On the capacity of bandlimited optical intensity channels with Gaussian noise
aa r X i v : . [ c s . I T ] F e b On the Capacity of Bandlimited Optical IntensityChannels with Gaussian Noise
Jing Zhou and Wenyi Zhang,
Senior Member, IEEE
Abstract —We determine lower and upper bounds on thecapacity of bandlimited optical intensity channels (BLOIC) withwhite Gaussian noise. Three types of input power constraintsare considered: 1) only an average power constraint, 2) only apeak power constraint, and 3) an average and a peak powerconstraint. Capacity lower bounds are derived by a two-stepprocess including 1) for each type of constraint, designing admis-sible pulse amplitude modulated input waveform ensembles, and2) lower bounding the maximum achievable information rates ofthe designed input ensembles. Capacity upper bounds are derivedby exercising constraint relaxations and utilizing known resultson discrete-time optical intensity channels. We obtain degrees-of-freedom-optimal (DOF-optimal) lower bounds which have thesame pre-log factor as the upper bounds, thereby characterizingthe high SNR capacity of BLOIC to within a finite gap. Wefurther derive intersymbol-interference-free (ISI-free) signalingbased lower bounds, which perform well for all practical SNRvalues. In particular, the ISI-free signaling based lower boundsoutperform the DOF-optimal lower bound when the SNR is below10 dB.
Index Terms —Bandlimited channel, channel capacity, intensitymodulation, optical wireless communications.
I. I
NTRODUCTION
A. Background and Related Work I NTENSITY modulation and direct detection (IM/DD) iswidely used in optical communications. In IM/DD, infor-mation is conveyed by the varying intensity of transmittedoptical signal, and the receiver detects the intensity of thereceived signal. There have been extensive studies on designand realization of IM/DD based optical wireless communica-tion systems (see [1], [2] and references therein). This paperfocuses on a simplified IM/DD channel model, known as theoptical intensity channel with Gaussian noise. This model issuitable for some kinds of IM/DD systems, e.g., indoor freespace optical communications [1], [3]. In optical intensitychannels, typically the average and/or peak optical power isconstrained due to safety reasons and practical considerations. This work was supported in part by the National Key Basic ResearchProgram of China under Grant 2013CB329205, by the National NaturalScience Foundation of China under Grant 61379003, and by the FundamentalResearch Funds for the Central Universities under Grants WK2100060020and WK3500000003.Jing Zhou is with the Department of Electronic Engineering and Infor-mation Science, University of Science and Technology of China, Hefei,China, and with the Air Force Aviation University, Zibo, China (e-mail:[email protected]).Wenyi Zhang is with the Key Laboratory of Wireless-Optical Communica-tions, Chinese Academy of Sciences, and with the Department of ElectronicEngineering and Information Science, University of Science and Technologyof China, Hefei, China (e-mail: [email protected]). Since the optical intensity is the optical power transferred per unit area, aconstraint on optical power is also a constraint on intensity. In this paper weuse ‘power’ to refer to the optical power unless otherwise specified.
Therefore the optical intensity channel we considered has twofundamental differences with electrical channel models likethe classical AWGN channel: 1) a nonnegativity constraint onthe input signal, 2) a different input cost metric. It is clearthat typical signaling methods for electrical channels cannotbe applied directly in optical intensity channels because ofthese differences.A number of information theoretic studies on optical in-tensity channels have been reported, mostly on discrete-timeoptical intensity channels (DTOIC); see, e.g., [3]–[8]. ForDTOIC with only an average power constraint, the exact ca-pacity is still unknown, whereas tight upper and lower boundshave been established. When the input is further boundedfrom above due to a peak power constraint, the optimal inputdistribution and the capacity can be numerically computed[9], [10]. Moreover, recent works have provided systematicresults on capacity-achieving input design for DTOIC [5],[7]. Only a few information theoretic studies have consideredcontinuous-time optical intensity channel models. In [11],sphere packing based capacity upper bounds for multicarrieroptical intensity channels (MCOIC) were established. Capacitybounds for bandlimited optical intensity channels (BLOIC)with an average power constraint were studied in [3]. Inanother aspect, the design of ISI-free signaling over BLOICwas studied in [12]–[14], while the information rate was notconsidered therein.
B. Channel Model and Motivation of Our Study
The BLOIC model considered in this paper is Y ( t ) = X ( t ) + Z ( t ) , X ( t ) ≥ (1)where X ( t ) is bandlimited to W , [ − W, W ] , Z ( t ) is whiteGaussian noise (the definition follows that in [15]) of one-sided power spectral density N with respect to W . Thebandwidth constraint of BLOIC is due to the optoelectroniccomponents and multipath distortion [1], [3]. In (1), withoutloss of generality we set the channel gain (including, e.g.,the responsivity of the photodiode and the optoelectronicconversion factor in IM/DD systems) to be unity, as in [3],[4]. The DTOIC Y [ n ] = X [ n ] + Z [ n ] , X [ n ] ≥ (2)is the discrete-time analog of (1), where Z [ n ] ∼ N(0 , σ ) ( N( a, b ) denotes Gaussian distribution with mean a and vari-ance b ) is independent and identically distributed (i.i.d.).To model the input constraint, we define the instantaneouspowers as P ( X ( t )) = x ( t ) and P ( X [ n ]) = x [ n ] for BLOIC and DTOIC, respectively. These definitions are different fromthose of the electrical power which is proportional to thesquared amplitude of the signal. Then the average power is E [ X [ n ]] in DTOIC. In BLOIC, the average power is definedas P { X } = lim T →∞ T E "Z T − T X ( t )d t . (3)Here we follow the definition of [15, Definition 14.6.1].For most classical channels such as the AWGN channel,the linear Gaussian channel or fading channels, we have thefollowing relationship between the capacity of the continuous-time bandlimited channel model and its discrete-time analogwith the same signal-to-noise ratio: C BL = C DT · W transmissions per second . (4)This relationship can be established by orthogonal transforms,e.g., by Nyquist rate signaling/sampling or more rigorously byKarhunen-Lo`eve expansion and the 2WT theorem. Moreover,signaling schemes designed based on discrete-time models canbe directly utilized in bandlimited communications by simplemodulation/demodulation methods. Because of these facts,many studies based on classical channels reasonably consideronly the discrete-time channel models, where the relationshipbetween the discrete-time channel inputs (i.e. x [ n ] ) and thecontinuous-time signal they represent (i.e. x ( t ) ) is x ( t ) = X n x [ n ] sin π (2 W t − n ) π (2 W t − n ) . (5)However, in this paper we emphasize that the aforemen-tioned equivalent relationship does not hold between DTOICand BLOIC. According to (5), it is easy to verify that theequivalent bandlimited waveform of a given nonnegative inputsequence is not necessarily nonnegative everywhere. In otherwords, it is possible that an admissible input sequence inDTOIC corresponds to an inadmissible input waveform inBLOIC. Then it is clear that the capacity of BLOIC can notbe obtained by solving the capacity problem of DTOIC andusing (4). Therefore, when a bandwidth constraint exists, usingDTOIC as the model of a continuous-time optical intensitychannel is an oversimplification of the capacity problem.To the best of our knowledge, [3] is the only informationtheoretic study directly on the BLOIC model (1). This studyis restricted to time-disjoint signaling (TDS) based on a