On Outage Behavior of Wideband Slow-Fading Channels
aa r X i v : . [ c s . I T ] O c t On Outage Behavior of Wideband Slow-FadingChannels
Wenyi Zhang and Urbashi MitraMing Hsieh Department of Electrical EngineeringUniversity of Southern California, Los Angeles, CA 90089Email: { wenyizha, ubli } @usc.edu . Abstract — This paper investigates point-to-point informationtransmission over a wideband slow-fading channel, modeledas an (asymptotically) large number of independent identicallydistributed parallel channels, with the random channel fadingrealizations remaining constant over the entire coding block. Onthe one hand, in the wideband limit the minimum achievableenergy per nat required for reliable transmission, as a ran-dom variable, converges in probability to certain deterministicquantity. On the other hand, the exponential decay rate of theoutage probability, termed as the wideband outage exponent,characterizes how the number of parallel channels, i.e. , the“bandwidth”, should asymptotically scale in order to achievea targeted outage probability at a targeted energy per nat. Weexamine two scenarios: when the transmitter has no channelstate information and adopts uniform transmit power allocationamong parallel channels; and when the transmitter is endowedwith an one-bit channel state feedback for each parallel channeland accordingly allocates its transmit power. For both scenarios,we evaluate the wideband minimum energy per nat and thewideband outage exponent, and discuss their implication forsystem performance.
I. I
NTRODUCTION
In wireless communication, it is a well-known fact thatspreading the transmit power across a large amount ofbandwidths provides an effective means of trading spectralefficiency for power efficiency. A popular trend in next-generation system standards is the proposal of utilizing wide-band transmission, in order to boost the system throughputwithout excess power expenditure in contrast to narrow-bandsystems. Besides single-carrier or impulse ultra-wideband(UWB) approaches, a commonly followed design principleis the multi-carrier solution like orthogonal frequency di-vision multiplexing (OFDM), which essentially decomposesa frequency-selective wideband channel into a series offrequency-nonselective narrow-band parallel channels, eachendowed with a small fraction of the total available trans-mit power. Assuming that the frequency-selective widebandchannel models a rich-scattering propagation environmentwith the number of independent propagation paths growinglinearly with the bandwidth, and that the carrier frequenciesof parallel channels are sufficiently separated with the effectof the multipath spread essentially eliminated, it is usuallya justified approximation that the decomposed narrow-bandparallel channels are frequency-nonselective and mutuallyindependent; see, e.g. , [1].Previous theoretical characterizations of wideband channels have primarily focused on the time-varying aspect. The keyperformance metrics thus are the ergodic achievable infor-mation rate and the ergodic capacity, under various channelsettings. Furthermore, there a highlighted issue is the lackof precise channel state information (CSI) at the receiver (letalone the transmitter), due to the time-varying nature of thechannel model. It has been persistently observed that the lackof precise receive CSI fundamentally affects the efficiencyof signaling schemes, in terms of both energy efficiency andspectral efficiency. There has been a vast body of literaturein this area, and we refer the reader to [2], [3], [4], [5] andreferences therein and thereafter for major results.For many practically important scenarios, such as typicalindoor channels, however, it has been reported from channelmeasurement campaigns that the channel response can usuallybe temporally quasi-static; see, e.g. , [6], [7]. That is, thechannel response can be approximated as to remain essentiallyconstant throughout the duration of transmitting an entirecoded message. In light of this observation, it is thus not onlya theoretically meaningful, but also a practically enlightening,exercise to investigate the wideband transmission problemfrom a non-ergodic perspective, for which a key performancemetric is the outage probability [8].At first glance, the outage probability of wideband sys-tems does not appear to be a well-posed problem. This isbecause in the wideband limit, the maximum achievable rate,or equivalently the minimum achievable energy per nat, asrandom variables, generally converge in probability to certaindeterministic quantities. More refined characterizations of theoutage probability, therefore, need to be studied. To thisend, we focus on the exponential decay rate of the outageprobability, for a given targeted energy per nat, as the channelbandwidth measured by the number of parallel channels,increases to infinity. In this paper, we term this exponentialdecay rate as the wideband outage exponent.For a fixed operational wideband transmission system, thewideband outage exponent is an incomplete performance de-scriptor because it does not