IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 62, NO. 12 .

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 62, NO. 12, DECEMBER 20166831Minimum Energy to Send k Bits OverMultiple-Antenna Fading ChannelsWei Yang, Member, IEEE, Giuseppe Durisi, Senior Member, IEEE, and Yury Polyanskiy, Senior Member, IEEEAbstract— This paper investigates the minimum energyrequired to transmit k information bits with a given reliabilityover a multiple-antenna Rayleigh block-fading channel, withand without channel state information (CSI) at the receiver.No feedback is assumed. It is well known that the ratio betweenthe minimum energy per bit and the noise level converges to 1.59 dB as k goes to infinity, regardless of whether CSI isavailable at the receiver or not. This paper shows that the lack ofCSI at the receiver causes a slowdown in the speed of convergenceto 1.59 dB as k compared with the case of perfectreceiver CSI. Specifically, we show that, in the no-CSI case, thegap to 1.59 dB is proportional to ((log k)/ k)1/3, whereas whenperfect CSI is available at the receiver, this gap is proportionalto 1/ k. In both cases, the gap to 1.59 dB is independent ofthe number of transmit antennas and of the channel’s coherencetime. Numerically, we observe that, when the receiver is equippedwith a single antenna, to achieve an energy per bit of 1.5 dB inthe no-CSI case, one needs to transmit at least 7 107 informationbits, whereas 6 104 bits suffice for the case of perfect CSI atthe receiver.Index Terms— Channel coding, energy efficiency, minimumenergy per bit, multiple-antenna fading channels, nonasymptoticanalysis.I. I NTRODUCTIONACLASSIC result in information theory is that, for a widerange of channels including AWGN channels and fadingchannels, the minimum energy per bit E b required for reliablecommunication satisfies [1], [2]Eb loge 2 1.59 dB.N0 min(1)Here, N0 is the noise power per complex degree of freedom.For fading channels, (1) holds regardless of whether theManuscript received July 14, 2015; revised May 19, 2016; acceptedSeptember 10, 2016. Date of publication October 6, 2016; date of currentversion November 18, 2016. This work was supported in part by the SwedishResearch Council under Grant 2012-4571, and in part by the National ScienceFoundation CAREER Award under Grant CCF-12-53205. This paper waspresented at the 2015 IEEE Information Theory Workshop and the 2015 IEEEInternational Symposium on Information Theory.W. Yang is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: weiy@princeton.edu).G. Durisi is with the Department of Signals and Systems, ChalmersUniversity of Technology, 41296 Gothenburg, Sweden (e-mail:durisi@chalmers.se).Y. Polyanskiy is with the Department of Electrical Engineering andComputer Science, Massachusetts Institute of Technology, Cambridge, MA02139 USA (e-mail: yp@mit.edu).Communicated by D. Tuninetti, Associate Editor for Communications.Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2016.2615629instantaneous fading realizations are known to the receiver ornot [2, Th. 1], [3].1The expression in (1) is asymptotic in several aspects: the blocklength n of each codeword is infinite; the number of information bits k, or equivalently, thenumber of messages M 2k is infinite; the error probability vanishes; the total energy E is infinite; E/n vanishes.For many channels, the limit in (1) does not change if we allowthe error probability to be positive. However, keeping any ofthe other parameters fixed results in a backoff from (1) [2],[4]–[8].In this paper, we study the maximum number of informationbits k that can be transmitted with a finite energy E and afixed error probability 0 over a multiple-input multipleoutput (MIMO) Rayleigh block-fading channel, when there isno constraint on the blocklength n. Equivalently, we determinethe minimum energy E required to transmit k information bitswith error probability . We consider two scenarios:1) neither the transmitter nor the receiver have a priorichannel state information (CSI);2) perfect CSI is available at the receiver (CSIR) and noCSI is available at the transmitter.Throughout the paper, we shall refer to these two scenarios asno-CSI case and perfect-CSIR case, respectively.Related work: For nonfading AWGN channels withunlimited blocklength, Polyanskiy et al. [8] showed that themaximum number of codewords M (E, ) that can be transmitted with energy E and error probability satisfies2 2E 1Elog M (E, ) log e Q ( ) log eN0N0E1 log O(1), E . (2)2N0Here, Q 1 (·) denotes the inverse of the Gaussian Q-function.The first term on the right-hand side (RHS) of (2) givesthe 1.59 dB limit. The second term captures the penaltydue to the stochastic variations of the channel. This termplays the same role as the channel dispersion in finiteblocklength analyses [7], [9]. In terms of the minimum energy1 Knowledge of the fading realizations at the transmitter may improve (1),because it enables the transmitter to signal on the channel maximumeigenvalue eigenspace [2].2 Unless otherwise indicated, the log and the exp functions are taken withrespect to an arbitrary fixed base.0018-9448 2016 IEEE. 