finite(typically low) dimensional signal space model over a finitetime interval. Using that model, the continuous-time channelis converted to a discrete-time vector Gaussian channel andthe input nonnegativity constraint is correspondingly convertedto an admissible region in the signal space. Capacity boundsare derived based on the converted vector Gaussian channelwith the admissible input region. It is thus clear that theupper bounds obtained therein are only information rate upper See [16], [17]. The 2WT theorem says that a channel with bandwidth W has essentially W T degrees of freedom (DOFs) in a length- T time interval,where T ≫ /W and DOF is defined as the dimension of the signal spacein that time interval. Note that the inadmissible waveforms can always be avoided in practiceby proper engineering design, but at the expense of performance, as shownlater. bounds of specific TDS schemes, rather than capacity upperbounds for the BLOIC. Moreover, due to the finite time lengthof the signal space model, the bandwidth constraint can onlybe approximately satisfied by permitting an ǫ -fractional out-of-band energy. Since the bandwidth is sensitive to ǫ , varying ǫ causes the achievable spectral efficiency to vary significantly.When ǫ tends to zero the achievable spectral efficiency tends tozero. Even for a fixed ǫ and given bandwidth, the signaling rateis limited due to the poor time-frequency concentration of therectangular basis function needed in the BLOIC signal spacemodel. So the available DOFs of the bandlimited channel ishardly exploited in the most efficient way. C. Our Contribution
In this paper, we study the fundamental limits of commu-nication over the BLOIC under different types of input con-straints. By designing admissible pulse amplitude modulated(PAM) input waveform ensembles with i.i.d. input symbolsand lower bounding their maximum achievable informationrates, we derive two kinds of capacity lower bounds: theDOF-optimal lower bounds and the ISI-free signaling basedlower bounds. The DOF-optimal lower bounds achieve theoptimal pre-log factor of the channel capacity by comparingwith capacity upper bounds derived from constraint relaxation.Thus we characterize the high-SNR capacity of the BLOIC towithin a finite gap. For example, the high-SNR asymptoticgap between the tightest lower and upper capacity bounds forthe only average power constrained case is 4.34 dB in SNR.The ISI-free signaling is preferred in practical communicationsystems because of its low detection complexity. We showthat the ISI-free signaling based lower bounds perform wellfor all practical SNR values, especially for low to moderateSNR. At high SNR, introducing a direct current (DC) bias inthe signal design is shown to be very helpful for boosting theinformation rate. We also study the effects of different peak-to-average-power ratios and different types of modulation pulses(i.e. shaping filters) on the capacity lower bounds, and giveseveral conjectures and discussions. All these results provideunderstanding on fundamental limits and signaling schemedesign for bandlimited communications using IM/DD.The remaining part of the paper is organized as follows.Section II presents our methods and main results. SectionIII gives comparisons among the capacity bounds based onnumerical results. Section IV provides some discussions andtwo conjectures.Throughout the paper the following notations are used: p X ( x ) denotes the probability density function (PDF) of X ; h ( · ) stands for the differential entropy, i.e. h ( X ) = − R ∞−∞ p X ( x ) log p X ( x )d x ; I ( Q , V ) , I ( X ; Y ) stands forthe mutual information between input X and output Y ofa channel with transition probability measure V when X has distribution Q ; H [ X ( t )] stands for the differential en-tropy per DOF of the bandlimited waveform ensemble X ( t ) ; I [ X ( t ); Y ( t )] stands for the mutual information per DOFbetween two bandlimited waveform ensembles X ( t ) and Y ( t ) ; C BA denotes the capacity of channel A under constraint B; R BA denotes the maximum achievable information rate of a TABLE IA
BBREVIATIONS
AP Average powerBLAWGN Bandlimited AWGNBLOIC Bandlimited optical intensity channelDC Direct currentDOF Degrees of freedomDTAWGN Discrete-time AWGNEPI Entropy power inequalityIM/DD Intensity modulation and direct detectioni.i.d. Independent and identically distributedISI Intersymbol interferenceMCOIC Multicarrier optical intensity channelPAM Pulse amplitude modulationPAPR Peak-to-average-power-ratioPC Power constraintPDF Probability density functionPL pulse Parametric linear pulsePNR Peak-to-noise ratioPP Peak powerPSWF Prolate spheroidal wave functionSC pulse Spectral-cosine pulseSNR Signal-to-noise ratioTDS Time-disjoint signaling constrained signaling scheme B over channel A. Boldfaceis used to denote matrices and vectors. Table I lists someabbreviations used in this paper.II. M
ETHODS AND R ESULTS
A. Preliminaries and Basic Methods
The different input power constraints in the BLOIC consid-ered in this paper are given in Table II. We use r to denote thepeak-to-average-power-ratio (PAPR), which is the ratio of themaximum allowed peak power (PP) to the maximum allowedaverage power (AP). We further use PC as a general notationfor these power constraints when a general discussion on themis needed. We denote a BLOIC under AP constraint as AP-BLOIC, and so on. For the DTOIC and the BLAWGN channelsimilar notations are used.In a bandlimited channel the input and output are randomwaveforms, while the input is drawn from a given waveformensemble. Following Shannon [18], we define the entropy perDOF of an input ensemble X ( t ) through the distribution ofits Nyquist sample sequence as H [ X ( t )] = lim N →∞ N + 1 Z p X ( x ) log 1 p X ( x ) d x (6) TABLE IID
IFFERENT I NPUT P OWER C ONSTRAINTS IN
BLOIC
Power Constraint Definition AP P { X } ≤ E , ≤ x ( t ) ≤ ∞ PP ≤ x ( t ) ≤ A PAPR P { X } ≤ E , ≤ x ( t ) ≤ r E where X = X − N , . . . , X , . . . , X N is the Nyquist samplesequence of X ( t ) . For example, for Z ( t ) defined in (1) wehave H [ Z ( t )] = log √ πeN W . The capacity of a bandlim-ited channel with input ensemble X ( t ) and output Y ( t ) isdefined as C = 2 W · max p X ( x ) I [ X ( t ); Y ( t )] (7)where I [ X ( t ); Y ( t )] is the mutual information per DOFbetween X ( t ) and Y ( t ) : I [ X ( t ); Y ( t )]= lim N →∞ N + 1 Z Z p X , Y ( x , y ) log p X , Y ( x , y ) p X ( x ) p Y ( y ) d x d y , (8)where X = X − N , . . . , X , . . . , X N and Y = Y − N , . . . , Y ,. . . , Y N , which are the Nyquist samples of X ( t ) and Y ( t ) ,respectively. The maximum achievable information rate R ofa specific signaling scheme has the same definition as C exceptthat the input ensemble X ( t ) must be generated using thatsignaling scheme.Our achievability results (lower bounds) for the BLOIC arederived by two basic steps:1) Designing an admissible input waveform ensemble sat-isfying certain constraints.2) Lower bounding the maximum achievable informationrate of the designed input ensemble.In particular, we design PAM input ensembles with i.i.d. inputsymbols as X PAM ( t ) = X i X i g ( t − iT s ) , X PAM ( t ) ≥ (9)to accomplish the first step, where the modulation pulse g ( t ) is a real L function (i.e. a finite-energy signal) bandlimitedto W . The design includes reasonable choices of the symbolrate /T s , the input symbol distribution p X ( x ) , and the pulse g ( t ) .Table III lists the pulses used in our results, including thesinc pulse, the S2 pulse [13], the spectral-cosine (SC) pulse,and the (first order) parametric linear (PL) pulse [19] (thedefinition of parameters S N and G will be given in (13)–(15)).Fig. 1 shows the Fourier transform of these pulses. To simplifythe proof of results we normalize the sinc, the S2, and the SCpulses to make them satisfy Z ∞−∞ g ( t )d t = 12 W , (10)and normalize the PL pulse to make it satisfy the followingdefinition.