provide the exact value of theoutage probability. However, the wideband outage exponentstill provides valuable insights into the system behavior in theregime of (asymptotically) small outage probabilities. Besidesits analytical tractability, it serves as a convenient footingfor comparing different coding schemes and different channelmodels. For two wideband transmission systems operating athe same level of energy per nat, if one of them has a widebandoutage exponent that is larger than the other’s, then we canexpect that for sufficiently large bandwidths, the first systemachieves smaller outage probabilities.Our analysis of the wideband outage exponent can beinterpreted as to yield two tradeoffs. On the one hand, it revealsa tradeoff between rate and reliability. This type of tradeoffis not entirely a new topic for fading channels. Specifically,it has been extensively studied in the context of narrow-bandhigh-power systems with multiple antennas, since the seminalwork of [10]. There, the rate is quantified by the multiplexinggain and the reliability by the diversity gain, thus the so-calledmultiplexing-diversity tradeoff indicates how the reliability isreduced as the rate increases. We note that, the diveristy orderis usually another “outage exponent”, because for typical slow-fading channels it essentially corresponds to the exponentialdecay rate of the outage probability as the signal-to-noise ratio(SNR) grows large. In contrast, the wideband outage exponentstudied in this paper is the exponential decay rate of the outageprobability as the number of parallel channels, i.e. , bandwidth,grows large.On the other hand, our analysis reveals a tradeoff be-tween energy efficiency and spectral efficiency. From a high-level perspective, the wideband outage exponent for wide-band quasi-static fading channels plays a similar role as thewideband slope [5] for wideband ergodic fading channels, inthat both asymptotically indicate the amount of bandwidthsrequired to support a targeted energy per nat. The widebandslope essentially corresponds to the relative (compared tothe first-order derivative) significance of the second-orderderivative of the fading-averaged achievable rate per narrow-band parallel channel, at vanishing power. Thus it captures theloss in energy efficiency due to large, yet still finite, operatingbandwidth. The wideband outage exponent essentially corre-sponds to the outage probability as an exponentially decreasingfunction of bandwidths, at a targeted energy efficiency. Thusit captures the required amount of bandwidth for attaining thetargeted energy efficiency, in order to satisfy a given reliabilityconstraint as specified by the outage probability.In the ergodic fading case, it suffices to separately codefor each single parallel channel thus convert the widebandproblem into a low-power problem; see, e.g. , [5]. This ap-proach is lossless in terms of achievable rate there becauseit exploits the temporal ergodicity of each parallel channel.In the quasi-static fading case under an outage probabilitycriterion, however, it is necessary to adopt joint coding acrossall the parallel channels. In contrast, the alternative approach,separate coding for each single parallel channel concatenatedwith a cross-channel erasure coding stage, can be substantiallysub-optimal due to the “hard thresholding” effect by outageevents for individual parallel channels. In the analysis of thejoint coding approach, we utilize some basic results in thetheory of large deviations. Such a tradeoff, in fact, has long been one of the central themes ininformation theory, as captured by the channel reliability function [9].
In the following, we briefly summarize the main results andobservations in this paper.1) For the case in which the transmitter has no CSI andadopts uniform transmit power allocation among parallelchannels, we derive general formulas for the widebandminimum energy per nat and the wideband outageexponent. It is shown that these quantities depend uponstatistics, specifically, the expectation and the logarith-mic moment generating function, of only the first-orderderivative of the per-channel achievable rate randomvariable at vanishing SNR.2) Applying the general formulas, we evaluate the wide-band outage exponent for scalar Rayleigh, Rician, andNakagami- m fading channels, as a function of the tar-geted energy per nat. Rician fading channels achievelarger wideband outage exponents than the Rayleighcase, but the gap becomes evident only when the line-of-sight component is dominant. The wideband outageexponent of Nakagami- m fading channels is precisely m times that of the Rayleigh case, implying that in thewideband regime an m -fold bandwidth savings can beexpected for Nakagami- m fading channels in contrast toRayleigh fading channels.3) We then evaluate the wideband outage exponent formultiple-input-multiple-output (MIMO) fading channels,with and without spatial antenna correlation, for spa-tially white and covariance-shaped circular-symmetriccomplex Gaussian inputs. For channels with spatiallyuncorrelated Rayleigh fading