Paper - Research Laboratory of Electronics - MIT
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004
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On the Performance of Peaky Capacity-Achieving Signaling on Multipath Fading Channels Desmond S. Lun, Student Member, IEEE, Muriel Médard, Senior Member, IEEE, and Ibrahim C. Abou-Faycal, Member, IEEE
Abstract—We analyze the error probability of peaky signaling on bandlimited multipath fading channels, the signaling strategy that achieves the capacity of such channels in the limit of infinite bandwidth under an average power constraint. We first derive an upper bound for general fading, then specialize to the case of Rayleigh fading, where we obtain upper and lower bounds that are exponentially tight and, therefore, yield the reliability function. These bounds constitute a strong coding theorem for the channel, as they not only delimit the range of achievable rates, but also give us a relationship among the error probability, data rate, bandwidth, peakiness, and fading parameters, such as the coherence time. They can be used to compare peaky signaling systems to other large bandwidth systems over fading channels, such as ultra-wideband radio and wideband code-division multiple access. We find that the error probability decreases slowly with ; under Rayleigh fading, the error probability the bandwidth varies roughly as , where 0. With parameters typical of indoor wireless situations, we study the behavior of the upper and lower bounds on the error probability and the reliability function numerically. Index Terms—Code-division multiple access (CDMA), error exponents, fading channels, ultra-wideband (UWB) radio, wideband.
I. INTRODUCTION
T
HE EMERGENCE of proposals for systems such as ultrawideband (UWB) radio and wideband code-division multiple access (CDMA) in recent years has brought about renewed interest in very-large-bandwidth fading channels. In this regime, it is known that transmission using “white-like” spread-spectrum signals (such as those used in direct-sequence (DS)-CDMA) performs poorly in terms of approaching capacity. Telatar and Tse [1] considered a finite number of time-varying paths, and demonstrated that the mutual information for white-like signals is inverselyproportional to the number of resolvable paths in the wideband limit. It follows that the mutual information is close to zero if the number of resolvable paths is large. Médard and Gallager [2] considered a channel that exhibits time and frequency decorre-
Paper approved by N. C. Beaulieu, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received March 26, 2003; revised August 19, 2003. This paper was presented in part at the Conference on Information Sciences and Systems, Princeton, NJ, March 2002; in part at the Vehicular Technology Conference, Vancouver, BC, Canada, September 2002; and in part at the International Symposium on Information Theory and its Applications, Xi’an, P.R. China, October 2002. D. S. Lun and M. Médard are with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:[email protected]; [email protected]). I. C. Abou-Faycal was with the Massachusetts Institute of Technology, Cambridge, MA 02139 USA. He is now with the Electrical and Computer Engineering Department, American University of Beirut, Beirut 1107 2020, Lebanon (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.829512
lation, and showed that the mutual information approaches zero with increasing bandwidth if spread-spectrum input signals are used. In their paper, spread-spectrum signals are characterized as those whose energy and fourth moment (kurtosis) in a fixed band scale inversely with the total bandwidth and the square of the total bandwidth, respectively; loosely speaking, those that spread energy more or less evenly over the entire available band. Subramanian and Hajek [3] instead considered a wide-sense stationary uncorrelated scattering fading channel with a constraint on the fourth moment of the output signal, and again found an inverse relation between capacity and bandwidth. Given an infinite band, capacity can instead be reached using peaky signaling, i.e., transmitting with a low-duty-cycle frequency-shift-keying (FSK) scheme. Such signaling is described as peaky because transmission energy is concentrated into narrow regions of time (using a low-duty cycle) and frequency (using FSK). The capacity reached by this scheme, which assumes no channel state information at either the receiver or the sender, is the same as that of the additive white Gaussian noise (AWGN) channel with a comparable received signal-to-noise ratio (SNR). This result was presented by Kennedy [4] and by Gallager [5, §8.6] for the case of Rayleigh fading, and most recently by Telatar and Tse [1] for general multipath fading. The particular peaky transmission scheme proposed in [1], which is summarized later, will form the basis of our paper and will be referred to as “capacity achieving,” since it achieves capacity in the wideband limit. In summary, using signals whose energy is spread evenly over a wide band results in poor performance with channels that exhibit time and frequency decorrelation (as is typical for fading channels). Intuitively, this is due to the fact that we are essentially transmitting over a large number of independent channels. If energy is spread evenly over the available band, then the ability to measure each channel decreases as the bandwidth increases and, despite the diversity gain, there is a severe performance degradation in the limit. It follows that, for sufficiently large bandwidth, spreading energy over that band in an even manner that keeps the fourth moment constrained, for example, with DS or related spread-spectrum techniques, is not advisable, and that peaky signaling should yield good performance. The bandwidth at which spreading begins to become detrimental, however, is not entirely clear, nor is the bandwidth at which peaky signaling begins to become advantageous. The former issue is addressed in [6] and [7], while we shall address the latter issue. In particular, we shall be able to assess the performance of the capacity-achieving peaky transmission scheme for a given,
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finite bandwidth, thus facilitating comparison with other transmission schemes on broadband fading channels. The result will, moreover, bring out the dependence of the error probability on the other characteristics of the scheme, such as its peakiness or peak power; an important implementation issue, since it is restricted by the physical limitations of the antenna and the power supply, as well as by safety regulations. We commence by covering the necessary background. We present the channel model in Section II and the capacity-achieving peaky transmission scheme in Section III. In Section IV, we analyze the probability of error of peaky signaling. We upper bound the probability of error, and observe that it vanishes for all rates under capacity as the bandwidth goes to infinity, thus yielding a strong coding theorem. We see that an upper bound that decays much faster as bandwidth increases can be obtained if the fading process is assumed to be Rayleigh, and that, under this additional assumption, a lower bound that is exponentially tight to the upper bound can be found. We find that, under Rayleigh fading, the error probability decreases slowly with the bandwidth , varying , where . This is consistent with recent roughly as results by Verdú [8] that show that approaching capacity is prohibitively expensive both in terms of spectral efficiency and peak-to-average power ratio in the absence of perfect channel side information at the receiver. In Section V, we illustrate the behavior of the bounds for a typical indoor wireless situation. II. CHANNEL MODEL that results from an input The channel-output waveform passed through a multipath fading channel is waveform generally given by (1) and are the gain where is the number of paths, is a and delay on the th path at time , respectively, and bandlimited white Gaussian noise process with power spectral . density and be the coherence time and delay spread of Let the fading channel, respectively. We assume that the processes and are constant and independent and identically distributed (i.i.d.) over time intervals of (i.e., block (i.e., an underspread channel). fading in time), and that Though real channels typically vary in a much more continuous manner and with some statistical correlation over time intervals greater than , the block-fading assumption is frequently used in the analysis of fading channels (see, for example, [6], [9], and [10]) as it greatly increases the tractability of the problem while capturing the essential time-varying quality embodied by channel coherence. The assumption of an underspread channel is not particularly restrictive, as typical delay spreads are on the order of microseconds, while typical coherence times are on the order of milliseconds for wireless channels. III. PEAKY SIGNALING We briefly describe the capacity-achieving peaky transmission scheme. It is shown in [1] that the scheme indeed achieves
all rates below capacity. More precisely, it is shown that all rates that satisfy (2) can be achieved with an arbitrarily small probability of error over a multipath fading channel with average power constraint , but no bandwidth constraint. Since the multipath fading channel can be viewed as the cascade of two channels, one that causes the multipath fading (and results in a signal with average power constrained to ) and a second that adds white Gaussian noise, it follows that the capacity of the infinite-bandwidth multipath fading channel cannot exceed that of the . Thus, infinite-bandwidth AWGN channel alone, i.e., recalling that the channel is assumed to be underspread, we conclude that rates very close to the capacity (which must lie and ) are achievable by between peaky signaling. . Consider the following codebook of size . Let The th code word is represented at baseband as a complex sinusoid of amplitude at frequency , i.e., (3) otherwise where the time duration of the signal is taken to be the coherence time . The frequency is chosen such that it is an , where ; therefore, the integer multiple of size of the codebook is directly related to the minimum transby . mission bandwidth required Let us consider the channel output over the interval . If the time axis at the receiver is shifted appropriately, then and are constant, owing to during this interval, the assumptions of the model, and we denote their values by and , respectively. Hence, by (1), the received signal when message is sent is
(4) is the complex fading where gain. We define signal power in the conventional sense as the received signal power, and thus normalize the channel gain so . that At the receiver, we form the correlator outputs (5) for
. Therefore (6)
where is a set of i.i.d. circularly symmetric complex Gaussian random variables, each satisfying .
