On the Energy Efficiency of LT Codes in Proactive Wireless Sensor Networks
Jamshid Abouei, J. David Brown, Konstantinos N. Plataniotis, Subbarayan Pasupathy
aa r X i v : . [ c s . I T ] D ec SUBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 On the Energy Efficiency of LT Codes inProactive Wireless Sensor Networks
Jamshid Abouei † , Member, IEEE,
J. David Brown †† , Konstantinos N. Plataniotis † , Senior Member, IEEE andSubbarayan Pasupathy † , Life Fellow, IEEE † The Edward S. Rogers Sr. Dept. of Electrical and Computer Engineering,University of Toronto, Toronto, Canada, Emails: { abouei, kostas, pas } @comm.utoronto.ca †† Ottawa, Canada, Tel: 613-236-1051, Email: david jw [email protected]
Abstract
This paper presents the first in-depth analysis on the energy efficiency of LT codes with Non Coherent M-aryFrequency Shift Keying (NC-MFSK), known as green modulation [1], in a proactive Wireless Sensor Network(WSN) over Rayleigh flat-fading channels with path-loss. We describe the proactive system model according toa pre-determined time-based process utilized in practical sensor nodes. The present analysis is based on realisticparameters including the effect of channel bandwidth used in the IEEE 802.15.4 standard, and the active modeduration. A comprehensive analysis, supported by some simulation studies on the probability mass function ofthe LT code rate and coding gain, shows that among uncoded NC-MFSK and various classical channel codingschemes, the optimized LT coded NC-MFSK is the most energy-efficient scheme for distance d greater than thepre-determined threshold level d T , where the optimization is performed over coding and modulation parameters. Inaddition, although uncoded NC-MFSK outperforms coded schemes for d < d T , the energy gap between LT codedand uncoded NC-MFSK is negligible for d < d T compared to the other coded schemes. These results come fromthe flexibility of the LT code to adjust its rate to suit instantaneous channel conditions, and suggest that LT codesare beneficial in practical low-power WSNs with dynamic position sensor nodes. Index Terms
Wireless sensor networks, energy efficiency, green modulation, LT codes. †† The second author completed his Ph.D. in ECE at University of Toronto.
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 I. I
NTRODUCTION
Wireless Sensor Networks (WSNs) have been recognized as a new generation of ubiquitous computingsystems to support a broad range of applications, including monitoring, health care and tracking environ-mental pollution levels. Minimizing the total energy consumption in both circuit components and RF signaltransmission is a crucial challenge in designing a WSN. Central to this study is to find energy-efficientmodulation and coding schemes in the physical layer of a WSN to prolong the sensor lifetime [2], [3]. Forthis purpose, energy-efficient modulation/coding schemes should be simple enough to be implemented bystate-of-the-art low-power technology, but still robust enough to provide the desired service. Furthermore,since sensor nodes frequently switch from sleep mode to active mode, modulation and coding circuitsshould have fast start-up times [4] along with the capability of transmitting packets during a pre-assignedtime slot before new sensed packets arrive. In addition, a WSN needs a powerful channel coding schemewhich protects transmitted data against the unpredictable and harsh nature of channels. Finally, sincecoding increases the required transmitted bandwidth, when considered independently of modulation, thebest tradeoff between energy-efficient modulation and coding for a given transmission bandwidth shouldbe considered as well. We refer to these low-complexity and low-energy consumption approaches inWSNs providing proper link reliability without increasing a given transmission bandwidth as
GreenModulation/Coding (GMC) schemes.There have been several recent works on the energy efficiency of various modulation and channelcoding schemes in WSNs (see e.g., [2], [5], [6]). Tang et al. [5] analyze the power efficiency of PulsePosition Modulation (PPM) and Frequency Shift Keying (FSK) in a WSN without considering the effectof channel coding. Under the assumption of the non-linear battery model, reference [5] shows that FSKis more power-efficient than PPM in sparse WSNs, while PPM may outperform FSK in dense WSNs.Reference [7] investigates the energy efficiency of BCH and convolutional codes with non-coherent FSKfor the optimal packet length in a point-to-point WSN. It is shown in [7] that BCH codes can improveenergy efficiency compared to the convolutional code for optimal fixed packet size. Reference [8] analyzesthe effect of different linear block channel codes with FSK modulation on energy consumptions of a low-power wireless embedded network when the number of hops increases. Liang et al. [9] investigate the
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 energy efficiency of uncoded NC-MFSK modulation scheme in a multiple-access WSN over Rayleighfading channels, where multiple senders transmit their data to a central node in a Frequency-DivisionMultiple Access (FDMA) fashion. Reference [10] presents the hardware implementation of the ForwardError Correction (FEC) encoder in IEEE 802.15.4 WSNs, which employs parallel just-in-time processingto achieve a low processing latency and energy consumption.Most of the pioneering works on energy-efficient modulation/coding, including research in [5], [7],[8], [11], [12], has focused only on minimizing the energy consumption of transmitting one bit, ignoringthe effect of bandwidth and transmission time duration. In a practical WSN however, it is shown thatminimizing the total energy consumption depends strongly on the active mode duration and the channelbandwidth. References [1], [2] and [6] address this issue in a point-to-point WSN, where a sensor nodetransmits an equal amount of data per time unit to a designated sink node. In [2], the authors considerthe optimal energy consumption per information bit as a function of modulation and coding parametersin a WSN over Additive White Gaussian Noise (AWGN) channels with path-loss. It is shown in [2] thatuncoded MQAM is more energy-efficient than uncoded MFSK for short-range applications. For higherdistance, however, using convolutional coded MFSK over AWGN is desirable. This line of work is furtherextended in [6] by evaluating the energy consumption per information bit of a WSN for Reed Solomon(RS) Codes and various modulation schemes over AWGN channels with path-loss. Also, the impact ofdifferent transmission distances on the energy consumption per information bit is investigated in [6]. In [2]and [6], the authors do not consider the effect of multi-path fading. Reference [1] addresses this problemin a similar WSN model as [2] and [6], and shows that among various sinusoidal carrier-based modulationschemes, Non-Coherent M-ary Frequency Shift Keying (NC-MFSK) with small order of constellation size M can be considered the most energy-efficient modulation in proactive WSNs over Rayleigh and Ricianfading channels. However, no channel coding scheme was considered in [1].More recently, the attention of researchers has been drawn to deploying rateless codes (e.g., LubyTransform (LT) code [13]) in WSNs due to the outstanding advantages of these codes in erasure channels .For instance in [14], the authors present a scheme for cooperative error control coding using ratelessand Low-Density