and spatially white inputs,the wideband outage exponent is essentially identical tothat of the scalar Nakagami- m fading case, with theparameter m equal to the product of the numbers oftransmit and receive antennas. For channels with spatialantenna correlation, a covariance-shaping problem needsto be solved for the input covariance matrix to maximizethe wideband outage exponent, for the given targetedenergy per nat. In general, the optimal input covariancematrix for minimizing the wideband minimum energyper nat may not be the optimal choice for maximizingthe wideband outage exponent, especially as the targetedenergy per nat gets large.4) We also examine the case in which the transmitter haspartial information of the channel state. We analyze aspecific transmission protocol in which the transmitterhas an one-bit information for each parallel channelindicating whether the squared fading coefficient (as-suming a scalar Rayleigh fading model) is beyond acertain threshold. Based upon such an one-bit feedback,the transmitter adaptively allocates its power among theparallel channels. We obtain the wideband minimumenergy per nat and the wideband outage exponent for thetransmission protocol. The wideband minimum energyper nat can be made arbitrarily small, by simply ad-justing the protocol parameters such that most (or evenall) transmit power is allocated for sufficiently strongarallel channels. The corresponding wideband outageexponent, however, quantifies the penalty in terms ofoutage probability, as the energy efficiency improves.This tradeoff is inevitable, because in order to achievea smaller value of the energy per nat reliably, moreparallel channels are needed, to statistically ensure thatthere exist sufficiently many strong parallel channels.When the transmission protocol employs on-off powerallocation, allocating transmit power only for sufficientlystrong parallel channels, the wideband outage exponentis reduced at an exponential speed as the widebandminimum energy per nat decreases at an inversely-linear speed, implying that the bandwidth needs to scaleexponentially, in order to maintain a fixed level of outageprobability asymptotically.In [11], the authors investigate the scaling of codeword errorprobabilities in low-power block Rayleigh fading channelswith uncorrelated multiple antennas. Their approach examinesthe codeword error probabilities for one parallel channel acrossseveral fading blocks, and simultaneously evaluates the fading-induced outage part and the noise-induced part of decodingerrors, finding that the former is usually dominant for typicalslow-fading channels. In this paper, our approach explicitlyconnects the parallel channel power to the bandwidth scaling,and exclusively examines the outage probability, assuming thatthe noise-induced error probability has been made arbitrarilysmall by coding over sufficiently long codewords, in lightof the quasi-static condition of the fading process. Such asimplification still captures the essence of quasi-static fadingchannels, and permits the analysis to yield additional insightsinto the system behavior, under more general settings assummarized above. It is worth noting that in [11], its ad hoc analysis for uncorrelated Rayleigh MIMO channels has alsoidentified the gain factor of the wideband outage exponent, asthe product of the numbers of transmit and receive antennas.Throughout this paper, we assume that the receiver pos-sesses perfect CSI thus can adopt coherent reception. Suchan assumption does not appear as a fundamental issue in thecontext of quasi-static fading channel model, and furthermore,in the low-power (per parallel channel) regime non-coherentreception usually attains first-order equivalence to coherentreception [5] thus would lead to the same wideband outageexponent. Regarding the transmit knowledge of CSI, however,we only consider the no-CSI or partial-CSI (specifically, one-bit CSI per parallel channel) case in this paper, and leavethe case of transmit full-CSI for future exploration. On onehand, the transmitter of a wideband system is unlikely to beendowed with precise channel state information due to itsimpractical demand on a high-capacity feedback link, so thecase of transmit full-CSI is a less imperative topic for practicalconsiderations. On the other hand, when the transmitter hasprecise knowledge of CSI, it can thus fully adapt its poweras well as rate allocation, under short/long-term power andfixed/variable rate constraints [12], [13]. A full account ofthese topics is beyond the scope of this paper and deservesa separate treatment. The remainder of this paper is organized as follows. SectionII formulates the general problem and establishes the generalformulas of the wideband minimum energy per nat and thewideband outage exponent. Section III specializes the generalresults to scalar channels, and Section IV addresses the effectsof multiple antennas and antenna correlation. Section V turnsto the partial transmit CSI case, solving the problem fora specific transmission protocol with one-bit channel statefeedback per parallel channel. Finally Section VI concludesthis paper. In this extended abstract, all the technical proofsare omitted and can be found in [14]. All the logarithms areto base e , and information is measured in units of nats.II. G ENERAL R ESULT
