The Impact of Hard-Decision Detection on the Energy Efficiency of Phase and Frequency Modulation

1 The Impact of Har d-Decision Detection on the Ener gy Ef ficienc y of Phase a nd Frequenc y Modulation Mustafa Cenk Gursoy Abstract — The central design challenge in next generation wireless systems is to hav e these systems operate at hi gh bandwidths and provide high d ata rates while b eing cognizant of the energy consumption lev els especially in mobile a pplications. Since communicating at very high data rates p rohibits obtaining high bit resolutions from the analog-to -digital ( A/D) con v erters, analysis of the energy efficiency under the assumption of hard-decision detection is called for to accurately predict the perfo rmance lev els. In this paper , transmission ov er the add itive white Gaussian noise (A WGN) channel, and coherent and noncoherent fading channels is considered, and th e impact of hard-decision detection on th e energy efficiency of phase and frequency modulation s is inv estigated. Energy efficiency is analyzed by studying the capacity of th ese modulation schemes and t he energy required t o send on e bit of information reliably in the lo w signal-to-noise ratio ( SNR ) regime. The capacity of hard-decision-d etected phase and frequency modulations is characterized at low SNR lev els t hrough closed-form expressions fo r the first and second deriv ativ es of the capacity at zero SNR . Subsequ ently , bit energy requirements in the low- SNR regime are identified. Th e increases in the bit energy in curred by hard-decision detection and ch annel fading ar e quantified. Moreo ver , practical design guidelines f or th e se lection of t he constellation size are drawn fr om the analysis of the spectral efficiency–bit energy tradeoff. Index T erms: Bit energy , spectral efficiency , A W GN channel, fading channels, phase-shift k eying, fre quency-shift keying, o n-off keying, hard-decision detection. I . I N T R O D U C T I O N Energy efficiency is of param ount importance in m any commun ication sy stems and particularly in mo bile wireless systems du e to the scarcity of en ergy resources. Energy efficiency can be m easured by the energy req uired to send one inf ormation bit reliably . It is well-k nown that for Gaussian channels sub ject to average power con straints, the minimum received b it energy normalized by th e noise spe ctral level is E b N 0 min = − 1 . 59 dB regar dless o f th e availability o f channel side infor mation (CSI) at the receiver ( see e. g., [ 1] – [5], and [8 ]). Golay [1] showed that this min imum bit energy can be achieved in the additive white Gaussian noise (A WGN) chan nel by pu lse p osition m odulatio n (PPM) with Mustaf a Cenk Gursoy is with the Depart ment of Electri cal Engineer- ing, Univ ersity of Nebraska- Lincoln, Lincoln, NE 68588 (e-mail : gu r- soy@en gr .unl .edu). This w ork w as support ed in pa rt by the NSF CAREER Grant CCF-0546384. The material in this paper was presente d in part at the IE EE Internat ional Symposium on Information Theory (ISIT), Nice, France, in June 2007, and at the IEEE Interna tional Symposium on Information Theory (ISIT), T oront o, Canada , in July 2008. vanishing duty cycle wh en th e receiver employs thre shold detection. Inde ed, T urin [2] proved that a ny orth ogona l M - ary modulation scheme with envelope detection at the receiver achieves the normalize d bit energy of − 1 . 59 dB in the A WGN channel as M → ∞ . It is fur ther shown in [ 3] and [4] that M -ary orth ogonal frequ ency-shift keying (FSK) achieves this min imum bit energy asym ptotically as M → ∞ also in nonco herent fading ch annels wh ere n either the receiv er nor the transmitter knows the fading coefficients. These studies demonstra te the asym ptotical high energy efficiency o f or- thogon al signalin g even wh en the receiver perf orms h ard- decision detection . As also well-k nown by now in the digital commun ications literature [23], these results are sho wn by proving that th e error probabilities of orthogonal sign aling can b e mad e arbitrarily sma ll as M → ∞ as long as the normalized bit energy (or e quiv alently SNR per b it) is gr eater than − 1 . 59 dB. As indicated by the un boun ded g rowth of M , the minimum b it energy is in g eneral achieved at infin ite bandwidth or eq uiv alently as the spectral efficiency ( rate in bits per second divided by b andwidth in Hertz) g oes to zero. Indeed fo r average p ower limited chann els, the b it ene rgy required fo r r eliable co mmun ication decreases monoton ically with incre asing b andwidth [6], [ 8]. This is th e fun damental bandwidth -power tradeoff. Recently , V erd ´ u [8] has offered a more sub tle analy sis of the tr adeoff of bit energy versus spec- tral efficiency . In this work, the wideband slop e, wh ich is the slope of the spectral efficiency c urve a t zero spectral ef ficiency , has emerged as a new analysis too l to measu re energy and bandwidth efficiency in the low-po wer regime. It is sho wn that if the receiver ha s perfect knowledge of the f ading coefficients, quaterna ry phase-shift keying (QPSK) is an o ptimally efficient modulatio n scheme ach ieving both the minimum bit en ergy of − 1 . 59 dB an d th e optimal wideb and slop e in the low SNR regime. This ind icates that besides orth ogona l s ignaling, phase modulatio n is also we ll-suited for en ergy efficient o peration . Howe ver , it should be noted that asympto tic efficiency of QPSK hold s under the assumption that the re ceiv er p erform s soft detec tion. V e rd ´ u [8] ha s also provided expression s for the minimum bit ene rgy an d wideband slope of the quantized QPSK. W e n ote that phase mod ulation is a widely u sed technique fo r infor mation transmission, an d the p erforma nce of coded p hase m odulatio n has been of interest in the research commun ity since the 1960s. One of the early works was condu cted in [1 0] where the cap acity and erro r exponents of a continuo us-phase m odulated system, in which the transmitted phase can assume any value in [ − π , π ) , is studied. More recent studies include [4], [9], and [11]– [14]. As discussed a bove, high energy efficiency r equires oper a- tion in the wideb and regime in which th e spectral efficiencies are low . This is is ach ieved by either decr easing the data rates or increasin g the bandwidth. If the system has large bandwidth , th en the data rates ar e hig h. For instance, if the total sign al p ower is P = 1 m W and the ba ndwidth is B = 1 GHz, then the capacity of the A WGN channel is C = B log 2  1 + P N 0 B  ≈ 27 . 