TABLE IIIL
IST OF THE P ULSES U SED
Name Notation and Definition Remarks
Sinc g sinc ( t ) = sinc (2 W t ) = sin 2 πWt πWt S N = ∞ , G = 1 S2 g △ ( t ) = ( sinc ( W t )) = sin πWt πWt ) S N = 1 , G = e SC g cos ( t ) = sinc (cid:0) W t − (cid:1) + sinc (cid:0) W t + (cid:1) = πWtπ ( − W t ) S N = π , G = PL g PL ( t ) = sinc (cid:16) Wt β (cid:17) sinc (cid:16) βWt β (cid:17) = sin (cid:16) β πWt (cid:17) sin (cid:16) β β πWt (cid:17) β (cid:16) πWt β (cid:17) , β ∈ (0 , f G PL ( f )( β = 0 . G sinc ( f ) G cos ( f ) G △ ( f )0 . /W / W − W − . W . W W Fig. 1. The Fourier transforms of the pulses used.
Definition 1 : A normalized Nyquist pulse g β ( t ) with roll offfactor β is a real L function which is bandlimited to W andsatisfies g β ( nT ) = δ [ n ] (11)where T = β W , and δ [ n ] is the unit impulse. Note : Letting G β ( f ) be the Fourier transform of g β ( t ) , itis easy to show that G β ( f ) satisfies G β (0) = Z ∞−∞ g β ( t )d t = T . (12)Our lower bounds can be categorized as DOF-optimal lowerbounds and ISI-free signaling based lower bounds, which aregiven in Sec. II-B and Sec. II-C, respectively. The basic ideafor deriving DOF-optimal lower bounds is due to Shannon’sderivation of capacity bounds for PP-BLAWGN channel in his1948 landmark paper [18]. The ISI-free signaling based lowerbounds are derived by designing admissible ISI-free signal-ing schemes and considering the capacity of the equivalentdiscrete-time memoryless channel models.We will present some general lower bounds which holdsfor all g ( t ) or p X ( x ) satisfying certain constraints, and thenuse some specific g ( t ) or p X ( x ) to get specific lower bounds. Some of the general lower bounds are given with respect totwo parameters which are G , exp W Z W log | W · G ( f ) | d f ! , (13) S ( τ ) , max t ∈ [0 ,τ ] ∞ X i = −∞ | g ( t − iτ ) | , (14)where G ( f ) is the Fourier transform of g ( t ) . General lowerbounds of this flavor were introduced for the PP-BLAWGNchannel in [20], where Shannon’s 1948 lower bound wastightened by optimizing the modulation pulse used. For brevitywe further define S N , S (cid:18) W (cid:19) , S β , S (cid:18) β W (cid:19) . (15)The converse results are given in Sec. II-D, where thebounding technique used is also based upon [18]. B. DOF-Optimal Capacity Lower Bounds
The following lemma from [18] plays a crucial role inderiving the results in this subsection.
Lemma 1 : If an ensemble of waveform X I ( t ) bandlimitedto W is filtered by G ( f ) , then the entropy per DOF of theoutput ensemble is H [ X O ( t )] = H [ X I ( t )] + 12 W Z W log | G ( f ) | d f . (16) Proof:
See [21, Chapter 6.4].All the results in this subsection are derived using i.i.d.Nyquist rate input ensembles as X PAM ( t ) = X i X i g (cid:18) t − i W (cid:19) , X PAM ( t ) ≥ . (17)A symbol rate no less than the Nyquist rate is necessary toexploit all the available DOFs of bandlimited channels in thehigh SNR regime [22]. Lemma 2 : The maximum achievable information rateachieved by the i.i.d. Nyquist rate ensemble X PAM ( t ) in (17)transmitted over the BLOIC can be lower bounded by R PAM ≥ W log (cid:18) G exp (2 h ( X ))2 πeN W (cid:19) (18) where G is defined as (13). Proof:
Consider an ergodic or cyclostationary ensembleof waveform X ( t ) bandlimited to W . For an additive noisechannel bandlimited to W the information rate R = 2 W · I [ X ( t ); Y ( t )]= 2 W · ( H [ Y ( t )] − H [ Y ( t ) | X ( t )])= 2 W · ( H [ X ( t ) + Z ( t )] − H [ Z ( t )]) (19)is achievable by using the ensemble X ( t ) . Using the vectorversion of the entropy power inequality (EPI) [23] e N h ( X + Y ) ≥ e N h ( X ) + e N h ( Y ) , (20)the information rate given by (19) can be lower bounded as R ≥ W log (cid:16) e H [ X ( t )] − H [ Z ( t )] (cid:17) = W log (cid:18) H [ X ( t )])2 πeN W (cid:19) . (21)Since X i is i.i.d., the ensemble X PAM ( t ) is cyclostationaryand the information rate (21) (replacing X ( t ) with X PAM ( t ) )is achievable. Now we evaluate H [ X PAM ( t )] . We note thatthe ensemble X PAM ( t ) as (17) can be obtained by filtering anideal bandlimited ergodic ensemble X sinc ( t ) = P i X i g sinc ( t − i/ W ) by W · G ( f ) . This is because if g sinc ( t ) is filtered by W · G ( f ) , the output is g ( t ) ( G sinc ( f ) · W · G ( f ) = G ( f ) as G sinc ( f ) equals / W within W ). Then by using Lemma1 and noting that H [ X sinc ( t )] = h ( X ) (we omit the index of { X i } since they are i.i.d.), we have H [ X PAM ( t )] = h ( X ) + log G . Combining this with (21) completes the proof. Theorem 1 : The capacity of the AP-BLOIC is lowerbounded by C APBLOIC ≥ W log (cid:18) h ( X ))2 πe N W (cid:19) (22)for any p X ( x ) satisfying E[ X ] = E and p X ( x ) = 0 for x < . Proof:
The proof is given in Appendix A.Theorem 1 is obtained by employing the pulse g △ ( t ) andnonnegative input symbols X i in (17) and lower bounding theinformation rate achieved. The maxentropic distribution of anonnegative random variable with a given expected value E isthe exponential distribution [23]: p X ( x ) = E − e − x / E , (23)whose differential entropy is h ( X ) = log e E . (24)Substituting (24) into (22) yields the following corollary. Corollary 1 : C APBLOIC ≥ W log (cid:18) πe E N W (cid:19) . (25)We call (25) the Exp-S2 lower bound since it is based on aninput ensemble using exponential symbol distribution and S2 Note that information measures of waveforms, H and I , have the sameproperties as the differential entropy h ( · ) and mutual information I ( X ; Y ) of scalar variables, respectively, if the limits in (6) and (8) exist. pulse. While (25) is the tightest bound we could obtain fromTheorem 1, the general bound (22) still has its own merit sinceit can be used to evaluate the performance of more practicalinput symbol distributions (e.g., a uniform distribution). Theorem 2 : The capacity of the PP-BLOIC is lower boundedby C PPBLOIC ≥ W log (cid:18) G πe S A N W (cid:19) (26)where G and S N are defined as (13) and (15), respectively,with respect to an arbitrary modulation pulse g ( t ) satisfying(10). Proof:
The proof is given in Appendix B.Theorem 2 is derived using an uniform input symbol dis-tribution which is the maxentropic distribution for boundedrandom variables without further constraint. If we employ g cos ( t ) as the modulation pulse and note that for g cos ( t ) we have S N = 4 /π and G = 1 / (see [20]), we get thefollowing corollary called the Unif-cos lower bound whichis a suboptimal example of (26). Corollary 2 : C PPBLOIC ≥ W log (cid:18) π e A N W (cid:19) . (27)Theorem 2 and Corollary 2 can be viewed as parallelresults of [20] (in which g cos ( t ) is proposed) on lowpass PP-BLAWGN channel. Theorem 3 : The capacity of the PAPR-BLOIC is lowerbounded by C PAPRBLOIC ( r ) ≥ W log (cid:18) G r exp (cid:16) S N − r S N+ rr µ (cid:17) πe S (cid:16) − e − µ µ (cid:17) E N W (cid:19) , r > W log (cid:16) G r πe S E N W (cid:17) , < r ≤ , (28)where r is the PAPR, G and S N are defined as (13) and (15),respectively, with respect to an arbitrary modulation pulse g ( t ) satisfying (10), and µ is the unique solution to S N − r S N + r r = 1 µ − e − µ − e − µ , (29) Proof:
The proof is given in Appendix C.In the proof of Theorem 3, we let the input symboldistribution be a truncated exponential distribution which isthe maxentropic distribution of a nonnegative random variablewith a given expected value and an upper bound. This distri-bution was used for bounding the capacity of the DTIOC in[4]. See Appendix C for details.
Note:
For the PAPR-DTOIC, when r ≤ the AP constraintbecomes inactive and the capacity is equal to that of thePP-DTOIC with the same PP constraint [4]. For the PAPR-BLOIC, however, it is nontrivial to find out the PAPR transi-tion point at which the AP constraint becomes inactive. Notethat r = 2 is only the transition point in (28), which is not thecapacity.By employing the pulse g △ ( t ) in (28) we get the followingspecific lower bound, called the TE-S2 (truncated-exponential-S2) lower bound, which is a suboptimal example of (28). Corollary 3 : C PAPRBLOIC ( r ) ≥ W log (cid:18) r e µ/r πe (cid:16) − e − µ µ (cid:17) E N W (cid:19) , r > W log (cid:16) r πe E N W (cid:17) , < r ≤ , (30)where r > , and µ is the unique solution to r = 1 µ − e − µ − e − µ . (31) Proof:
The proof is given in Appendix C.
C. ISI-Free Signaling based Capacity Lower Bounds
The results in this subsection are given in the form oflower bounds on R PC, IFSBLOIC , which is the maximum achievableinformation rate of ISI-free signaling over the PC-BLOIC. Ofcourse, they are also lower bounds on the capacity of the PC-BLOIC.ISI-free signaling avoids ISI by using modulation pulses thatsatisfy the Nyquist criterion [15]. It may use Nyquist pulses(e.g., raised-cosine pulse) and a direct-sampling detector, oralternatively use the so-called T -orthogonal pulses (e.g., rootraised-cosine pulse) and a matched filter detector. ISI-freesignaling achieves the Nyquist rate only when the sinc pulseis used.For the BLOIC, [12] recognizes two important facts on ISI-free signaling using nonnegative pulses:1) ISI-free signaling is impossible when a matched filterreceiver is used.2) ISI-free signaling is possible when a direct-samplingreceiver is used. The maximum symbol rate is a halfof the Nyquist rate, achieved by employing the pulse g △ ( t ) .The derivations of all the bounds in the following two theo-rems use the second fact, i.e., employing i.i.d. PAM signalingas (9) with modulation pulse g △ ( t ) and letting T s = T = W . Theorem 4 : The maximum achievable information rates ofISI-free signaling over the AP-BLOIC, the PP-BLOIC, andthe PAPR-BLOIC, are lower bounded by (32), (33), and (34),respectively: R AP , IFSBLOIC ≥ W (cid:18) e π E N W (cid:19) ; (32) R PP , IFSBLOIC ≥ W (cid:18) πe A N W (cid:19) ; (33) R PAPR , IFSBLOIC ≥ W log (cid:18) r e µ/r πe (cid:16) − e − µ µ (cid:17) E N W (cid:19) , r > W log (cid:16) r πe E N W (cid:17) , < r ≤ , (34)where r is the PAPR, µ is the unique solution to (31). Proof:
The proof is given in Appendix D.Similar to the DOF-optimal bounds, the derivation of thebounds in Theorem 4 uses the maxentropic input symbol distri-butions for each type of constraint. We call (32), (33), and (34)the Exp-S2-IFS lower bound, the Unif-S2-IFS lower bound, and the TE-S2-IFS lower bound, respectively. The followingresult (called the Geom-S2-IFS lower bound), however, usesa geometry distribution which has been proposed in [6] forbounding the capacity of the DTOIC.
Theorem 5 : R AP , IFSBLOIC ≥ W · max l I ( Q g ( l ) , V ) (35)where Q g ( l ) is a geometric distribution with PDF p X ( x, l ) = ∞ X i =0 ll + E (cid:18) E l + E (cid:19) i δ ( x − il ) , l > , (36)and V is the transition probability of the channel Y = X + Z ,where Z ∼ N (0 , N W ) . Proof:
The proof is given in Appendix E.The following result is based on DC-aided ISI-free signalingover the BLOIC [13] whose symbol rate can surpass a halfof, and even approach, the Nyquist rate.