Consider K parallel channels C k , k = 1 , . . . , K . We assumethat all the channels have identical statistics, characterized bythe transition probability distribution p ( y | x, s ) , where x ∈ X is the input, y ∈ Y the output, and s ∈ S the channel state. X , Y , and S are appropriately defined measurable spaces. Forsimplicity, in this paper we further assume that the K parallelchannels are mutually independent, and thus the channel states S k , k = 1 , . . . , K , are also mutually independent. The quasi-static condition indicates that each S k remains constant overthe entire coding block. So for each coding block, we caninterpret { S k } Kk =1 as K independent and identically distributed(i.i.d.) realizations of some channel state distribution p S ( s ) .Regarding the channel state information, we assume thatthe receiver has perfect knowledge of S k , k = 1 , . . . , K ,but the transmitter has no knowledge of them for lack ofan adequate feedback link. In Section V, we will endow thetransmitter with an one-bit channel state feedback for eachparallel channel.By defining a cost function for each channel input x ∈ X , c ( x ) ≥ , we can associate with an input distribution p X ( x ) an average cost c X = E [ c ( X )] . If we impose a total averagecost constraint ρ , and choose an input distribution satisfying c X = ρ/K for each parallel channel C k , k = 1 , . . . , K , thenfrom the coding theorem for parallel channels (see, e.g. , [9]),the total achievable rate conditioned upon the channel staterealizations as we code jointly over the K parallel channels is R ( K, ρ ) := K X k =1 I ( X k ; Y k , S k )= K X k =1 I ( X k ; Y k | S k ) , (1)where X k and Y k denote the input and output of channel C k .Here we utilize the fact that the channel input is independentof the channel state, due to the lack of transmit CSI. Note that X k incurs the average cost ρ/K , for k = 1 , . . . , K . The per-channel mutual information I ( X k ; Y k | S k ) is a random variableinduced by S k , thus we denote it by J ( ρ/K, S k ) in the sequel.The total achievable rate then can be rewritten as R ( K, ρ ) = K X k =1 J ( ρ/K, S k ) , (2)hich is the sum of K i.i.d. random variables. We note thatit is definitely legitimate to let parallel channels be allocateddifferent average costs, under the total average cost constraint ρ . In this paper, for simplicity, we only consider the uniformcost allocation, except in Section V, where the one-bit channelstate feedback leads to a certain capability of adaptive powerallocation. Such a uniform cost allocation is also in spirit withthe basic philosophy of wideband systems, which advocatespreading the energy across large bandwidths [15].Letting K grow from one to infinity, and specifying ac-cordingly the input distributions indexed by K , we obtain asequence of achievable rate random variables { R ( K, ρ ) } ∞ K =1 .We then use outage probability to characterize the systemperformance. We specify a target rate r ≥ in nats, thenthe outage probability O ( K, ρ, r ) is O ( K, ρ, r ) := Pr [ R ( K, ρ ) ≤ r ] . (3)In this paper, we are primarily interested in the widebandasymptotic behavior of O ( K, ρ, r ) . By introducing the averagecost per nat, η := ρr , (4)we can characterize the wideband asymptotic decay rate of O ( K, ρ, r ) by the wideband outage exponent E ( η ) := lim K →∞ − log O ( K, ρ, r ) K . (5)The following proposition gives the general formula of E ( η ) ,under mild technical conditions. Proposition 1:
Suppose that the following conditions hold:(i) For every s ∈ S , the function J ( γ, s ) /γ is nonnegativeand monotonically non-increasing with γ > , and the limit ˙ J (0 , s ) := lim γ ↓ γ J ( γ, s ) (6)exists as an extended number on R + ∪ { , ∞} ;(ii) The limit ˙ J (0 , S ) , as a random variable induced by S ,has its logarithmic moment generating function Λ( λ ) := log E h exp( λ ˙ J (0 , S )) i (7)defined as an essentially smooth and lower semi-continuousfunction on λ ∈ R .The wideband outage exponent of the wideband communi-cation system is then given by E ( η ) = sup λ ≤ (cid:26) λη − Λ( λ ) (cid:27) , (8)for every η ≥ ¯ η , where ¯ η := 1 E [ ˙ J (0 , S )] (9)is the minimum achievable wideband energy per nat of thegiven coding scheme. The corresponding cost per bit can be easily obtained by shifting downthe cost per nat by ln 2 = − . dB. An intuitive way of understanding the proposition is asfollows. For sufficiently large K , we may approximate theper-channel achievable rate J ( ρ/K, S k ) by its first-order ex-pansion term ˙ J (0 , S k ) · ( ρ/K ) , assuming that all the higher-order terms can be safely ignored. Such an approximationimmediately leads to R ( K, ρ ) ≈ ρK K X k =1 ˙ J (0 , S k ) , (10)and the wideband outage exponent readily follows fromCram´er’s theorem [16].III. S CALAR F ADING
