Analysis of Fixed Outage Transmission Schemes: A Finer Look at the Full Multiplexing Point
aa r X i v : . [ c s . I T ] O c t Analysis of Fixed Outage Transmission Schemes: AFiner Look at the Full Multiplexing Point
Peng Wu and Nihar Jindal
Department of Electrical and Computer EngineeringUniversity of MinnesotaEmail: pengwu, [email protected]
Abstract — This paper studies the performance of transmissionschemes that have rate that increases with average SNR whilemaintaining a fixed outage probability. This is in contrast to theclassical Zheng-Tse diversity-multiplexing tradeoff (DMT) thatfocuses on increasing rate and decreasing outage probability.Three different systems are explored: antenna diversity systems,time/frequency diversity systems, and automatic repeat request(ARQ) systems. In order to accurately study performance inthe fixed outage setting, it is necesary to go beyond the coarse,asymptotic multiplexing gain metric. In the case of antennadiversity and time/frequency diversity, an affine approximationto high SNR outage capacity (i.e., multiplexing gain plus apower/rate offset) accurately describes performance and showsthe very significant benefits of diversity. ARQ is also seen toprovide a significant performance advantage, but even an affineapproximation to outage capacity is unable to capture thisadvantage and outage capacity must be directly studied in thenon-asymptotic regime.
I. I
NTRODUCTION
In many time-varying communication systems, the receiverhas accurate instantaneous channel state information (CSI),generally acquired from received pilot symbols, while thetransmitter only knows the channel statistics (e.g., averagereceived SNR and the fading distribution) but has no instanta-neous CSI. This could be the case if, for example, the channelcoherence time is long enough to allow for receiver training(over a reasonably small fraction of the coherence time) butthis information cannot be fed back to the transmitter becausethe feedback delay is too large relative to the coherence time.Performance in such a setting is generally dictated by fadingand the relevant performance metric is known to be the outageprobability , which is the probability that the instantaneousmutual information is smaller than the transmission rate,because this quantity reasonably approximates the probabilityof decoder (frame) error if a strong channel code is used [1].Such systems have traditionally been studied by consideringoutage probability versus average SNR for a fixed transmissionrate, leading to measures such as diversity order (generallydefined as the slope of the outage vs. SNR curve in log-scale).In modern communication systems, however, transmissionrate is generally adjusted according to the average SNR (viaadaptive modulation and coding) and thus systems need to bestudied at various rates and SNR levels. The seminal work ofZheng and Tse [2] took precisely this viewpoint in introducingthe diversity-multiplexing tradeoff (DMT). Loosely speaking,the DMT framework considers the performance of a family of codes indexed by average SNR such that the coding rateincreases as r log SN R , and the outage/error probability ofthe code decreases approximately as
SN R − d . The quantity r is the multiplexing gain while d is the diversity order (of thefamily of codes, not of a particular code). The DMT region isthe set of ( r, d ) pairs achievable by any family of codes, andcan be simply quantified in terms of the number of transmitand receive antennas, N t and N r respectively, and the receiverstrategy for MIMO systems.Over the past few years the DMT framework has become thebenchmark for comparing different transmission strategies fordifferent systems (MIMO, cooperative transmission, multipleaccess channel). Although the DMT framework has beenincredibly useful in this role by providing a meaningful andtractable metric to compare different schemes that simultane-ously achieve increasing rate and decreasing outage proba-bility , one very important paradigm not sufficiently capturedby the DMT are codes that achieve increasing rate and fixedoutage probability .Families of codes that achieve a fixed rather than decreasingoutage are important because they are used in a number ofimportant wireless systems, most prominently in the cellulardomain. In this setting, as the average SNR of a user increases(i.e., as a user moves closer to the base station), it is desirableto use the additional SNR to increase rate but not to decreaseoutage probability (i.e., packet error rate); indeed, many sys-tems continually adjust rate precisely to maintain a target errorprobability (e.g., − ). In a voice system, for example, thevoice decoder may be able to provide sufficient quality if nomore than of packets are received incorrectly and thusthere is no benefit to decrease