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ENERGY-AWARE AD HOC WIRELESS NETWORKS RECENT RESULTS ON THE CAPACITY OF WIDEBAND CHANNELS IN THE LOW-POWER REGIME SERGIO VERDU, PRINCETON UNIVERSITY β ABSTRACT three primary performance parameters, band- 6 width, power, and data rate (bits per second), This article gives a brief overview of recent must respect the fundamental information theo- 5 results on the limits of reliable communication in retic limit characterized by the region of achiev- the low-power wideband regime from the stand- able spectral efficiency (data rate divided by 4 point of information theory. Results on channel bandwidth) and energy per bit (power divided by (b/s/Hz) capacity and optimum signaling are reviewed for data rate). In information theory, the “wide- 3 both point-to-point channels and multiuser chan- band” communication regime refers to the Spectral efficiency 2 nels. region in the maximal spectral efficiency vs. energy per bit curve where the spectral efficiency 1 INTRODUCTION is low. Consequently, highly spectrally efficient broadband channels with high signal-to-noise The technology of wideband communication is ratio (SNR) lie outside the scope of our review. –2.5 0 2.5 gaining ground in applications such as wireless The main driving forces behind the interest in personal communications, satellite, and sensor wideband communications are: The tradeoff of networks. • Ability to transmit at an energy per informa- The interest in wideband channels was tion bit close to the minimum spectral efficiency vs. spurred mid-20th century by the antijamming • Diversity against frequency-selective fading capabilities of spread spectrum gained at the • Ease of multiplexing/multiaccess energy-per expense of spectral inefficiency. Ever since the • Ability to coexist with other systems using the 1960s, power-limited deep space communication same band information bit is the has emerged as one of the main beneficiaries of Technological advances in very large scale advances in binary error correcting coding tech- integration (VLSI), error control coding, signal key measure of the nology, as exemplified most notably by convolu- processing, and synchronization make informa- tional, turbo, and low-density parity check codes. tion-theoretic fundamental limits on channel capacity of channels The success of direct-sequence spread-spectrum capacity increasingly relevant to communications code-division multiple access (CDMA)-based engineering. The low spectral efficiency typical of in the power-limited second-generation wireless telephony, developed wideband systems does not imply that the com- in the late 1980s, led a decade later to the adop- munication is wasteful of channel resources or regime. Many tion of CDMA (in a wider-band format) for that the system operates far from channel capaci- third-generation wireless personal communica- ty. It all depends on how far from the fundamen- important communi- tions [1, 2]. Recently, two major high-speed tal trade-off of spectral efficiency vs. energy per wireless systems based on orthogonal multiplex- bit the system operates. Furthermore, in multius- cation channels ing coupled with spread-spectrum countermea- er channels, the spectral efficiency achieved by sures against out-of-cell interference have been any one user may be small but the sum of the operate in the region developed: time-division multiple access data rates may actually be near capacity. (TDMA)/direct sequence [3] and orthogonal fre- Although the capacity of wideband channels of low spectral quency-division multiple access (OFDMA)/fre- has been studied since the inception of informa- quency hopping [4]. The increasingly popular tion theory, the last few years have seen a flurry efficiency in which is unlicensed industrial, scientific, and medical of activity in the field leading to a body of results bands are devoted to low-power spread-spec- with interesting practical implications. The pur- close to its minimum trum systems. Other examples of wideband chan- pose of this article is to give a brief overview of nels receiving increasing attention are low-power those recent information theoretic results on the value for reliable low-data-rate sensor networks envisioned for capacity of wideband channels.1 Major issues of future civilian and military applications, and interest in this context are communication in the transmission. communication based on ultra-wideband pulses low-power regime (i.e., near minimum energy (spanning from DC to gigahertz). per bit), spread-spectrum multi-access, the effect For reliable communication, the choice of the of fading, optimum signaling, and power control. 