Generator-Matrix (LDGM) codes in a multiple relay WSN. However, investigating the UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 energy efficiency of rateless codes in WSNs with green modulations over realistic fading channel modelshas received little attention. To the best of our knowledge, there is no existing analysis on the energyefficiency of rateless coded modulation that considers the effect of channel bandwidth and active modeduration on the total energy consumption in a practical proactive WSN. This paper addresses this problemand presents the first in-depth analysis of the energy efficiency of LT codes with NC-MFSK (known asgreen modulation) as described in [1]. The present analysis is based on a realistic model in proactiveWSNs operating in a Rayleigh flat-fading channel with path-loss. In addition, we obtain numerically theprobability mass function of the LT code rate and the corresponding coding gain, and study their effectson the energy efficiency of the WSN. This study uses the classical BCH and convolutional codes (asreference codes), utilized in IEEE standards, for comparative evaluation. Experimental results show thatthe optimized LT coded NC-MFSK is the most energy-efficient scheme for distance d greater than thethreshold level d T . In addition, although uncoded NC-MFSK outperforms coded schemes for d < d T , theenergy gap between LT coded and uncoded NC-MFSK is negligible for d < d T compared to the othercoded schemes. This result comes from the simplicity and flexibility of the LT codes, and suggests thatLT codes are beneficial in practical low-power WSNs with dynamic position sensor nodes.The rest of the paper is organized as follows. In Section II, the proactive system model over a realisticwireless channel model is described. The energy consumption of uncoded NC-MFSK modulation schemeis analyzed in Section III. Design of LT codes and the energy efficiency of the LT coded NC-MFSK arepresented in Section IV. In addition, the energy efficiency of some classical channel codes are studied inthis section. Section V provides some numerical evaluations using realistic models to confirm our analysis.Also, some design guidelines for using LT codes in practical WSN applications are presented. Finally inSection VI, an overview of the results and conclusions are presented.For convenience, we provide a list of key mathematical symbols used in this paper in Table I.II. S YSTEM M ODEL AND A SSUMPTIONS
In this work, we consider a proactive wireless sensor system, in which a sensor node continuouslysamples the environment and transmits an equal amount of data per time unit to a designated sink node.Such a proactive sensor system is typical of many environmental applications such as sensing temperature,
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 TABLE IL
IST OF N OTATIONS B Channel bandwidth d Transmission distance E t Energy of uncoded transmitted signal E N Total energy consumption for uncoded case h i Fading channel coefficient for symbol i L d Channel gain factor with distance dM Constellation size N Number of sensed message n Codeword block length O ( x ) Output-node degree distribution P c Circuit power consumption P t Power of transmitted signal P b Bit error rate P R ( ℓ ) pmf of LT code rate R c Code rate T ac Active mode duration T s Symbol duration η Path-loss exponent Ω E ˆ | h i | ˜ γ i Instantaneous SNR Υ c Coding gain humidity, level of contamination, etc [15]. For this proactive system, the sensor and sink nodes synchronizewith one another and operate in a real time-based process as depicted in Fig 1. During active mode duration T ac , the analog signal sensed by the sensor is first digitized by an Analog-to-Digital Converter(ADC), and an N -bit message sequence M N , ( m , m , ..., m N ) is generated, where N is assumedto be fixed, and m i ∈ { , } , i = 1 , , ..., N . The bit stream is then sent to the channel encoder. Theencoding process begins by dividing the uncoded message M N into blocks of equal length denoted by B j , ( m ( j − k +1 , ..., m jk ) , j = 1 , ..., Nk , where k is the length of any particular B j , and N is assumed tobe divisible by k . Each block B j is encoded by a pre-determined channel coding scheme to generate a UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 t sl ac T fi ac T ac sl T fi sl T N T Active ModeActive Mode
Fig. 1. A practical multi-mode operation in a proactive WSN. coded bit stream C j , ( a ( j − n +1 , ..., a jn ) , j = 1 , ..., Nk , with block length n , where n is either a fixed value(e.g., for block and convolutional codes) or a random variable (e.g., for LT codes).The coded stream is then modulated by an NC-MFSK scheme and transmitted to a designated sinknode. Finally, the sensor node returns to sleep mode , and all the circuits of the transceiver are shutdown forsleep mode duration T sl for energy saving. We denote T tr as the transient mode duration consisting of theswitching time from sleep mode to active mode (i.e., T sl → ac ) plus the switching time from active mode tosleep mode (i.e., T ac → sl ), where T ac → sl is short enough compared to T sl → ac to be negligible. Furthermore,when the sensor switches from sleep mode to active mode to send data, a significant amount of power isconsumed for starting up the transmitter, while the power consumption during T ac → sl is negligible. Underthe above considerations, the sensor/sink nodes have to process one entire N -bit message M N during ≤ T ac ≤ T N − T tr , before a new sensed packet arrives, where T N , T tr + T ac + T sl is fixed, and T tr ≈ T sl → ac .Since sensor nodes in a typical WSN are densely deployed, the distance between nodes is normally short.Thus, the circuit power consumption in a WSN is comparable to the output transmit power consumption.We denote the total circuit power consumption as P c , P ct + P cr , where P ct and P cr represent thecircuit power consumptions for sensor and sink nodes, respectively. In addition, the power consumptionof RF signal transmission in the sensor node is denoted by P t . Taking these into account, the total energyconsumption during the active mode period, denoted by E ac , is given by E ac = ( P c + P t ) T ac . Also, theenergy consumption in the sleep mode period, denoted by E sl , is given by E sl = P sl T sl , where P sl is thecorresponding power consumption. It is worth mentioning that during the sleep mode interval, the leakagecurrent coming from the CMOS circuits embedded in the sensor node is a dominant factor in P sl . Clearly, UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 higher sleep mode duration increases the energy consumption E sl due to increasing leakage current aswell as T sl . Present state-of-the art technology aims to keep a low sleep mode leakage current no largerthan the battery leakage current, which results in P sl much smaller than the power consumption in activemode [16]. For this reason, we assume that P sl ≈ . As a result, we have the following definition. Definition 1 (Performance Metric):
The energy efficiency, referred to as the performance metric of theproposed WSN, can be measured by the total energy consumption in each period T N corresponding to N -bit message M N as follows: E N = ( P c + P t ) T ac + P tr T tr , (1) where P tr is the circuit power consumption during the transient mode period. We use (1) to investigate and compare the energy efficiency of uncoded and coded NC-MFSK for variouschannel coding schemes.
