IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 6, JUNE 2002 1319Spectral Efficiency in the Wideband RegimeSergio Verdú, Fellow, IEEEInvited PaperAbstract—The tradeoff of spectral efficiency (b/s/Hz) versus en-ergy-per-information bit is the key measure of channel capacity inthe wideband power-limited regime. This paper finds the funda-mental bandwidth–power tradeoff of a general class of channels inthe wideband regime characterized by low, but nonzero, spectralefficiency and energy per bit close to the minimum value requiredfor reliable communication. A new criterion for optimality of sig-naling in the wideband regime is proposed, which, in contrast tothe traditional criterion, is meaningful for finite-bandwidth com-munication.Index Terms—Antenna arrays, channel capacity, fading chan-nels, noisy channels, spectral efficiency, wideband regime.I. INTRODUCTIONSHORTLY after “A Mathematical Theory of Communica-tion,” Claude Shannon [1] pointed out that as the bandwidthtends to infinity, the channel capacity of an ideal bandlimited ad-ditive white Gaussian noise (AWGN) channel approaches(b/s) (1)whereis the received power and is the one-sided noisespectral level. Since capacity is monotonically increasing withbandwidth, the right-hand side of (1) is the maximum rateachievable with power. Moreover, communicating at rate ,the received signal energy per information bit is equal to(2)and since the maximum value ofis the right-hand side of(1), the minimum received signal energy per information bit re-quired for reliable communication satisfies1.59 dB (3)Gaussian inputs are not mandatory to achieve (1). In 1948,Shannon [2] had already noticed that for low signal-to-noiseratios, binary antipodal inputs are as good as Gaussian inputsin the sense that the ratio of mutual information to capacityapproaches unity. Since then, this criterion (which can berephrased as the input attaining the derivative of capacityat zero signal-to-noise ratio) has traditionally been adoptedManuscript received April 18, 2001; revised January 22, 2002.The author is with the Department Electrical Engineering, Princeton Univer-sity, Princeton, NJ 08544 USA (e-mail: [email protected]).Communicated by S. Shamai, Guest Editor.Publisher Item Identifier S 0018-9448(02)04027-0.as a synonym of asymptotic optimality in the low-powerregime. Using this criterion, Golay [3] showed that (1) canbe approached by on–off keying (pulse position modulation)with very low duty cycle, a signaling strategy whose errorprobability was analyzed in [4].Enter fading. Jacobs [5] and Pierce [6] noticed not only that(1) is achieved if all the energy is concentrated in one message-dependent frequency slot, but also that the limiting rate in (1)is unexpectedly robust: it is achievable even if the orthogonalsignaling undergoes fading which is unknown to the receiver (aresult popularized by Kennedy [7]). Since only one frequency(or time) slot carries energy, this type of orthogonal signalingnot only is extremely “peaky” but requires that the number ofslots grows exponentially with the number of information bits.Doppler spread or a limitation in the peakiness of the orthog-onal signaling can be modeled by letting the signal-dependentwaveform at the receiver have a given power spectral density, shifted in frequency by an amount that depends on themessage, with different shifts sufficiently far apart to maintainorthogonality. In this case, the infinite-bandwidth achievablerate was obtained by Viterbi [8]As we sawin(3), determining the infinite-bandwidthcapacityis equivalent to finding the minimum energy per bit required totransmit information reliably. To obtain this quantity, we canchoose to maximize the information per unit energy in con-trast to the standard Shannon setting in which the informationper degree of freedom is maximized. Motivated by the opti-mality of on–off signaling in the infinite-bandwidth limit, Gal-lager [9] found the exact reliability function in the setting of abinary-input channel where information is normalized, not toblocklength, but to the number of “’s” contained in the code-word. More generally, we can pose the “capacity per unit cost”problem where an arbitrary cost function is defined on the inputalphabet [10]. An important class of cost functions are thosewhich, like energy, assign a zero cost to one of the input sym-bols. For those cost functions, the capacity per unit cost not onlyis equal to the derivative at zero cost of the Shannon capacity butadmits a simple formula [10]. Even in this more general setting,capacity per unit cost is achieved by on–off signaling with van-ishing duty cycle.A wide variety of digital communication systems (particu-larly in wireless, satellite, deep-space, and sensor networks)operate in the power-limited region where both spectralefficiency (rate in bits per second divided by bandwidth in0018-9448/02$17.00 © 2002 IEEE1320 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 6, JUNE 2002hertz) and energy-per-bit are relatively low. The widebandregime is an attractive choice because of power savings, ease ofmultiaccess, ability of overlay with other systems, and diversityagainst frequency-selective fading. The information-theoreticanalysis of those channels, in addition to leading to the mostefficient bandwidth utilization, reveals design insights on goodsignaling strategies.From the existing results we could draw the following con-clusions about signaling and capacity in the wideband regime.• On–off signaling approaches capacity as the duty cyclevanishes.• The derivative at zero signal-to-noise ratio of the Shannoncapacity determines the wideband fundamental limits.• Capacity is not affected by fading.• Receiver knowledge of channel fade coefficients is use-less.• An input whose mutual information achieves the deriva-tive of capacity at zero signal-to-noise ratio is widebandoptimal.These conclusions and design guidelines have been drawn inthe literature in the natural belief that infinite-bandwidth limitsare representative of the large (but finite) bandwidth regime ofinterest in practice. However, in this paper, we show that thoseconclusions are misguided as long as the allowed bandwidthis finite, regardless of how large it is. Indeed, operation in theregime of low spectral efficiency does not imply disregard forthe bandwidth required by the system. Thus, we will see thatdesign guidelines obtained by infinite-bandwidth analyses neednot