Theorem 6 : The maximum achievable information rates ofISI-free signaling over the AP-BLOIC, PP-BLOIC, and PAPR-BLOIC are lower bounded by (37) (38), and (39), respectively: R AP, IFSBLOIC ≥ sup β ∈ (0 , W β log S β πe E N W ! ; (37) R PP, IFSBLOIC ≥ sup β ∈ (0 , W β log S β πe A N W ! ; (38) R PAPR, IFSBLOIC ≥ sup β ∈ (0 , W β log (cid:18) r exp (cid:16) S β − r S β + rr µ (cid:17) S β πe (cid:16) − e − µ µ (cid:17) E N W (cid:19) , r > β ∈ (0 , W β log (cid:16) r S β πe E N W (cid:17) , < r ≤ , (39)where the parameter S β , defined as (15), is determined bythe normalized Nyquist pulse g β ( t ) used, and µ is the uniquesolution to S β − r S β + r r = 1 µ − e − µ − e − µ . (40) Proof:
The proof is given in Appendix F.By using PL pulse as g β ( t ) in Theorem 6, numerical lowerbounds are given in Sec. III. The bounds obtained from (37)and (38) are called the Unif-PL-IFS lower bounds and thebound obtained from (39) is called the TE-PL-IFS lowerbound. The reason for choosing the PL pulse is as follows.In bias-aided ISI-free signaling over the BLOIC, there isa tradeoff between the required DC bias and the achievedsymbol rate: a higher symbol rate requires a larger DC bias(and a larger power cost). This leads to a tradeoff betweenpower and DOF, see Fig. 5 in Sec. III. When the rate of ISI-free signaling is close to the Nyquist rate, the required DCbias increases sharply (cf. Fig. 4 of [13]). Achieving ISI-freesignaling at exactly the Nyquist rate is impossible because itrequires an infinitely large DC bias. A general analysis onthe optimal DC bias-symbol rate tradeoff of arbitrary Nyquistpulses is difficult. But for certain kind of parametric pulses(e.g., raised cosine pulse with a roll off factor β ) this tradeoff TABLE IVU
PPER B OUNDS AND R ELATED
DTOIC R
ESULTS
Upper bounds Related DTOIC result
SP UB3, Fig. 2 [6, (11)]Dual UB, Fig. 2 [4, (28)]SP UB1 and SP UB2, Fig. 6 [7, Theorem 1]Dual UB1, Fig. 6 [4, (19)]Dual UB2, Fig. 6 [4, (20)]Dual UB1, Fig. 7 [4, (11)]Dual UB2, Fig. 7 [4, (12)] has been numerically characterized in [13] in which the PLpulse was shown to be a good choice in a variety of pulses.
D. Capacity Upper Bounds
The following lemma holds for all the input power con-straints given in Table II.
Lemma 3 : The capacity of the PC-BLOIC with bandwidth W is upper bounded by C PCBLOIC ≤ C PC ,σ = N W DTOIC · W transmissions per second (41)where C PCDTOIC is the capacity of the DTOIC under the sametype of constraint with equal parameters as the PC-BLOIC.
Proof:
The proof is given in Appendix G.Combining Lemma 3 and known capacity upper bounds forthe AP-DTOIC, the following capacity upper bounds for theAP-BLOIC are obtained.
Theorem 7 : The capacity of the AP-BLOIC is upperbounded by the following two bounds simultaneously: C APBLOIC ≤ W log e π (cid:18) E√ N W + 2 (cid:19) ! , (42) C APBLOIC ≤ sup α ∈ [0 , W (cid:18) log (cid:18) e π E N W (cid:19) α − log (1 − α ) − α α α (cid:19) . (43) Proof:
By Lemma 3 and the upper bounds for the AP-DTOIC from [3, eqn. (21)] (implicitly given therein) and [7,eqn. (1)], (42) and (43) are obtained, respectively.
Remark 1 : More capacity upper bounds for the AP-BLOICcan be obtained by other capacity upper bounds for the AP-DTOIC. Parallel results of Theorem 7 for the PP- and PAPR-BLOIC can be similarly obtained. Since the mathematicalexpressions of these results can be written out directly basedon the corresponding DTOIC results, we only give themnumerically in Sec. III, and list them in Table IV. Accordingto the type of the related DTOIC results, we categorize theseupper bounds as sphere-packing based ones (SP UB) andduality based ones (Dual UB). III. C
OMPARISONS OF B OUNDS
In this section we give numerical evaluation of our results.In all figures the SNR and PNR of the BLOIC is defined asSNR = E√ N W and PNR = A√ N W , respectively.In Fig. 2 our main results on the AP-BLOIC are plotted. Athigh SNR, it is shown that the Exp-S2 lower bound (25) andthe upper bound (42) are the tightest lower and upper bounds,respectively. Moreover, they have the same asymptotic slopeand the high SNR asymptotic gap between them is 4.34 dBin SNR or 2.89 bit/s/Hz in spectral efficiency.For comparison, the information rate bounds in [3] for aspecific TDS scheme called 3-PSWF, whose lower bound isthe best among all examples in [3], are also plotted in Fig. 2. Itis clear that the lower bound increases very slowly with SNRand the upper bound is not a capacity upper bound for theAP-BLOIC. Moreover, a high SNR asymptotic upper boundfor the AP-BLOIC based on the result of [11] is also shownand the details about this bound are given in Sec. IV.An important observation is that ISI-free signaling performswell for all practical SNR values (e.g. in Fig. 2 we show theSNR range [ − , in dB). At low to moderate SNR, all theISI-free signaling based lower bounds outperform the Exp-S2lower bound (25), and the Geom-S2-IFS lower bound (35) isthe tightest one. At high SNR, the Unif-PL-IFS lower bound(37) obtained by DC bias aided ISI-free signaling achievesinformation rates close to the best known capacity lower boundobtained without ISI-free constraint.Fig. 3 and Fig. 4 show the AP-BLOIC capacity lowerbounds in the low to moderate SNR regime. The informationrates of regular PAM constellations based (TDS) schemes of[3] are also given for comparison. Only when the SNR isbelow 0 dB, the TDS schemes may have similar informationrates compared with some of our lower bounds. The lowerbound (35) stands out from all the results at low to moderateSNR.Fig. 5 shows the information rates of DC bias aided ISI-free signaling using g PL ( t ) under different roll-off factors (i.e.the RHS of (37) excluding the supremum operation), and atradeoff between low-SNR and high-SNR information rates fora given β is clear. As practical systems always use a fixed β ,a carefully chosen β (typically from 0.15 to 0.4) may balancethe performance for most practical SNR values.Fig. 6 and Fig. 7 show the capacity bounds for the PP- andPAPR-BLOIC (where r = 2 . ), respectively. The behavior ofthese bounds are similar to that in the AP-BLOIC case. Notethat at low SNR (33) and (34) are equal to (38) and (39),respectively.In Fig. 8 we show the lower bounds on the capacity of thePAPR-BLOIC given by (30) for different PAPR values, whereall the bounds are derived using the S2 pulse. Meanwhile, (25)is given as a benchmark since it can be viewed as the case of r = ∞ , noting that as r → ∞ , µ tends to r , and the RHS of(30) monotonically increases and tends to the RHS of (25). For A . -fractional bandwidth definition is used in the evaluation of theperformance of 3-PSWF (also the TDS based PAM results in Fig. 3 and Fig.4). Although the PSWF who achieve the best time-frequency concentration areused, the total DOF efficiency is dominated by the rectangular basis functionwhich always exists in TDS signal space models. -5 0 5 10 15 20 SNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) SP UB1 (42)SP UB2 (43)SP UB3Dual UBExp-S2 LB (25)Exp-S2-IFS LB (32)Geom-S2-IFS LB (35)Unif-PL-IFS LB (37)Asymptotic UB [11]3PSWF UB [3]3PSWF LB [3]