In this section, we apply Proposition 1 to scalar fadingchannels. For each parallel channel, say, C k , k = 1 , . . . , K ,the channel input-output relationship is Y k = H k X k + Z k , (11)where the complex-valued input X k has an average powerconstraint ρ/K , and the additive white noise Z k is circular-symmetric complex Gaussian with Z k ∼ CN (0 , . Thechannel state S k in Section II here corresponds to the fadingcoefficient H k . Let the distribution of X k be CN (0 , ρ/K ) , wehave ˙ J (0 , H k ) = lim γ ↓ γ J ( γ, H k ) = | H k | . (12)Therefore Proposition 1 asserts that the wideband minimumenergy per nat is ¯ η = 1 / E [ | H | ] , and for every η ≥ ¯ η , thewideband outage exponent is E ( η ) = sup λ ≤ (cid:26) λη − log E (cid:2) exp( λ | H | ) (cid:3)(cid:27) . (13)In this section, without loss of generality, we normalize thefading coefficient H k such that E [ | H k | ] = 1 , so that ¯ η = 1 .We further examine the wideband outage exponent (13) bythree case studies: Rayleigh, Rician, and Nakagami- m . Rayleigh fading:
The squared fading coefficient | H | issimply an exponential random variable, therefore the logarith-mic moment generating function is Λ( λ ) = − log(1 − λ ) , andthe wideband outage exponent is easily evaluated as E ( η ) = 1 η − η ) , (14)for every η ≥ . Rician fading:
We assume that the fading coefficient is H ∼ CN ( κ, − κ ) , where κ ∈ (0 , specifies the line-of-sight (LOS) component. Hence | H | / ( − κ ) is a standardnoncentral chi-square random variable, with degree of freedom In fact, more general input distributions, such as symmetric phase-shiftkeying (PSK) [17] and low duty-cycle on-off keying (OOK) even with non-coherent detection [18], can also achieve the same limit function ˙ J (0 , s ) asthat of the Gaussian inputs; see [5] for further elaboration. and non-centrality parameter κ / (1 − κ ) . After manipu-lations it can be evaluated that the wideband outage exponentis E ( η ) = 1(1 − κ ) η + κ − κ − s κ (1 − κ ) η + log (1 − κ ) η " s κ (1 − κ ) η (15)for every η ≥ . When κ = 0 the Rician fading casereduces to the Rayleigh fading case. Numerically we canobserve that E ( η ) becomes larger as κ increases. This isintuitively reasonable, because as κ increases the channelbecomes the more like a non-faded Gaussian-noise channel.We also observe that, such an increase in E ( η ) with κ israther abrupt. For κ ≤ . , the gap between the E ( η ) curvesis numerically negligible; but as κ ≥ . , the gap becomessignificant. Nakagami- m fading: In the Nakagami- m fading case, thesquared fading coefficient | H | is a Gamma distributedrandom variable with shape parameter m ≥ / and scaleparameter /m , i.e. , the probability density function of | H | is p ( a ) = a m − exp( − ma ) m m / Γ( m ) for a ≥ . Thelogarithmic moment generating function is therefore Λ( λ ) = − m · log(1 − λ/m ) , for λ < m . Consequently, the widebandoutage exponent is easily evaluated as E ( η ) = m · (cid:20) η − η ) (cid:21) , (16)for every η ≥ . Comparing with the Rayleigh fading case, weobserve that the wideband outage exponent of the Nakagami- m fading channel is exactly m times that of the Rayleighfading channel. So for m > , the requirement on bandwidthsof Nakagami- m fading channels is less stringent than thatof Rayleigh fading channels. More precisely, in the regimeof small outage probabilities, Nakagami- m fading exhibits anapproximate m -fold savings in bandwidths. For / ≤ m < ,Rayleigh fading channels become more bandwidth-efficient,though. Such an m -scaling property quantitatively reflects thequalitative intuition that the Nakagami- m fading exhibits lessuncertainty as the fading figure m increases.IV. M ULTIPLE A NTENNAS
In this section, we consider channels with multiple inputsand outputs. For each parallel channel, say, C k , k = 1 , . . . , K ,the channel equation is Y k = H k X k + Z k . (17)The complex-valued vector input X k ∈ C n t × has an averagepower constraint E [ X † k X k ] = ρK . (18)The temporally i.i.d. additive noise Z k ∈ C n r × is alsospatially white, with Z k ∼ CN ( , I n r × n r ) . The fading matrix The Rayleigh fading case corresponds to m = 1 . H k ∈ C n r × n t is characterized by some appropriate probabilitydistribution. For instance, a commonly used fading model isthat the elements of H k are i.i.d. circular-symmetric complexGaussian; see, e.g. , [19].Let the input distribution be X k ∼ CN (0 , ( ρ/K ) Σ ) , wherethe covariance matrix Σ ∈ R n t × n t satisfies tr Σ = 1 ,where tr( · ) denotes the matrix trace operator. The resultingachievable rate for parallel channel C k hence is J ( ρ/K, H k ) = log det (cid:16) I n r × n r + ρK H k ΣH † k (cid:17) , (19)and as K → ∞ , we get ˙ J (0 , H k ) = 1 ρ lim K →∞ K · J ( ρ/K, H k )= tr h H k ΣH † k i . (20)From (20) we can then apply Proposition 1 to evaluate thewideband minimum energy per nat and the wideband outageexponent, for different channel models and input covariancestructures. A. Spatially White Inputs
If we let the input covariance matrix be Σ = (1 /n t ) · I n t × n t ,then we further simplify (20) to ˙ J (0 , H k ) = 1 n t tr h H k H † k i = 1 n t X i,j | H k ( i, j ) | , (21)where H k ( i, j ) denotes the i th-row j th-column element of H k . Spatially white Rayleigh fading:
