Modulation Selection from a Battery Power Efficiency Perspective

IEEE TRANSACTIONS ON COUNICATIONS, VOL. 58, NO. 7, JULY 21 197 odulation Selection fro a Battery Power Efficiency Perspective Dongliang Duan, Student eber, IEEE, Fengzhong Qu, eber, IEEE, Liuqing Yang, Senior eber, IEEE, Ananthra Swai, Fellow, IEEE, and Jose C. Principe, Fellow, IEEE Abstract In this paper, we copare the battery power efficiencies of various pulse-based odulations widely adopted for their low coplexity. Taking into account circuit odules and battery iperfectness, we establish siple closed-for analytical forulas which can be used to conveniently deterine the relative preference between arbitrary pulse-based odulation pairs in ters of their actual average battery energy consuption. Index Ters Battery power efficiency BPE), pulse-based odulations, nonlinear battery odel. I. INTRODUCTION BATTERY power efficiency BPE) is a critical factor in odern wireless counication systes, especially for wireless sensor networks WSNs) driven by nodes with nonrenewable batteries 1]. ost literature on iproving syste power efficiency treats the batteries as ideal and linear, by assuing that all the battery s capacity can be fully utilized see e.g., 2], 4], 6]). However, the epirically extracted battery odels show that the actual battery discharge is a nonlinear process 5], 9]. Accordingly, using those nonlinear battery odels has the potential to significantly iprove the syste lifetie 3]. The power efficiencies of odulations are already widely studied and well understood 7, Chapter 5]. However, these analyses often neglect the circuit syste and battery inefficiency and the results are all based on the spectral efficiency rather than the battery power efficiency. In 11], a ore realistic odel for the transceiver nodes is proposed to study the BPEs of pulse-position odulation PP) and frequencyshift keying FSK). In this odel, circuit power consuptions, inefficiencies of power aplifier PA) and DC/DC converter and nonlinear property of the battery power dissipation are considered. Under these ore realistic considerations, 11] shows that, though considered to have the sae spectral efficiency 7], the BPEs of PP and FSK are actually very different and their relative BPE heavily relies upon the counication range and the syste design criterion. Later, this Paper approved by D. I. Ki, the Editor for Spread Spectru Transission and Access of the IEEE Counications Society. anuscript received August 27, 28; revised April 11, 29, Septeber 21, 29, and Deceber 1, 29. D. Duan, F. Qu, L. Yang, and J. C. Principe are with the Departent of Electrical and Coputer Engineering, University of Florida, P.O. Box 11613, Gainesville, FL 32611, USA e-ail: {ddl85, jiqufz}@ufl.edu, lqyang@ece.ufl.edu, principe@cnel.ufl.edu). A. Swai is with the Ary Research Laboratory, 28 Powder ill Road, Adelphi, D 2783 e-ail: a.swai@ieee.org). Parts of the results in this paper have been presented at IEEE Wireless Counications & Networking Conference, Budapest, Hungary, April 5-8, 29. This work is in part supported by Office of Naval Research under grant #N14-7-1-868, and Ary Research Office under Contract #W911NF- 7-D-1. Digital Object Identifier 1.119/TCO.21.7.8443 9-6778/1$25. c 21 IEEE sae odel is also adopted in 8] to analyze the BPEs of PP and on-off keying OOK) with iperfect channel state inforation. However, all these works heavily rely on the nuerical analysis and no closed-for relationship has been established between the BPE coparison results and various syste and odulation paraeters. oreover, these results are also liited to certain odulations and lack generality. In this paper, we develop a general analytical fraework to copare the BPEs between any pair of pulse-based odulations in explicit closed for. To do that, we take the difference between the actual average battery energy consuption AABEC) per bit of different odulations. We show that the excessive battery energy consuption due to the nonlinear discharge process can be expressed explicitly as a function of the pulse shape and energy strength. In addition, we prove that this function is quadratic in the K-th power of the counication range, where K is the channel pathloss exponent. Based on properties of quadratic functions, we also exhaustively discuss all possible coparison outcoes. Specific coparison exaples are also presented to validate our analysis and to illustrate our general result. II. SYSTE ODEL A. Coparison etric In this paper, our etric of interest is battery power