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
The message is then repeated over disjoint time intervals to obtain time diversity. Hence, denoting the th correlator output , we have at interval by
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bandwidths are required for good performance, not to mention very complex encoding and decoding schemes. A. Strong Coding Theorem for Multipath Fading Channels
(7) and , where, owing to the block for is a sequence of i.i.d. complex random fading assumption, variables (with no particular distribution) and is a set of i.i.d. circularly symmetric complex Gaussian random variables of unit variance. We construct the decision variables
Recall the transmission scheme described in Section III. Owing to symmetry, we can assume without loss of generality that the transmitted message is 1. An error occurs if or if for some . Let be the event , and let be the event for . Then, denoting the error probability by , we have (11)
(8) For notational convenience, we define
and use the following threshold decoding rule. Let
(12)
(9) (where is an arbitrary parameter) be the threshold. If exceeds for one value of only, then we estimate ; otherwise, we declare an error. We use threshold decoding primarily for its simplicity. Note that the decoding rule is noncoherent, measuring only the energy of the received signal. We transmit using the above scheme for a fraction of time , and then transmit nothing for the remainder of the time. Observe and an energy of to transmit that it takes a time of nats of information. Therefore, one codeword that conveys the average power is , and the rate is given by
An upper bound to is applied to obtain
(13) where (14) Thus
(10) IV. PROBABILITY OF ERROR WITH PEAKY SIGNALING We now derive upper and lower bounds on the probability of error of the capacity-achieving scheme described in Section III. These bounds decay to zero with increasing transmission bandwidth for all rates under capacity, thus yielding a “strong” coding theorem that differs from the “weak” coding theorem of [1], in that it not only delimits the range of achievable rates, but also brings out the relationship among the error probability, data rate, and parameters of the transmission scheme. We emphasize that the lower bound applies to the probability of error of the capacity-achieving scheme and not to general transmission schemes over fading channels. Therefore, there may exist transmission schemes that achieve better performance than the lower bound. Nevertheless, the lower bound is useful, as it gives us a notion of the tightness of the upper bound. It is worth mentioning that the capacity and error exponents associated with a fading channel have been studied previously by Telatar [11], though the model used was very different. Telatar built upon Gallager’s results for energy-limited channels [12], where the channel is modeled as discrete time and discrete input. He showed that, using random block codes and 0–1 signaling, the capacity of the Rayleigh fading channel is the same as that of the AWGN channel in the limit of large bandwidth and large SNR, and examined the rate at which this limiting behavior is approached. He found that the rate of this approach is very slow, and therefore, under the conditions imposed by the energy-limited formulation, inordinately large
is found in [1]. The Chernoff bound
(15) Therefore, in order to upper bound , it remains only to find an upper bound on . As a first step, we can upper bound using the Chebyshev inequality. Theorem 1 (Strong Coding Theorem for Multipath Fading Channels): Let a multipath fading channel have coherence time , delay spread , and Gaussian noise of power spectral den. Let be the variance of the complex sity fading gains. Then there exists a transmission scheme of average , bandwidth , and rate with power , peak power probability of error satisfying
(16) is in the half-open interval , , and . , the Moreover, for all rates less than probability of error can be made to vanish as . Proof: We use the transmission scheme described in Section III. We observe that, in this scheme where
(17)
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and
sufficiently large , the first additive term dominates, which . The theorem, however, holds for any distridecays as ; so it holds regardless of the number of paths and bution of the statistical behavior of the gains and delays on those paths. If is available, a better some information on the statistics of bound could be obtained by determining the statistics of , through which depend on those of (24)
(18) Keeping in mind that forward to verify that
and
are independent, it is straight-
(19) hence
We therefore specialize to the case of Rayleigh fading, where a large number of independent paths is present, resulting in a . circularly symmetric complex Gaussian distribution for Theorem 2 (Strong Coding Theorem for Rayleigh Fading Channels): Let a Rayleigh fading channel have coherence time , delay spread , and Gaussian noise of power spectral . Then there exists a transmission scheme of density , bandwidth , and rate average power , peak power with probability of error satisfying (25) where
(20) and substituting for
(26)
using (10) gives (21)
Recalling (17) for the expectation of we have
, given that
,
is in the half-open interval , , and . , the Moreover, for all rates less than probability of error can be made to vanish as . Proof: We again use the transmission scheme described in Section III. We have
(22)
(27)
We now apply the Chebyshev inequality to obtain by the Chernoff bound. Then, using (8), we obtain
(23) The upper bound (16) follows by substituting the upper bound on given by (23), and the upper bound on given by (15) combined with (14) into (11), and minimizing over its domain to set the optimal threshold. The second part of the theorem follows by noting that the first term of the upper bound, being the upper bound on , for all , while the second term of the vanishes as , can be made upper bound, being the upper bound on to vanish as only for , as established in [1].