The squared fading coef-ficients H k ( i, j ) , i = 1 , . . . , n r , j = 1 , . . . , n t , are n t n r i.i.d.exponential random variables, therefore their sum is a chi-square random variable, or, more generally speaking, a Gammarandom variable. Consequently, the wideband outage exponentis essentially (except scaling) identical to the scalar Nakagami- m fading case in Section III, i.e. , E ( η ) = n t n r · (cid:20) n r η − n r η ) (cid:21) , (22)for every η ≥ ¯ η := 1 /n r . So besides the n r -fold decrease inthe wideband minimum energy per nat, the multiple antennasfurther boost the wideband outage exponent by a factor of n t n r , the product of the numbers of transmit and receiveantennas, in contrast to the scalar Rayleigh fading case. Itis interesting to note that the gain factor n t n r also appearsas the maximum achievable diversity order in the high-SNRasymptotic regime [10]. Spatially correlated Rayleigh fading:
In this case, we as-sume that each H k ( i, j ) follows the marginal distribution CN (0 , , and that the vectorization of H † k , V k := vec( H † k ) , isa correlated circular-symmetric complex Gaussian vector witha covariance matrix Ψ ∈ R n t n r × . It then follows that themoment generating function of ˙ J (0 , H k ) is E [exp( λ ˙ J (0 , H k ))] = (cid:20) det (cid:18) I − λn t Ψ (cid:19)(cid:21) − , (23)or λ < n t /µ max ( Ψ ) , where µ max ( · ) is the maximumeigenvalue of the operand matrix. Consequently, the widebandoutage exponent is given by E ( η ) = sup λ ≤ (cid:26) λη + log det (cid:18) I − λn t Ψ (cid:19)(cid:27) , (24)for every η ≥ ¯ η = 1 /n r . B. Input Covariance Shaping for Spatially Correlated Fading
When Ψ = I , there exists spatial correlation among thetransmit/receive antennas. Despite its simplicity, the spatiallywhite input X ∼ CN (cid:16) , ρn t K I (cid:17) generally even does notoptimize the wideband minimum energy per nat among allpossible inputs, let alone the wideband outage exponent. Aswill be shown, for spatially correlated fading channels, theoptimal input covariance actually depends upon the targetedenergy efficiency η .Utilizing the prior development in this section, we have thefollowing result: Proposition 2:
For a spatially correlated fading channelwith the fading matrix vectorization V k := vec( H † k ) ∼ CN ( , Ψ ) , suppose that the input covariance matrix is ( ρ/K ) Σ . Then the resulting wideband minimum energy pernat ¯ η and the wideband outage exponent E ( η ) are respectivelygiven by ¯ η = 1tr (( I n r × n r ⊗ Σ ) Ψ ) , E ( η ) = sup λ ≤ (cid:26) λη + log det ( I − λ ( I n r × n r ⊗ Σ ) Ψ ) (cid:27) (25)for every η ≥ ¯ η , where ⊗ denotes the Kronecker product. A covariance-shaping problem:
In light of Proposition 2, wecan formulate the following input covariance shaping problemto maximize the resulting wideband outage exponent for agiven targeted energy per nat η : max Σ E ( η ) , s.t. ¯ η ≤ η, Σ is positive semi-definite, and tr Σ = 1 . Here we note that ¯ η also depends upon the choice of Σ .Although its log-determinant part is concave in Σ , the function E ( η ) itself is generally not concave due to the supremumoperation. Therefore the covariance-shaping problem may notbe easily computed for general spatial correlation Ψ .V. O NE -B IT C HANNEL S TATE F EEDBACK
Since fading channels exhibit abundant diversity in variousforms, even a limited amount of channel state informationvia feedback enables the transmitter to adapt its power andrate allocation such that the resulting performance can bedramatically improved; see, e.g. , [21] and references therein.The benefit becomes the most significant for wideband sys-tems. It has been shown that [22], wideband systems withmerely one-bit (per parallel channel) CSI at the transmitterachieve essentially the same performance as wideband systemswith full transmit CSI, in the sense that their achievableinformation rates can both be made arbitrarily large with their ratio asymptotically approaching one. Even though such adivergent property of information rates heavily depends uponthe underlying channel fading model with an infinite supportset (see [23, Sec. II.A] for some comments on this issue), itis still a reasonable expectation that the asymptotic results,if appropriately interpreted, will provide useful insights intoreal-world practice.In this section, we proceed to investigate the effect of limitedtransmit CSI, from the perspective of the wideband outageexponent. For concreteness, we analyze a specific two-levelpower allocation scheme with one-bit channel state feedback,for the scalar Rayleigh fading channel model. It is conceptuallystraightforward to formulate similar problems for systems withfiner power allocation levels and channel state quantizationlevels, or with more general fading models. However, therethe analytical approach quickly turns out to be intractable andunilluminating, and instead a heavy dose of numerical recipebecomes necessary, which is beyond the scope of this paper.