Channel Model:
The choice of low transmission power in WSNs results in several consequences to thechannel model. It is shown by Friis [17] that a low transmission power implies a small range. For short-range transmission scenarios, the root mean square (rms) delay spread is in the range of nanoseconds [18]which is small compared to symbol durations for modulated signals. For instance, the channel bandwidthand the corresponding symbol duration considered in the IEEE 802.15.4 standard are B = 62 . KHz and T s = 16 µ s, respectively [19, p. 49], while the rms delay spread in indoor environments are in the rangeof 70-150 ns [20]. Thus, it is reasonable to expect a flat-fading channel model for WSNs. In addition,many transmission environments include significant obstacle and structural interference by obstacles (suchas wall, doors, furniture, etc), which leads to reduced Line-Of-Sight (LOS) components. This behaviorsuggests a Rayleigh fading channel model. Under the above considerations, the channel model betweenthe sensor and sink nodes is assumed to be Rayleigh flat-fading with path-loss, which is a feasible modelin static WSNs [5], [11]. For this model, we assume that the channel is constant during the transmissionof a codeword, but may vary from one codeword to another. We denote the fading channel coefficientcorresponding to an arbitrary transmitted symbol i as h i , where the amplitude (cid:12)(cid:12) h i (cid:12)(cid:12) is Rayleigh distributedwith probability density function (pdf) given according to f | h i | ( r ) = r Ω e − r , r ≥ , where Ω , E [ | h i | ] UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 (e.g., pp. 767-768 of [21]). This results in | h i | being chi-square distributed with 2 degrees of freedom,where f | h i | ( r ) = e − r Ω , r ≥ .To model the path-loss of a link where the transmitter and receiver are separated by distance d , letdenote P t and P r as the transmitted and the received signal powers, respectively. For a η th -power path-loss channel, the channel gain factor is given by L d , P t P r = M l d η L , where M l is the gain marginwhich accounts for the effects of hardware process variations, background noise and L , (4 π ) G t G r λ is thegain factor at d = 1 meter which is specified by the transmitter and receiver antenna gains G t and G r ,and wavelength λ (e.g., [2], [11], Ch. 4 of [22] and [23]). As a result, when both fading and path-lossare considered, the instantaneous channel coefficient corresponding to an arbitrary symbol i becomes G i , h i √L d . Denoting x i ( t ) as the RF transmitted signal with energy E t , the received signal at the sinknode is given by y i ( t ) = G i x i ( t ) + n i ( t ) , where n i ( t ) is AWGN at the sink node with two-sided powerspectral density given by N . Under the above considerations, the instantaneous Signal-to-Noise Ratio(SNR), denoted by γ i , corresponding to an arbitrary symbol i can be computed as γ i = | G i | E t N . Under theassumption of a Rayleigh fading channel model, γ i is chi-square distributed with 2 degrees of freedomand with pdf f γ ( γ i ) = γ e − γi ¯ γ , where ¯ γ , E [ | G i | ] E t N = Ω L d E t N denotes the average received SNR.III. E NERGY C ONSUMPTION OF U NCODED
NC-MFSK M
ODULATION
We first consider uncoded MFSK modulation in the proposed proactive WSN, where M orthogonalcarriers can be mapped into b , log M bits. Denoting E t as the uncoded MFSK transmit energy per symbolwith symbol duration T s , the transmitted signal from the sensor node is given by x i ( t ) = q E t T s cos(2 π ( f + i ∆ f ) t ) , i = 0 , , ...M − , where f is the first carrier frequency in the MFSK modulator and ∆ f = T s is the minimum carrier separation in the non-coherent case. Thus, the channel bandwidth is obtained as B ≈ M × ∆ f , which is assumed to be a fixed value. Since we have b bits during each symbol period T s ,we can write T ac = Nb T s = M NB log M . (2)Recalling that B and N are fixed, an increase M results in an increase in T ac . However, as illustrated inFig. 1, the maximum value for T ac is bounded by T N − T tr . Thus, the maximum constellation size M , UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 denoted by M max , b max , for uncoded MFSK is calculated by bmax b max = BN ( T N − T tr ) . It is shown in [1]that the transmit energy consumption per each symbol for an uncoded NC-MFSK is obtained as E t , P t T s ≈ (cid:20)(cid:16) − (1 − P s ) M − (cid:17) − − (cid:21) L d N Ω (3) ( a ) = − (cid:18) − M − M P b (cid:19) M − ! − − L d N Ω , (4)where ( a ) comes from the fact that the relationship between the average Symbol Error Rate (SER) P s andthe average Bit Error Rate (BER) P b of MFSK is given by P s = M − M P b [21, p. 262]. Using (2), theoutput energy consumption of transmitting N -bit during T ac of an uncoded NC-MFSK is then computedas P t T ac = T ac T s E t ≈ − (cid:18) − M − M P b (cid:19) M − ! − − L d N Ω N log M . (5)For the sensor node with uncoded MFSK, we denote the power consumption of frequency synthesizer,filters and power amplifier as P Sy , P F ilt and P Amp , respectively. In this case, the power consumption ofthe sensor circuitry with uncoded MFSK can be obtained as P ct = P Sy + P F ilt + P Amp , (6)where P Amp = α P t , in which α is determined based on the type (or equivalently drain efficiency) ofpower amplifier. For instance, for a class B power amplifier, α = 0 . [2], [5]. It is shown in [1] that thepower consumption of the sink circuitry with uncoded NC-MFSK scheme can be obtained as P cr = P LNA + M × ( P F ilr + P ED ) + P IF A + P ADC , (7)where P LNA , P F ilr , P ED , P IF A and P ADC denote the power consumption of Low-Noise Amplifier (LNA),filters, envelop detector, IF amplifier and ADC, respectively. In addition, it is shown that the powerconsumption during transition mode period T tr is governed by the frequency synthesizer [4]. Thus, theenergy consumption during T tr is obtained as P tr T tr = 1 . P Sy T tr [18]. Substituting (2)-(7) in (1), thetotal energy consumption of an uncoded NC-MFSK scheme for transmitting N -bit information in eachperiod T N , under the constraint M ≤ M max and for a given P b is obtained as E N = (1+ α ) − (cid:18) − M − M P b (cid:19) M − ! − − L d N Ω N log M +( P c −P Amp ) M NB log M +1 . P Sy T tr . (8) UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 It is shown in [1] that the above uncoded NC-MFSK is more energy-efficient than other sinusoidalcarrier-based modulation schemes, and is a good option for low-power and low data rate WSN applications.For energy optimal designs, however, the impact of channel coding on the energy efficiency of the proposedWSN must be considered as well. It is a well known fact that channel coding is a classical approach usedto improve the link reliability along with the transmitter energy saving due to providing the coding gain[21]. However, the energy saving comes at the cost of extra energy spent in transmitting the redundantbits in codewords as well as the additional energy consumption in the process of encoding/decoding. Fora specific transmission distance d , if these extra energy consumptions outweigh the transmit energy savingdue to the channel coding, the coded system would not be energy-efficient compared with an uncodedsystem. In the subsequent sections, we will argue the above problem and determine at what distance useof specific channel coding becomes energy-efficient compared to uncoded systems. In particular, we willshow in Section V that the LT coded NC-MFSK surpasses this distance constraint in the proposed WSN.IV. E NERGY C ONSUMPTION A NALYSIS OF
LT C
ODED
NC-MFSKIn this section, we present the first in-depth analysis on the energy efficiency of LT coded NC-MFSKfor the proposed proactive WSN. To get more insight into how channel coding affects the circuit and RFsignal energy consumptions in the system, we modify the energy concepts in Section III, in particular,the total energy consumption expression in (8) based on the coding gain and code rate. To address thisproblem and for the purpose of comparative evaluation, we first start with classical BCH and convolutionalchannel codes (referred to as fixed-rate codes), that are widely utilized in IEEE standards [24], [25]. Wefurther present the first study on the tradeoff between LT code rate and coding gain required to achieve acertain BER, and the effect of this tradeoff on the total energy consumption of LT coded NC-MFSK fordifferent transmission distances . BCH Codes:
In the BCH ( n, k, t ) code with up to t -error correction capability, each k -bit message B j ∈ M N is encoded into nR BCc valid codewords C j , ( a ( j − n +1 , ..., a jn ) , j = 1 , ..., Nk with block length n , where R BCc , kn is the BCH code rate. For this code, n = 2 m − bits, where m ≥ , t < m − andthe number of parity-check bits is upper bounded by n − k ≤ mt . It is worth mentioning that the number In the sequel and for simplicity of notation, we use the superscripts ‘BC’, ‘CC’ and ‘LT’ for BCH, convolutional and LT codes, respectively.