9 Gb its/s 1 . If the bandwidth is increased to B = 10 GHz, the capacity becomes 245.7 Gbits/s. Similarly , high rates are also achieved in fading chan nels when the av ailable bandwidth is large. For instance, in cu rrent practical app lications, wideband CDMA and ultrawideban d systems offer hig h data rate ser vices by using lar ge b andwidth s [27]. Add itionally , op erating at high b andwidths and providing high data rates wh ile conserving the energy in mo bile ap plica- tions are the key fea tures of next g eneration wireless systems which have the g oal of offering mobile multimedia access. For instance, on e o f the features o f fo urth g eneration (4 G) systems will be th e ab ility to supp ort mu ltimedia serv ices at low transmission cost [27, Chap. 23, available on line]. On the other hand, at these very hig h transmission rates, obtaining high bit resolutions from A/D converters may eith er be n ot possible o r prohib iti vely expensive. Therefor e, in such cases, the pe rfor- mance of soft detection will be a lo ose u pper boun d on the actual system perfo rmance, and analysis un der the assumption of hard-d ecision d etection will provide mor e faithful resu lts. Moreover , even if the data r ates are not high, h ard-dec ision detection of the received sig nals migh t be prefer red when reduction in the computa tional burden is required [23]. Such a requirem ent, for instance, may be enforce d in sensor networks that consist of lo w-cost, low-power , small- sized sensor nod es [15]. The refore, it is timely and practically relevant to stud y the energy ef ficiency of phase and freq uency mo dulations in the wideband regime when the receiver perfo rms hard-decision detection. The contributions of this p aper ar e the following: 1) W e ob tain closed-form expressions for the first and second deriv ati ves at zero SNR of the hard-dec ision- detected PSK capacity fo r arbitrary modulatio n size M . 2) W e find the bit energy required at ze ro s pectral effi ciency and wideban d slope when PSK is emp loyed at the transmitter . The analysis is initially perfor med for n on- coheren t fading chann els, and sub sequently sp ecialized to the A WGN and coheren t fading channels. W e quan tify the incre ase in the bit energy req uirements due to h ard- decision detection and chann el fading. 3) W e stu dy the energy ef ficiency of hard-decision-d etected on-off frequen cy-shift k eying (OOFSK) m odulation which is a general or thogon al signaling scheme that combines orthog onal FSK an d on-off keying (OOK) and introdu ces peakedness in both tim e and frequen cy . W e show that the b it energy re quiremen ts grow without bound with de creasing SNR if the peakedness in both time and fre quency is limited . W e identify the imp act upon the energy efficiency of the n umber o f orthog onal 1 W e have assumed that N 0 = 4 × 10 − 21 W/Hz [24]. frequen cies, M , an d the du ty cycle of OO K. W e prove a sufficient co ndition on how fast the duty cycle has to vanish with decreasin g S NR in order to app roach the fundam ental bit energy limit o f − 1 . 59 d B. The organization o f the rest of the paper is as fo llows. In Section II, we de scribe the c hannel model. The energy efficiency of p hase mod ulation is inv estigated in Section II I. M -ary OOFSK mo dulation and its special case M -a ry FSK modulatio n are co nsidered in Section I V. Section V in cludes our conclu sions. I I . C H A N N E L M O D E L W e co nsider the following chan nel mod el r k = h k s x k + n k k = 1 , 2 , 3 . . . (1) where x k is the discrete inpu t, s x k is the transmitted signal when th e input is x k , and r k is the received signal durin g the th e k th symbol d uration. h k is th e ch annel gain . h k is a fixed constant in unfaded A WGN channels, while in flat fading channels, h k denotes the fading coef ficient. { n k } is a sequence of indep endent an d identically distributed (i.i.d.) zero-m ean circularly symmetric Gaussian random vectors with co variance matrix E { nn † } = N 0 I w here I den otes the iden tity matrix. W e assume that the system has an av erage energy constrain t of E {k s x k k 2 } ≤ E ∀ k . At the tr ansmitter, if M -ar y PSK modulation is em ployed for transm ission, the discrete inp ut, x k , takes values fro m { 1 , . . . , M } . If x k = m , then the transmitted signal in the k th symbol dura tion is s x k = s m = √ E e j θ m (2) where θ m = 2 π ( m − 1) M for m = 1 , . . . , M , is o ne o f the M phases av ailable in the constellation. I n th e case of PSK modulatio n, since s x k is a one-dimensiona l comp lex po int, we opted to no t use the boldface repr esentation. Accordin gly , the o utput r k and the noise n k are one complex-d imensional points. T he rece iv er is assumed to perf orm hard-d ecision detection. Therefore, each rece iv ed signal r k is mapped to o ne of the poin ts in the co nstellation set { √ E e j 2 π ( m − 1) / M , m = 1 , . . . , M } b efore going thro ugh th e deco der . W e assume that maximum likelihood decision r ule is used at the detecto r . In [21], we have in troduce d the o n-off frequ ency-shift key- ing (OOFSK) mo dulation by overlaying frequen cy-shift ke ying (FSK) o n on-off ke ying ( OOK). In M -ary OOFSK mo dulation , the transmitter either sends no sign al with pro bability 1 − ν or sends one o f M ortho gonal FSK signals each with prob ability ν / M . Hence, ν ∈ (0 , 1] can be seen a s th e du ty cycle of th e transmission. In this case, the d iscrete inpu t takes values from x k ∈ { 0 , 1 , 2 , . . . , M } . If x k = 0 , then there is no tran smission and the g eometric r epresentation o f th e tran smitted signal is the M - complex dimension al vector s 0 = (0 , 0 , . . . , 0 ) . On the other han d, if x k = m 6 = 0 , an FSK signa l is sent an d the geometric representatio n is given by s x k = s m = ( s m, 1 , s m, 2 , . . . , s m,M ) m = 1 , 2 , . . . , M , ( 3) where s m,m = p E /ν e j θ m and s m,i = 0 for i 6 = m . The phases θ m can be ar bitrary . Note that in M -a ry OOFSK 2 modulatio n, we hav e M + 1 po ssible inp ut signals inclu ding the no sig nal transmission. Therefor e, n o signal tra nsmission being a part of the mo dulation also con veys a message to the r eceiv er . While FSK signals h av e ene rgy E /ν , the average energy o f OOFSK mod ulation is E . Hence, the peak- to-average power r atio of signaling is 1 /ν . In the O OFSK transmission and reception m odel, th e r eceiv ed signal r k and n oise n k are also M -dimen sional. It is assumed that th e r eceiv er per forms energy detection on the received vecto r r k . Fin ally , n ote that OOFSK is a general ortho gonal signaling for mat and specializes to regular orthogo nal FSK if ν = 1 , and to OOK if M = 1 and ν 6 = 1 . W e remark that in both PSK and OOFSK cases, the channel, after hard -decision de tection, can be regard ed as a discre te channel with finitely many inpu ts and outputs. Hencefo rth, capacity and achiev able rate expressions thro ughou t the paper will be obtain ed con sidering these d iscrete c