A. Transmission Protocol and Wideband Minimum Energy perNat
Consider the Rayleigh fading model as in Section III. Foreach parallel channel, the receiver informs the transmitter anone-bit feedback F k = (cid:26) if | H k | ≤ τ otherwise , (26)where τ > is a protocol parameter. Since | H k | is anexponential random variable, the probabilities of F k are p :=Pr[ F k = 0] = 1 − exp( − τ ) and p := Pr[ F k = 1] = exp( − τ ) .Denote the number of parallel channels with F · = 0 by K , which is a binomial random variable, or, the sum of K i.i.d. Bernoulli random variables. Denote ( K − K ) by K , which is also binomial. For each parallel channel with F · = 0 , we allocate an input power ( g /K ) ρ ; and for eachwith F · = 1 , we allocate an input power ( g /K ) ρ . Here theprotocol parameters g ∈ [0 , and g = 1 − g characterizethe power allocation scheme. By noting that the K parallelchannels are mutually independent, we may write the totalachievable rate as R ( K, ρ ) = P Kj =1 log (cid:16) g ρK | ˜ H (1) j | (cid:17) if K = 0 P K i =1 log (cid:16) g ρK | ˜ H (0) i | (cid:17) + P K j =1 log (cid:16) g ρK | ˜ H (1) j | (cid:17) if K = 0 or K P Ki =1 log (cid:16) g ρK | ˜ H (0) i | (cid:17) if K = K (27)where, K is a binomial random variable with Pr[ K = k ] = C kK p k p K − k , for k = 0 , , . . . , K ; (28) | ˜ H (0) i | , i = 1 , . . . , K , are samples from a sequence ofi.i.d. random variables with a (truncated exponential) densityfunction f ( x ) = exp( − x ) /p , for x ∈ [0 , τ ] ; and similarly, | ˜ H (1) j | , j = 1 , . . . , K = K − K , are samples from a If K = 0 or K = 0 , then the “reserved” power g ρ or g ρ is notactually utilized and thus wasted. In this paper, we do not allow the transmitterto adaptively adjust g according to the realization of K , for analyticalsimplicity. equence of i.i.d. random variables with f ( x ) = exp( − x ) /p ,for x ∈ ( τ, ∞ ) . In the sequel, for notational convenience, wedenote | ˜ H (0) i | by A (0) i , and | ˜ H (1) j | by A (1) j .As K → ∞ , the total rate R ( K, ρ ) converges in probabilityto a deterministic quantity, which in turn gives the widebandminimum energy per nat, as established by the followingproposition. Proposition 3:
For a given transmission protocol with one-bit channel state feedback, as K → ∞ , it achieves thewideband minimum energy per nat ¯ η = (cid:20) τ + 1 − g τ − exp( − τ ) (cid:21) − . (29) B. Wideband Outage Exponent: On-Off Power Allocation
From Proposition 3, it appears always beneficial to increase τ , because this leads to small ¯ η which implies improved energyefficiency in the wideband limit. However, this perspectiveoverlooks the outage probability. Indeed, as will be shownshortly, increasing τ generally has a rather detrimental impactupon the wideband outage exponent. Therefore a tradeoff ex-ists between ¯ η and E ( η ) , leveraged by the protocol parameters τ and g . In this subsection, we single out the special case ofon-off power allocation with g = 0 , i.e. , the transmitter onlyuses those parallel channels with squared fading coefficientgreater than τ . In the next subsection, based upon the keyinsights obtained from the on-off power allocation case, wewill further establish general results for ≤ g ≤ .Specifying g = 0 in Proposition 3, we get ¯ η = 1 / ( τ + 1) ,in order words, the achievable rate converges to ( τ +1) ρ in thewideband limit. To evaluate E ( η ) , it is useful to first performthe following qualitative reasoning. Roughly speaking, as longas there are sufficiently many parallel channels with F · = 1 ,then the achievable rate will be bound to exceed τ ρ , becausethe on-off power allocation effectively converts the originalwideband Rayleigh fading channel into a wideband fadingchannel with squared fading coefficients always larger than τ .So the “threshold rate” τ ρ , or equivalently, the “threshold η ” /τ , has a special role in the on-off power allocation scheme.For rates below this threshold rate, the dominant outage eventoccurs when the number of “on” parallel channels, K , doesnot grow to infinity with K . Following this line of thought,we can establish the following proposition characterizing thewideband outage exponent. Proposition 4:
For a given transmission protocol with one-bit channel state feedback and on-off power allocation, thewideband outage exponent is given by E ( η ) = τ + (1 − x ∗ ) [1 /η − τ − − log (1 /η − τ )] − x ∗ log( e τ − − H b ( x ∗ ) if / ( τ + 1) ≤ η ≤ /ττ − log( e τ − if η > /τ (30)where x ∗ = ( e τ −
1) exp(1 /η − τ − e τ −
1) exp(1 /η − τ −