efficiency which we will siply call efficiency. Hence, we utilize the average actual battery energy consuption per bit as an indicator. The saller it is, the ore efficient the odulation is; and vice versa. B. General Pulse-based odulation odel In this category of odulation schees, the inforation bits are encoded in various characteristics of the transitted pulse, such as the pulse presence, position, and shape etc. Here, we define a general odel for the pulse-based odulation with the following paraeters: 1) the pulse shape pt); 2) the pulse energy E p ; 3) pulse duration T p ; 4) deodulation duration T d depending on the odulation schee; and 5) sybol duration T s. C. Path-Loss Channel Gain The channel can be either deterinistic or Rayleigh fading with additive white Gaussian noise AWGN). In either case, the channel gain factor Gd) depends on the transceiver distance d and is given by 1, Chapter 4]: Gd) =P s /P r = l G 1 d K,whereP s and P r are the transitted and received power of the signal, and the reaining paraeters are defined

198 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 58, NO. 7, JULY 21 in Table I. Accordingly, the relationship between the energy at the transitter E and the energy at the receiver E r is: E/E r = P s /P r = Gd) = l G 1 d K. 1) D. Nonlinear Battery odel As introduced in 11], the nonlinear behavior of the battery discharge process can be captured by P = I ax Vi I in μi) fi)di, where P is the average power consuption of the battery over a battery discharge process, V is the battery voltage, fi) is the density function of the battery discharge current profile during tie period of interest t in,t ax ], μi) is the battery efficiency factor 5] and I ax and I in are respectively the axiu and iniu affordable discharge currents. To facilitate our ensuing analysis, we define the instantaneous power consuption at tie t as P t) = Vit)/μit)). Then, the average power consuption of the battery over the discharge interval t in,t ax ] can be alternatively expressed as: tax tax Vit) P = P t)dt = dt, 2) t in t in μit)) where ω is a positive paraeter. To describe the relationship between the battery efficiency factor μi) and the discharge current i, we adopt the epirical forula obtained in 5]: μi) =1 ωi. 3) III. AVERAGE ACTUAL BATTERY ENERGY CONSUPTION ANALYSIS A. Transitter Battery Energy Consuption We obtain the actual battery energy consuption for transitting a single pulse as the following lea: Lea 1 The total battery energy consuption for transitting a single pulse pt) with duration T p and energy E p is approxiately: E t = ωγ p1 + α) 2 V 2 Ep 2 + 1+α E p + P ct T p, 4) with paraeters defined in Table I. Proof: See Appendix A. In 4), and α ters reflect the influence of the inefficiency of DC/DC converter 1 and the extra PA power loss on the battery energy consuption, respectively. Lea 1 shows that the total battery energy consuption can be decoposed into three parts: 1) The first ter in 4) refers to the excess power loss due to the nonlinear battery discharge process. This ter is proportional to the square of the energy of the transitted signal. In addition, this ter is only affected by the pulse shape through a scaling factor γ p. Notice that, though γ p can be atheatically interpreted as the energy of p t), itisnot the actual energy consuption of a DC battery with a constant voltage V. As detailed in 1 DC/DC converter atches the voltage level of the battery and the syste circuit. If not used, =1,otherwise<1. l TABLE I NOTATIONS channel link argin G 1 gain factor at d =1 K μi) α pt) γ p E p E P ct P cr k 2 k 1 k r 1 r 2 path-loss exponent battery efficiency factor μi) =1 ωi transfer efficiency of the DC/DC converter extra power loss factor of the PA transitted pulse ) 2 dt pt) pt) dt pulse energy average actual battery energy consuption AABEC) per bit transitter circuit power receiver circuit power 2 l G2 1 ω1+α)2 V 2 log 2 l G 1 1+α) ) γp,e2 pr, ) Epr, P ct Tp,+Pcr T d, k 1 ) 2 4k 2 k k1 + k 1 ) 2 4k 2 k 2k 2 k1 k 1 ) 2 4k2 k Appendix A, the scaling factor γ p is the result of noralization of the un-aplified pulse wavefor p t) in order to ensure a constant pure without any circuit energy consuption) and ideal without any battery nonlinearity) battery energy consuption independent of the actual shape the pulse takes. Finally, this ter also depends on the battery paraeter ω, which captures the nonlinear feature of the battery. 2) The second ter in 4) refers to the energy carried by the transitted signal. If there were not effects of the DC/DC converter via ) and the PA via α), it would be exactly the energy of the transitted pulse. 3) The third ter in 4) refers to the circuit energy consuption. It depends on the power of the circuit and the pulse duration T p. B. Receiver Battery Energy Consuption At the receiving node, there is no PA but a low noise aplifier LNA) with nearly constant power consuption. Thus, the current I r = P cr /V ) where P cr is the circuit power consuption at the receiver. In general, I r is very sall, so μi r )=1 ωi r μ ax =111]. The receiving circuit needs to be turned on for the deodulation duration T d. Hence, the total battery energy consuption of the receiving