hannels. I I I . E N E R G Y E FFI C I E N C Y O F P H A S E M O D U L A T I O N A. Noncoher ent Rician F ad ing Chann els In this section , we study the perfor mance of ph ase modu- lated signals when they are hard -decision detected. W e initially consider transmission of PSK signals over non coheren t Rician channels in which neither receiv er n or transmitter k nows the fading coefficients. Results fo r this channe l are su bsequently specialized to obtain th e perfor mance re sults of PSK in un - faded A WGN ch annels and coh erent fadin g channels. Hence, we fir st assume that the fading coefficients { h k } , who se realizations are unk nown at the transmitter and r eceiver d ue to the no ncoher ence a ssumption, ar e i.i.d. prop er complex Gaussian rand om variables with m ean E { h k } = d 6 = 0 and variance E {| h k − d | 2 } = γ 2 . W e further assume that th e channel statistics, and hen ce d and γ 2 , ar e known bo th at the transmitter and r eceiv er . Note that d 6 = 0 is r equired beca use phase cannot b e used to transmit inf ormation in a no ncoher ent Rayleigh fading ch annel wh ere d = 0 . In th e no ncoher ent Rician chann el mo del, the co nditional probab ility d ensity fun ction (pd f) o f the chann el output given the input is a co nditionally com plex Gaussian pdf and is gi ven by 2 f r | s m ( r | s m ) = 1 π ( γ 2 | s m | 2 + N 0 ) e − | r − ds m | 2 γ 2 | s m | 2 + N 0 . (4) Recall that { s m = √ E e j θ m } are the PSK signals and h ence | s m | = √ E for all m = 1 , . . . , M . Due to this constant magnitud e prop erty , it ca n be ea sily shown th at the max imum likelihood d etector selects s k as the transmitted signal if 3 Re ( rs ∗ k ) > Re ( rs ∗ i ) ∀ i 6 = k (5) where s ∗ k is th e complex conju gate of s k , an d Re () denotes the operation that selects the r eal part. W e denote the signal at th e output of the detector by y and assume that y ∈ { 1 , . . . , M } . Note that y = l fo r l = 1 , . . . , M means tha t the d etected 2 Since the channel is memoryle ss, we hencefo rth, wit hout loss of genera lity , drop the ti me index k in the equat ions for the sake of simpl ificati on. 3 The decision rule is obtained when we assume, wit hout loss of generalit y , that d = | d | . signal is √ E e j 2 π ( l − 1) / M . Under the decision ru le (5), the decision region for y = l is the two-dimensional region D l =  r = | r | e j θ : (2 l − 3) π M ≤ θ < (2 l − 1) π M  (6) for l = 1 , 2 , . . . , M . With har d-decision d etection at the receiver , the resulting ch annel is a symmetric, discrete, mem- oryless chann el with inpu t x ∈ { 1 , . . . , M } and outp ut y ∈ { 1 , . . . , M } . The tran sition p robab ilities are given by P l,m = P ( y = l | x = m ) (7) = P  (2 l − 3) π M ≤ θ < (2 l − 1) π M | x = m  (8) = Z (2 l − 1) π M (2 l − 3) π M f θ | s m ( θ | s m ) dθ (9) where f θ | s m ( θ | s m ) is the conditional probability den sity func- tion of the ph ase of th e received signal given th at the inpu t is x = m , and h ence the transmitted sign al is s m . It is well- known that th e capacity of this symmetr ic channel is achieved by equ iprobab le inputs an d the r esulting cap acity expression [25] is 4 C M ,nc ( SNR ) = lo g M − H ( y | x = 1) (10) = log M + M X l =1 P l, 1 log P l, 1 (11) where SNR = E N 0 , H ( · ) is the en tropy f unction, and P l, 1 = P ( y = l | x = 1) . In order to evaluate the capacity o f general M -ary PSK transmission with a hard-dec ision detector, th e transition p robab ilities P l, 1 = P ( y = l | x = 1) = Z (2 l − 1) π M (2 l − 3) π M f θ | s 1 ( θ | s 1 ) dθ (12) should be computed. Starting fr om (4) and noting that the condition al joint mag nitude and phase distribution is gi ven by f | r | ,θ | s 1 ( | r | , θ | s 1 ) = | r | π ( γ 2 | s 1 | 2 + N 0 ) e − | r | 2 + | d | 2 | s 1 | 2 − 2 | r || d || s 1 | cos θ γ 2 | s 1 | 2 + N 0 (13) where, witho ut loss of gen erality , we hav e assumed tha t d = | d | , we can easily find that for θ ∈ [0 , 2 π ) , f θ | s 1 ( θ | s 1 ) is giv en by (14) on the next page where Q ( x ) = R ∞ x 1 √ 2 π e − t 2 / 2 dt is the Gaussian Q -fun ction 5 . Since f θ | s 1 is r ather complicated , closed-for m capacity expr essions in term s of Q -f unctions a re av ailable only for the spe cial cases of M = 2 and 4: C 2 ,nc ( SNR ) = log 2 − h Q s 2 | d | 2 SNR γ 2 SNR + 1 !! , and (1 5) C 4 ,nc ( SNR ) = 2 C 2 ,nc  SNR 2  (16) where h ( x ) = − x log x − (1 − x ) log(1 − x ) is the b inary 4 Throughout the paper , log is used to denote the logari thm to the base e , i.e., the natural logarit hm. Additionall y , the subscrip t ”nc” in C M ,nc signifies the nonco herent channel. 5 See also [9 ] and references therein for a similar formula of the phase probabil ity density function deri ve d for the A WGN channel. 3 f θ | s 1 ( θ | s 1 ) = 1 2 π e − | d | 2 SNR γ 2 SNR +1 + s | d | 2 SNR π ( γ 2 SNR + 1) cos θ e − | d | 2 SNR γ 2 SNR +1 sin 2 θ 1 − Q s 2 | d | 2 SNR γ 2 SNR + 1 cos 2 θ !! (14) entropy fu nction. For the o ther cases, the cha nnel capa city can only be foun d th rough numerical integration and co mputation . On the othe r hand, the beh avior of the capacity in th e low- SNR regime can be accur ately pred icted thro ugh the secon d- order T aylor series expansion of the capacity 6 : C M ,nc ( SNR ) = ˙ C M ,nc (0) SNR + ¨ C M ,nc (0) SNR 2 2 + o ( SNR 2 ) where ˙ C M ,nc (0) an d ¨ C M ,nc (0) de note th e first an d sec- ond d eriv ati ves, r espectiv ely , o f the channel capacity (in nats/symbol) with respec t to SNR at SNR = 0 . In the fo llowing result, we p rovide closed-for m expressions for these deriv a- ti ves. No te that the wideb and regime in which SNR per unit bandwidth is small can equivalently be regarded as the low- SNR regime. Theor em 1 : The first and second de riv ati ves of C M ,nc ( SNR ) in nats per symb ol at SNR = 0 are given by ˙ C M ,nc (0) = ( 2 | d | 2 π M = 2 M 2 | d | 2 4 π sin 2 π M M ≥ 3 , and ¨ C M ,nc (0) =          8 3 π  1 π − 1  | d | 4 − 4 | d | 2 γ 2 π M = 2 ∞ M = 3 4 3 π  1 π − 1  | d | 4 − 4 | d | 2 γ 2 π M = 4 ψ ( M ) | d | 4 − | d | 2 γ 2 2 π M 2 sin 2 π M M ≥ 5 (17) respectively , where ψ ( M ) = M 2 16 π 2  (2 − π ) sin 2 2 π M + ( M 2 − 4 π ) sin 4 π M − 2 M sin 2 π M sin 2 π M  . (18) Pr oof : See Appendix A. The derivati ve expressions in ( 17) are in closed-fo rm and can be comp uted easily . There fore, the low- SNR approximatio n of the capacity of M -ar y PSK can be read ily obtained from C M ,nc ( SNR ) ≈ ˙ C M ,nc (0) SNR + ¨ C M ,nc (0) SNR 2 2 . The following corollary provide s th e asym ptotic beh avior as M → ∞ . In th is asympto tic regime, the tran smitted signal is the contin uous phase which can take any value in [ − π , π ) . Cor ollary 1: In the lim it a s M → ∞ , the fir st an d second deriv ati ves of the capacity at zero SNR converge to lim