outage below . Therefore,serving each user at the largest rate that maintains outageminimizes per-user resource consumption (i.e., time/frequencyslots) and thereby maximizes system capacity.In order to accurately study fixed-outage schemes, it is nec-essary to directly study the manner in which outage capacityscales with SNR. We denote outage capacity as R ( P, ǫ ) , wherethis quantity is the rate that achieves an outage probability of ǫ at an average SNR of P . In the context of the DMT, fixedoutage systems (for any ǫ > ) achieve zero diversity ( d = 0 )and thus can achieve the maximum multiplexing gain. In otherwords, the DMT tells us that R ( P, ǫ ) ≈ r max log P for any ǫ > , where r max is the maximum multiplexing gain, but can-not provide a more precise characterization than multiplexing multiplexing gain(r) d i v e r s i t y ga i n ( d ) × × (a) DMT Regions P(dB) C ε ( bp s / H z ) C ε × C ε × (b) R ( P, ǫ ) for ǫ = . Fig. 1. Full multiplexing performance in the the diversity-multiplexingtradeoff gain (or pre-log factor). In many scenarios of interest, such acharacterization is not sufficient to meaningfully characterizeperformance.To further illustrate the need to directly study outage ca-pacity, let us consider a simple example. For a × systemthe maximum diversity order is d ∗ ( r ) = 1 − r , while fora × ( N t = 1 , N r = 2 ) system d ∗ ( r ) = 2(1 − r ) [2].The DMT regions for both systems are shown in Fig. 1 (a).Because min( N t , N r ) = 1 both regions share the same fullmultiplexing point ( r = 1 , d = 0 ), a DMT-based comparisonwould indicate that the systems are equivalent in a fixed outagesetting. However, the plot of R ( P, ǫ ) for ǫ = . in Fig. 1 (b)shows that there is a huge power gap ( . dB) between thetwo systems; it clearly is not sufficient to consider only themultiplexing gain, which is the slope of the R ( P, ǫ ) curve.Motivated by this example, one step in the right direction isto consider affine rather than linear approximations to outagecapacity (at high SNR): R ( P, ǫ ) = r max log P + O (1) , (1)where the non-vanishing O (1) term, which depends on ǫ and the system configuration (i.e., N t and N r ), is capableof capturing power/rate offsets such as that seen in Fig. 1(b). This affine approximation, first proposed in [3], has beenextremely useful in analysis of the ergodic capacity of CDMAsystems [3] and MIMO systems [4] [5]. More recently, theaffine approximation has also been employed to study theoutage capacity of MIMO systems at asymptotically high SNR[6]. In [6], an expression for the constant term in (1) is givenin terms of the statistics of the channel matrix (more precisely,in terms of the distribution of the determinant of HH H where H is the channel matrix). A. Contribution of Work
In this paper, we first consider the case of antenna diversity(SIMO or MISO; Section III) and show that fixed outagecapacity can be exactly specified in terms of the inverse ofthe fading CDF. Although this result can be viewed as aspecial case of [6] (and also appears in [8, Section 5.4]), it isuseful to consider this base case to more precisely illustrate theimportance of fixed-outage analysis. Next we consider systemswith time and/or frequency diversity (Section IV), which aremodeled as block-fading channels, again in the context of the affine approximation to outage capacity. Finally, we considerthe performance of systems using hybrid ARQ (automaticrepeat request) for incremental redundancy as well as chasecombining. Unlike antenna or time/frequency diversity, thebenefit of H-ARQ vanishes at asymptotically high SNR; inother words, the high-SNR affine approximation to outagecapacity is unaffected by H-ARQ. However, H-ARQ doesprovide a very significant advantage of low and moderateSNR’s. In order to understand these gains, we directly studyoutage capacity at finite SNR’s.II. S
YSTEM M ODEL
We consider a block-fading channel, denoted by H , which israndomly drawn according to a known probability distribution(e.g., spatially white Rayleigh fading) and then fixed forthe duration of a codeword. Furthermore, the receiver isassumed to have perfect channel state information (CSI) butthe transmitter has no instantaneous knowledge of the channelrealization and is only aware of the probability distribution.The received signal y is given by: y = √ P Hx + z , (2)where the input x is constrained to have unit norm, z isthe complex Gaussian noise and P represents the (average)received SNR. We consider cases where the input is Gaussianand further specify its structure where needed.The outage probability is the probability that the instanta-neous mutual information is smaller than the transmission rate R : P out ( R, P ) = P [ I ( X ; Y | H , P ) < R ] , (3)and the outage capacity R ( P, ǫ ) is defined as the maximumrate that achieves the desired outage probability: R ( P, ǫ ) , sup P out ( R,P ) ≤ ǫ R. (4)Note that this quantity is essentially the same as the ǫ -capacitydefined by Verdu and Han [7] .III. A NTENNA D IVERSITY