DUAN et al.: ODULATION SELECTION FRO A BATTERY POWER EFFICIENCY PERSPECTIVE 199 node is: E r = P cr T d. 5) C. Average Actual Battery Energy Consuption of odulation Schees Taking into account the path-loss effect of the channel in Section II-C, we establish the relationship between the AABEC of the odulation schee and the transission distance as follows: Theore 1 The AABEC per bit for a odulation schee is a quadratic function of d K, where d is the transission distance and K is the path loss exponent, i.e.: E d) =k 2 d K ) 2 + k1 d K + k 6) with paraeters defined in Table I. Proof: See Appendix B. IV. GENERAL COPARISON RESULT With Theore 1, we will now look at the energy consuption difference for different odulation schees to see how their energy consuptions copare when transitting at the sae rates while achieving identical syste perforance. The AABEC difference between any two pulse-based odulations is given by: E D d) =k2 2 k1 2 ) d K) 2 +k 2 1 k1 1 )dk +k 2 k1 ) = k2 d K ) 2 7) + k 1 dk + k, where the subscript D eans difference and the superscripts 1 and 2 each refers to a odulation schee in coparison. It is evident according to Theore 1 that the AABEC difference is in a siple quadratic for in d K and the sign of a quadratic function can be deterined by the discriinant = ) k1 2 4k 2 k and the roots when > ) ) k ) r 1 = k1 + 2 1 4k 2 k / ) and r2 = ) k ) k1 2 1 4k 2 k / ). Fro the properties of quadratic functions, we can obtain the following result: Result 1 The relative efficiency between two odulation schees ust follow one and only one of the following cases: C1 One odulation schee is always ore efficient than the other. This occurs when r 1 and r 2 are both negative. C2 One odulation schee is ore efficient when d<d c and the other is ore efficient when d > d c,where d c = axr 1,r 2 ). This occurs when > with r 1 r 2 <. C3 One odulation schee is ore efficient when d d c1,d c2 ), and the other is ore efficient otherwise, where d c1 = inr 1,r 2 ) and d c2 =axr 1,r 2 ).This occurs when > with r 1 and r 2 both positive. Here, d c s refer to critical distances at which the relative battery efficiency of odulations changes sign. TABLE II SIULATION PARAETERS ω =.5 T p =1.33 1 4 s α =.33 μ in =.5 K =3 N /2= 171dB/Hz G 1 =27dB l =4dB V =3.7V P cr =52.5W P ct = 15.8W =.8 dc ) 35 3 25 2 15 1 1-3 1-4 BER P e) 1-5 32 16 Fig. 1. Coparison results of -PP and OOK under deterinistic AWGN channel as a function of PP odulation size and BER requireent. Below the surface: OOK-advantageous region; above the surface: PPadvantageous region. V. NUERICAL ANALYSIS WITH SPECIFIC COPARISON PAIRS Fro Section IV, we see that all possible fors of BPE coparison result in an explicit closed for. In this section, the nuerical analysis for specific coparison pairs are obtained through Result 1. The paraeters used in the nuerical plots are listed in Table II 2], 11]. For fair coparisons, we ensure that each odulation pair has identical bandwidth occupancy and bandwidth efficiency while achieving the sae bit error rate BER) perforance. A. OOK and -PP under Deterinistic AWGN Channel As discussed in 8], OOK and -PP need to use the sae pulse pt) with duration T p for sae bandwidth occupancy and OOK needs to be duty cycled to guarantee the sae bandwidth efficiency. When copared with -PP, the OOK duty cycling factor is log 2 and thus for OOK, Td O = T p = log 2 T s O. For -PP, T d P = Ts P = T p. Without loss of generality, we use the rectangular pulse in base band. Hence, γpr, O =and γpr,1 O = γpr, P =1/T p for any. In AWGN channels, the sybol energies Eb O and Eb P to achieve certain prescribed BER P e are readily available through existing forulas 7]. Plugging in all the paraeters in Table I, we will find that the coparison results falls under C2 in Result 1. Accordingly, the preference region is plotted in Fig. 1. The closed-for expressions for the critical transission distances are neat and convenient to use in real applications. However, in the derivation process, we used the approxiation ωit) 1, where it) is the instantaneous current drawn 8 4 2