Now, since we have assumed that the fading is Rayleigh, the are i.i.d. circularly symmetric complex Gaussian random variables, so the are i.i.d. exponentially . Hence distributed random variables with mean
B. Strong Coding Theorem for Rayleigh Fading Channels
The supremum is achieved by so we have
Although Theorem 1 does indeed give a valid upper bound on the error probability, the bound decays very slowly in . For
(28)
(29) ,
(30)
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
where
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C. Lower Bound on the Error Probability (31)
By substituting (9) and (10), we can write (32)
We have arrived at an exponential upper bound on the probability of error of the capacity-achieving scheme. There is, however, little information on the tightness of the bound. We therefore derive a lower bound on the probability of error and compare it with the upper bound. Theorem 3: The transmission scheme described in Section III has a probability of error under Rayleigh fading that satisfies
where (33) to be the upper For notational convenience, we define given by (32), and to be the upper bound on bound on given by (13), viz.
(40) and are given by (33) and (14), respecwhere and approach zero with intively. The quantities creasing and can be taken as
(34) (41) for
. Since these upper bounds are valid for any , we have, by choosing optimally
(42) (35)
It is evident upon differentiation that, as functions of , is strictly decreasing, while is strictly increasing. In adwhen dition,
Proof: Having made the Rayleigh fading assumption, the and can, in fact, be evaluated exactly probabilities , and are both random since, conditioned upon degrees of freedom. So, using the cumulative variables with distribution function (CDF) for random variables, we have
(43)
(36) which is in the interval
if where
is defined by (31). Similarly
(37) (44) Therefore, given that (37) is satisfied, we can upper bound (35) by
Because and we arrive at the inequalities
are both positive for all ,
(45) (38) and where Observe that we can write where
is given by (26). (46) (39)
for all , over Hence, since its domain of definition, given by (37). The interval (37) grows as decreases, approaching the interval that encompasses all rates under capacity as approaches zero. Thus, there exists such that is positive as long as the rate does not exceed capacity (i.e., (2) is satisfied), and such that the bound approaches infinity. on given by (38) vanishes as
by taking only one of the summation terms. We apply Stirling’s formula to bound the factorial function, and obtain the lower bounds (47) and (48)
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where and are given by (41) and (42), respectively. Having found lower bounds on and , we are now in a position to derive a lower bound on the error probability . We commence with the following observation:
(49) . Using the lower bounds by the independence of the events and and the previously obtained upper bounds, (30) on and (13), we straightforwardly obtain (40), thus completing the proof. It is worth noting from the preceding proof that the upper and are exponentially tight in the and lower bounds on sense that the exponents are arbitrarily close for sufficiently sufficiently large. Indeed, we large or, equivalently, for have (50) for . We now turn our attention to the exponential dependence of , which, we recall, is directly related to error probability on the bandwidth. We define the reliability function of the Rayleigh fading channel using peaky signaling with duty factor as (51) analogously to the treatment of random block coding over discrete memoryless channels by Gallager [5, §5.8]. It represents the true exponential dependence of the error probability on for sufficiently large. Theorem 4: The reliability function of the Rayleigh fading channel using peaky signaling with duty factor is equal to , the error exponent of the upper bound, for all nonnegative rates . Proof: Applying l’Hôpital’s rule to the lower bound (40) yields
Fig. 1. Reliability function E (R; ) as a function of R for = 10 = 10 (dashed), and = 10 (dotted).
(solid),
and, again by applying l’Hôpital’s rule, we see that . By noting that is a probability and is, therefore, at most one, the reverse inequality follows. Hence (55) Thus we see that the upper and lower bounds on are exponentially tight. is monotonically inOwing to the fact that the function creasing (56) and, since we have established that all , we can interchange the yielding
and
is such that where given by (36). Hence
converges for operators,
(57) and is
(58)
(52) . Recall from Section IV-B that, for before optimization over , we have the following upper bound on the error probability:
(53) from which the reverse inequality to (52) follows straightforwardly. , we use For
(54)
V. NUMERICAL RESULTS We now proceed to evaluate the quantities derived in the previous sections for particular parameter choices. We choose fading parameters that are typical for very-high frequency s and transmission in an indoor environment. Let s. We set , so nat/s (1.44 bits/s). Suppose the peak-to-average power constraint , which restricts the duty factor to be at least is . This choice of peak-to-average power constraint is not unreasonable; if the average power of the transmitter were 1 mW, for example, then the constraint implies that its peak power could not exceed 0.25 W. The behavior of the reliability function (26) for various values of the duty factor is shown in Fig. 1. Note the rapid decay of
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
Fig. 2. Optimal duty factor as a function of R.
Fig. 3. Error probability as a function of bandwidth at an E =N of 13 dB; upper bound for the multipath fading channel (solid), upper bound for the Rayleigh fading channel (dashed), and lower bound for the Rayleigh fading channel (dotted). The bandwidth W is in Hertz.
the reliability function. We therefore expect that the minimum bandwidth required to achieve a particular performance to increase very rapidly as the rate approaches capacity. It is also evident that smaller values of the duty factor are required to achieve higher rates, though the optimal for a given rate is not immediately apparent. This optimization can be performed numerically, and the result is shown in Fig. 2. As expected, we see that the optimal duty factor gradually decreases to zero as capacity is approached. More surprising, however, is the fact that, even for to maximize very low rates, it is necessary that the reliability function, which translates to a peak power that is approximately 2000 times larger than the average. Hence, it is reasonable to suppose that, under a constraint of 250 on the peak-to-average power, the optimal performance is achieved by . the peakiest signal possible. Thus, we set We now turn to investigating the interplay among the physical parameters of interest. The upper and lower bounds are shown as of 13 dB (i.e., a rate functions of the bandwidth at a fixed of 0.035 nats/s or 0.050 bits/s) in Fig. 3. Note how the bounds for some , decay slowly with bandwidth, roughly as as expected from the properties that we have derived for them. In Fig. 4, we fix the bandwidth at 1 GHz and show the upper . The general upper and lower bounds as functions of bound (16) is shown to quantify exactly how much looser it is than (25) under the additional assumption of Rayleigh fading. For the lower bound, we use (54). The minimizations over for the general bound and for the lower bound are performed numerically. Of the three bounds, it is the upper bound under the Rayleigh fading assumption that is the most interesting. The general upper
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Fig. 4. Error probability as a function of E =N at a bandwidth of 1 GHz; upper bound for the multipath fading channel (solid), upper bound for the Rayleigh fading channel (dashed), and lower bound for the Rayleigh fading channel (dotted).
2
Fig. 5. Bandwidth required as a function of E =N for = 4 10 and error probability bounds of 10 (solid), 10 (dashed), and 10 (dotted). The bandwidth W is in Hertz.