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6832IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 62, NO. 12, DECEMBER 2016per bit E b (k, ) necessary to transmit k bits with error probability , (2) implies that, for large E, E b (k, )2 loge 2 1Q ( ) loge 2 (3)N0k i.e., that the gap to 1.59 dB is proportional to 1/ k. Theasymptotic expansion (2) is established in [8] by showing thatin the limit E a nonasymptotic achievability boundand a nonasymptotic converse bound match up to third order.The achievability bound is obtained by computing the errorprobability under maximum-likelihood decoding of a codebook consisting of M orthogonal codewords (e.g., uncodedM-ary pulse-position modulation (PPM)). The converse boundfollows from the meta-converse theorem [7, Th. 27] withauxiliary distribution chosen equal to the noise distribution.Kostina et al. [10] generalized (2) to the setting of joint sourceand channel coding, and characterized the minimum energyrequired to reproduce k source samples with a given fidelityafter transmission over an AWGN channel.Moving to fading channels, for the case of no CSI, flashsignalling [2, Definition 2] (i.e., peaky signals) must be usedto reach the 1.59 dB limit [2]. In the presence of a peakpower constraint, (1) can not be achieved [11]–[14]. Verdú [2]studied the rate of convergence of the minimum energy per bitto 1.59 dB as the spectral efficiency vanishes. He showedthat, differently from the perfect-CSIR case, in the no-CSI casethe 1.59 dB limit is approached with zero wideband slope.Namely, the slope of the spectral-efficiency versus energy-perbit function at 1.59 dB is zero. This implies that operatingclose to the 1.59 dB limit is very expensive in terms ofbandwidth in the no-CSI case. For the scenario of finiteblocklength n, fixed energy budget E, and fixed probabilityof error , bounds and approximations on the maximumchannel coding rate over fading channels (under various CSIassumptions) are reported in [15]–[21].Contributions: Focusing on the regime of unlimitedblocklength, but finite energy E, and finite error probability ,we provide upper and lower bounds on the maximum numberof codewords M (E, ) that can be transmitted over anm t m r MIMO Rayleigh block-fading channel with channel’scoherence interval of n c symbols. For the no-CSI case, weshow that for every (0, 1/2)log M (E, ) 2/3 m r E 1/3mr Em r E 1log log e V0 ·Q ( )N0N0N0 2/3E log log E, E (4) O(log E)2/3where 2 1/3 V0 12 1/3 (log e)2/3 .3(5)Note that the asymptotic expansion (4) does not depend on thenumber of transmit antennas m t and the channel’s coherenceinterval n c . The fact that the first term does not depend on n cand m t follows directly from [2, Eq. (52)] by noting that anm t m r block-fading MIMO channel with coherence intervaln c is equivalent to an m t n c m r n c memoryless MIMO fadingchannel with block-diagonal channel matrix [2, p. 1339]. Ourresult (4) shows that the same holds for the second termin the expansion of log M (E, ) for E . In terms ofminimum (received) energy per bit E b (k, ), (4) implies that,for large E,3 E b (k, )loge k 1/3 1 2/3Q ( ) loge 2 V0 ·(loge 2)4/3N0k(6)i.e., the gap to 1.59 dB is proportional to ((loge k)/k)1/3 .We establish (4) by analyzing in the limit E an achievability bound and a converse bound. Theachievability bound follows from a nonasymptoticextension of Verdú’s capacity-per-unit-cost achievabilityscheme [5, pp. 1023–1024]. This scheme relies on acodebook consisting of the concatenation of uncoded PPMand a repetition code, and on a decoder that performsbinary hypothesis testing. The converse bound relies on themeta-converse theorem [7, Th. 31] with auxiliary distributionchosen as in the AWGN case. The resulting bound involvesan optimization over the infinite-dimensional space of inputcodewords (recall that in our setup there is no constrainton the blocklength n). By exploiting the Gaussianity ofthe fading process, we show that this infinite-dimensionaloptimization problem can be reduced to a three-dimensionalone. The tools needed to establish this result are theones developed by Abbe et al. [22] to prove Telatar’sminimum outage probability conjecture for multiple-inputsingle-output (MISO) Rayleigh-fading channels. Indeed, bothproblems involve the optimization of quantiles of a weightedconvolution of exponential distributions.The asymptotic analysis of achievability and conversebounds reveals the following tension: on the one hand, onewould like to make the codewords peaky to overcome lackof channel knowledge; on the other hand, one would like tospread the energy of the codewords uniformly over multiplecoherence intervals to mitigate the stochastic variations in thereceived signal energy due to the fading.For the case of perfect CSIR, we prove that for every (0, 1/2) mr E2m r E 1 log e Q ( ) log elog M (E, ) N0N0 mr E1 O log E , E . log2N0(7)Note that the asymptotic expansion (7) is also independentof the number of transmit antennas m t and the channel’scoherence interval n c . Furthermore, apart from an energynormalization resulting from the array gain, this asymptoticexpansion coincides with the one given in (2) for the AWGNcase up to a O log E term. In terms of minimum (received)3 By considering the received energy per bit instead of the transmit energyper bit, we account for the array gain resulting from the use of multiple receiveantennas.