191 IEEE TRANSACTIONS ON COUNICATIONS, VOL. 58, NO. 7, JULY 21 dc ) 35 3 25 2 15 1 with approx. w/o approx. =16 =2 =4 5 1 5 1 4 1 3 BER P e) Fig. 2. Coparison of d c obtained by our closed for expression and via nuerical search. dc ) 1 8 6 4 2 1 3 1 4 BER P e) 1 5 2 4 8 Fig. 3. Coparison results of -FSK and -PP under Rayleigh fading channel as a function of odulation size and BER requireent. Below the surface: PP-advantageous region; above the surface: FSK-advantageous region. fro the battery. To verify the validity of this approxiation, fro Fig. 2, we see that our results with approxiation solid line with circle) are quite accurate when copared with the results obtained by nuerical search without any approxiation dotted line with star). B. -FSK and -PP under Rayleigh Fading Channel Treating the carriers with different frequencies as pulse shapers, we can consider -FSK as a pulse-based odulation schee with inforation ebedded in the shape of the pulse. The signal pulse transitted for sybol {, 1,..., 1} is p F t) =sin2πf c +2 1) f)t),t,t s ],where f c is the central carrier frequency and f is the frequency spacing. For -PP, all possible signals utilize the sae pulse shaper and the inforation is conveyed by the signal position. To copare with -FSK, we take the pulse in pass band, then the basic pulse is p P t) =sin2πf c t with t,t s / ], 16 32 where f c is the carrier frequency. Correspondingly, the pulse transitted for sybol is p P t) ) =pp t Ts. It is shown in 11] that with f =1/T s, -FSK and - PP have identical bandwidth occupancy and bandwidth efficiency. Hence, no duty cycling is needed. For both odulation schees, Td F = T d P = T s. To copare these two odulation schees, we need to first obtain the pulse shape factors γp F and γp P. According to Section II-B and Lea 1, for -FSK:γp, F = Ts sin2πfc+2 1) f)t)]2 dt = Ts Ts sin2πfc+2 1) f)t] dt]2 2 / T s ) 2 2π = 2π 2 T s. Siilarly, for -PP, we have γp, P = 2π2 2π2 T s/ = T s = γp, F. PP and FSK have the sae required energy at the receiver, i.e., Epr F = EP pr = E sr, wheree sr is the required average sybol energy to get the objective BER perforance P e.in Rayleigh fading channels, this value can be obtained fro the forula given in 11]. Substituting all these paraeters into Table I, we obtain k2 FvP =1 ) 2π2 2 l G2 1 ω1+α)2 T sv 2 log 2 E2 sr <,kfvp 1 =,k FvP = 1 P ctt s 2 log 2 >. Accordingly, the coparison between PP and FSK also falls under C2 in Result 1 and the preference region is plotted in Fig. 3. Interestingly, although ωit) 1 as used in our analysis and verified in Section V-A, the coparison between -PP and -FSK here shows that this slight nonlinearity of the battery is actually not negligible. If no battery nonlinearity were considered, then we would expect that -PP is always preferred since it costs less transitter circuit power consuption. However, coparison result shows that -FSK is preferred when transission distance is sufficiently large. In addition, for ost counication systes in real applications, the critical transission distances given by Fig. 1 and Fig. 3 are within the typical transission distance for exaple, typical transission distance of wireless sensor networks is around 1 to 1 eters). This confirs that our neat analytical forulas have practical values in real applications. VI. CONCLUSIONS In this paper, we copared pulse-based odulations fro the battery power efficiency perspective. First, we adopted the realistic nonlinear battery odel and took into account all coon power consuption factors in the syste odel. Then, we analyzed the battery power efficiency between arbitrary odulation pairs and found that the coparison results can be characterized by the transission distances in an explicit closed for. The nuerical coparison exaples are presented to validate our analysis and to illustrate our result. APPENDIX A PROOF OF LEA 1 Denote the output current as pt). 2. Then, at the output stage, the total energy consuption E o can be obtained as E o = T p V pt) dt. Due to the inefficiency of the DC/DC 2 In this paper, we assue PA of type AB is used. Thus the output current has the sae shape as the pulse pt). If PA of other type is used, the current ay take a distorted shape p t). Accordingly, the γ p in this lea should be replaced by γ p calculated by the distorted pulse shape p t).