Fig. 6. Duty factor as a function of E =N for a bandwidth of 10 GHz and error probability bounds of 10 (solid), 10 (dashed), and 10 (dotted).
bound is too loose to allow us to consider any remotely feasible settings, and lower bounds on the error probability relate to the best possible performance, which is usually of less interest than the worst possible performance (related to upper bounds on the error probability). In addition, the upper bound under the Rayleigh fading assumption has a simple exponential expression that the other two bounds do not have. Thus, we concentrate on this upper bound and examine the relationship between (Fig. 5), and between the duty factor bandwidth and and (Fig. 6) for fixed target error probabilities. The plot of the duty factor as a function of tells us how peaky the signal needs to be to achieve a particular probability of error for and bandwidth, and therefore, the peak power a given required. For all of the preceding plots, we have taken the peak ; this plot gives us a notion of power limitation to be (or decrease in rate) that would be necesthe increase in sary to maintain performance if the peak power limitation were could be decreased if lower (and, conversely, how much the peak power limitation were higher).
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VI. CONCLUSION Explicit upper and lower bounds on the probability of error of a peaky transmission scheme that achieves the capacity of the multipath fading channel in the limit of infinite bandwidth were calculated. The upper bounds can be made to decay to zero as the bandwidth goes to infinity for all rates below capacity, thus yielding a strong coding theorem similar to those derived by Gallager for discrete-time memoryless channels. Specifically, an upper bound for general fading and upper and lower bounds under the additional assumption of Rayleigh fading were obtained. It was shown that, in the latter case, the upper bound decays much faster as bandwidth increases than in the former, and that the upper and lower bounds are exponentially tight, yielding the true exponential dependence of the error probability, or the reliability function. These bounds allow us to assess the performance of the scheme for a given finite bandwidth and peak power constraint, and give us a notion of how quickly the error probability decays to zero as the bandwidth approaches infinity, and of the importance of the various parameters relevant to determining this rate of decay. It was found that, under Rayleigh fading, the error probability decreases rather slowly with the , where . This bandwidth , varying roughly as finding is not surprising in light of recent results by Verdú [8] that show that approaching capacity is very costly both in terms of spectral efficiency and peak-to-average power ratio in the absence of perfect channel side information at the receiver. The interaction among the error probability, bandwidth, data rate, and peakiness of the transmission scheme for a typical indoor wireless scenario was investigated. Finally, note that this scheme and the analysis of the error probability can be extended to a multiple-access situation. Since transmission takes place at a low duty cycle, multiple users can be multiplexed by time division. If the users are cooperating, then it is clear that noninterfering users can be supported for a given value of the duty factor . If they are not cooperating, then a term due to interference from other users can be incorporated into our existing expressions for the upper bound on the probability of error. The probability of two time slots overlapping, however, is dominated by , as approaches zero when there are users. Hence, unless the duty factor is very minute (at most, on the order of the target error probability), such a naïve noncooperative scheme is unlikely to yield good performance. REFERENCES [1] I. E. Telatar and D. N. Tse, “Capacity and mutual information of wideband multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 1384–1400, July 2000. [2] M. Médard and R. G. Gallager, “Bandwidth scaling for fading multipath channels,” IEEE Trans. Inform. Theory, vol. 48, pp. 840–852, Apr. 2002. [3] V. G. Subramanian and B. Hajek, “Broadband fading channels: Signal burstiness and capacity,” IEEE Trans. Inform. Theory, vol. 48, pp. 809–827, Apr. 2002. [4] R. S. Kennedy, Fading Dispersive Communication Channels. New York: Wiley Interscience, 1969. [5] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968. [6] M. Médard and D. N. C. Tse, “Spreading in block-fading channels,” in Proc. Asilomar Conf. Signals, Systems, and Computers, vol. 2, 2000, pp. 1598–1602. [7] C. Zheng, “Optimum spreading bandwidth for DS-CDMA on time and frequency fading channels,” Master’s thesis, Mass. Inst. Technol., Cambridge, MA, 2002.
[8] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans. Inform. Theory, vol. 48, pp. 1319–1343, June 2002. [9] S. Shamai (Shitz) and T. L. Marzetta, “Multiuser capacity in block fading with no channel state information,” IEEE Trans. Inform. Theory, vol. 48, pp. 938–942, Apr. 2002. [10] G. Kaplan and S. Shamai (Shitz), “Error exponents and outage probabilities for the block-fading Gaussian channel,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, London, U. K., Sept. 1991, pp. 329–334. [11] I. E. Telatar, “Coding and multiaccess for the energy limited Rayleigh fading channel,” Master’s thesis, Mass. Inst. Technol., Cambridge, MA, 1988. [12] R. G. Gallager, “Energy Limited Channels: Coding, Multiaccess and Spread Spectrum,” Mass. Inst. Technol. LIDS, Tech. Rep. LIDS-P-1714, 1987.
Desmond S. Lun (S’01) received the B.Sc. and B.E. (Hons.) degrees from the University of Melbourne, Melbourne, Australia, in 2001, and the S.M. degree in 2002 in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, where he is currently working toward the Ph.D. degree in the Department of Electrical Engineering and Computer Science. His research interests include fading channels and network coding.
Muriel Médard (S’91–M’95–SM’00) received the B.S. degrees in electrical engineering and computer science and in mathematics in 1989, a B.S. degree in humanities in 1990, a M.S. degree in electrical engineering in 1991, and a Sc.D. degree in electrical engineering in 1995, all from the Massachusetts Institute of Technology (MIT), Cambridge. She is currently a Harold E. and Esther Edgerton Associate Professor in the Electrical Engineering and Computer Science Department at MIT and a member of the Laboratory for Information and Decision Systems (LIDS). She was previously an Assistant Professor in the Electrical and Computer Engineering Department and a member of the Coordinated Science Laboratory at the University of Illinois, Urbana-Champaign. From 1995 to 1998, she was a Staff Member at the MIT Lincoln Laboratory in the Optical Communications and the Advanced Networking Groups. Her research interests are in the areas of reliable communications, particularly for optical and wireless networks. She serves as an Associate Editor for the OSA Journal of Optical Networking. Dr. Médard serves as an Associate Editor for the Optical Communications and Networking Series of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and as an Associate Editor in Communications for the IEEE TRANSACTIONS ON INFORMATION THEORY. She has served as a Guest Editor for the IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY. She received a National Science Foundation CAREER award in 2000. She was awarded the IEEE Leon K. Kirchmayer Prize Paper Award in 2002. She was coawarded the Best Paper Award with G. Weichenberg and V. Chan at the Fourth International Workshop on the Design of Reliable Communication Networks (DRCN 2003), Banff, AB, Canada, October 2003.
Ibrahim C. Abou-Faycal (S’96–M’00) received the Ingenieur des Arts et Manufactures diploma from Ecole Centrale Paris, Paris, France, and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology (MIT), Cambridge, in 1996 and 2001, respectively. From 2001 until 2003 he was with the Department of Electrical Engineering and Computer Science at MIT as a Postdoctoral Lecturer, and since September 2003, he has been with the Electrical and Computer Engineering Department at the American University of Beirut, Beirut, Lebanon. His research interests are in information theory, digital communications, stochastic systems and optimization.