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 M-ary Modulation b T Bit duration Symbol duration
MTT bs log = EncoderBlock Code (n,k) M-ary Modulation
Bit duration b T Bit duration bb TnkT = ~ Symbol duration csbs
RTMTT == log~~ nkR c = b R (bits/sec) MRR bs log = (sym/sec) b R (bits/sec) MRR bs log~~ = (sym/sec) bb RknR = ~ (bits/sec) Fig. 2. A schematic diagram of the bandwidth expansion in an arbitrary block code ( n, k ) with M-ary modulation scheme. of transmitted bits in each period T N is increased from N -bit uncoded message to NR BCc = Nk n bits codedone. To compute the total energy consumption of coded scheme, we use the fact that channel coding canreduce the required average SNR value to achieve a given BER. Taking this into account, the proposedWSN with BCH codes benefits in transmission energy saving specified by E BCt = E t Υ BCc , where Υ BCc ≥ is the coding gain of BCH coded NC-MFSK. Table II displays the coding gain of some BCH ( n, k, t ) codes with NC-MFSK scheme over a Rayleigh flat-fading channel for different values of M and given P b = (10 − , − ) . For these results, a hard-decision decoding algorithm is considered. It is seen fromTable II that there is a tradeoff between coding gain and decoder complexity. In fact, achieving a highercoding gain for given M , requires a more complex decoding process, (i.e., higher t ) with more circuitpower consumption.It should be noted that the cost of the energy savings of using BCH codes is the bandwidth expansion BR BCc as depicted in Fig. 2. In order to keep the bandwidth of the coded system the same as that of theuncoded case, we must keep the information transmission rate constant, i.e., the symbol duration T s ofuncoded and coded NC-MFSK would be the same. Thus, we can drop the superscript “BC” in T s for thecoded case. However, the active mode duration increases from T ac = Nb T s in the uncoded system to T BCac = NbR
BCc T s = T ac R BCc (9)for the BCH coded case. Thus, one would assume that the total time T N increases to T N R BCc for the codedscenario. It should be noted that the maximum constellation size M , denoted by M max , b max , for the Denoting ¯ γ = Ω L d E t N and ¯ γ BC = Ω L d E BCt N as the average SNR of uncoded and BCH coded schemes, respectively, the BCH coding gain (expressed in dB) is defined as the difference between the values of ¯ γ and ¯ γ BC required to achieve a certain BER, where E BCt = E t Υ BCc . UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 TABLE IIC
ODING GAIN ( dB ) OF BCH ( n, k, t ) AND CONVOLUTIONAL CODED
NC-MFSK
OVER R AYLEIGH FLAT - FADING CHANNEL FOR
BER = (10 − , − ) .BCH Code R BCc
M=2 M=4 M=8 M=16 M=32 M=64BCH (7 , , (2 . , .
8) (0 . , .
4) (0 . , .
2) (0 . , .
0) (0 . , .
0) (0 . , . BCH (15 , , (1 . , .
6) (0 . , .
3) (0 . , .
0) (0 . , .
0) (0 . , .
0) (0 . , . BCH (15 , , (2 . , .
3) (2 . , .
3) (0 . , .
0) (0 . , .
4) (0 . , .
0) (0 . , . BCH (15 , , (4 . , .
6) (2 . , .
9) (2 . , .
1) (1 . , .
60) (0 . , .
8) (0 . , . BCH (31 , , (1 . , .
5) (0 . , .
2) (0 . , .
0) (0 . , .
0) (0 . , .
0) (0 . , . BCH (31 , , (2 . , .
9) (1 . , .
0) (0 . , .
8) (0 . , .
2) (0 . , .
0) (0 . , . BCH (31 , , (2 . , .
1) (2 . , .
2) (1 . , .
6) (1 . , .
4) (0 . , .
7) (0 . , . BCH (31 , , (4 . , .
4) (3 . , .
2) (2 . , .
3) (2 . , .
1) (1 . , .
0) (1 . , . BCH (31 , , (5 . , .
9) (4 . , .
8) (3 . , .
8) (3 . , .
3) (2 . , .
8) (2 . , . Convolutional Code R CCc
M=2 M=4 M=8 M=16 M=32 M=64trel (6 , [53 75]) (3 . , .
6) (2 . , .
1) (2 . , .
3) (1 . , .
0) (1 . , .
5) (1 . , . trel (7 , [133 171]) (4 . , .
7) (3 . , .
5) (2 . , .
4) (1 . , .
0) (1 . , .
6) (1 . , . trel (7 , [133 165 171]) (5 . , .
4) (4 . , .
1) (3 . , .
9) (3 . , .
3) (2 . , .
8) (2 . , . trel ([4 3] , [4 5 17; 7 4 2]) (2 . , .
6) (1 . , .
7) (0 . , .
1) (0 . , .
6) (0 . , .
5) (0 . , . trel ([5 4] , [23 35 0; 0 5 13]) (2 . , .
5) (1 . , .
4) (1 . , .
8) (1 . , .
2) (0 . , .
9) (0 . , . coded NC-MFSK is calculated by bmax b max = BR BCc N ( T N R BCc − T tr ) , which is approximately the same as that ofthe uncoded case.Now, we are ready to compute the total energy consumption in the case of BCH coded NC-MFSK. SinceBCH codes are implemented using Linear-Feedback Shift Register (LFSR) circuits, the BCH encoder canbe assumed to have negligible energy consumption. Thus, the energy cost of the sensor circuity with BCHcoded NC-MFSK scheme is approximately the same as that of uncoded one. Also, the energy consumptionof an BCH decoder is negligible compared to the other circuit components in the sink node, as shownin Appendix I. Substituting (4) in E BCt = E t Υ BCc , and using (6), (7) and (9), the total energy consumptionof transmitting NR BCc bits in each period T N R BCC for a BCH coded NC-MFSK scheme, and for a given P b is UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 obtained as E BCN = (1 + α ) − (cid:18) − M − M P b (cid:19) M − ! − − L d N ΩΥ BCc NR BCc log M +( P c − P Amp ) M NBR
BCc log M + 1 . P Sy T tr , (10)under the constraint M ≤ M max . Convolutional Codes:
A convolutional code ( n, k, L ) is commonly specified by the number of input bits k , the number of output bits n , and the constraint length L . As with BCH codes, the rate of a convolutionalcode is given by the ratio R CCc , kn . Since k and n are small integers (typically from 1 to 8) with k < n ,the convolutional encoder is extremely simple to implement and can be assumed to have negligible energyconsumption. With a similar argument as for BCH codes, the convolutional coded NC-MFSK providesan energy saving compared to the uncoded system, which is specified by E CCt = E t Υ CCc . Table II givesthe coding gain Υ CCc of some practical convolutional codes used in IEEE standards with an NC-MFSKmodulation scheme, over a Rayleigh flat-fading channel for different constellation size M and given P b = (10 − , − ) . For these results, a hard-decision Viterbi decoding algorithm is considered. It is seenfrom Table II that for a given M , the convolutional codes with lower rates and higher constraint lengthsachieve greater coding gains. Also, in contrast to [2], where the authors assume a fixed convolutionalcoding gain for every value of M , it is observed that the coding gain of convolutional coded NC-MFSKis a monotonically decreasing function of M .It should be noted that the energy efficiency analysis presented for BCH codes, in particular derivingthe total energy consumption E CCN , is valid for convolution codes. Thus, by substituting (4) in E CCt = E t Υ CCc ,and using (6), (7) and T CCac = NbR
CCc T s , the total energy consumption of transmitting NR CCc bits in eachperiod T N R CCc for a convolutional coded NC-MFSK scheme, achieving a certain P b , is obtained as E CCN = (1 + α ) − (cid:18) − M − M P b (cid:19) M − ! − − L d N ΩΥ CCc NR CCc log M +( P c − P Amp ) M NBR
CCc log M + 1 . P Sy T tr . (11) The constraint length L represents the number of bits in the encoder memory that affect the generation of the n output bits. UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 k Bit Message Sequence m m m a a a Encoded Bit Stream a a a a a m m m Fig. 3. Bipartite graph of an LT code with k = 6 and n = 8 . LT Codes:
LT codes are the first class of Fountain codes (designed for erasure channels) that arenear optimal erasure correcting codes [13]. The traditional schemes for data transmission across erasurechannels use continuous two-way communication protocols, meaning that if the receiver can not decodethe received packet correctly, asks the transmitter (via a feedback channel) to send the packet again. Thisprocess continues until all the packets have been decoded successfully. Fountain codes in general, andLT codes in particular, surpass the above feedback channel problem by adopting an essentially one-waycommunication approach.LT codes are usually specified jointly by two parameters k (number of input bits) and O ( x ) (the output-node degree distribution). The encoding process begins by dividing the original message M N into blocksof equal length. Without loss of generality and for ease of our analysis, we use index j = 1 , meaninga single k -bit message B , ( m , m , ..., m k ) ∈ M N is encoded to codeword C , ( a , a , ..., a n ) . Eachsingle coded bit a i is generated based on the encoding protocol proposed in [13]: i ) randomly choosea degree ≤ D ≤ k from a priori known degree distribution O ( x ) , ii ) using a uniform distribution,randomly choose D distinct input bits, and calculate the encoded bit a i as the XOR-sum of these D bits. The above encoding process defines a sparse bipartite graph connecting encoded (or equivalentlyoutput) nodes to input nodes (see, e.g., Fig. 3). It is seen that the LT encoding process is extremelysimple and has very low energy consumption. Unlike classical linear block and convolutional codes, inwhich the codeword block length is fixed, for the above LT code, n is a variable parameter, resulting ina random variable LT code rate R LTc , kn . More precisely, a n ∈ C is the last bit generated at the output If the mean degree D is significantly smaller than k , then the graph is sparse. UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 of LT encoder before receiving the acknowledgement signal from the sink node indicating termination ofa successful decoding process. This inherent property of LT codes means they can vary their codewordblock lengths to adapt to any wireless channel condition.We now turn our attention to the LT code design in the proposed WSN. As seen in the above encodingprocess, the output-node degree distribution is the most influential factor in the LT code design. On the onehand, we need high degree D in order to ensure that there are no unconnected input nodes in the graph.On the other hand, we need low degree D in order to keep the number of XOR-sum modules for decodingsmall. The latter characteristic means the WSN has less complexity and lower power consumption. Todescribe the output-node degree distribution used in this work, let µ i , i = 1 , ..., k , denote the probability thatan output node has degree i . Following the notation of [26], the output-node degree distribution of an LTcode has the polynomial form O ( x ) , P ki =1 µ i x i with the property that O (1) = P ki =1 µ i = 1 . Typically,optimizing the output-node degree distribution for a specific wireless channel model is a crucial task indesigning LT codes. The original output-node degree distribution for LT codes, namely the Robust Solitondistribution [13], intended for erasure channels, is not optimal for an error-channel and has poor errorcorrecting capability. In fact, for wireless fading channels, it is still an open problem, what the “optimal” O ( x ) is. In this work, we use the following output-node degree distribution which was optimized for aBSC using a hard-decision decoder [27]: O ( x ) = 0 . x + 0 . x + 0 . x + 0 . x + 0 . x + 0 . x +0 . x + 0 . x . (12)The LT decoder at the sink node can recover the original k -bit message B with high probability afterreceiving any (1 + ǫ ) k bits in its buffer, where ǫ depends upon the LT code design [26]. For this recoveryprocess, the LT decoder needs to correctly reconstruct the bipartite graph of an LT code. Clearly, thisrequires perfect synchronization between encoder and decoder, i.e., the LT decoder would need to knowexactly the randomly generated degree D for encoding original D bits. One practical approach suitablefor the proposed WSN model is that the LT encoder and decoder use identical pseudo-random generatorswith a common seed value which may reduce the complexity further. In this work, we assume that thesink node recovers k -bit message B using a simple hard-decision “ ternary message passing ” decoder in UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 a nearly identical manner to the “ Algorithm E ” decoder in [28] for Low-Density Parity-Check (LDPC)codes . The main reason for using the ternary decoder here is to make a fair comparison to the BCHand convolutional codes, since those codes involved a hard-decision prior to decoding. Also, the degreedistribution O ( x ) in (12) was optimized for a ternary decoder in a BSC and we are aware of no better O ( x ) for the ternary decoder in Rayleigh fading channels.To analyze the total energy consumption of transmitting NR LTc bits during active mode period for LTcoded NC-MFSK scheme, one would compute the LT code rate and the corresponding LT coding gain.Let us begin with the case of asymptotic LT code rate, where the number of input bits k goes to infinity. Itis shown that the LT code rate is obtained asymptotically as a fixed value of R LTc ≈ O ave I ave for large valuesof k , where O ave and I ave represent the average degree of the output and the input nodes in the bipartitegraph, respectively (see Appendix II for the proof). When using finite- k LT codes in a fading channel, theinstantaneous SNR changes from one codeword to the next. Consequently, the rate of the LT code for anyblock can be chosen to achieve the desired performance for that block (i.e., we can collect a sufficientnumber of bits to achieve a desired coded BER). This means that for any given average SNR, the LTcode rate is described by either a probability mass function (pmf) or a probability density function (pdf)denoted by P R ( ℓ ) . Because it is difficult to get a closed-form expression of P R ( ℓ ) , we use a discretizednumerical method to calculate the pmf P R ( ℓ ) , Pr { R LTc = ℓ } , ≤ ℓ ≤ , for different values of M .Table III presents the pmfs of LT code rate for M = 2 and various average SNR over a Rayleigh fadingchannel model. To gain more insight into these results, we plot the pmfs of the LT code rate in Fig. 4for the case of M = 2 . It is observed that for lower average SNRs the pmfs are larger in the lower rateregimes (i.e., the pmfs spend more time in the low rate region). Also, all pmfs exhibit quite a spike forthe highest rate, which makes sense since once the instantaneous SNR hits a certain critical value, thecodes will always decode with a high rate. Also, Fig. 5 illustrates the pmf of LT code rates for variousconstellation size M and for average SNR equal to 16 dB. It can be seen that as M increases, the rateof the LT code tends to have a pmf with larger values in the lower rate regions. Description of the ternary message passing decoding is out of scope of this work, and the reader is referred to Chapter 4 in [27] formore details.