DUAN et al.: ODULATION SELECTION FRO A BATTERY POWER EFFICIENCY PERSPECTIVE 1911 converter and the extra power loss of the power aplifier, the output pulse energy is E p = E o /1 + α). Define p t) = pt)/ T p pt) dt, thenpt) = 1+α)Ep V p t). In addition to the current induced by the transitted wavefor, the circuit power consuption will also induce a current I ct = P ct /V ). Thus, the instantaneous current running through the battery is it) = pt) + I ct. Fro 2), we can obtain the total battery energy consuption in one pulse duration as follows: E t = = V P t)dt = it) dt V 1 ωit) Vit) μit)) dt it)1 + ωit))dt = V pt) + I ct)1+ω pt) +I ct)] dt ] = V 1+2ωI ct) pt) dt+1+ωi ct) I ctdt+ω pt)) 2 dt ] V pt) dt + I ctdt + ω pt)) 2 dt 1 + α)ep = V + Pct V V Tp + ω 1 + α)2 Ep 2 ] γ 2 V 2 p = ωγp1 + α)2 2 V Ep 2 + 1+α E p + Pct Tp, where substitution γ p := T p p t)) 2 dt is used. In the above derivation, the first approxiation coes fro the fact that practically ωit) 1, and thus 1/1 ωit)) 1+ωit), and the second approxiation follows fro ωi ct 1. APPENDIX B PROOF OF THEORE 1 Fro 4) and 5), we see that the total battery energy consuption for one pulse counication is: E p =E t +E r = ωγ p1 + α) 2 V 2 Ep 2 + 1+α E p + P ctt p +P cr T d As introduced in Section II-B, a odulation schee with size will choose fro pulses to transit log 2 bits of inforation, so the average actual battery energy consuption per bit transission is: where the subscript {, 1,..., 1} refers to the sybol transitted. Notice that fro 1), E p, = l G 1 d K E pr,, thus: )] l 2 G 2 E = 1ω1+α) 2 γ p,e V 2 pr, 2 d 2K log 2 + l G 11+α) + Pct )] E pr, d K Tp,+Pcr T d,. This is a quadratic function of d K and the coefficients are as stated in Theore 1. REFERENCES 1] I. F. Akyildiz, W. Su, Y. Sankarasubraania, and E. Cayirci, A survey on sensor networks," IEEE Coun. ag., vol. 4, no. 8, pp. 12-114, Aug. 22. 2] S. Cui, A. J. Goldsith, and A. Bahai, Energy-constranined odulation optiization," IEEE Trans. Coun., vol. 4, no. 5, pp. 2349-236, Sep. 22. 3] K. Lahiri, A. Raghunathan, S. Dey, and D. Panigrahi, Battery-driven syste design: a new frontier in low power design," in Proc. Intnl. Conf. VLSI Design, Bangalore, India, Jan. 22, pp. 261-267. 4] J. N. Lanean, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: efficient protocals and outage behavior," IEEE Trans. Inf. Theory, vol. 5, no. 12, pp. 362-38, Dec. 24. 5]. Pedra and Q. Wu, Battery-powered digital COS design," IEEE Trans. VLSI Syst., vol. 1, no. 5, pp. 61-67, Oct. 22. 6] Y. Prakash and S. K. S. Gupta, Tie to failure estiation for batteries in portable systes," in Proc. Wireless Coun. Netw. Conf., New Orleans, LA, ar. 23, pp. 212-217. 7] J. Proakis, Digital Counications, 4th edition. New York: cgraw- Hill, Feb. 21. 8] F. Qu, D. Duan, L. Yang, and A. Swai, Signaling with iperfect channel state inforation: a battery power efficiency coparison," IEEE Trans. Signal Process., vol. 56, no. 9, pp. 4486-4495, Sep. 28. 9] D. Rakhatov and S. Vrudhula, Tie to failure estiation for batteries in portable systes," in Proc. International Syp. Low Power Electron. Design, Huntington Beach, CA, USA, Aug. 21, pp. 88-91. 1] T. S. Rappaport, Wireless Counications: Principles and Practice, 2nd edition. Prentice-Hall, 22. 11] Q. Tang, L. Yang, G. B. Giannakis, and T. Qin, Battery power efficiency of PP and FSK in wireless sensor networks," IEEE Trans. Coun., vol. 6, no. 4, pp. 138-1319, Apr. 27. 1 1 E = E p, log 2 1 = log 2 ωγp, 1 + α) 2 V 2 Ep, 2 +1+α E p,+ P ] ctt p, +P cr T d,