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On the Performance of Peaky Capacity-Achieving Signaling on Multipath Fading Channels Desmond S. Lun, Student Member, IEEE, Muriel Médard, Senior Member, IEEE, and Ibrahim C. Abou-Faycal, Member, IEEE
Abstract—We analyze the error probability of peaky signaling on bandlimited multipath fading channels, the signaling strategy that achieves the capacity of such channels in the limit of infinite bandwidth under an average power constraint. We first derive an upper bound for general fading, then specialize to the case of Rayleigh fading, where we obtain upper and lower bounds that are exponentially tight and, therefore, yield the reliability function. These bounds constitute a strong coding theorem for the channel, as they not only delimit the range of achievable rates, but also give us a relationship among the error probability, data rate, bandwidth, peakiness, and fading parameters, such as the coherence time. They can be used to compare peaky signaling systems to other large bandwidth systems over fading channels, such as ultra-wideband radio and wideband code-division multiple access. We find that the error probability decreases slowly with ; under Rayleigh fading, the error probability the bandwidth varies roughly as , where 0. With parameters typical of indoor wireless situations, we study the behavior of the upper and lower bounds on the error probability and the reliability function numerically. Index Terms—Code-division multiple access (CDMA), error exponents, fading channels, ultra-wideband (UWB) radio, wideband.
I. INTRODUCTION
T
HE EMERGENCE of proposals for systems such as ultrawideband (UWB) radio and wideband code-division multiple access (CDMA) in recent years has brought about renewed interest in very-large-bandwidth fading channels. In this regime, it is known that transmission using “white-like” spread-spectrum signals (such as those used in direct-sequence (DS)-CDMA) performs poorly in terms of approaching capacity. Telatar and Tse [1] considered a finite number of time-varying paths, and demonstrated that the mutual information for white-like signals is inverselyproportional to the number of resolvable paths in the wideband limit. It follows that the mutual information is close to zero if the number of resolvable paths is large. Médard and Gallager [2] considered a channel that exhibits time and frequency decorre-
Paper approved by N. C. Beaulieu, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received March 26, 2003; revised August 19, 2003. This paper was presented in part at the Conference on Information Sciences and Systems, Princeton, NJ, March 2002; in part at the Vehicular Technology Conference, Vancouver, BC, Canada, September 2002; and in part at the International Symposium on Information Theory and its Applications, Xi’an, P.R. China, October 2002. D. S. Lun and M. Médard are with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:[email protected]; [email protected]). I. C. Abou-Faycal was with the Massachusetts Institute of Technology, Cambridge, MA 02139 USA. He is now with the Electrical and Computer Engineering Department, American University of Beirut, Beirut 1107 2020, Lebanon (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.829512
lation, and showed that the mutual information approaches zero with increasing bandwidth if spread-spectrum input signals are used. In their paper, spread-spectrum signals are characterized as those whose energy and fourth moment (kurtosis) in a fixed band scale inversely with the total bandwidth and the square of the total bandwidth, respectively; loosely speaking, those that spread energy more or less evenly over the entire available band. Subramanian and Hajek [3] instead considered a wide-sense stationary uncorrelated scattering fading channel with a constraint on the fourth moment of the output signal, and again found an inverse relation between capacity and bandwidth. Given an infinite band, capacity can instead be reached using peaky signaling, i.e., transmitting with a low-duty-cycle frequency-shift-keying (FSK) scheme. Such signaling is described as peaky because transmission energy is concentrated into narrow regions of time (using a low-duty cycle) and frequency (using FSK). The capacity reached by this scheme, which assumes no channel state information at either the receiver or the sender, is the same as that of the additive white Gaussian noise (AWGN) channel with a comparable received signal-to-noise ratio (SNR). This result was presented by Kennedy [4] and by Gallager [5, §8.6] for the case of Rayleigh fading, and most recently by Telatar and Tse [1] for general multipath fading. The particular peaky transmission scheme proposed in [1], which is summarized later, will form the basis of our paper and will be referred to as “capacity achieving,” since it achieves capacity in the wideband limit. In summary, using signals whose energy is spread evenly over a wide band results in poor performance with channels that exhibit time and frequency decorrelation (as is typical for fading channels). Intuitively, this is due to the fact that we are essentially transmitting over a large number of independent channels. If energy is spread evenly over the available band, then the ability to measure each channel decreases as the bandwidth increases and, despite the diversity gain, there is a severe performance degradation in the limit. It follows that, for sufficiently large bandwidth, spreading energy over that band in an even manner that keeps the fourth moment constrained, for example, with DS or related spread-spectrum techniques, is not advisable, and that peaky signaling should yield good performance. The bandwidth at which spreading begins to become detrimental, however, is not entirely clear, nor is the bandwidth at which peaky signaling begins to become advantageous. The former issue is addressed in [6] and [7], while we shall address the latter issue. In particular, we shall be able to assess the performance of the capacity-achieving peaky transmission scheme for a given,
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finite bandwidth, thus facilitating comparison with other transmission schemes on broadband fading channels. The result will, moreover, bring out the dependence of the error probability on the other characteristics of the scheme, such as its peakiness or peak power; an important implementation issue, since it is restricted by the physical limitations of the antenna and the power supply, as well as by safety regulations. We commence by covering the necessary background. We present the channel model in Section II and the capacity-achieving peaky transmission scheme in Section III. In Section IV, we analyze the probability of error of peaky signaling. We upper bound the probability of error, and observe that it vanishes for all rates under capacity as the bandwidth goes to infinity, thus yielding a strong coding theorem. We see that an upper bound that decays much faster as bandwidth increases can be obtained if the fading process is assumed to be Rayleigh, and that, under this additional assumption, a lower bound that is exponentially tight to the upper bound can be found. We find that, under Rayleigh fading, the error probability decreases slowly with the bandwidth , varying , where . This is consistent with recent roughly as results by Verdú [8] that show that approaching capacity is prohibitively expensive both in terms of spectral efficiency and peak-to-average power ratio in the absence of perfect channel side information at the receiver. In Section V, we illustrate the behavior of the bounds for a typical indoor wireless situation. II. CHANNEL MODEL that results from an input The channel-output waveform passed through a multipath fading channel is waveform generally given by (1) and are the gain where is the number of paths, is a and delay on the th path at time , respectively, and bandlimited white Gaussian noise process with power spectral . density and be the coherence time and delay spread of Let the fading channel, respectively. We assume that the processes and are constant and independent and identically distributed (i.i.d.) over time intervals of (i.e., block (i.e., an underspread channel). fading in time), and that Though real channels typically vary in a much more continuous manner and with some statistical correlation over time intervals greater than , the block-fading assumption is frequently used in the analysis of fading channels (see, for example, [6], [9], and [10]) as it greatly increases the tractability of the problem while capturing the essential time-varying quality embodied by channel coherence. The assumption of an underspread channel is not particularly restrictive, as typical delay spreads are on the order of microseconds, while typical coherence times are on the order of milliseconds for wireless channels. III. PEAKY SIGNALING We briefly describe the capacity-achieving peaky transmission scheme. It is shown in [1] that the scheme indeed achieves
all rates below capacity. More precisely, it is shown that all rates that satisfy (2) can be achieved with an arbitrarily small probability of error over a multipath fading channel with average power constraint , but no bandwidth constraint. Since the multipath fading channel can be viewed as the cascade of two channels, one that causes the multipath fading (and results in a signal with average power constrained to ) and a second that adds white Gaussian noise, it follows that the capacity of the infinite-bandwidth multipath fading channel cannot exceed that of the . Thus, infinite-bandwidth AWGN channel alone, i.e., recalling that the channel is assumed to be underspread, we conclude that rates very close to the capacity (which must lie and ) are achievable by between peaky signaling. . Consider the following codebook of size . Let The th code word is represented at baseband as a complex sinusoid of amplitude at frequency , i.e., (3) otherwise where the time duration of the signal is taken to be the coherence time . The frequency is chosen such that it is an , where ; therefore, the integer multiple of size of the codebook is directly related to the minimum transby . mission bandwidth required Let us consider the channel output over the interval . If the time axis at the receiver is shifted appropriately, then and are constant, owing to during this interval, the assumptions of the model, and we denote their values by and , respectively. Hence, by (1), the received signal when message is sent is
(4) is the complex fading where gain. We define signal power in the conventional sense as the received signal power, and thus normalize the channel gain so . that At the receiver, we form the correlator outputs (5) for
. Therefore (6)
where is a set of i.i.d. circularly symmetric complex Gaussian random variables, each satisfying .