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 TABLE IIIP
ROBABILITY MASS FUNCTION OF LT CODE RATE OVER R AYLEIGH FADING CHANNEL FOR VARIOUS AVERAGE
SNR
AND M = 2 .Code pmf pmf pmf pmf pmf pmf pmf pmf pmf pmf pmfRate (6 dB) (8 dB) (10 dB) (12 dB) (14 dB) (16 dB) (18 dB) (20 dB) (22 dB) (24 dB) (26 dB)0.95 0.0101 0.0521 0.1380 0.3067 0.4742 0.6244 0.7429 0.8290 0.8884 0.9281 0.95400.90 0.0180 0.0514 0.0975 0.0978 0.0906 0.0728 0.0536 0.0372 0.0249 0.0164 0.01060.85 0.0222 0.0472 0.0841 0.0656 0.0563 0.0431 0.0307 0.0209 0.0138 0.0090 0.00580.80 0.0310 0.0540 0.0718 0.0613 0.0499 0.0370 0.0239 0.0174 0.0114 0.0070 0.00470.75 0.0442 0.0649 0.0608 0.0617 0.0382 0.0348 0.0189 0.0159 0.0104 0.0054 0.00430.70 0.0438 0.0563 0.0506 0.0466 0.0302 0.0248 0.0148 0.0111 0.0072 0.0043 0.00290.65 0.0375 0.0439 0.0409 0.0331 0.0244 0.0169 0.0114 0.0075 0.0048 0.0031 0.00200.60 0.0370 0.0405 0.0362 0.0285 0.0206 0.0142 0.0095 0.0062 0.0040 0.0026 0.00160.55 0.0355 0.0367 0.0317 0.0243 0.0174 0.0119 0.0079 0.0051 0.0033 0.0021 0.00130.50 0.0373 0.0369 0.0309 0.0233 0.0164 0.0111 0.0073 0.0048 0.0030 0.0019 0.00120.45 0.0345 0.0327 0.0286 0.0197 0.0158 0.0093 0.0065 0.0039 0.0025 0.0016 0.00100.40 0.0397 0.0362 0.0278 0.0210 0.0145 0.0097 0.0061 0.0041 0.0026 0.0016 0.00100.35 0.0395 0.0346 0.0269 0.0193 0.0132 0.0088 0.0057 0.0037 0.0023 0.0015 0.00090.30 0.0421 0.0356 0.0270 0.0192 0.0130 0.0086 0.0056 0.0036 0.0023 0.0015 0.00090.25 0.0440 0.0360 0.0278 0.0187 0.0126 0.0083 0.0054 0.0034 0.0022 0.0014 0.00090.20 0.0496 0.0393 0.0286 0.0197 0.0132 0.0086 0.0056 0.0035 0.0023 0.0014 0.00090.15 0.0541 0.0414 0.0295 0.0201 0.0133 0.0087 0.0056 0.0035 0.0023 0.0014 0.00090.10 0.0658 0.0486 0.0338 0.0227 0.0149 0.0097 0.0062 0.0039 0.0025 0.0016 0.00100.00 0.3138 0.2115 0.1392 0.0903 0.0579 0.0369 0.0235 0.0149 0.0094 0.0059 0.0038 Table IV illustrates the average LT code rates and the corresponding coding gains of LT coded NC-MFSK using O ( x ) in (12), for M = 2 , , , and given P b = 10 − . The average rate for a certainaverage SNR is obtained by integrating the pmf over the rates from to . It is observed that the LT codeis able to provide a huge coding gain Υ LTc given P b = 10 − , but this gain comes at the expense of a verylow average code rate, which means many additional code bits need to be sent. This results in higherenergy consumption per information bit. An interesting point extracted from Table IV is the flexibility ofthe LT code to adjust its rate (and its corresponding coding gain) to suit instantaneous channel conditionsin WSNs. For instance in the case of favorable channel conditions, the LT coded NC-MFSK is able to UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 −3 −2 −1 LT Code Rate p m f SNR ave =10 dBSNR ave =14 dBSNR ave =18 dBSNR ave =24 dB
Fig. 4. The pmf of LT code rate for various average SNR and M=2. achieve R LTc ≈ with Υ LTc ≈ dB, which is similar to the case of uncoded NC-MFSK, i.e., n = k . Inaddition, by comparing the results in Table IV with those in Table II for BCH and convolutional codes,one observes that LT codes outperform the other coding schemes in energy saving at comparable rates.The effect of LT code rate flexibility on the total energy consumption is also observed in the simulationresults in the subsequent section.Unlike BCH and convolution codes in which the active mode duration of coded NC-MFSK is fixed,for the LT coded NC-MFSK, we have non-fixed values for T LTac = NbR
LTc T s . With a similar argument asfor BCH and convolutional codes, the total energy consumption of transmitting NR LTc bits for a given P b is obtained as a function of the random variable R LTc as follows: E LTN = (1 + α ) − (cid:18) − M − M P b (cid:19) M − ! − − L d N ΩΥ LTc NR LTc log M +( P c − P Amp ) M NBR
LTc log M + 1 . P Sy T tr , (13)where the goal is to minimize the average E LTN over R LTc . UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 −3 −2 −1 LT Code Rate p m f M=2M=4M=8M=16
Fig. 5. The pmf of LT code rate for various constellation size M and average SNR=16 dB. V. N
UMERICAL R ESULTS
In this section, we present some numerical evaluations using realistic parameters from the IEEE802.15.4 standard and state-of-the art technology to confirm the energy efficiency analysis of uncoded andcoded NC-MFSK modulation schemes discussed in Sections III and IV. We assume that the NC-MFSKmodulation scheme operates in the f = G t = G r = 5 dBi. Thus for f = L (dB) ,
10 log (cid:16) (4 π ) G t G r λ (cid:17) ≈ dB, where λ , × f = 0 . meters. Weassume that in each period T N , the sensed data frame size N = 1024 bytes (or equivalently N = 8192 bits)is generated, where T N is assumed to be 1.4 seconds. The channel bandwidth is assumed to be B = 62 . KHz, according to IEEE 802.15.4 [19, p. 49]. From bmax b max = BN ( T N − T tr ) , we find that M max ≈ (orequivalently b max ≈ ) for NC-MFSK. The power consumption of the LNA and IF amplifier are considered9 mw [30] and 3 mw [2], [5], respectively. The power consumption of the frequency synthesizer is supposedto be 10 mw [4]. Table V summarizes the system parameters for simulation. The results in Tables II-IVare also used to compare the energy efficiency of uncoded and coded NC-MFSK schemes. UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 TABLE IVA
VERAGE LT CODE RATE AND C ODING GAIN OF LT CODED
NC-MFSK
OVER R AYLEIGH FLAT - FADING MODEL FOR P b = 10 − AND
M=2,4,8,16.M=2 M=4 M=8 M=16Average Average Coding Average Average Coding Average Coding Average CodingSNR (dB) Code Rate Gain (dB) SNR (dB) Code Rate Gain (dB) Code Rate Gain (dB) Code Rate Gain (dB)5 0.2560 25 0 0.0028 33.87 0.0012 36.46 0.0012 38.486 0.3174 24 2 0.0140 31.87 0.0024 34.46 0.0012 36.487 0.3819 23 4 0.0460 29.87 0.0095 32.46 0.0021 34.488 0.4475 22 6 0.1100 27.87 0.0330 30.46 0.0086 32.489 0.5120 21 8 0.2100 25.87 0.0870 28.46 0.0320 30.4810 0.5738 20 10 0.3300 23.87 0.1800 26.46 0.0870 28.4811 0.6315 19 12 0.4600 21.87 0.3000 24.46 0.1800 26.4812 0.6840 18 14 0.5900 19.87 0.4400 22.46 0.3100 24.4813 0.7307 17 16 0.7000 17.87 0.5700 20.46 0.4500 22.4814 0.7716 16 18 0.7800 15.87 0.6800 18.46 0.5800 20.4815 0.8067 15 20 0.8500 13.87 0.7700 16.46 0.6900 18.4816 0.8365 14 22 0.8900 11.87 0.8400 14.46 0.7800 16.4817 0.8614 13 24 0.9200 9.87 0.8800 12.46 0.8400 14.4818 0.8821 12 26 0.9400 7.87 0.9100 10.46 0.8800 12.4819 0.8991 11 28 0.9500 5.87 0.9300 8.46 0.9200 10.4820 0.9130 10 30 0.9600 3.87 0.9500 6.46 0.9400 8.4822 0.9333 8 32 0.9600 1.87 0.9500 4.46 0.9500 6.4824 0.9466 6 34 0.9600 -0.13 0.9600 2.46 0.9500 4.4826 0.9551 4 36 0.9700 -2.13 0.9600 0.46 0.9600 2.4828 0.9606 2 38 0.9700 -4.13 0.9700 -1.54 0.9600 0.4830 0.9640 0 40 0.9700 -6.13 0.9700 -3.54 0.9700 -1.52TABLE VS