M →∞ ˙ C M ,nc (0) = π | d | 2 4 and (19) lim M →∞ ¨ C M ,nc (0) = ( π 2 − 8 π + 8) | d | 4 16 − | d | 2 γ 2 π 2 . (20) In the low-power regime, th e trade off between b it energy and spectr al efficiency is a key measure of perf ormanc e [8]. The n ormalized energy per b it can be obtained f rom E b N 0 = 6 It can be easily seen from the smoothness and boundedness of f θ | s 1 in (14) that C M ,nc ( SNR ) is cont inuous and dif ferenti able in SNR. SNR log 2 C ( SNR ) where C ( SNR ) is the channel capacity in nats/symbol. The maxim um ach iev able spectr al efficiency in bits/s/Hz is giv en b y C  E b N 0  = C ( SNR ) log 2 e b its/s/Hz if we, with out loss of g enerality , assume that one symb ol o ccupies a 1s × 1Hz time-fr equency slot. T wo imp ortant notio ns regarding the spectral-efficiency/bit-energy tradeo ff in the low power regime are the bit-e nergy required at zero sp ectral efficiency and wideband slope, given by E b N 0     C =0 = log 2 ˙ C (0) , and S 0 = 2( ˙ C (0)) 2 − ¨ C (0) , (21) respectively . The wide band slope, S 0 , provide s the slope of the spectral ef ficiency cu rve C ( E b / N 0 ) at zero spectral efficiency [ 8]. Therefo re, E b N 0    C =0 and S 0 constitute a linear approx imation o f th e spe ctral efficiency cu rve in the low- SNR regime, i.e., C  E b N 0  = S 0 10 log 10 2 E b N 0     dB − E b N 0     C =0 ,dB ! + ǫ (22) where E b N 0    dB = 10 log 10 E b N 0 and ǫ = o  E b N 0 − E b N 0    C =0  , and char acterize the spectr al-efficiency/bit-energy tradeo ff at low spectr al ef ficiencies. Hence, these quan tities enable u s to analyze the en ergy efficiency and inves tigate the interac - tions b etween spec tral an d energy efficiencies in the low- SNR regime. Depen ding on ly o n ˙ C (0) an d ¨ C (0) , th e bit energy at zero spectral efficiency and wideban d slop e achieved by M - ary PSK signals can be readily obtained by using th e formulas in (1 7). Note that in the noncoh erent Rician fading c hannel, the r eceived b it energy is E r b,nc N 0 = ( | d | 2 + γ 2 ) SNR log 2 C M ,nc ( SNR ) . Cor ollary 2: The received bit en ergy a t zero spectral effi- ciency and wid eband slope achieved by M -ary PSK signaling in th e nonc oherent Rician fading chan nel are given by E r b,nc N 0     C =0 =  π 2  1 + 1 K  log 2 M = 2 4 π M 2 sin 2 π M  1 + 1 K  log 2 M ≥ 3 and S 0 ,nc =            3 π − 1+ 3 π 2 K M = 2 0 M = 3 6 π − 1+ 3 π K M = 4 M 4 8 π 2 sin 4 π M − ψ ( M )+ 1 2 π K M 2 sin 2 π M M ≥ 5 , (23) respectively , where ψ ( M ) is g iv en in ( 18), and K = | d | 2 γ 2 is the Rician factor . As it will be e vident in num erical results, generally th e 4 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 E b /N 0 (dB) Spectral Efficiency (bits/s/Hz) M = 3 M = 2 M = 4 M = 8 M = 10 M = 16 M = 32 QPSK, soft M = 1024 Fig. 1. Spectra l effici enc y C ( E b / N 0 ) vs. bit energy E b / N 0 for hard- decision detec ted M -ary PSK wit h M = 2 , 3 , 4 , 8 , 10 , 16 , 32 , 1024 and soft-detect ed QPSK in the noncohe rent Rician fading channel w ith Rician factor K = | d | 2 γ 2 = 1 . E r b,nc N 0    C =0 is the minim um bit energy required for reliable transmission whe n M 6 = 3 . On the oth er hand , the minimum bit energy is achieved at a n onzero spectral efficiency wh en M = 3 . Note tha t th is b ehavior is no t exh ibited when 3- PSK signals are so ft-detected [9]. Hence , this result is tightly linked to correct- detection and erro r probab ilities which are in general function s of the distances in the signal con stellation. Note fu rther that at suf ficiently low SNR s, 3- PSK perform s worse than 2 -PSK (i.e., BPSK), indicating that the decrea se in the signal distanc e fro m 2 √ E in 2-PSK to √ 3 E in 3-PSK has a more domin ating effect in th e low- SNR r egime tha n th e increase in the constellation size M from 2 to 3. Figure 1 plots the spectral efficiency curves as a fu nction of the b it energy for h ard-dec ision detected PSK with different constellation sizes in the non coheren t Rician fading chann el with R ician factor K = 1 . As observed in this figur e, the informa tion-theo retic analysis conducted in this paper provides se veral practical design guidelin es. W e note that alth ough hard-d ecision d etected 2-PSK and 4- PSK achieve the same minimum b it en ergy of 3.38 d B at zero spectral efficiency , 4-PSK is more efficient at low but nonzero spec tral efficiency values due to its wideband slope bein g twice that of 2-PSK. In the r ange of spectral efficiency values considered in th e figure, 3- PSK perf orms worse th an both 2 -PSK and 4- PSK. 3-PSK achieves its minim um bit en ergy of 4.039 dB at 0 .0101 bits/s/Hz. Op eration b elow this lev el of spectral efficiency should be a voided as it only increases the energy requir ements. W e further observe that increasing the constellation size to 8 pr ovides much im provement over 4-PSK. 8- PSK ac hieves a minimu m bit ene rgy of 2 . 692 dB. No te from (23) that E r b,nc N 0    C =0 is inv ersely p ropo rtional to M 2 sin 2 π M for M ≥ 3 . Here, we see two compe ting factors. As M increases, the term M 2 increases a nd tend s to d ecrease th e bit energy requiremen t while the ter m sin 2 π M decreases due to a decrease in the minimum distance , which is p ropor tional to sin π M in M -PSK constellation. Hence, when we increase M from 4 to 8, M 2 is the dominant term and we n ote significan t gains. As M is further increased, sin 2 π M acts more strong ly to of fset the gains from M 2 and we see diminishing r eturns. For instance, ther e is little to be gained by increasing the constellation size more than 32 as 32-PSK achieves a m inimum bit energy of 2 . 482 dB and the min imum bit energy as M → ∞ is 2 . 4 68 dB. W e find that the wideban d slop es of hard- decision d etected PSK with M = 8,10 ,16,3 2, and 1024 are 0 .571, 0.584 , 0.599, 0.607, and 0.609 , respectively . The similarity of th e wid eband slope values is also appar ent in the figur e. No te that the wid eband slope of 3-PSK, as pre dicted, is 0 . For com parison, th e sp ectral efficiency of so ft-detected QPSK is also provid ed in Fig. 1. It has been shown in [1 3] that und er the pe ak constraint | x i | 2 ≤ E , the bit energy required at zero sp ectral efficiency and wideban d slop e in the nonco herent Rician fadin g ch annel with Rician factor K ar e E b N 0    C =0 =  1 + 1 K  log 2 a nd S 0 = 2 K 2 (1+ K ) 2 , r espectively . It is also proven that soft-dete cted QPSK is optimally efficient achieving these values. Note that when K = 1 , the bit energy at zero spectral efficiency is 1.4 18 dB which is also observed in Fig. 1. Note that even as M → ∞ , hard -decision detection presents a loss of 2.4 68 - 1.41 8 = 1.0 5 dB in the minimum bit energy . B. A WGN Channe ls Note that the noncoher ent Rician fading chan nel, in which we have E { h