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
The message is then repeated over disjoint time intervals to obtain time diversity. Hence, denoting the th correlator output , we have at interval by
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bandwidths are required for good performance, not to mention very complex encoding and decoding schemes. A. Strong Coding Theorem for Multipath Fading Channels
(7) and , where, owing to the block for is a sequence of i.i.d. complex random fading assumption, variables (with no particular distribution) and is a set of i.i.d. circularly symmetric complex Gaussian random variables of unit variance. We construct the decision variables
Recall the transmission scheme described in Section III. Owing to symmetry, we can assume without loss of generality that the transmitted message is 1. An error occurs if or if for some . Let be the event , and let be the event for . Then, denoting the error probability by , we have (11)
(8) For notational convenience, we define
and use the following threshold decoding rule. Let
(12)
(9) (where is an arbitrary parameter) be the threshold. If exceeds for one value of only, then we estimate ; otherwise, we declare an error. We use threshold decoding primarily for its simplicity. Note that the decoding rule is noncoherent, measuring only the energy of the received signal. We transmit using the above scheme for a fraction of time , and then transmit nothing for the remainder of the time. Observe and an energy of to transmit that it takes a time of nats of information. Therefore, one codeword that conveys the average power is , and the rate is given by
An upper bound to is applied to obtain
(13) where (14) Thus
(10) IV. PROBABILITY OF ERROR WITH PEAKY SIGNALING We now derive upper and lower bounds on the probability of error of the capacity-achieving scheme described in Section III. These bounds decay to zero with increasing transmission bandwidth for all rates under capacity, thus yielding a “strong” coding theorem that differs from the “weak” coding theorem of [1], in that it not only delimits the range of achievable rates, but also brings out the relationship among the error probability, data rate, and parameters of the transmission scheme. We emphasize that the lower bound applies to the probability of error of the capacity-achieving scheme and not to general transmission schemes over fading channels. Therefore, there may exist transmission schemes that achieve better performance than the lower bound. Nevertheless, the lower bound is useful, as it gives us a notion of the tightness of the upper bound. It is worth mentioning that the capacity and error exponents associated with a fading channel have been studied previously by Telatar [11], though the model used was very different. Telatar built upon Gallager’s results for energy-limited channels [12], where the channel is modeled as discrete time and discrete input. He showed that, using random block codes and 0–1 signaling, the capacity of the Rayleigh fading channel is the same as that of the AWGN channel in the limit of large bandwidth and large SNR, and examined the rate at which this limiting behavior is approached. He found that the rate of this approach is very slow, and therefore, under the conditions imposed by the energy-limited formulation, inordinately large
is found in [1]. The Chernoff bound
(15) Therefore, in order to upper bound , it remains only to find an upper bound on . As a first step, we can upper bound using the Chebyshev inequality. Theorem 1 (Strong Coding Theorem for Multipath Fading Channels): Let a multipath fading channel have coherence time , delay spread , and Gaussian noise of power spectral den. Let be the variance of the complex sity fading gains. Then there exists a transmission scheme of average , bandwidth , and rate with power , peak power probability of error satisfying
(16) is in the half-open interval , , and . , the Moreover, for all rates less than probability of error can be made to vanish as . Proof: We use the transmission scheme described in Section III. We observe that, in this scheme where
(17)
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and
sufficiently large , the first additive term dominates, which . The theorem, however, holds for any distridecays as ; so it holds regardless of the number of paths and bution of the statistical behavior of the gains and delays on those paths. If is available, a better some information on the statistics of bound could be obtained by determining the statistics of , through which depend on those of (24)
(18) Keeping in mind that forward to verify that
and
are independent, it is straight-
(19) hence
We therefore specialize to the case of Rayleigh fading, where a large number of independent paths is present, resulting in a . circularly symmetric complex Gaussian distribution for Theorem 2 (Strong Coding Theorem for Rayleigh Fading Channels): Let a Rayleigh fading channel have coherence time , delay spread , and Gaussian noise of power spectral . Then there exists a transmission scheme of density , bandwidth , and rate average power , peak power with probability of error satisfying (25) where
(20) and substituting for
(26)
using (10) gives (21)
Recalling (17) for the expectation of we have
, given that
,
is in the half-open interval , , and . , the Moreover, for all rates less than probability of error can be made to vanish as . Proof: We again use the transmission scheme described in Section III. We have
(22)
(27)
We now apply the Chebyshev inequality to obtain by the Chernoff bound. Then, using (8), we obtain
(23) The upper bound (16) follows by substituting the upper bound on given by (23), and the upper bound on given by (15) combined with (14) into (11), and minimizing over its domain to set the optimal threshold. The second part of the theorem follows by noting that the first term of the upper bound, being the upper bound on , for all , while the second term of the vanishes as , can be made upper bound, being the upper bound on to vanish as only for , as established in [1].