Fig. 2. Upper bounds (UB) and lower bounds (LB) for the capacity of the AP-BLOIC. -6 -4 -2 0 2 4 6 8
SNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) Exp-S2 LB (25)Exp-S2-IFS LB(32)Geom-S2-IFS LB (35)3PSWF [3]Rect. PAM [3]
Fig. 3. Lower bounds on C APBLOIC : moderate SNR region. our bounds, it is shown that the PAPR constraint only causessome SNR loss. Moreover, an example of the capacity lowerbound of the BLOIC with input constraint P { X } ≤ E , ≤ x ( t ) ≤ A (called AP-PP-BLOIC, where A is a constant) isgiven, denoted as AP-PP LB. It is obtained by setting PAPR r = A / E in Corollary 3 for each SNR. The value of A isset to be 10 dB higher than the noise variance. When the SNRis relatively low, the PAPR is large so that the bound is close -12 -10 -8 -6 -4 -2 0 SNR (dB) -3 -2.5 -1.5 -1 -0.5 I n f o r m a ti on R a t e ( b it/ s / H z ) Exp-S2 LB (25)Exp-S2-IFS LB (32)3PSWF [3]Rect. PAM [3]2PAM TDS4PAM TDS
Fig. 4. Lower bounds on C APBLOIC : low SNR region. to the Exp-S2 LB. When the SNR exceeds 7 dB the boundstops increasing.IV. D
ISCUSSIONS AND C ONJECTURES
This section discusses further improvement of our lower andupper bounds, since there are still considerable gaps betweenthem. We believe that new bounding techniques are needed
SNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) Exp-S2 LB (25) β =0.05 β =0.15 β =0.4 β =0.6 β =0.8 β =1 Fig. 5. Information rates of DC bias aided ISI-free signaling using g PL ( t ) under different roll-off factors. PNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) SP UB1SP UB2Dual UB1Dual UB2Unif-cos LB (27)Unif-S2-IFS LB (33)Unif-PL-IFS LB (38)
Fig. 6. Upper and lower bounds on the capacity of the PP-BLOIC. to tighten the gap. We give two conjectures which considerslower and upper bounding, respectively.Our DOF-optimal capacity lower bounds have a generalform of C PCBLOIC ≥ W log (cid:18) η E N W (cid:19) . (44)In Theorem 2 and Theorem 3, η can be maximized by findingout the optimal g ( t ) that maximizes G / S N . However, this isstill an open problem. Moreover, the optimal g ( t ) for thePAPR-BLOIC may vary for different PAPR values. In Fig.9, the values of η in the lower bounds on the capacity of thePAPR-BLOIC obtained by using g △ ( t ) and g cos ( t ) in Theorem3 are given for different PAPR values. It is shown that when r is smaller than 2.7, g cos ( t ) is better, and otherwise g △ ( t ) is better. For large PAPR values, finding out a pulse whichachieves larger η than that obtained by g △ ( t ) is difficult, -5 0 5 10 15 20 SNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) SP UB1 for PP-BLOICSP UB2 for PP-BLOICSP UB (39) for AP-BLOICDual UB1Dual UB2TE-cos LBTE-S2-IFS LB (34)TE-PL-IFS LB (39)
Fig. 7. Upper and lower bounds on the capacity of the PAPR-BLOIC, r =2 . . SNR (dB) I n f o r m a ti on R a t e ( b it/ s / H z ) Exp-S2 LB (25), (r= ∞ )TE-S2 LB (30), r=4TE-S2 LB (30), r=3TE-S2 LB (30), r=2.5TE-S2 LB (30), r=2TE-S2 LB (30), r=1.5TE-S2 LB (30), r=1AP-PP LB (PP =10dB) Fig. 8. Lower bounds on the capacity of the PAPR-BLOIC and the AP-PP-BLOIC. because a G ( f ) with relatively large G always has largesidelobes in the time domain, which also causes large S . Dueto the time-frequency uncertainty, we cannot make G / S N verylarge. In summary, we have the following conjecture: Conjecture 1 : The high-SNR asymptotic capacity expressionof i.i.d. Nyquist rate PAM signaling over the AP-BLOIC,denoted as C AP, i.i.d. NRPBLOIC , satisfies lim
SNR →∞ (cid:26) C AP, i.i.d. NRPBLOIC − W log (cid:18) πe E N W (cid:19)(cid:27) = 0 . (45)For the capacity of the AP-DTOIC, the high-SNR asymp-totically tight bounds reported in [3] (implicitly) and in [4],[6] imply that lim SNR →∞ (cid:26) C APDTOIC − log λ E σ (cid:27) = 0 (46) PAPR η Using g △ ( t )Using g cos( t ) Fig. 9. Behavior of η for two specific modulation pulses in the PAPR-BLOIC. where λ = p e π . Our tightest capacity bounds for the AP-BLOIC at high SNR, (25) and (42), have the same pre-logfactor but different pre-SNR factors: W log (cid:18) λ E√ N W + 2 λ (cid:19) ≥ C APBLOIC ≥ W log λe E√ N W . (47)We may thus expect that lim
SNR →∞ (cid:26) C APBLOIC − W log ρλ E√ N W (cid:27) = 0 . (48)where ρ ≤ is a factor determining the high SNR capacityof the AP-BLOIC. The existence and value of the factor ρ in(48) is of interest to us because if ρ is strictly less than one,then we can conclude that when we simplify the BLOIC to aDTOIC which transmits W times per second, there does exista penalty on capacity because of the fundamental distinctionbetween the BLOIC and the DTOIC. We have the followingconjecture. Conjecture 2 : The factor ρ in (48) exists and satisfies ρ < . Remark 2:
A possible way of settling Conjecture 2 is usingthe sphere packing based upper bounding technique in [11],by which we can get a high-SNR asymptotic upper bound onthe capacity of the AP-BLOIC as lim
SNR →∞ (cid:26) C APBLOIC − W log (cid:18) lim K →∞ K (cid:0) Vol(Υ K ) (cid:1) K πe E N W (cid:19) (cid:27) ≤ , (49)where Υ K is the admissible region of length- K input symbolsequences as Υ K = (cid:26) [ c k ] Kk =1 : 12 + Re " K X k =1 c k exp − j kπW tK ≥ , c k ∈ C (cid:27) , (50) and Vol(Υ K ) is its volume. Conjecture 2 can be proved ifwe can show that lim K →∞ K (cid:0) Vol(Υ K ) (cid:1) K < e / . However,a direct calculation of this limit or even its upper bound isnontrivial. In [11] it was proved that Υ K is a subset of a K -dimensional trigonometric moment space M K whose volumehas been determined to satisfy lim K →∞ K (cid:0) Vol( M K ) (cid:1) K <πe/ . Thus we have the asymptotic upper bound shown inFig. 2, based on (49). Unfortunately this is not enough forsettling Conjecture 2. Note:
In fact, Υ K is the admissible region of the input ofthe AP-MCOIC with K subcarriers and a nominal bandwidth W . As K → ∞ , [11] shows that the high-SNR asymptoticcapacity of the AP-MCOIC is upper bounded by W log E N W . We note that each asymptotic result obtained by (49) is alsoa high-SNR asymptotic upper bound on the capacity of theAP-BLOIC, although it is obtained from considering the AP-MCOIC. The interpretation is as follows. For a fixed W , K tending to infinity is equivalent to the length of a W -intervalsample sequence of a block of input of the MCOIC tendingto infinity. Meanwhile, the out-of-band energy of the MCOICin the sense of nominal bandwidth decreases to zero, and thetime domain W -interval samples of the MCOIC reduce to theNyquist samples. In summary, as K tends to infinity, the limitof the admissible region of the input of the AP-MCOIC tendsto the admissible region of the input of the AP-BLOIC withbandwidth W . So the capacity of the AP-MCOIC convergesto the capacity of the AP-BLOIC whose bandwidth is equalto W . A PPENDIX
AConsider an i.i.d. Nyquist rate PAM ensemble using g △ ( t ) as X △ ( t ) = X i X i g △ (cid:18) t − i W (cid:19) = X i X i sin πW ( t − i/ W )2( πW ( t − i/ W )) . (51)Let the input symbols be i.i.d. nonnegative (which guaranteesthe nonnegativity of X △ ( t ) since g △ ( t ) is nonnegative) andlet E[ X ] be equal to E . Then the AP of X △ ( t ) is P { X △ } = lim T →∞ T E "Z T − T ∞ X i = −∞ X i sin πW ( t − iT s )2 ( πW ( t − iT s )) d t = lim N →∞ N T s Z NT s − NT s ∞ X i = −∞ E [ X i ] sin πW ( t − iT s )2 ( πW ( t − iT s )) d t = E · lim N →∞ N T s N − X n = − N Z ( n +1) T s nT s ∞ X i = −∞ sin πW ( t − iT s )2 ( πW ( t − iT s )) d t = E · lim N →∞ N T s N − X n = − N ∞ X i = −∞ Z ( n +1) T s nT s sin πW ( t − iT s )2 ( πW ( t − iT s )) d t The expression of this result is different from [11, (56)] (when the channelgain is normalized to one) because [11] used a nonstandard definition of thepower spectral density of the white Gaussian noise. = E · lim N →∞ N T s N − X n = − N ∞ X i = −∞ Z ( n +1 − i ) T s ( n − i ) T s sin πW t ′ πW t ′ ) d t ′ = E · lim N →∞ N T s N Z ∞−∞ sin πW t πW t ) d t = E · T s · W = E , (52)where the second equality follows by the linearity of ex-pectation and letting T = N T s , the third by dividing theinterval of integration, the fourth by noting that the infinitesum converges for t ∈ R , the fifth by defining t ′ , t − iT s ,and the sixth by noting that the integrals over all the intervals [( n − i ) T s , ( n + 1 − i ) T s ) sum to the integral over ( −∞ , ∞ ) .Thus the AP constraint is satisfied, and X △ ( t ) is admissiblefor the AP-BLOIC. Since G △ ( f ) = ( W W −| f | W , | f | ≤ W , otherwise, (53)we have G = exp W Z W log | W · G △ ( f ) | d f ! = exp W Z W log (cid:18) W − fW (cid:19) d f ! = 1 e . (54)Substituting (54) into (18) completes the proof of Theorem 1.A PPENDIX
BConsider the i.i.d. Nyquist rate PAM ensemble X PAM ( t ) as(17). By letting X i be bounded within [ − A S N , A S N ] , we canmake the ensemble satisfy − A ≤ X PAM ( t ) ≤ A , accordingto the definition of S N . Let a DC bias D = A / be addedto X PAM ( t ) , and thus the nonnegativity and PP constraints ofthe PP-BLOIC are both satisfied. The maximum differentialentropy of X i is h ( X ) = log AS N and is obtained by letting X i be uniformly distributed. Using Lemma 2, (26) is obtainedimmediately. A PPENDIX
CFor the case of r > , let us consider the i.i.d. Nyquist ratePAM ensemble X PAM ( t ) as (17). Assume the inputs X i to bebounded within [0 , L ] and let E[ X i ] = L/ν , ν ≥ , where ν is the PAPR of X i . The maxentropic distribution of X i is atruncated exponential distribution according to [4, eqn. (42)], p X ( x ) = 1 L µ − e − µ e − µxL , ≤ x ≤ L, (55)where µ is the unique solution of ν = 1 µ − e − µ − e − µ . (56)The differential entropy of (55) is h ( X ) = µν + log (cid:18) L − e − µ µ (cid:19) . (57) Then the maximum achievable information rate of X PAM ( t ) can be lower bounded by R [ X ( t )] ≥ W log G ν e µ/ν πe (cid:18) − e − µ µ (cid:19) L ν N W ! , (58)which follows from Lemma 2 by using the distribution (55).We now design X PAM ( t ) to be an admissible input ensembleof the PAPR-BLOIC with PAPR r and convert (58) into acapacity lower bound of the PAPR-BLOIC with parameters r , E , G , and S N . We first shift the distribution of X i by − L sothat X i is distributed in (cid:2) − L , L (cid:3) . Now its mean becomes E[ X i ] = Lν − L . (59)The corresponding i.i.d. Nyquist rate PAM ensemble, denotedas X ( t ) , satisfies − S N L ≤ X ( t ) ≤ S N L . (60)By adding a DC bias D = S N L on X ( t ) , we obtain anadmissible waveform ensemble X PAM ( t ) that satisfies ≤ X ( t ) ≤ S N L . The PP constraint is satisfied by letting S N L be equal to r E . Since the mean of X ( t ) is equal to E[ X i ] ,i.e. the mean of the input symbol (cf. the derivation of (52)),the AP of X PAM ( t ) is E = D + E[ X i ] = 2 − ν + ν S N ν L. (61)The PAPR of X PAM ( t ) is r if we set ν as ν = 2 r S N − r S N + r . (62)According to (62) and S N L = r E , we can convert (58) into ageneral capacity lower bound of the PAPR-BLOIC as the caseof r > in Theorem 3.For the case of < r ≤ , consider the designed inputensemble X PAM ( t ) (after DC bias being added) in the proofof Theorem 2 in Appendix B. Apparently the AP of X PAM ( t ) is A / . If we let A = r E , the AP of X PAM ( t ) is then r E / which is smaller than E since r < . So X PAM ( t ) is alsoadmissible for the PAPR-BLOIC with r < and r E = A .Replacing A in (26) by r E , the case of < r < is obtainedand the proof of Theorem 3 is completed.Corollary 3 follows immediately from Theorem 3 by letting g ( t ) be g △ ( t ) and noting that in this case G = 1 /e (see (54))and S N = 1 (this was implicitly shown in [18]).A PPENDIX