k } = d and E {| h k − d | 2 } = γ 2 , specializes to th e A WGN chann el if we assume γ 2 = 0 . With this assumption, the fadin g coefficients beco me determ inistic, i.e., h k = d , and the c hannel mod el is now r k = ds x k + n k where th e ch annel gain is d . Note also that whe n we ha ve γ 2 = 0 , (4) becomes the cond itional de nsity f unction of the ou tput giv en the input in the A WGN ch annel. Moreover, the maxim um likelihood decision rule an d decision regions f or the A WGN chan nel ar e the same as in (5) and (6), respe ctiv ely . Assuming fu rther that d = 1 leads to the stan dard unfaded Gaussian ch annel with unit chan nel g ain, i.e., the in put-ou tput relation becomes r k = s x k + n k . Based on the ab ove observations, we im mediately have the f ollowing Corollary . Cor ollary 3: F or the A WGN ch annel with channel ga in d , the first and second deriv ati ves of the PSK capacity at SNR = 0 are gi ven by the expressions in ( 17) if we let γ 2 = 0 . Furthermo re, the b it energy , E r b N 0    C =0 = | d | 2 SNR log 2 C M ( SNR )    C =0 , an d wideband slope expr essions are obtained if we let γ 2 = 0 and hence K → ∞ in the fo rmulas in (23). Remark: W e sh ould no te that the first deriv ati ve of the capacity o f PSK in the A WGN ch annel h as previously been giv en in [7] th rough the bit energy expression s. In add ition, V erd ´ u in [8] has pr ovided the second derivati v e expression for the sp ecial case of M = 4 . Fig. 2 plots the spectr al efficiency cu rves as a function of the bit en ergy for h ard-d ecision detected M -ary PSK fo r various values of M an d soft-de tected QPSK in the A WGN channel. Con clusions similar to those giv en fo r Fig. 1 also apply for Fig. 2. Th e main d ifference be tween the fig ures is 5 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 E b /N 0 (dB) Spectral efficiency (bits/s/Hz) M = 2 M = 3 M = 4 M = 8 M = 10 M = 16 M = 32 M = 1024 QPSK,soft −1.59 Fig. 2. Spectra l effici enc y C ( E b / N 0 ) vs. bit energy E b / N 0 for hard- decision detec ted M -ary PSK wit h M = 2 , 3 , 4 , 8 , 10 , 16 , 32 , 1024 and soft detected QPSK in the A WGN channel. that substantially lower bit energies are need ed in the A WGN channel. For instance, 2- an d 4-PSK now achieve a minimum bit energy of 0. 369 dB while 8- PSK attains − 0 . 318 d B. As M → ∞ , the m inimum bit energy go es to − 0 . 54 2 d B. W e note that higher energy requ irements in the non cohere nt Rician channel is due to fading and no t kn owing th e chann el. C. Coher ent F ad ing Channe ls In co herent fading channels, the fading co efficients { h k } are assumed to be p erfectly known at the receiver . W e a ssume that no such knowledge is av ailable at the transmitter . The only req uirements on the fading coefficients are that th eir variations are ergod ic and they h av e finite second mom ents. Hence, indep endence o f the ran dom variables { h k } is no longer imposed. Due to th e p resence o f r eceiver ch annel side informa tion (CSI), maximum likelihoo d detection is the scaled nearest point detection . In this case, th e average capacity is C M ,c ( SNR ) = lo g M + M X l =1 E h { P l, 1 ,h log P l, 1 ,h } (24) where P l, 1 ,h = P ( y = l | x = 1 , h ) = Z (2 l − 1) π M (2 l − 3) π M f θ | s 1 ,h ( θ | s 1 , h ) dθ and f θ | s 1 ,h ( θ | s 1 , h ) is g iv en in (25) on the next pa ge with the definition SNR = E / N 0 . Note that if we assum e γ 2 = 0 an d replace d by the rand om chan nel gain h k in the nonco herent Rician fading ch annel, we o btain the mo del for co herent fading channels. Hence, similarly as in Section II I-B, results for coheren t channe ls can be obtaine d easily by specializing those for the nonc oherent Rician chan nel. Since we are interested in th e average cap acity (2 4), expressions will in v olve th e expected values o f the ran dom g ain h . Hence, we have the following Corollary to The orem 1. Cor ollary 4: The first and second deriv ati ves of C M ,c ( SNR ) in n ats per symb ol at SNR = 0 are o btained by assuming in (17) γ 2 = 0 , rep lacing d by h , and taking the expectation o f the te rms that in volve h . The r esulting expressions are ˙ C M ,c (0) =  2 π E {| h | 2 } M = 2 M 2 4 π sin 2 π M E {| h | 2 } M ≥ 3 , and ¨ C M ,c (0) =        8 3 π  1 π − 1  E {| h | 4 } M = 2 ∞ M = 3 4 3 π  1 π − 1  E {| h | 4 } M = 4 ψ ( M ) E {| h | 4 } M ≥ 5 (26) respectively , where ψ ( M ) is g iv en in (18). Note that the first and second d eriv ati ves of the capacity at z ero SNR are essentially equal to the scaled versions of those obtained in the A WGN channel with d = 1 . The scale factors are E {| h | 2 } and E {| h | 4 } for the first an d secon d deriv ati ves, respectively . In the coheren t fading case, we can define the rece i ved bit e nergy as E r b,c N 0 = E {| h | 2 } SNR log 2 C M ( SNR ) since E {| h | 2 } SNR is the av erage received signal-to-n oise ratio. It immediately fo llows fro m Corollary 4 that E r b / N 0 | C =0 in the coheren t fadin g channel is the same as th at in the A WGN channel. On th e o ther hand, the wideb and slope is scaled by ( E {| h | 2 } ) 2 /E {| h | 4 } . Fig. 3 plots th e spectral efficiency curves as a fu nction o f b it e nergy for hard-de cision detected M -a ry PSK and soft d etected QPSK in the co herent Rayleigh fading channel. Comparison of Fig. 2 and Fig. 3 reveals that th e bit energy le vels required at zero spectral efficiency ar e indeed the same for both cases. Howe ver , th e presence of fadin g in duces a per forman ce penalty b y r educing the wid eband slope with a factor of E {| h | 2 } 2 /E {| h | 4 } = 1 / 2 . T herefo re, at low but nonzer o spectral efficiencies, the same bit energy as in the A WGN channel can be achieved at the cost of red uced spectral efficiency . I V . E N E R G Y E FFI C I E N C Y O F O RT H O G O N A L S I G N A L I N G As discussed in Section I, orthog onal signa ling is o ptimally energy efficient in th e infinite ban dwidth regime even if the receiver perfor ms hard -decision detection. For instance, PPM with vanishing d uty cycle or M -ary FSK as M → ∞ achie ves the minimum bit energy of − 1 . 