Now, since we have assumed that the fading is Rayleigh, the are i.i.d. circularly symmetric complex Gaussian random variables, so the are i.i.d. exponentially . Hence distributed random variables with mean
B. Strong Coding Theorem for Rayleigh Fading Channels
The supremum is achieved by so we have
Although Theorem 1 does indeed give a valid upper bound on the error probability, the bound decays very slowly in . For
(28)
(29) ,
(30)
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
where
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C. Lower Bound on the Error Probability (31)
By substituting (9) and (10), we can write (32)
We have arrived at an exponential upper bound on the probability of error of the capacity-achieving scheme. There is, however, little information on the tightness of the bound. We therefore derive a lower bound on the probability of error and compare it with the upper bound. Theorem 3: The transmission scheme described in Section III has a probability of error under Rayleigh fading that satisfies
where (33) to be the upper For notational convenience, we define given by (32), and to be the upper bound on bound on given by (13), viz.
(40) and are given by (33) and (14), respecwhere and approach zero with intively. The quantities creasing and can be taken as
(34) (41) for
. Since these upper bounds are valid for any , we have, by choosing optimally
(42) (35)
It is evident upon differentiation that, as functions of , is strictly decreasing, while is strictly increasing. In adwhen dition,
Proof: Having made the Rayleigh fading assumption, the and can, in fact, be evaluated exactly probabilities , and are both random since, conditioned upon degrees of freedom. So, using the cumulative variables with distribution function (CDF) for random variables, we have
(43)
(36) which is in the interval
if where
is defined by (31). Similarly
(37) (44) Therefore, given that (37) is satisfied, we can upper bound (35) by
Because and we arrive at the inequalities
are both positive for all ,
(45) (38) and where Observe that we can write where
is given by (26). (46) (39)
for all , over Hence, since its domain of definition, given by (37). The interval (37) grows as decreases, approaching the interval that encompasses all rates under capacity as approaches zero. Thus, there exists such that is positive as long as the rate does not exceed capacity (i.e., (2) is satisfied), and such that the bound approaches infinity. on given by (38) vanishes as
by taking only one of the summation terms. We apply Stirling’s formula to bound the factorial function, and obtain the lower bounds (47) and (48)
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where and are given by (41) and (42), respectively. Having found lower bounds on and , we are now in a position to derive a lower bound on the error probability . We commence with the following observation:
(49) . Using the lower bounds by the independence of the events and and the previously obtained upper bounds, (30) on and (13), we straightforwardly obtain (40), thus completing the proof. It is worth noting from the preceding proof that the upper and are exponentially tight in the and lower bounds on sense that the exponents are arbitrarily close for sufficiently sufficiently large. Indeed, we large or, equivalently, for have (50) for . We now turn our attention to the exponential dependence of , which, we recall, is directly related to error probability on the bandwidth. We define the reliability function of the Rayleigh fading channel using peaky signaling with duty factor as (51) analogously to the treatment of random block coding over discrete memoryless channels by Gallager [5, §5.8]. It represents the true exponential dependence of the error probability on for sufficiently large. Theorem 4: The reliability function of the Rayleigh fading channel using peaky signaling with duty factor is equal to , the error exponent of the upper bound, for all nonnegative rates . Proof: Applying l’Hôpital’s rule to the lower bound (40) yields
Fig. 1. Reliability function E (R; ) as a function of R for = 10 = 10 (dashed), and = 10 (dotted).
(solid),
and, again by applying l’Hôpital’s rule, we see that . By noting that is a probability and is, therefore, at most one, the reverse inequality follows. Hence (55) Thus we see that the upper and lower bounds on are exponentially tight. is monotonically inOwing to the fact that the function creasing (56) and, since we have established that all , we can interchange the yielding
and
is such that where given by (36). Hence
converges for operators,
(57) and is
(58)
(52) . Recall from Section IV-B that, for before optimization over , we have the following upper bound on the error probability:
(53) from which the reverse inequality to (52) follows straightforwardly. , we use For
(54)
V. NUMERICAL RESULTS We now proceed to evaluate the quantities derived in the previous sections for particular parameter choices. We choose fading parameters that are typical for very-high frequency s and transmission in an indoor environment. Let s. We set , so nat/s (1.44 bits/s). Suppose the peak-to-average power constraint , which restricts the duty factor to be at least is . This choice of peak-to-average power constraint is not unreasonable; if the average power of the transmitter were 1 mW, for example, then the constraint implies that its peak power could not exceed 0.25 W. The behavior of the reliability function (26) for various values of the duty factor is shown in Fig. 1. Note the rapid decay of
LUN et al.: ON THE PERFORMANCE OF PEAKY CAPACITY-ACHIEVING SIGNALING ON MULTIPATH FADING CHANNELS
Fig. 2. Optimal duty factor as a function of R.
Fig. 3. Error probability as a function of bandwidth at an E =N of 13 dB; upper bound for the multipath fading channel (solid), upper bound for the Rayleigh fading channel (dashed), and lower bound for the Rayleigh fading channel (dotted). The bandwidth W is in Hertz.
the reliability function. We therefore expect that the minimum bandwidth required to achieve a particular performance to increase very rapidly as the rate approaches capacity. It is also evident that smaller values of the duty factor are required to achieve higher rates, though the optimal for a given rate is not immediately apparent. This optimization can be performed numerically, and the result is shown in Fig. 2. As expected, we see that the optimal duty factor gradually decreases to zero as capacity is approached. More surprising, however, is the fact that, even for to maximize very low rates, it is necessary that the reliability function, which translates to a peak power that is approximately 2000 times larger than the average. Hence, it is reasonable to suppose that, under a constraint of 250 on the peak-to-average power, the optimal performance is achieved by . the peakiest signal possible. Thus, we set We now turn to investigating the interplay among the physical parameters of interest. The upper and lower bounds are shown as of 13 dB (i.e., a rate functions of the bandwidth at a fixed of 0.035 nats/s or 0.050 bits/s) in Fig. 3. Note how the bounds for some , decay slowly with bandwidth, roughly as as expected from the properties that we have derived for them. In Fig. 4, we fix the bandwidth at 1 GHz and show the upper . The general upper and lower bounds as functions of bound (16) is shown to quantify exactly how much looser it is than (25) under the additional assumption of Rayleigh fading. For the lower bound, we use (54). The minimizations over for the general bound and for the lower bound are performed numerically. Of the three bounds, it is the upper bound under the Rayleigh fading assumption that is the most interesting. The general upper
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Fig. 4. Error probability as a function of E =N at a bandwidth of 1 GHz; upper bound for the multipath fading channel (solid), upper bound for the Rayleigh fading channel (dashed), and lower bound for the Rayleigh fading channel (dotted).
2
Fig. 5. Bandwidth required as a function of E =N for = 4 10 and error probability bounds of 10 (solid), 10 (dashed), and 10 (dotted). The bandwidth W is in Hertz.