YSTEM E VALUATION P ARAMETERS B = 62 . KHz N = − dB P ADC = 7 mw M l = 40 dB P Sy = 10 mw P LNA = 9 mw L = 30 dB P F ilt = 2 . mw P ED = 3 mw η = 3 . P F ilr = 2 . mw P IF A = 3 mw Ω = 1 T N = 1 . sec T tr = 5 µs UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 To t a l E n e r g y C on s u m p t i on ( d B m J ) UncodedBCH CodeConvolutional CodeLT Code d ’T d T d < d T d > d T Fig. 6. Total energy consumption of optimized coded and uncoded NC-MFSK versus d for P b = 10 − . Fig. 6 shows the total energy consumption versus distance d for the optimized BCH, convolutionaland LT coded NC-MFSK schemes, compared to the optimized uncoded NC-MFSK for P b = 10 − . Theoptimization is done over M and the parameters of coding scheme. Simulation results show that for d less than the threshold level d T ≈ m, the total energy consumption of optimized uncoded NC-MFSK isless than that of the coded NC-MFSK schemes. However, the energy gap between LT coded and uncodedNC-MFSK is negligible compared to the other coded schemes as expected. For d > d T , the LT codedNC-MFSK scheme is more energy-efficient than uncoded and other coded NC-MFSK schemes. Also, it isobserved that the energy gap between LT and convolutional coded NC-MFSK increases when the distance d grows. This result comes from the high coding gain capability of LT codes which confirms our analysisin Section IV. The threshold level d T (for LT code) or d ′ T (for BCH and convolutional codes) are obtainedwhen the total energy consumptions of coded and uncoded systems become equal. For instance, using L d = M l d η L , and the equality between (8) and (11) for uncoded and convolutional coded NC-MFSK, UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 we have d ′ T = M Ω( P c − P Amp )(1 + α ) − (cid:18) − M − M P b (cid:19) M − ! − − L M l Υ CCc (1 − R CCc )Υ CCc R CCc − η . (14)It should be noted that the above threshold level imposes a constraint on the design of the physicallayer of some wireless sensor networking applications, in particular dynamic WSNs. To obtain moreinsight into this issue, let assume that the location of the sensor node is changed every T d ≫ T c timeunit, where T c is the channel coherence time. For the moment, let us assume that the sensor node aimsto choose either a fixed-rate coded or an uncoded NC-MFSK based on the distance between sensor andsink nodes. According to the results in Fig. 6, it is revealed that using fixed-rate channel coding is notenergy efficient for short distance transmission (i.e., d < d ′ T ), while for d > d ′ T , convolutional codedNC-MFSK is more energy-efficient than other schemes. For this configuration, the sensor node must havethe capability of an adaptive coding scheme for each distance d . However, as discussed previously, the LTcodes can adjust their rates for each channel condition and have (with a good approximation) minimumenergy consumption for every distance d . This indicates that LT codes can surpass the above distanceconstraint for WSN applications with dynamic position sensor nodes over Rayleigh fading channels. Thischaracteristic of LT codes results in reducing the complexity of the network design as well. Of interest isthe strong benefits of using LT coded NC-MFSK compared with the coded modulation schemes in [2],[6]. In contrast to classical fixed-rate codes used in [2], [6], the LT codes can vary their block lengths toadapt to any channel condition in each distance d . Unlike [2] and [6], where the authors consider fixed-ratecodes over an AWGN channel model, we considered a Rayleigh fading channel which is a general modelin practical WSNs. The simplicity and flexibility advantages of LT codes with an NC-MFSK scheme makethem the preferable choice for wireless sensor networks, in particular for WSNs with dynamic positionsensor nodes. UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 VI. C
ONCLUSION
In this paper, we analyzed the energy efficiency of LT coded NC-MFSK in a proactive WSN overRayleigh fading channels with path-loss. It was shown that the energy efficiency of LT codes is similarto that of uncoded NC-MFSK scheme for d < d T , while for d > d T , LT coded NC-MFSK outperformsother uncoded and coded schemes, from the energy efficiency point of view. This result follows from theflexibility of the LT code to adjust its rate and the corresponding LT coding gain to suit instantaneouschannel conditions for any transmission distance d . This rate flexibility offers strong benefits in using LTcodes in practical WSNs with dynamic distance and position sensors. In such systems and for every valueof distance d , LT codes can adjust their rates to achieve a certain BER with low energy consumption.The importance of our scheme is that it avoids some of the problems inherent in adaptive coding orIncremental Redundancy (IR) systems (channel feedback, large buffers, or multiple decodings), as wellas the coding design challenge for fixed-rate codes used in WSNs with dynamic position sensor nodes.The simplicity and flexibility advantages of LT codes make the LT code with NC-MFSK modulation canbe considered as a Green Modulation/Coding (GMC) scheme in dynamic WSNs.A
PPENDIX IE NERGY C ONSUMPTION OF
BCH D
ECODER
For the sink circuitry, we have an extra energy cost due to the decoding process in a coded NC-MFSKscheme. It is shown in [18, p. 160] that the energy consumption of an BCH ( n, k, t ) decoder per codeword,denoted by ˜ E BCDec , is computed as ˜ E BCDec = (2 nt + 2 t ) ( E add + E mult ) , where E add and E mult represent theenergy consumptions of adder and multiplier in unit of W/MHz, respectively. For instance, the energyconsumption of 0.5 µ W/MHz indicates that the consumed energy per clock cycle is 0.5 pJ. Thus, iffor decoding of each bit we consider 20 clock cycles, the energy consumption is 10 pJ/bit. For thiscase, the total energy consumption a BCH ( n, k, t ) decoder to recover N -bit message M N is obtainedas E BCDec = Nk ˜ E BCDec = Nk (2 nt + 2 t ) ( E add + E mult ) . It is shown in [31] that the energy consumption peraddition or multiplication operation is on the order of pJ per bit. According to the values of n and t inTable II, one can with a good approximation assume that the energy consumption of an BCH decoder is UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 negligible compared to the other circuit components in the sink node.A PPENDIX
