I Introduction
In Part IV of his 1948 landmark paper [1], Shannon introduced a bandlimited Gaussian channel (BLGC) as
(1) 
where is bandlimited to Hz, and is additive white Gaussian noise (AWGN) of power spectral density on . Two types of input constraint were considered: 1) an average power (AP) constraint , and 2) a peak power (PP) or amplitude constraint as . For brevity we denote these two cases as APBLGC and PPBLGC, respectively, and use similar notations for other kinds of channels subsequently. For an APBLGC of signaltonoiseratio , Shannon established the famous formula
(2) 
But for PPBLGC, Shannon only provided upper and lower bounds on its capacity, and almost no further results have been known since then. An exception is [2], which improved Shannon’s capacity lower bound for PPBLGC by 0.567 dB.
A related topic is the capacity problem of the PP constrained discretetime Gaussian channel (DTGC) as
(3) 
where the channel input satisfies (possibly combined with an AP constraint as ), and the Gaussian noise is memoryless. The capacity of PPDTGC can be numerically evaluated using techniques initiated in [3]. However, PPDTGC only models intersymbolinterferencefree (ISIfree) transmission, which cannot achieve optimal bandwidth efficiency in BLGC with PP constraint. The reason is that, for a Nyquist shaping pulse, a smaller excess bandwidth causes larger sidelobes, whose superposition sharply increases the peak of continuoustime signal waveform as the excess bandwidth tends to zero [4].
In this paper, we revisit the continuoustime scenario and consider an extended version of PPBLGC, namely, BLGC with a peaktoaveragepowerratio (PAPR) constraint. In Sec. II we establish a general capacity lower bound for PAPRBLGC by lower bounding the information rate of constrained pulse amplitude modulation (PAM) with independent and identically distributed (i.i.d.). input symbols. The bound has a form similar to (2) except for a preSNR factor determined by the shaping pulse. We maximize the preSNR factor by optimizing the pulse, and the tightest lower bound obtained outperforms the result on PPBLGC in [2] by 0.926 dB. Moreover, our bound reduces to (2) when the PAPR constraint tends to infinity. Capacity upper bound for PAPRBLGC is also provided based on Shannon’s approach for PPBLGC in [1]. In Sec. III, using the same techniques, we study bandlimited optical intensity channels (BLOIC) [4, 5, 6], improve capacity lower bounds in [6] at high SNR, and disprove a conjecture therein. Our result also outperforms the capacity lower bound of APBLOIC in [7] obtained by a sphere packing argument. Finally, Sec. IV concludes the paper.
Ii Capacity Bounds for PAPRBLGC
Following [1], the capacity of a BLGC is defined as
(4) 
where
(5) 
is the mutual information per degreeoffreedom (DoF) between
and , and are Nyquist sample sequences of and , respectively. For a PAPRBLGC, the constraint on the input waveform is(6) 
where . We refer to (6) as a PAPR constraint .
In [1], by deriving high and low peaktonoise ratio (PNR) asymptotic results, Shannon showed that the capacity of PPBLGC with can be expressed as
(7) 
where is an unknown prePNR factor satisfying . When is sufficiently small, the PP constraint dominates and the PAPRBLGC behaves like a PPBLGC. When tends to infinity the PAPRBLGC behaves like an APBLGC. So we can infer that the capacity of PAPRBLGC can be written as
(8) 
The preSNR factor is a nondecreasing function of satisfying and .
Iia Capacity Lower Bound
We derive capacity lower bounds for PAPRBLGC by lower bounding the information rate of PAM waveform ensemble like
(9) 
under the PAPR constraint, where the shaping pulse is an (i.e., finiteenergy) funtion normalized as
(10) 
We denote the Fourier transform of
by . Let the input symbols be i.i.d. satisfying , . To achieve the maximum prelog factor, we let be bandlimited to and . Since the power of an i.i.d. PAM ensemble as (9) is given by [References, Sec. 14.5.1], to meet the AP constraint in (6) we let . To meet the PP constraint in (6), first we let satisfy . Define(11) 
which evaluates the maximum possible superposition of peak caused by pulse shaping. Then and the PP constraint in (6) is equivalent to .
We use the following lemma to derive our lower bound. The lemma is essentially due to Shannon [1], and can also be proved using Szegö’s theorem [2].
Lemma 1: The achievable information rate of the i.i.d. PAM ensemble (9) is lower bounded by
(12) 
where is the differential entropy of , and
(13) 