Fig. 6. Duty factor as a function of E =N for a bandwidth of 10 GHz and error probability bounds of 10 (solid), 10 (dashed), and 10 (dotted).
bound is too loose to allow us to consider any remotely feasible settings, and lower bounds on the error probability relate to the best possible performance, which is usually of less interest than the worst possible performance (related to upper bounds on the error probability). In addition, the upper bound under the Rayleigh fading assumption has a simple exponential expression that the other two bounds do not have. Thus, we concentrate on this upper bound and examine the relationship between (Fig. 5), and between the duty factor bandwidth and and (Fig. 6) for fixed target error probabilities. The plot of the duty factor as a function of tells us how peaky the signal needs to be to achieve a particular probability of error for and bandwidth, and therefore, the peak power a given required. For all of the preceding plots, we have taken the peak ; this plot gives us a notion of power limitation to be (or decrease in rate) that would be necesthe increase in sary to maintain performance if the peak power limitation were could be decreased if lower (and, conversely, how much the peak power limitation were higher).
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VI. CONCLUSION Explicit upper and lower bounds on the probability of error of a peaky transmission scheme that achieves the capacity of the multipath fading channel in the limit of infinite bandwidth were calculated. The upper bounds can be made to decay to zero as the bandwidth goes to infinity for all rates below capacity, thus yielding a strong coding theorem similar to those derived by Gallager for discrete-time memoryless channels. Specifically, an upper bound for general fading and upper and lower bounds under the additional assumption of Rayleigh fading were obtained. It was shown that, in the latter case, the upper bound decays much faster as bandwidth increases than in the former, and that the upper and lower bounds are exponentially tight, yielding the true exponential dependence of the error probability, or the reliability function. These bounds allow us to assess the performance of the scheme for a given finite bandwidth and peak power constraint, and give us a notion of how quickly the error probability decays to zero as the bandwidth approaches infinity, and of the importance of the various parameters relevant to determining this rate of decay. It was found that, under Rayleigh fading, the error probability decreases rather slowly with the , where . This bandwidth , varying roughly as finding is not surprising in light of recent results by Verdú [8] that show that approaching capacity is very costly both in terms of spectral efficiency and peak-to-average power ratio in the absence of perfect channel side information at the receiver. The interaction among the error probability, bandwidth, data rate, and peakiness of the transmission scheme for a typical indoor wireless scenario was investigated. Finally, note that this scheme and the analysis of the error probability can be extended to a multiple-access situation. Since transmission takes place at a low duty cycle, multiple users can be multiplexed by time division. If the users are cooperating, then it is clear that noninterfering users can be supported for a given value of the duty factor . If they are not cooperating, then a term due to interference from other users can be incorporated into our existing expressions for the upper bound on the probability of error. The probability of two time slots overlapping, however, is dominated by , as approaches zero when there are users. Hence, unless the duty factor is very minute (at most, on the order of the target error probability), such a naïve noncooperative scheme is unlikely to yield good performance. REFERENCES [1] I. E. Telatar and D. N. Tse, “Capacity and mutual information of wideband multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 1384–1400, July 2000. [2] M. Médard and R. G. Gallager, “Bandwidth scaling for fading multipath channels,” IEEE Trans. Inform. Theory, vol. 48, pp. 840–852, Apr. 2002. [3] V. G. Subramanian and B. Hajek, “Broadband fading channels: Signal burstiness and capacity,” IEEE Trans. Inform. Theory, vol. 48, pp. 809–827, Apr. 2002. [4] R. S. Kennedy, Fading Dispersive Communication Channels. New York: Wiley Interscience, 1969. [5] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968. [6] M. Médard and D. N. C. Tse, “Spreading in block-fading channels,” in Proc. Asilomar Conf. Signals, Systems, and Computers, vol. 2, 2000, pp. 1598–1602. [7] C. Zheng, “Optimum spreading bandwidth for DS-CDMA on time and frequency fading channels,” Master’s thesis, Mass. Inst. Technol., Cambridge, MA, 2002.
[8] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans. Inform. Theory, vol. 48, pp. 1319–1343, June 2002. [9] S. Shamai (Shitz) and T. L. Marzetta, “Multiuser capacity in block fading with no channel state information,” IEEE Trans. Inform. Theory, vol. 48, pp. 938–942, Apr. 2002. [10] G. Kaplan and S. Shamai (Shitz), “Error exponents and outage probabilities for the block-fading Gaussian channel,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, London, U. K., Sept. 1991, pp. 329–334. [11] I. E. Telatar, “Coding and multiaccess for the energy limited Rayleigh fading channel,” Master’s thesis, Mass. Inst. Technol., Cambridge, MA, 1988. [12] R. G. Gallager, “Energy Limited Channels: Coding, Multiaccess and Spread Spectrum,” Mass. Inst. Technol. LIDS, Tech. Rep. LIDS-P-1714, 1987.
Desmond S. Lun (S’01) received the B.Sc. and B.E. (Hons.) degrees from the University of Melbourne, Melbourne, Australia, in 2001, and the S.M. degree in 2002 in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, where he is currently working toward the Ph.D. degree in the Department of Electrical Engineering and Computer Science. His research interests include fading channels and network coding.
Muriel Médard (S’91–M’95–SM’00) received the B.S. degrees in electrical engineering and computer science and in mathematics in 1989, a B.S. degree in humanities in 1990, a M.S. degree in electrical engineering in 1991, and a Sc.D. degree in electrical engineering in 1995, all from the Massachusetts Institute of Technology (MIT), Cambridge. She is currently a Harold E. and Esther Edgerton Associate Professor in the Electrical Engineering and Computer Science Department at MIT and a member of the Laboratory for Information and Decision Systems (LIDS). She was previously an Assistant Professor in the Electrical and Computer Engineering Department and a member of the Coordinated Science Laboratory at the University of Illinois, Urbana-Champaign. From 1995 to 1998, she was a Staff Member at the MIT Lincoln Laboratory in the Optical Communications and the Advanced Networking Groups. Her research interests are in the areas of reliable communications, particularly for optical and wireless networks. She serves as an Associate Editor for the OSA Journal of Optical Networking. Dr. Médard serves as an Associate Editor for the Optical Communications and Networking Series of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and as an Associate Editor in Communications for the IEEE TRANSACTIONS ON INFORMATION THEORY. She has served as a Guest Editor for the IEEE JOURNAL OF LIGHTWAVE TECHNOLOGY. She received a National Science Foundation CAREER award in 2000. She was awarded the IEEE Leon K. Kirchmayer Prize Paper Award in 2002. She was coawarded the Best Paper Award with G. Weichenberg and V. Chan at the Fourth International Workshop on the Design of Reliable Communication Networks (DRCN 2003), Banff, AB, Canada, October 2003.
Ibrahim C. Abou-Faycal (S’96–M’00) received the Ingenieur des Arts et Manufactures diploma from Ecole Centrale Paris, Paris, France, and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology (MIT), Cambridge, in 1996 and 2001, respectively. From 2001 until 2003 he was with the Department of Electrical Engineering and Computer Science at MIT as a Postdoctoral Lecturer, and since September 2003, he has been with the Electrical and Computer Engineering Department at the American University of Beirut, Beirut, Lebanon. His research interests are in information theory, digital communications, stochastic systems and optimization.