59 dB. In this section, we analyze the non-asympto tic energy efficiency of orthogon al signaling. W e consider on-o ff FSK ( OOFSK) mo dulation in which FSK is combin ed with o n-off keying (or equi va- lently PPM) and peakedn ess is intro duced in both time an d frequen cy . The study of OO FSK mo dulation enables us to provide a gener al un ified analysis of ortho gonal signaling as OOFSK can be redu ced to OOK and FSK with the app ropriate choice of param eters. A. OOFSK Modula tion 1) A WGN Channels: In this section, we consider the trans- mission o f OOFSK signals. W e again assume that the receiv ed signal is hard-d ecision d etected at the receiv er . In [20] an d 6 f θ | s 1 ,h ( θ | s 1 , h ) = 1 2 π e −| h | 2 SNR + r | h | 2 SNR π cos θ e −| h | 2 SNR sin 2 θ  1 − Q ( p 2 | h | 2 SNR cos 2 θ )  (25) −1.5 −1 −0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 E b /N 0 (dB) Spectral efficiency (bits/s/Hz) M = 1024 M = 32 M = 16 M = 10 M = 8 M = 4 M = 3 M = 2 QPSK,soft −1.59 Fig. 3. Spectra l effici enc y C ( E b / N 0 ) vs. bit energy E b / N 0 for hard- decision detec ted M -ary PSK wit h M = 2 , 3 , 4 , 8 , 10 , 16 , 32 , 1024 and soft detected QPSK in the coherent Rayleigh fading channel . [22], max imum a posteriori pro bability (MAP) detection ru le for OOFSK m odulation is id entified and th e er ror p robability expressions are o btained. W e in itially consider th e A WGN channel a s th e results for this chann el will immediately imp ly similar conclusions f or fadin g c hannels. T he optimal detection rule in th e A WGN chan nel is given by the following: s i for i 6 = 0 is detected if | r i | 2 > | r j | 2 ∀ j 6 = i and | r i | 2 > τ (27) where τ = ( [ I − 1 0 ( ξ ) ] 2 4 α 2 ξ ≥ 1 0 ξ < 1 , ξ = M (1 − ν ) e α 2 ν , a nd α 2 = SNR ν . Above, r i is the i th compon ent of the received vector r . s 0 is de tected if | r i | 2 < τ ∀ i . No te th at since s 0 = (0 , . . . , 0) , d etection of s 0 is essentially the detectio n of no transmission. Note furthe r that the d etection rule in (27) togeth er with the rule fo r s 0 can be regarded as energy detection. After detection , th e channe l can no w be seen as a discrete ch annel with M + 1 inputs and M + 1 o utputs. From the err or pr obability ana lysis in [20] and [22], we h av e the expressions in (2 8) through (3 1) on the n ext p age for the tran sition pr obabilities in the A WGN chan nel. In these expressions, Q 1 ( · , · ) is the Marcum Q -fun ction [1 6], and I − 1 0 is th e functio nal in verse o f the zeroth o rder mod ified Bessel function o f the first kind . T he rates achiev ed by th e M - ary OOFSK mod ulation with du ty cycle ν an d eq uipro bable FSK signals is gi ven by (33) on the next p age. If M -ary OOFSK sig nals have a symbol dur ation of T , the bandwidth requirem ent is M T and the spectral effi ciency is given by I M ( SNR ,ν ) T M T = I M ( SNR ,ν ) M bits/s/Hz. It is shown in [ 21] that in the A WGN ch annel, the first deriv ati ve of the cap acity of soft- detected OOFSK is zer o at SNR = 0 . For the sake of comp leteness, we provide this re sult below . Theor em 2 : The first d eriv ati ve of the capacity at zero SNR achieved by soft-de tected M - ary OOFSK signaling with a fixed du ty factor ν ∈ (0 , 1] over th e A WGN channel is ze ro, and hen ce th e bit energy req uired at zero spectral efficiency is infinite. Pr oof : See [2 1]. Since hard-de cision detection does n ot increase the capacity , we immediately have the following Corollary to The orem 2. Cor ollary 5: The first derivati v e at zero SNR o f the achiev- able rates of h ard-d ecision-detected M - ary OOFSK tr ansmis- sion with a fixed duty factor ν ∈ (0 , 1] over the A WGN channel is zero i. e., ˙ I M (0 , ν ) = 0 , a nd hen ce the bit en ergy required at zero spe ctral efficiency is in finite, i.e., E b N 0     I =0 = log 2 ˙ I M (0 , ν ) = ∞ . (34) On the o ther han d, we kn ow from [1] and [8] that if the duty cycle ν vanishes simultan eously with SNR , the min imum bit energy of − 1 . 59 dB can be ach iev ed. The following result identifies the rate at which ν should decrease as SNR gets smaller . Theor em 3 : Ass ume that ν = SNR (1+ ǫ ) log 1 SNR for SNR < 1 and for some ǫ > 0 . Then, we have lim ǫ → 0 lim SNR → 0 I M ( SNR , ν ) SNR = 1 (35) and hence lim ǫ → 0 lim SNR → 0 SNR log 2 I M ( SNR , ν ) = log 2 = − 1 . 59 dB . (36) Pr oof: Note that as SNR → 0 , ν → 0 and α 2 = SNR ν = (1 + ǫ ) log 1 SNR → ∞ . I t can also be seen that ξ → ∞ an d τ → ∞ as SNR dim inishes. Fro m ( 28), we immediately note that P 0 . 0 → 1 and P l, 0 → 0 for l = 2 , . . . , M . In ( 29), all th e terms in the summation other than for n = 0 v anishes b ecause α 2 → ∞ . There fore, in o rder for P l,l for l = 1 , . . . , M to approa ch 1, we need Q 1 ( √ 2 α, √ 2 τ ) → 1 . Also note that if Q 1 ( √ 2 α, √ 2 τ ) → 1 , then we c an observe fro m (3 0) and (31) that P 0 ,l → 0 and P l,m → 0 . Hence, e ventually all crossover e rror probab ilities will vanish and correct detection probab ilities will be 1. In [16], it is sho wn that Q 1 ( a, aζ ) ≥ 1 − ζ 1 − ζ e − a 2 (1 − ζ ) 2 2 0 ≤ ζ < 1 . From this lower bou nd we can immediately see that lim SNR → 0 Q 1 ( √ 2 α, √ 2 τ ) = 1 if lim SNR → 0 τ α 2 < 1 . No te th at b oth α 2 and τ grow without bound as SNR → 0 . Recall that τ = [ I − 1 0 ( ξ ) ] 2 4 α 2 . Equiv alently , we have I 0 ( √ 4 α 2 τ ) = ξ = M (1 − ν ) e α 2 ν . Using the asympto tic form I 0 ( x ) = 1 √ 2 π x e x + O  1 x 3 / 2  [26] for large x , we can easily show that lim SNR → 0 τ α 2 =  1+ ǫ/ 2 1+ ǫ  2 < 1 ∀ ǫ > 0 if 7 P 0 , 0 = (1 − e − τ ) M , and P l, 0 = 1 M (1 − (1 − e − τ ) M ) for l = 1 , 2 , . . . , M , (28) P l,l = M − 1 X n =0 ( − 1) n n + 1  M − 1 n  e − n n +1 α 2 Q 1 r 2 n + 1 α, p 2( n + 1) τ ! for l = 1 , 2 , . . . , M , (29) P 0 ,l = (1 − e − τ ) M − 1  1 − Q 1  √ 2 α, √ 2 τ  for l = 1 , 2 , . . . , M , (30) P l,m = 1 M − 1 (1 − P m,m − P 0 ,m ) for all l 6 = 0 , m 6 = 0 , a nd l 6 = m (31) I M ( SNR , ν ) = H ( y ) − H ( y | x ) (32) = − ((1 − ν ) P 0 , 0 + ν P 0 , 1 ) log ((1 − ν ) P 0 , 0 + ν P 0 , 1 ) − M  (1 − ν ) P 1 , 0 + ν M P 1 , 1 + ( M − 1) ν M P 1 , 2  log  (1 − ν ) P 1 , 0 + ν M P 1 , 1 + ( M − 1) ν M P 1 , 2  + (1 − ν ) ( P 0 , 0 log P 0 , 0 + M P 1 , 0 log P 1 , 0 ) + ν ( P 0 , 1 log P 0 , 1 + P 1 , 1 log P 1 , 1 + ( M − 1) P 2 , 1 log P 2 , 1 ) . (33) ν = SNR (1+ ǫ ) log 1 SNR . Therefore, if ν decays at this ra te, the error probabilities g o to zero . It can then be shown that lim SNR → 0 I M ( SNR ,ν ) SNR = 1 1+ ǫ . Since results hold for any ǫ > 0 , letting ǫ → 0 g iv es the desired result.  