1) + 1 /η − τ , (31) and H b ( x ) := − x log x − (1 − x ) log(1 − x ) is the binaryentropy function.The wideband outage exponent E ( η ) is monotonically non-decreasing with η . For small and large values of τ , the large- η portion of E ( η ) , τ − log( e τ − , exhibits the followingbehavior: τ − log( e τ − ∼ log(1 /τ ) , τ ≪ (32) τ − log( e τ − ∼ exp( − τ ) , τ ≫ . (33)For small τ , the wideband outage exponent slowly growslarge for η > /τ . This has been observed for the feedback-less case in Section III, which may be roughly viewed as thelimiting case for τ → . For large τ , the wideband outageexponent rapidly (at an exponential speed) decreases to zero.Such a decrease may be interpreted as the penalty for thebenefit in ¯ η . Figure 1 displays curves of E ( η ) as a function of η , for τ = 0 . , . , , . The tradeoff between ¯ η and E ( η ) isevidently visible. −5 0 5 10 1500.20.40.60.811.21.41.61.82 τ = 0.25, 0.5,1, 2 PSfrag replacements η (dB) E ( η ) Fig. 1. The wideband outage exponent E ( η ) as a function of the energy pernat η for the on-off power allocation scheme, for different values of τ . Thedashed-dot curve is the upper envelope of the E ( η ) over all τ . For a given targeted η , it is then important to adjust τ to maximize the resulting E ( η ) . This maximum widebandoutage exponent can be obtained by plotting all the E ( η ) versus η curves for different values of τ , then taking theirupper envelop. Analytically, from (30) we can find that theupper envelope exactly corresponds to those turning points (1 /τ, τ − log( e τ − in the first quadrant, and Figure 1 alsodisplays this upper envelope in dashed-dot curve. For example,in order to achieve η = 1 = 0 dB, from the dashed-dot curvewe find that the on-off power allocation scheme should have τ = 1 , yielding the maximum wideband outage exponent E (1) ≈ . . C. Wideband Outage Exponent: General Case
For general transmission protocols with g > , we cananalogously identify a threshold rate as in the on-off powerallocation case. Following similar reasoning as in Section V-B,we can find that the threshold rate is in general given by r c :=(1 − g ) τ ρ . Below this threshold rate, the dominant outagevent occurs when K does not grow to infinity with K . Thefollowing proposition gives the wideband outage exponent forthe general case. Proposition 5:
For a given transmission protocol with one-bit channel state feedback, the wideband outage exponent isgiven by E ( η ) = min n inf x ∈ (0 , ˜ E ( η, x ) , ˜ E ( η ) o if ¯ η ≤ η ≤ − g ) τ ˜ E ( η ) if η > − g ) τ , (34)where ˜ E ( η ):= sup λ ≤ (cid:26) λη + log(1 − g λ ) − log h − e − (1 − g λ ) τ i(cid:27) , ˜ E ( η, x ):= sup λ ≤ (cid:26) λη − x log h e (1 − g λ/x ) τ − i + x log( x − g λ )+(1 − x ) log[1 − x − (1 − g ) λ ] + (1 − λ ) τ o . For a given targeted energy per nat η > and a giventransmission protocol, we can then utilized Proposition 5to compute the resulting wideband outage exponent E ( η ) .Furthermore, by optimizing over the protocol parameters τ and g , we can compute the maximum achievable E ( η ) .Interestingly, numerical results show that the maximum E ( η ) isachieved by the optimal on-off power allocation scheme with g = 0 and τ = 1 /η . Although we still lack an analyticalproof for this claim, it is tempting to conjecture that theon-off power allocation scheme with τ = 1 /η maximizesthe wideband outage exponent among all the transmissionprotocols as described in Section V-A.VI. C ONCLUSION
Motivated by wideband communication in certain slow-fading propagation environments such as indoor systems, weinvestigate a mathematical model of wideband quasi-staticfading channels, focusing on its asymptotic behavior of chan-nel outage probability in the wideband regime. This work iscomplementary to the vast body of prior research on ergodicwideband channels, which are more suitable for modelinglong-term system throughputs in time-varying propagtion en-vironments.The main finding in this paper essentially reiterates asimple theme that, a communication system is unlikely tosimultaneously maintain both high rate and high reliability, or,both high energy efficiency and high spectral efficiency. As thetargeted energy efficiency approaches the minimum achievableenergy per nat in the wideband limit, the outage probabilityincreases, as quantitatively captured by the wideband outageexponent. Such tradeoffs can be leveraged by several factors,like channel fading models, deployment of multiple antennas,antenna spatial correlation, and partial transmit knowledgeof the channel state information. In this paper we illustratethe effect of these factors through a series of representative case studies, and the accumulated insights may serve asuseful guidelines for performing first-cut analysis of widebandtransmission systems.A
CKNOWLEDGEMENT
This work was supported in part by NSF NRT ANI-0335302, NSF ITR CCF-0313392, and NSF OCE-0520324.R
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