40 1070-9916/02/$17.00 © 2002 IEEE IEEE Wireless Communications • August 2002 THE LOW POWER REGIME 0.2 From Shannon’s 1948 formula for the capacity of AWGN the ideal bandlimited additive white noise chan- 0.18 nel, it follows that the received energy per bit 0.16 (ratio of power to data rate) divided by the one- sided noise spectral level N0 required to achieve 0.14 spectral efficiency equal to C b/s/Hz is given by 0.12 Rayleigh Rayleigh known channel unknown channel ErC21− b ()C = . (1) 0.1 N0 C 0.08 As the bandwidth grows and the spectral efficiency vanishes, C→ 0, it can be seen from Eq. 1 0.06 that the minimum received energy necessary to Spectral efficiency (b/s/Hz) transmit 1 bit of information reliably (by means of 0.04 channel codes with growing blocklength) satisfies 0.02 Er b ==−ln2159 .dB . 0 (2) –2 1.59–1 0 1 2 3 4 N0 min Eb/N0 (dB) Operation at very small SNR per degree of I freedom allows maintaining Eb/N0 close to its Figure 1. Spectral efficiency of the AWGN channel and the Rayleigh flat fad- minimum value at the expense of low (but nonze- ing channel with and without receiver knowledge of fading coefficients. ro) spectral efficiency (bits per second per hertz). Subsequent work in the 1960s showed that Eq. 2 is not only a feature of the ideal band-limited If the receiver knows the fading coefficients additive white noise channel, but also holds when and coherent communication is feasible, quadra- the channel is subject to frequency-flat fading, ture phase shift keying (QPSK) has been shown even if the fading coefficients are unknown at the [6] to attain optimal spectral efficiency in the receiver [5]. It has been shown recently [6] that wideband regime, whereas in those conditions on- Eq. 2 is actually a feature of any fading channel off signaling is distinctly suboptimal, requiring as long as the background noise is Gaussian. It is more than six times the minimum bandwidth. The indeed possible to obtain a lower value of minimum bandwidth turns out to be proportional 2 r to the kurtosis of the fading amplitude distribu- Eb tion [6]. Thus, Rayleigh fading requires twice the N0 min bandwidth required in the absence of fading, and the bandwidth required with log-normal fading if the additive noise is non-Gaussian [6, 7]. grows exponentially with the dB variance. Up until recently, the only information-theo- In the noncoherent regime where the channel retic guidance on the capabilities of wideband is not fully known at the receiver, attaining chan- communication was offered by the infinite band- nel capacity at low power requires flash signal- width capacity (or equivalently, minimum energy ing, an asymptotic form of on-off signaling where per bit) results (e.g., [5–8]). In particular, [7] the on level has unbounded energy and the duty gives a formula computable even in cases where cycle is vanishingly small [6]. Furthermore, the the Shannon capacity is unknown. It is natural to lack of precise knowledge of channel coefficients extrapolate information-theoretic results makes operation near obtained for the infinite bandwidth (zero spec- r tral efficiency) regime to the wideband regime. Eb This leads to conclusions such as: N0 min • Wideband capacity is not affected by fading. • Receiver knowledge of fading coefficients can- prohibitively expensive in terms of bandwidth not increase capacity, and consequently there is and complexity. no capacity loss due to noncoherent reception. Figure 1 compares the spectral efficiency of • On-off signaling such as pulse position modu- the unfaded (additive white Gaussian noise) lation is near optimal. channel with the spectral efficiency of the However, it has been shown recently [6] that Rayleigh channel in the wideband regime. If the these conclusions are unwarranted. The crucial channel coefficients are known at the receiver, it point is that operation in the regime of low spec- can be seen that the slope of the spectral effi- tral efficiency does not imply disregard for the ciency curve is half of that in the unfaded case. bandwidth required by the system. On the con- If the channel coefficients are unknown at the trary, with the possible exception of ultra wide- receiver [9], despite what may appear from Fig. band pulse communication, given a certain 1, Eq. 2 still holds. However, the slope is equal 1 Seventy percent of the power and data rate, it is of interest to minimize to zero, and nonnegligible spectral efficiency references in this article the required bandwidth (to the extent allowed by requires much higher levels of energy per bit. have appeared within the the given complexity constraints). Even if we One of the conclusions drawn from the results last three years. operate at an energy per bit close to Eq. 2, the of [6] is that the traditional paradigm of the voice- bandwidth required to achieve a given data rate band telephone channel that maximizes data rate 2 Fourth moment divided turns out to be very sensitive to receiver side for given power and bandwidth may be an invita- by second moment information and to the nature of the fading. tion to inefficient design in the wideband regime. squared. IEEE Wireless Communications • August 2002 41 From the standpoint A more sensible approach is to minimize band- Since the pioneering information-theoretic width for a given rate and power.

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