S2 







RC 


PL 



BTN 

“Better than Nyquist” pulse [11].  
ICIT 


We tighten (12) by finding a maxentropic distribution for under the constraints and . According to [References, Theorem 12.1.1] we obtain the following solution.
1) When
, the maxentropic distribution is a truncated Gaussian distribution with probability density function (PDF)
(14) 
where ,
(15) 
and the parameter is the unique solution of
(16) 
The corresponding differential entropy is
(17) 
2) When
, the maxentropic distribution is a uniform distribution with PDF
(18) 
and the corresponding differential entropy is
(19) 
Substituting (17) and (19) into (12) yields the following result.
IiB Pulse Optimization
The general lower bound (20) can be evaluated using specific pulses. Table I lists the pulses considered in this paper. Fig. 1 shows the Fourier transform of the pulses used in PAPRBLGC, all of which are normalized to satisfy (10). Since no analytic solution for (16) is known, for our result, the preSNR factor (21) is evaluated numerically. Although the exact value of is difficult to find analytically for arbitrary pulses, the sidelobes of the pulses in Table I decay asymptotically as or faster as . This makes reliable numerical evaluation of possible. For parametric pulses such as the RC pulse, we optimize it over to maximize for each .
In Fig. 2, our numerical results for the preSNR factor lower bound (21) is provided. The BTN pulse shows the best performance. Fig. 5 provides the optimal value of for the BTN pulse and the PL pulse for each (the step size for optimizing is 0.01). Fig. 5 provides the preSNR factor lower bounds obtained by using the BTN pulses with some specific values of , and the best lower bound obtained by optimizing the BTN pulse over (denoted “BTNOPT LB”). The optimal value decreases as the PAPR constraint increases. As , it appears that tends to zero for both the BTN pulse and the PL pulse, i.e. both pulses tend to the sinc pulse. The following result shows that the lower bounds for the BTN, PL and RC pulses in Fig. 2 tend to one as .
Proposition 2: For the PAPRBLGC, as grows without bound, there exists a parametric pulse such that tends to one and (20) tends to (2).
Outline of Proof: By substituting (15) into (16) we obtain
(22) 
By noting that the RHS of (22) is continuous for and , we can prove that when is finite, if grows without bound, then will also grow without bound, and . Thus when , we have for a given . Using a parametric pulse which tends to the sinc pulse as (e.g., the BTN pulse), we can make arbitrarily close to one as grows without bound.
By replacing in the second case of Proposition 1 with , we revisit a capacity lower bound for PPBLGC established in [2], which is equivalent to
(23) 
where is the prePNR factor in (7). Thus we can compare our results with known capacity lower bounds for PPBLGC. Using the S2 pulse, in [1] it was shown that
(24) 
Using the SC pulse, in [2]
(25) 
was obtained, which improved (24) by 0.567 dB. According to Fig. 2, both the BTN and PL pulses outperform the SC pulse. In particular, employing the BTN pulse we obtain
(26) 
This tightens the existing PPBLGC result (25) by dB.
Remark: In PPBLGC, the achievable rate using a specific pulse is determined by , which is proportional to the slope of the corresponding curve in Fig. 5 when . But for PAPRBLGC, when , a pulse with larger is not necessarily better. Nevertheless, if two pulses have the same (), we always prefer the one that with smaller (larger ).
IiC Capacity Upper Bound
In [1], by proving an asymptotic upper bound for the PPBLGC as , Shannon essentially established the following general upper bound for the capacity of bandlimited channels with an amplitude constraint.
Lemma 2: Let the input of a BLGC be constrained by an amplitude constraint , where . The capacity of this channel is upper bounded by
(27) 
where is the capacity of a DTGC as (3) satisfying and .