IIA
SYMPTOTIC
LT C
ODE R ATE
The proof of the remark is straightforward using the notation of [32] and the bipartite graph conceptsin graph theory. Obviously, the output-node degree distribution O ( x ) induces a distribution on the inputnodes in the bipartite graph. Thus, in the asymptotic case of k → ∞ , we have the input-node degreedistribution defined as I ( x ) , P ∞ i =1 ν i x i , where ν i denotes the probability that an input node has a degree i . In this case, the average degree of the input and output nodes are computed as d I ( x ) dx (cid:12)(cid:12) x =1 , I ave and d O ( x ) dx (cid:12)(cid:12) x =1 , O ave , respectively. Thus, the number of edges exiting the input nodes of the bipartite graph,in the asymptotic case of k → ∞ , is k I ave , which must be equal to n O ave , the number of edges enteringthe output nodes in the graph. As a results, the asymptotic LT code rate is obtained as R LTc = kn ≈ O ave I ave ,which is a deterministic value for given O ( x ) and I ( x ) .R EFERENCES [1] J. Abouei, K. N. Plataniotis, and S. Pasupathy, “Green modulation in proactive wireless sensor networks,”
Submitted to IEEETransactions on Wireless Communications , Sept. 2009.[2] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained modulation optimization,”
IEEE Trans. on Wireless Commun. , vol. 4, no.5, pp. 2349–2360, Sept. 2005.[3] S. Howard, K. Iniewski, and C. Schlegel, “Error control coding in low-power wireless sensor networks: when is ECC energy-efficient?,”
EURASIP Journal of Wireless Communications and Networking, , , no. 2, April 2006.[4] A. Y. Wang, S.-H. Cho, C. G. Sodini, and A. P. Chandrakasan, “Energy efficient modulation and MAC for asymmetric RF microsensorsystems,” in
Proc. of International Symposium on Low Power Electronics and Design (ISLPED’01) . Huntington Beach, Calif, USA,Aug. 2001, pp. 106–111.[5] Q. Tang, L. Yang, G. B. Giannakis, and T. Qin, “Battery power efficiency of PPM and FSK in wireless sensor networks,”
IEEE Trans.on Wireless Commun , vol. 6, no. 4, pp. 1308–1319, April 2007.[6] S. Chouhan, R. Bose, and M. Balakrishnan, “Integrated energy analysis of error correcting codes and modulation for energy efficientwireless sensor nodes,”
IEEE Trans. on Wireless Commun. , vol. 8, no. 10, pp. 5348–5355, Oct. 2009.[7] Y. Sankarasubramaniam, I. F. Akyildiz, and S. W. McLaughlin, “Energy efficiency based packet size optimization in wireless sensornetworks,” in
Proc. of IEEE International Workshop on Sensor Network Protocols and Applications , 2003.[8] H. Karvonen, Z. Shelby, and C. Pomalaza-Raez, “Coding for energy efficient wireless embedded networks,” in
Proc. of IEEEInternational Workshop on Wireless Ad-Hoc Networks , June 2004, pp. 300–304.
UBMITTED TO IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, DECEMBER 2009 [9] X. Liang, W. Li, and T. A. Gulliver, “Energy efficient modulation design for wireless sensor networks,” in Proc. IEEE Pacific RimConf. on Commun., Computers and Signal Processing (PACRIM’07) , Aug. 2007, pp. 98–101.[10] L. Li, R. G. Maunder, B. M. Al-Hashimi, and L. Hanzo, “An energy-efficient error correction scheme for IEEE 802.15.4 wirelesssensor networks,”
Submitted to IEEE Transactions on Circuits and Systems II , 2009.[11] F. Qu, D. Duan, L. Yang, and A. Swami, “Signaling with imperfect channel state information: A battery power efficiency comparison,”
IEEE Trans. on Signal Processing , vol. 56, no. 9, pp. 4486–4495, Sep. 2008.[12] B. Shen and A. Abedi, “Error correction in heterogeneous wireless sensor networks,” in
Proc. of 24th IEEE Biennial Symposium onCommunication . Kingston, Canada, June 2008.[13] M. Luby, “LT codes,” in
Proc. of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS) , 2002, pp. 271–280.[14] A. W. Eckford, J. P. K. Chu, and R. S. Adve, “Low-complexity cooperative coding for sensor networks using rateless and LDGMcodes,” in
Proc. of IEEE International Conference on Communications (ICC’06) . Istanbul, Turkey, June 2006, pp. 1537–1542.[15] C. M. Cordeiro and D. P. Agrawal,
Ad Hoc and Sensor Networks: Theory and Applications , World Scientific Publishing, 2006.[16] S. Mingoo, S. Hanson, D. Sylvester, and D. Blaauw, “Analysis and optimization of sleep modes in subthreshold circuit design,” in
Proc. 44th ACM/IEEE Design Automation Conference , June 2007, pp. 694–699.[17] H. T. Friis, “A note on a simple transnission formula,” in
Proc. IRE , 1946, vol. 34, pp. 245–256.[18] H. Karl and A. Willig,
Protocols and Architectures for Wireless Sensor Networks , John Wiley and Sons Inc., first edition, 2005.[19] IEEE Standards, “Part 15.4: Wireless Medium Access control (MAC) and Physical Layer (PHY) Specifications for Low-Rate WirelessPersonal Area Networks (WPANs),” in
IEEE 802.15.4 Standards , Sept. 2006.[20] L. Barclay,
Propagation of Radiowaves , The Institution of Electrical Engineers, London, second edition, 2003.[21] J. G. Proakis,
Digital Communications , New York: McGraw-Hill, forth edition, 2001.[22] T. S. Rappaport,
Wireless Communications: Principles and Practice , Englewood Cliffs, NJ: Prentice-Hall, second edition, 2002.[23] R. Min and A. Chadrakasan, “A framework for energy-scalable communication in high-density wireless networks,” in
Proc. ofInternational Syposium on Low Power Electronics Design , 2002.[24] IEEE P802.15-09-0329-00-0006, “IEEE P802.15 WG for Wireless Personal Area Networks,” May 2009.[25] IEEE Standards, “Part 16: Air interface for fixed and mobile broadband wireless access systems,” in
IEEE 802.16.e Standards , 2005.[26] A. Shokrollahi, “Raptor codes,”
IEEE Trans. on Inform. Theory , vol. 52, no. 6, pp. 2551–2567, June 2006.[27] J. D. Brown,
Adaptive Demodulation Using Rateless Erasure Codes , Ph.D. Thesis, University of Toronto, 2008.[28] T. J. Richardson and R. L. Urbanke, “The capacity of low density parity-check codes under message-passing decoding,”
IEEE Trans.on Inform. Theory , vol. 47, no. 2, pp. 599–618, Feb. 2001.[29] “Range extension for IEEE 802.15.4 and ZigBee applications,”
FreeScale Semiconductor, Application Note , Feb. 2007.[30] A. Bevilacqua and A. M. Niknejad, “An ultrawideband CMOS low-noise amplifier for 3.1-10.6 GHz wireless receivers,”
IEEE Journalof Solid-State Circuits , vol. 39, no. 12, pp. 2259–2268, Dec. 2004.[31] P. Meir, R. A. Rutenbar, and L. R. Carley, “Exploring multiplier architecture and layout for low power,” in
Proc. of IEEE CustomIntegrated Circuits Conference , 1996, pp. 513–516.[32] O. Etesami and A. Shokrollahi, “Raptor codes on binary memoryless symmetric channels,”