We begin by examining antenna diversity, which is one ofthe commonly employed diversity techniques. The results ofthis section are a special case of [6], and precisely the sameanalysis appears in [8, Section 5.4]; thus this section shouldbe treated as background material.If the transmitter has N t > antennas while N r = 1 ,the mutual information for a spatially white Gaussian input(components of x are iid Gaussian with variance N t ) is log (cid:16) || H || PN t (cid:17) and therefore the outage probability isgiven by: P out ( R, P ) = P (cid:20) log (cid:18) || H || PN t (cid:19) < R (cid:21) . (5) In some cases we compute outage probability assuming the input x is Gaussian and spatially white, while the precise definition of ǫ -capacityrequires an explicit optimization over the input distribution. This choice ofinput is easily shown to be optimal when N t = 1 , but is not necessarilyoptimal for N t > . y setting this quantity to ǫ and solving for R , we get: C N t × ǫ ( P ) = log (cid:18) F − || H || ( ǫ ) PN t (cid:19) , (6)where F − || H || ( · ) is the inverse of the CDF of random variable || H || . In iid Rayleigh fading the components of H are iid CN (0 , and thus || H || is chi-square with N t degrees offreedom and has the following CDF: F || H || ( x ) = 1 − e − x N t X k =1 x k − ( k − . (7)If N t = 1 the channel gain is exponential and the inverse CDFcan be written in closed form to yield: C × ǫ ( P ) = log (cid:18) (cid:18) − ǫ (cid:19) P (cid:19) , (8)while for N t > the inverse CDF needs to be computednumerically.It can be convenient to relate the outage capacity to theAWGN capacity at SNR P : C AW GN ( P ) = log (1 + P ) : C ǫ ( P ) = C AW GN (Γ ǫ P ) = log (1 + Γ ǫ P ) (9)where the gap to capacity is: Γ N t × ǫ = F − || H || ( ǫ ) N t . (10)For small ǫ we can approximate the CDF of || H || as F || H || ( x ) ≈ x Nt N t ! and therefore the gap can be approximatedas: Γ N t × ǫ ≈ ǫ Nt ( N t !) Nt N t . (11)In the case of receive diversity ( N t = 1 , N r > ) theachieved mutual information is log (cid:0) || H || P (cid:1) , becauseusing optimal maximum-ratio transmission prevents the powerloss experienced with transmit diversity, and therefore: C × N r ǫ ( P ) = log (cid:16) F − || H || ( ǫ ) P (cid:17) , (12)where || H || is chi-square with N r degrees of freedom.As expected, there is a log ( N t ) dB power gap between C × N r ǫ ( P ) and C N t × ǫ ( P ) .In Fig. 2 the outage capacity of × , × , and × systems are plotted for ǫ = 0 . . The capacity gap for the × system is approximately 20 dB ( Γ ǫ = − ln(1 − ǫ ) ≈ ǫ ),while it is about 11.5 dB for the × system ( Γ ǫ ≈ p ǫ ).Fixed outage analysis very clearly illustrates the advantage ofantenna diversity.IV. T IME / F
REQUENCY D IVERSITY
Another very common method of realizing diversity isthrough time or frequency, i.e., by coding across multiplecoherence times/bands. If a codeword spans L coherencebands (in time and/or frequency), the outage probability isgiven by [8, Equation 5.83]: P out ( R, P ) = P " L L X i =1 log (1 + P | h i | ) < R (13) P(dB) C apa c i t y ( bp s / H z ) C ε × C ε × C ε × C awgn1 × Fig. 2. C awgn and C ǫ (bps/Hz) vs P (dB), ǫ = 0 . where h i is the channel gain over the i -th band, each h i is unitvariance complex Gaussian (Rayleigh fading), and h , . . . , h L are assumed to be iid. It is important to note that this outageprobability expression approximates the performance of astrong channel code that is interleaved across the L bands,and not that of a sub-optimal repetition code.For notational convenience we define the function G L ( R ) to be equal to the outage expression in (16). In terms of thisfunction C ǫ ( P ) = G − L ( ǫ ) . (14)Although we cannot reach a closed form for G − L ( ǫ ) , thisquantity can be computed numerically by noting that C ǫ ( P ) is equal to the R that satisfies: ǫ = Z Z · · · Z L P Li =1 log (1+ x i ) We first investigate incremental redundancy techniques, inwhich the the transmitter sends additional parity bits (rather than retransmitting the same packet) whenever a NACK isreceived. In this setting it has been shown that the total mutualinformation is the sum of the mutual information receivedin each ARQ round, and that decoding is possible once the accumulated mutual information is larger than the numberof information bits [9]. In other words, the number of ARQrounds X is the smallest number l such that: l X i =1 log (1 + P | h i | ) > R. (19)The constraint caps this quantity by L , and an outage occurswhenever the mutual information after L rounds is smallerthan R : P out ( R ) = P " L X i =1 log (1 + P | h i | ) < R . (20)For simplicity we consider single antenna systems ( N t = N r = 1 ), and use h i to denote the channel during the i -thARQ round. In the following sections we consider the casewhere the channel is iid across ARQ rounds. Similar to thedefinition in Section IV, here we use G IR,i ( R ) to denote theprobability that the sum of mutual information is less than thefirst round rate R after i rounds. Then, R = G − IR,L ( ǫ ) . Goback to the definition of η , we have η IR = C IR,Lǫ = RE [ X ]= G − IR,L ( ǫ )1 + P L − i =1 G IR,i ( G − IR,L ( ǫ )) (21)It is useful to compare performance to a system withoutARQ that always codes over the L available slots (whereasARQ allows for early completion), which precisely corre-sponds to L -order time/freq diversity (Section IV). Afterproperly normalizing rates, we get: C IR,Lǫ C nARQǫ = LE [ X ] (22)where C nARQǫ is the outage capacity of a corresponding noARQ protocol. Since L ≥ E [ X ] , then C IR,Lǫ ≥ C nARQǫ (23)Actually, the quantity LE [ X ] determines the advantage of ARQ,and it is not difficult to show the following limit: lim P →∞ E [ X ] = L (24)This shows that the effect of ARQ vanishes at asymptoticallyhigh SNR because the expected number of ARQ roundsconverges to the maximum L. Indeed, it can further be shownthat the rate advantage of ARQ also vanishes at asymptoticallyhigh SNR: lim P →∞ [ C IR,Lǫ ( P ) − C nARQǫ ( P )] = 0 (25)In other words, the high SNR affine approximation is thesame regardless of whether ARQ is used. On the other 10 20 30 40 5002468101214 P(dB) C ε ( bp s / H z ) ARQ and non−ARQ,L=1ARQ,L=2 no ARQ,L=2ARQ,L=3no ARQ,L=3 Fig. 4. Outage capacity for the IR strategy and the no ARQ strategy in theiid Rayleigh fading channel, ǫ = 0 . hand, the number of expected ARQ rounds is less than L atasymptotically low SNR.Based on these asymptotic results one might conclude thatARQ provides a benefit only at relatively low SNR’s. However,numerical results indicate that the high SNR asymptotics kickin only for very large SNR’s. Indeed, ARQ does achieve asignificant advantage for a relatively large range of SNR’s. InFig. 4 the outage capacity is shown for ǫ = 0 . and L = 1 , and . Note that 2 rounds of ARQ provide a significant poweradvantage relative to no ARQ up to approximately 30 dB,while the advantage lasts past 40 dB for L = 3 . Asymptoticmeasures such as multiplexing gain and rate/power offset areclearly misleading in this context. B. Chase Combining If chase combining is used, the transmitter retransmits thesame packet whenever a NACK is received and the receiverperforms maximal-ratio-combining (MRC) on all receivedpackets.. As a result, SNR rather the mutual information isaccumulated over ARQ rounds and the outage probability isgiven by: P out ( R ) = P " log (1 + P L X i =1 | h i | ) < R (26)Note that this strategy essentially allows a repetition code tobe used up to L times. A straightforward derivation shows theoutage capacity of CC in the iid Rayleigh fading channel is: C CC,Lǫ ( P ) = R − e − R − P ) P L − k =1 ( L − k ) (2 R − k − P ( k − (27)where R has to be obtained from G − CC,L ( ǫ ) numerically.Chase combining indeed provides some advantage at low andmoderate SNR, but performs poorly at high SNR becauseof the sub-optimality of the repetition codes. In Fig. 5 wecompare the performance of IR, CC and no ARQ strategy for L = 4 . We see that CC performance reasonably at low SNRbut trails off at high SNR. P(dB) C ε ( bp s / H z ) no ARQ,L=1 CC,ARQ,L=4no ARQ,L=4IR,ARQ,L=4 Fig. 5. Comparison of the outage capacity among IR strategy, CC strategyand no ARQ strategy, ǫ = 0 . VI. C ONCLUSION In this paper we have studied open-loop communicationsystems under the assumption that rate is adjusted such thata fixed outage probability is maintained. The over-archingtakeaways of this work are two-fold. First, we have arguedthat schemes that increase rate but have a fixed rather thandecreasing outage probability may be more practically relevantthan the increasing rate/decreasing outage schemes addressedby the diversity-multiplexing tradeoff. Second, we have shownthat asymptotic measures should be used very carefully inanalysis of fixed-outage systems. Multiplexing gain is certainlytoo coarse in this context, while high SNR rate/power offsetsare sometimes meaningful (antenna diversity, time/frequencydiversity) but can also be misleading in other settings (e.g.,ARQ systems) due to their asymptotic nature.We hope this paper establishes a more meaningful and prac-tically relevant framework by which different communicationtechniques, such as partial channel feedback and relaying, canbe studied by the research community.R EFERENCES[1] G. Caire, G. Taricco and E. Biglieri,“Optimum power control over fadingchannels,” IEEE Trans. Inf. 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