W e note tha t Zheng et al. hav e shown in [1 9] that th e low SNR capacity of unk nown Rayleig h fading channel can be approa ched by on-o ff keying if log 1 SNR log log 1 SNR ≤ α 2 ≤ lo g 1 SNR . W e see a similar beh avior her e when FSK sign als ar e sent over the A WGN cha nnel and energy detecte d. 2) F ading Channels: In coheren t f ading chan nels where the receiver has p erfect knowledge of the fading co efficients, the transition pr obabilities are the same as those in (28)-(31) with the only difference th at we now h ave α 2 = SNR ν | h | 2 . As a result, the achievable rates I M ( SNR , ν, | h | 2 ) are also depen- dent o n the fading co efficients and average achievable rates are obtaine d by findin g the expe cted value I M ,c ( SNR , ν ) = E | h | 2 { I M ( SNR , ν, | h | 2 ) } . In noncoh erent R ician fading channe ls with E { h } = d and E {| h − d | 2 } = γ 2 , the transition p robab ilities [20], [22] are given by ( 37) thro ugh (39) on the n ext page. I n these expressions, τ =  Φ − 1 ( ξ ) ξ ≥ 1 0 ξ < 1 where Φ( x ) = e α 2 γ 2 x 1+ α 2 γ 2 I 0 2 p x α 2 | d | 2 1 + α 2 γ 2 ! , and (40) ξ = M (1 − ν ) ν (1 + α 2 γ 2 ) e α 2 | d | 2 1+ α 2 γ 2 . (41) The achiev able rates, I M ,nc ( SNR , ν ) , can be obtaine d from (33). Since the presen ce o f fadin g un known at the transmitter does not improve th e perfor mance, we re adily conclude that the bit energy req uirements in fading chan nels still grow without bound with vanishing SNR . Cor ollary 6: The first der iv ati ves at zero SNR of the a chiev- able rates I M ,c ( SNR , ν ) and I M ,nc ( SNR , ν ) are eq ual to zer o, i.e., ˙ I M ,c (0 , ν ) = ˙ I M ,nc (0 , ν ) = 0 . Therefore, the bit en- ergy req uired at zero spectr al e fficiency is infinite in b oth coheren t and n oncoh erent fading chann els, i.e., E b,c N 0    I =0 = E b,nc N 0    I =0 = ∞ . On th e other h and, in non coheren t fading cha nnels, if | d | ≥ 1 , th en following the same step s as in the pr oof o f Th eorem 3, we can show tha t the minimum bit energy of − 1 . 59 dB is achieved as SNR → 0 if ν = SNR (1+ ǫ ) log 1 SNR . B. FSK Modulatio n Recall that if we set ν = 1 in OOFSK mod ulation, we recover the regular FSK mod ulation. Similarly , choosing ν = 1 in th e decision rules an d transition pro babilities leads to the co rrespon ding expressions for FSK. For instance , when ν = 1 , τ = 0 in the decision rule ( 27) of OO FSK modulatio n. There fore, s i is declared a s the d etected signal if the i th compon ent of the received vector r has the largest energy , i.e., | r i | 2 > | r j | 2 ∀ j 6 = i. This is the well-k nown nonco herent d etection o f FSK signals. Fu rthermo re, T heorem 2 and Corollaries 5 and 6 are valid for all ν ∈ ( 0 , 1 ] and hence for ν = 1 a s well. There fore, the same con clusions are au tomatically d rawn for FSK modulatio n. Hence, altho ugh FSK is energy efficient asymptotically as M → ∞ , op erating at very lo w SNR le vels with fixed M is extremely energy inefficient as th e bit energy requireme nt increases withou t bound with decreasing SNR . As a result, th e minimum bit energy is achieved at a nonzero spectral efficiency , th e value of w hich can be foun d thro ugh num erical analysis. W e finally note tha t when FSK m odulation is co nsidered , the achievable rates are indeed the capacity of FSK mo dulation as it is well- known that h ard-de cision detection cap acity is achieved with equipro bable signals. C. Nume rical Results In th is section , we pr ovide n umerical r esults an d in itially concentr ate on FSK mo dulation due to its widespread an d 8 P 0 , 0 = (1 − e − τ ) M , and P l, 0 = 1 M (1 − (1 − e − τ ) M ) for l = 1 , 2 , . . . , M , (37) P l,l = M − 1 X n =0 ( − 1) n  M − 1 n  e − nα 2 | d | 2 n (1+ γ 2 α 2 )+1 n (1 + γ 2 α 2 ) + 1 Q 1 s 2 α 2 | d | 2 (1 + γ 2 α 2 )( n (1 + γ 2 α 2 ) + 1) , s 2( n (1 + γ 2 α 2 ) + 1) τ (1 + γ 2 α 2 ) ! for l 6 = 0 , P 0 ,l = (1 − e − τ ) M − 1 1 − Q 1 s 2 α 2 | d | 2 1 + γ 2 α 2 , r 2 τ 1 + γ 2 α 2 !! for l = 1 , 2 , . . . , M , (38) P l,m = 1 M − 1 (1 − P m,m − P 0 ,m ) for all l 6 = 0 , m 6 = 0 , a nd l 6 = m (39) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2 3 4 5 6 7 8 9 10 11 E b /N 0 (dB) Spectral efficiency (bits/s/Hz) M = 48 M = 32 M = 16 M = 8 M = 4 M = 2 Fig. 4. Bit energ y E b / N 0 vs. Spectral ef ficienc y C ( E b / N 0 ) for energy- detec ted M -ary FSK in the A WGN channel. frequen t use. Fig. 4 p lots the bit energy E b / N 0 curves as a function of spectral efficienc y for M -ar y FSK in the A WGN channel f or d ifferent values of M . In all cases, we ob serve that the m inimum b it energy is achieved at a non zero spectr al efficiency C ∗ , and the bit energy re quiremen ts increase to in- finity as sp ectral efficiency d ecreases to zer o. Hence, op eration below C ∗ should b e a voided. Another observation is that the minimum bit energy and the spectra l ef ficiency value at which the min imum is achieved de crease with in creasing M . For instance, whe n M = 2 , th e min imum bit energy is 7.821 dB and is ach iev ed at C ∗ = 0 . 251 bits/s/Hz. If the value of M is increased to 48, the m inimum bit ene rgy decreases to 2.6 17 dB an d is now attained at C ∗ = 0 . 074 bits/s/Hz. Another fact is th at as M increases, the minimum bit energy is achieved at a higher SNR value. In deed, we can show that lim ǫ → 0 M →∞ C M ( SNR ) SNR     SNR =(1+ ǫ ) l og M = lim ǫ → 0 M →∞ C M ((1 + ǫ ) log M ) (1 + ǫ ) log M (42) = lim ǫ → 0 1 1 + ǫ lim M →∞ P 1 , 1 = 1 . (43) Hence, if SNR gr ows logarithmically with increa sing M , the bit energy E b N 0 = SNR log 2 C M ( SNR ) approa ches log 2 = − 1 . 59 dB. T he 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2 3 4 5 6 7 8 9 10 11 12 Spectral efficiency (bits/s/Hz) E b /N 0 (dB) M = 48 M = 32 M = 16 M = 8 M = 4 M = 2 Fig. 5. Bit energy E b / N 0 vs. Spectral ef ficienc y C ( E b / N 0 ) for ener gy- detec ted M -ary FSK in the coherent Ricia n fading channe l with Rician fact or K = 1 . proof o f ( 43) is o mitted bec ause T urin [2] h as alread y shown that − 1 . 59 dB is achieved if the signa l duration increases as log M , which in tu rn increases the SNR loga rithmically in M . Figures 5 an d 6 p lot th e bit e nergy cu rves for M - ary FSK transmission over coher ent and n oncoh erent Rician fading channels. As pred icted, the bit en ergy levels for all values of M in crease withou t bound as th e spectral efficiency decreases to zero. Due to the presence of fading, the minimu m bit energies h av e increased with respect to tho se achieved in the A WGN chann el. For instance, when M = 48 , the minimum bit energies are n ow E b / N 0 min = 3 . 4 5 dB in th e coheren t Rician fading cha nnel an d E b / N 0 min = 4 . 2 3 dB in the nonco herent Ri cian fading chann el. W e again observe that the minimum bit energy dec reases with increasing M . Fig. 7 p rovides the minimu m bit energy values as a fu nction of M in the A WGN and nonco herent Rician fadin g ch annels with different Rician factors. I n all cases, the minim um bit e nergy decreases with increasing M . Howe ver , Fig. 7 in dicates that approa ching − 1 . 59 d B is very slow and dem anding in M . I n this figure, we also note the e nergy penalty due to the presence of unknown fading. But, as the Rician factor K increases, the no ncoher ent Rician chann el app roaches to the A WGN channel and so do the min imum bit e nergy r equireme nts. Figures 8 and 9 plot the spectral efficiencies and average 9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2 3 4 5 6 7 8 9 10 11 12 Spectral efficiency (bits/s/Hz) E b /N 0 (dB) M = 48 M = 32 M = 16 M = 8 M = 4 M = 2 Fig. 6. Bit energ y E b / N 0 vs. Spectral ef ficienc y C ( E b / N 0 ) for energy- detec ted M -ary FSK in the noncoheren t Ricia n fadi ng channe l with Rician fac tor K = 1 . 