for wireless microsensor networks, IEEE Trans. Wireless Commun. (2002) 660–670. [4] O. Younis, S. Fahmy, Distributed clustering in ad-hoc sensor networks: a hybrid, energy-efficient approach, in: Proceedings of the 23rd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), 2004. [5] C. Schurgers, V. Tsiatsis, S. Ganeriwal, M. Srivastava, Optimizing sensor networks in the energy–latency–density design space, IEEE Trans. Mobile Computing (2002) 70–80. [6] P.Y. Massaad, M. Medard, L. Zheng, Impact of processing energy on the capacity of wireless channels, in: Proceedings of the International Symposium on Information Theory and its Applications (ISITA), 2004. [7] S. Cui, A.J. Goldsmith, A. Bahai, Energy-contrained modulation optimization, IEEE Trans. Wireless Commun. 4 (8) (2005) 2349– 2360. [8] J. Ammer, J. Rabaey, The energy-per-useful-bit metric for evaluating and optimizing sensor network physical layers, in: 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and Networks (SECON ’06), 2006. [9] P. Lettieri, M.B. Srivastava, Adaptive frame length control for improving wireless link throughput, range, and energy efficiency, in: Proceedings of INFOCOM ’98, 1998. [10] T. Yoo, R.J. Lavery, A. Goldsmith, D.J. Goodman, Throughput optimization using adaptive techniques, Technical Report, 2004. [11] V. Rodriguez, R. Mathar, Generalised link-layer adaptation with higher-layer criteria for energy-constrained and energy-sufficient data terminals, in: Proceedings of the 7th International Symposium on Wireless Communication Systems (ISWCS), York, 2010. [12] Y. Sankarasubramaniam, I. Akyildiz, S. McLaughlin, Energy efficiency based packet size optimization in wireless sensor networks, in: Proceedings of the First IEEE International Workshop on Sensor Network Protocols and Applications, 2003. [13] R.K. Reddy, S. De, H.M. Gupta, Joint control of transmit power and frame size for energy-optimized data transfer in wireless sensor networks, IEICE Trans. Commun. E93-B (8) (2010) 2043–2052. [14] A. Wang, S. Cho, C.G. Sodini, A.P. Chandrakasan, Energy efficient modulation and MAC for asymmetric RF microsensor systems, in: Proceedings of the 2001 international symposium on Low power electronics and design, 2001. [15] J. Deng, Y. Han, P. Chen, P. Varshney, Optimum transmission range for wireless ad hoc networks, in: Proceedings of Wireless Communications and Networking Conference (WCNC), 2004. [16] V. Rodoplu, T. Meng, Minimum energy mobile wireless networks, IEEE J. Sel. Areas Commun. 17 (8) (1999) 1333–1344. [17] C. Chien, M. Srivastava, R.. Jain, P. Lettieri, V. Aggarwal, R. Sternowski, Adaptive radio for multimedia wireless links, IEEE J. Sel. Areas Commun. 17 (5) (1999) 793–813. [18] M. Holland, T. Wang, B. Tavli, A. Seyedi, W. Heinzelman, Optimizing physical layer parameters for wireless sensor networks, ACM Trans. Sensor Networks 7 (4) (2011) 28–48. [19] S. Verdu, Spectral efficiency in the wideband regime, IEEE Trans. Inform. Theory 48 (6) (2002) 1319–1343. [20] S. Cui, A.J. Goldsmith, A. Bahai, Power Estimation for Viterbi Decoders, Wireless Systems Lab, Stanford University, 2003. [21] S.P. Boyd, L. Vandenberghe, Convex Optimization, first ed., Cambridge University Press, Cambridge, UK, 2004. [22] MoteIV, tmote Sky Data Sheet, 2007. .

1 MATLAB code for the model described in this paper is available at http://www.ece.rochester.edu/research/wcng/.

T. Wang et al. / Ad Hoc Networks 10 (2012) 569–585 Tianqi Wang received the B.E. degree in 2004 in Communications Engineering, from Beijing University of Posts and Telecommunications, Beijing, China, and the M.Eng. degree in 2007 in Electrical and Computer Engineering, from Memorial University of Newfoundland, St. John’s, Canada. He is currently working toward a Ph.D. degree at the Department of Electrical and Computer Engineering, University of Rochester, USA. His current research interests include wireless communications and crosslayer design in wireless sensor networks.

Wendi Heinzelman is an associate professor in the Department of Electricals and Computer Engineering at the University of Rochester, and she holds a secondary appointment as an associate professor in the Department of Computer Science. Dr. Heinzelman also currently serves as Dean of Graduate Studies for Arts, Sciences and Engineering at the University of Rochester. Dr. Heinzelman received a B.S. degree in Electrical Engineering from Cornell University in 1995 and M.S. and Ph.D. degrees in Electrical Engineering and Computer Science from MIT in 1997 and 2000, respectively. Her current research interests lie in the areas of wireless communications

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and networking, mobile computing, and multimedia communication. Dr. Heinzelman received the NSF CAREER award in 2005 for her research on cross-layer architectures for wireless sensor networks, and she received the ONR Young Investigator Award in 2005 for her work on balancing resource utilization in wireless sensor networks. She is an Associate Editor for the IEEE Transactions on Mobile Computing, an Associate Editor for the ACM Transactions on Sensor Networks and an Associate Editor for Elsevier Ad Hoc Networks Journal. Dr. Heinzelman is a senior member of the IEEE and the ACM, and she is co-founder of the N^2 Women (Networking Networking Women) group.

Alireza Seyedi (S’95, M’04, SM’10) received his B.S. degree from Sharif University of Technology, Tehran, Iran, in 1997 and his M.S. and Ph.D. degrees from Rensselaer Polytechnic Institute, Troy, NY, in 2000 and 2004, respectively, all in electrical engineering. Since 2007 he has been with the Faculty of the Electrical and Computer Engineering Department at the University of Rochester, where he currently is an Assistant Professor (Research). Prior to that, he was with Philips Research North America. His primary research interests are in the areas of Communications, Control, and their convergence. He is a Senior Member of the IEEE Communications, Control and Signal Processing Societies.