DConsider the following input ensemble which achieves ISI-free signaling according to [12]: X IFS △ ( t ) = X i X i sin πW ( t − i/W )( πW ( t − i/W )) , (63)where the pulse used is a scaling of g △ ( t ) with a factor of twoso as to satisfy Definition 1. Letting the input symbols { X i } bei.i.d. and satisfy E[ X ] = E , then the AP of the input ensembleis P { X } = E (cf. the derivation of (52), noting that thetime domain integral of the pulse used here is /W ). Denote the equivalent discrete-time memoryless channel of ISI-freesignaling over the BLOIC using (63), which is obtained from W -interval sampling at the receiver, as Y [ i ] = X [ i ] + Z [ i ] . (64)The value of g (0) determines the noiseless sample values whenthe corresponding input symbols are given. Since · g △ (0) = 1 ,the noiseless samples satisfy X [ i ] ≡ X i . The variance ofthe noise samples can be minimized by an ideal bandlimitedfiltering over W on the received noisy waveform, not affectingthe value of X [ i ] in (64) and preserving the memorylessproperty of noise samples. So the obtained noise samples Z [ i ] are i.i.d. with variance N W . Then if we let X i beexponentially distributed as (23), we have R [ X IFS △ ( t )] = W · I ( X [ i ] , Y [ i ])= W · ( h ( Y [ i ]) − h ( Z [ i ])) ≥ W (cid:16) e h ( X i ) − h ( Z [ i ]) (cid:17) = W (cid:18) e π E N W (cid:19) , (65)where the inequality follows from the EPI, and (32) is ob-tained.The proofs of (33), the first case of (34), and the secondcase of (34) are similar except that we let X i be uniformlydistributed in [0 , A ] , truncated-exponentially distributed in [0 , r E ] (whose PDF can be obtained from (55) by replacing L with r E and replacing ν with r ), and uniformly distributed in [0 , r E ] , respectively. A PPENDIX
EThe proof of Theorem 5 can be done by a variation on theproof of Theorem 4 as follows. Note that (64) can be viewedas a DTOIC where Z i ∼ N (0 , N W ) , transmitting only W times per second. So we have R AP , IFSBLOIC ≥ C AP , σ = N W DTOIC · W transmissions per second . (66)Now all capacity lower bounds of the AP-DTOIC can beused to derive a lower bound for the maximum achievableinformation rate of ISI-free signaling over the AP-BLOIC. Thetightest known capacity lower bound of the DTOIC is in [6],where a geometric distribution as (36) is used as the inputdistribution. The parameter l (space between mass points ofgeometric distribution) in (36) is optimized for each SNR tomaximize the information rate achieved by such lower bound.That information rate is thus max l I ( Q g ( l ) , V ) . The proof ofTheorem 5 is then completed by replacing C AP , σ = N W DTOIC in(66) by max l I ( Q g ( l ) , V ) .A PPENDIX
FConsider the i.i.d PAM ensemble X PAM ( t ) as (9) in whichwe let g ( t ) be a Nyquist pulse satisfying Definition 1 and T s be β W so that the ISI-free property is achieved. Let the inputsymbols X i be uniformly distributed in [ − L/ , L/ . Since E[ X i ] = 0 , the mean of the ensemble obtained is also zero(cf. the derivation in (52)). According to the definition of S ( τ ) , we have − S β L ≤ X PAM ( t ) ≤ S β L . So the DC bias neededis S β L , and the mean (AP) of X PAM ( t ) becomes E = S β L , (67)with that DC bias.Consider the equivalent discrete-time channel of such ISI-free signaling, denoted as Y [ i ] = X [ i ]+ Z [ i ] (cf. the derivationof (65)). Since X [ i ] is uniformly distributed, from (67) wehave h ( X [ i ]) = log L = log 2 ES β . (68)So the maximum achievable information rate can be lowerbounded as (cf. (65)) R = 2 W β · I ( X [ i ] , Y [ i ]) ≥ W β log (cid:16) e h ( X [ i ]) − h ( Z [ i ]) (cid:17) = W β log S β πe E N W ! , (69)where the inequality follows from the EPI, and (37) is obtainedimmediately. This completes the proof of (37).Since the input ensemble obtained in the proof of (37) isbounded in [0 , S β L ] , by letting L = AS β the same input en-semble is admissible for the PP-BLOIC. Combining L = AS β ,(67), and (69) we obtain (38).Consider a truncated exponential distribution Q TE whosePDF is as (55) except letting S β L = r E , ν = 2 r S β − r S β + r . (70)From the proof of Theorem 3, we know that if we use such aninput symbol distribution to replace the uniform distributionused in the proof of (37), the obtained input ensemble isadmissible for the PAPR-BLOIC (i.e. the AP equals E and thePAPR equals r ) after adding a minimum required DC bias.Replace the h ( X [ i ]) in (69) by the differential entropy of Q TE (which is obtained by combining (70) and (57)) we obtain thefirst case of (39). For the second case we alternatively usean uniform distribution Q U in h , r ES β i which guarantees theobtained input ensemble satisfying the PP constraint. Notingthat the AP constraint is also satisfied, the second case of(39) can be obtained from replacing the h ( X [ i ]) in (69) bythe differential entropy of Q U . This completes the proof ofTheorem 6. A PPENDIX
GConsider a bandlimited channel as Y ( t ) = X ( t ) + Z ( t ) (where Z ( t ) is defined as in (1)), denoted as channel A, witha relaxed version of input power constraint PC as follows:1) the nonnegativity and PP constraints on X ( t ) hold only at t = n/ W, ∀ n ∈ Z , i.e. on a sequence of Nyquist samplepoints; 2) the AP constraint is defined like that in the BLOIC.Obviously, the capacity of this channel, denoted as C PC-relaxedA ,is an upper bound of C PCBLOIC . Each input ensemble of channelA, denoted as X A ( t ) , can be viewed as an i.i.d. Nyquist rate PAM ensemble as (17) where g ( t ) = g sinc ( t ) and theinput symbols { X i } are just the Nyquist samples of X A ( t ) .By matched filtering X A ( t ) with W · G sinc ( f ) and samplingat Nyquist intervals (an information lossless procedure), anequivalent model of channel A is obtained as the followingdiscrete-time memoryless channel: Y [ i ] = X [ i ] + Z [ i ] , (71)where X [ i ] = X i , and Z [ i ] ∼ N (0 , N W ) . This channel is aDTOIC with the same type of constraints and correspondingparameters per the constraint PC of channel A. Since thischannel transmits W symbols per second, we have C PCBLOIC ≤ C
PC-relaxedA = C PC ,σ = N W DTOIC · W transmissions per second . (72)This completes the proof of Lemma 3.A CKNOWLEDGEMENTS
The authors would like to thank the anonymous reviewersfor their comments which helped to significantly improve themanuscript. R
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