100 200 300 400 500 600 700 800 900 1000 1 2 3 4 5 6 7 8 M E b /N 0 min AWGN K = 16 K = 9 K = 4 K = 1 K = 0 Fig. 7. Minimum bit energy E b / N 0 min vs. M for M -ary FSK in the A WGN channel and noncoherent Rician fading channel s with Ricia n factors K = 0 , 1 , 4 , 9 , 16 . received SNR values at wh ich E b / N 0 min is achieved as a function of M . As we have also ob served in Figs. 4 an d 6, we see in Fig. 8 that the spectral efficiency at which E b / N 0 min is achieved decr eases with increasing M . From Fig. 8, we further note th at the required spectral efficiencies are lower and hence the bandw idth r equirem ents are hig her in noncoh erent fading channels. In Fig. 9, we ob serve that the SNR lev els at which E b / N 0 min is ac hieved increases with incr easing M . As predicted by (43), SNR increases logarithmically with M in the A WGN chann el. Similar rates of increase are a lso no ted for the non coheren t fading chann el. Figs. 10 and 1 1 plot the bit en ergies as a function of spectral efficiency o f 8-OOFSK with different d uty cycle factor s in the A WGN and n oncohe rent Rayleigh fading chann els. W e immediately observe that decre asing th e duty cycle ν lowers 100 200 300 400 500 600 700 800 900 1000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 M Spectral Efficiency at which E b /N 0 min is achieved K = 9 K = 4 K = 1 K = 0 K = 16 AWGN Fig. 8. S pectra l ef ficienc y at which E b / N 0 min is achie v ed vs. M for M - ary FSK in the A WGN channel and noncoherent Ric ian fading channels wit h Ricia n factors K = 0 , 1 , 4 , 9 , 16 . 0 100 200 300 400 500 600 700 800 900 1000 2 3 4 5 6 7 8 9 10 11 M SNR at which E b /N 0 min is achieved K = 0 K = 9 K = 4 K = 1 AWGN K = 16 Fig. 9. SNR at whi ch E b / N 0 min is achie ved vs. M for M -ary FSK in th e A WGN channel and noncoher ent Rici an fadi ng channels with Rician factors K = 0 , 1 , 4 , 9 , 16 . the m inimum bit energy . Hence, in creasing th e signa l peaked- ness in the time domain improves the en ergy efficiency . In the A WGN chann el, while regular 8-FSK (8-OOFSK with ν = 1 ) has E b / N 0 min = 4 . 08 dB, 8 -OOFSK with ν = 0 . 01 h as E b / N 0 min = 2 . 017 dB. Howe ver , this energy gain is obtain ed at the cost o f increased p eak-to-average ratio. W e also n ote that unknown f ading again induces a energy penalty with respect to that achieved in the A WGN chann el as observed by com paring Figs. 10 and 11. V . C O N C L U S I O N In this paper, we have analyzed the imp act of ha rd-decision detection on the energy efficiency of phase mo dulation and frequen cy modula tion to gether with o n-off keying. W e have obtained closed-for m expressions for the first and second 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 5 6 7 8 Spectral efficiency (bits/s/Hz) E b /N 0 (dB) ν = 0.001 ν = 0.01 ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.8 ν = 1 Fig. 10. Bit energy E b / N 0 vs. Spectral effic ienc y C ( E b / N 0 ) for 8-OOFSK in the A WGN channel. The duty cyc le values are ν = 1 , 0 . 8 , 0 . 5 , 0 . 3 , 0 , 1 , 0 . 01 and 0.001. 0 0.05 0.1 0.15 0.2 0.25 0.3 4 5 6 7 8 9 10 Spectral efficiency (bits/s/Hz) E b /N 0 (dB) ν = 1 ν = 0.01 ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.8 Fig. 11. Bit ene rgy E b / N 0 vs. Spect ral effic ienc y C ( E b / N 0 ) for 8-OOFSK in the noncoherent Rayleigh fading channel. The duty cycle v alue s are ν = 1 , 0 . 8 , 0 . 5 , 0 . 3 , 0 , 1 , 0 . 01 and 0.001. deriv ati ves of the M -ary PSK cap acity in A WGN, coheren t fading, and non coheren t Rician fading chan nels. Subsequently , we have found closed-form expressions for the bit ene rgy required at zero spe ctral efficiency and wideban d slope, and quantified the lo ss in energy efficiency incur red by hard- decision detection and chan nel fading. Th e inefficiency of 3-PSK at very low SNR s is n oted. W e have also considered energy detected M - ary OOFSK transmission over the A WGN and fading channels. W e h av e shown that bit energy requir e- ments g row without bou nd as SNR vanishes for any fixed duty cycle value. Results are easily sp ecialized to FSK mo dulation as well. Thro ugh n umerical results, we have inv estigated the value of th e minimum bit energy fo r different values of M in various chann els. W e have shown th rough numerica l results that the minimum bit energy decreases with decreasing du ty cycle and incr easing M . W e have proved that if the duty cycle decreases as SNR log 1 SNR , the minim um bit energy of − 1 . 5 9 dB can be approach ed. A P P E N D I X A. Pr oof of Theo r em 1 The main approach is to obtain ˙ C M ,nc (0) and ¨ C M ,nc (0) by fir st find ing the deriv ativ es o f th e transition pro babilities { P l, 1 } . This can be accom plished by finding the first and second der iv ati ves of f θ | s 1 with respect to SNR . Howev er , the presence of q | d | 2 SNR π ( γ 2 SNR +1) in the seco nd term of (14) complicates this appro ach becau se it leads to the result that d f θ | s 1 d SNR    SNR =0 = ∞ . I n o rder to circu mvent this pro blem, we define the new v ariable a = √ SNR and consider the condition al density e xpression in (44) on th e next page. No w , the deriv ativ e expressions in (45) and (46) on the next pag e ev aluated at a = 0 can easily be verified. Using the derivati v es o f P l, 1 and per formin g several algeb raic op erations, we a rrive to the following T aylor expansion o f C M ,nc ( a ) at a = 0 : C M ,nc ( a ) = φ 1 ( M ) a 2 + φ 2 ( M ) a 3 + φ 3 ( M ) a 4 + o ( a 4 ) (4 7) = φ 1 ( M ) SNR + φ 2 ( M ) SNR 3 / 2 + φ 3 ( M ) SNR 2 + o ( SNR 2 ) (48) where (48) follows due to the fact that a = √ SNR . In the above expansion, φ 1 ( M ) , φ 2 ( M ) , and φ 3 ( M ) ar e given by ( 49)– (51) o n the next pa ge. W e immed iately co nclude fr om (48) that ˙ C M ,nc (0) = φ 1 ( M ) . No te that the exp ansion includ es the term SNR 3 / 2 which imp lies th at ¨ C M ,nc (0) = ±∞ for all M . Howev er , it can be easily seen that φ 2 ( M ) = 0 fo r all M 6 = 3 , an d at M = 3 , φ 2 (3) = 0 . 1718 | d | 3 . Theref ore, while ¨ C 3 ,nc (0) = ∞ , ¨ C M ,nc (0) = 2 φ 3 ( M ) fo r M 6 = 3 . Fur ther algebraic steps and simplification yield (17).  R E F E R E N C E S [1] M. J. E. Golay , “Note on the theoreti cal effic ienc y of information recept ion with PPM, ” P r oc. IRE , vol. 37, pp. 1031 , Sept. 1949. [2] G. L. T urin, “The asymptotic behavior of ideal M -ary s ystems, ” Pr oc. IRE , vo l. 47, pp. 93-94 , Jan. 1959. [3] I. Ja cobs, “The asympt otic behavi or of incohere nt M -ary communi cation s systems, ” Proc. IEE E , vol. 51, pp. 251-2 52, Jan. 1963. [4] J. N. Pierce, “Ultimat e performance of M -ary transmissions on fadi ng channe ls, ” IEEE T r ans. Inform. Theory , vol. 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