By combining this and the numerical computation technique in [3], we provide numerical upper bound (denoted “UB”) for the PAPRBLGC in Fig. 5. It is shown that the gap between our capacity upper and lower bounds is still large for small , and the gap decreases as increases. For example, when , the high SNR gap is about 2.30 dB; see Fig. 5. As the gap diminishes, as indicated by Proposition 2.
Iii Capacity Bounds for PAPRBLOIC
In intensity modulation and direct detection based optical communications, a wellknown channel model is the BLOIC,
(28) 
where is the transmitted optical intensity (i.e., the optical power transferred per unit area), which is bandlimited to due to the limited available bandwidth of optoelectronic components, and is the same as that in (1). We consider a PAPR constraint on the input optical power as
(29) 
The capacity of the BLOIC has been studied in the literature; see [5, 7, 6] and references therein. In particular, capacity bounds for the PAPRBLOIC was derived in [6] using the bounding techniques in Sec. II, and the lower bound is like
(30) 
where , and the preOSNR factor is given by
(31) 
where and are defined as (11) and (13), respectively, with respect to an pulse normalized as
(32) 
and is the unique solution of
(33) 
In [6], although the benefit of pulse optimization had been recognized, only the S2 and SC pulses were used to evaluate the lower bound (30). Now we further consider the BTN and the ICIT pulses in Table I. Fig. 8 shows the spectra of these pulses which are normalized as (32) (note that in Fig. 1 the normalization is as (10)). Fig. 8 gives the corresponding results on the preOSNR factor, and Fig. 8 gives the corresponding optimal parameters. Compared to the results in Fig 2, the preOSNR factor lower bounds in Fig. 8 behave differently. The ICIT based lower bound performs the best at low PAPR (although it performs almost the same as the BTN based one when ), but it begins decreasing at . The BTN based lower bound decreases slower at high PAPR region, but our further calculation reveals that in fact it tends to zero as . The S2 based lower bound is the only nondecreasing bound which tends to . Since a PAM waveform ensemble with a given PAPR is still admissible in channels with higher PAPR constraints, the preOSNR factor of capacity, , is always nondecreasing with . So our optimal lower bound in Fig. 8 is also nondecreasing:
(34) 
It was conjectured in [6] that at high OSNR, the maximum achievable preOSNR factor using i.i.d. PAM ensemble in the APBLOIC is , which is achieved using the S2 pulse. The present results show that the ICIT and BTN pulses perform better (but the input alphabet for must be carefully chosen), and the former one achieves a preOSNR factor 0.06606, which outperforms by 0.26 dB. Moreover, this result also outperforms the result obtained by a sphere packing argument in [7], which shows that for the APBLOIC (i.e., PAPRBLOIC as ).
Iv Conclusion
This paper studies the capacity of continuoustime Gaussian channels with given bandwidth under a PAPR constraint, namely, both a PP constraint and an AP constraint, connected by their ratio . Such a combination of constraints is more relevant to practical systems than a single constraint. For both BLGC and BLOIC, by numerically evaluating information rates of i.i.d. PAM ensembles with optimized pulses, we obtain new achievable rate results which outperform existing ones. However, for both types of channels, the optimal pulse for a given PAPR is still unknown. When is relatively small, the gaps between capacity upper and lower bounds are still considerable. These problems are left to future study.
Acknowledgment
This work was supported in part by the Key Research Program of Frontier Sciences of CAS under Grant QYZDYSSWJSC003, by the National Natural Science Foundation of China under Grant 61722114, and by the Fundamental Research Funds for the Central Universities under Grants WK3500000003 and WK3500000005.
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