Uniform Power Allocation with Thresholding over Rayleigh Slow Fading Channels with QAM Inputs

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1 Uniform Power Allocation with Thresholding over ayleigh Slow Fading Channels with QA Inputs Hwanjoon (Eddy) Kwon, Young-Han Kim, and haskar D. ao Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 99-, USA, Abstract In this paper, we consider the power allocation problem that minimizes the outage probability for ayleigh slow fading channels with equiprobable QA inputs. We focus on the uniform power allocation with thresholding (UPAT) policy that assigns nonzero constant power only to a subset of the subchannels. This simple suboptimal policy can significantly alleviate the feedback overhead and the complexity compared to the optimal mercury/water-filling (WF) solution. Through asymptotic analysis and numerical simulations, we first show that the optimal UPAT, namely, the UPAT with the optimal threshold, performs close to WF if the constellation size is large enough that log, where is the fixed target transmission rate. This condition log turns out to define a natural system operating point. As we show through numerical results, if log, both WF and the optimal UPAT perform poorly due to having too small and their performance can be significantly improved by using a larger. From these results, we conclude that for a given target transmission rate, the optimal UPAT performs close to WF as long as the constellation size is chosen appropriately not to limit the performance. Index Terms Constellation size, fading, mercury/water-filling, outage probability, power allocation, QA. I. INTODUCTION The performance of a communication system over a fading channel can be substantially improved by adapting transmit power according to the channel gains [] [8]. If the channel is partitioned into independent AWGN subchannels, the waterfilling power allocation [] along with a Gaussian input distribution maximizes the mutual information (I) of the channel. In contrast, for non-gaussian inputs such as pulse amplitude modulation (PA) and quadrature amplitude modulation (QA), the mercury/water-filling (WF) power allocation maximizes the I []. However, the amount of overhead to enable transmit power adaptation is often significant in practice [], []. In addition, the WF power allocation involves the inverse minimum mean-square error (SE) functions [], an exact implementation of which can be excessively complex in practice []. To overcome these difficulties, a simple suboptimal power allocation policy, uniform power allocation with thresholding (UPAT) that assigns nonzero constant power only to a subset of the subchannels, has received much attention [] [], due to the remarkably relaxed overhead requirements [], [] as well as the simple transceiver structure []. The work of. ao was supported by the National Science Foundation Grant No. CCF-. uch effort has been devoted to developing simple methods to determine the threshold value (or equivalently, the subset of the subchannels to which nonzero constant power is assigned) for UPAT and to analyzing the resulting performance [] []. However, most of the existing results have focused on Gaussian inputs over ergodic (or fast fading) channels and thus the insight into practical system design has been rather limited. oreover, the UPAT scheme with the optimal threshold, henceforth referred to as the optimal UPAT, has not been well studied. In this paper, we study the outage performance of the optimal UPAT for ayleigh slow fading channels with equiprobable QA inputs. Note that it is mathematically challenging to derive the exact performance results, since the performance metric involves the I of an AWGN channel with QA input which does not have a closed-form expression. In our work, we rely on asymptotic analysis, engineering intuition, and comprehensive numerical work. Through asymptotic analysis and numerical simulations, we first show that the optimal UPAT performs close to WF if the constellation size is large enough that log, where is the fixed target transmission rate. This condition log turns out to define a natural system operating point. As we show through numerical results, if log, both WF and the optimal UPAT perform poorly due to having too small and their performance can be significantly improved by using a larger. From these results, we conclude that for a given target transmission rate, the optimal UPAT performs close to WF as long as the constellation size is chosen appropriately not to limit the performance. The rest of the paper is organized as follows. Section II presents the system model and the performance metric. Section III formulates the outage probability with WF and the optimal UPAT. Section IV discusses the outage performance. Throughout, for a vector a =(a,..., a n ), a n n i= a i. Component-wise inequalities are denoted by and. II. SYSTE ODEL AND PEFOANCE ETIC A. System odel Consider transmission over a nonergodic block-fading channel [], also called slow fading channel, consisting of blocks of L channel uses, where block i =,,..., undergoes a random channel gain H i that is constant during the block and is independently and identically distributed (i.i.d.) across the blocks. Assume that the channel inputs to the blocks are //$. IEEE

2 independently and uniformly distributed over the standard - QA [9] constellation set S, where j : j =,,...} and s S s =. Suppose that H i } i= is known to the transmitter so that transmit power for each block can be adapted to the channel strength, subject to an average power constraint P. To describe a power allocation scheme, it is convenient to define P H i which indicates the SN in block i with the uniform power allocation (UPA). Let γ = } i=. Then, a power allocation scheme is described as p(γ; ) =p i (γ; )} i=, where p i(γ; ) indicates the normalized transmit power of block i, i.e., p(γ; ). Note that p(γ; ) depends on H i } i=, P, and through γ. The channel output vector Y i C L in block i, isgivenby Y i = H i pi (γ; )P S i + Z i, i =,,...,, () where S i S L is the -QA channel input vector and Z i N C (, I) is the Gaussian channel noise vector. Note that p i (γ; ) corresponds to the instantaneous SN in block i. Throughout, we assume the ayleigh fading H i N C (, ) and therefore the probability density function (pdf) of is given by f γi (ξ) = P e P ξ,ξ.. Performance etric: Outage Probability ecall that the I of the AWGN channel Y = ρs + Z under the uniform distribution of the input S over S is [] (ρ) =log [ ( log s S E Z )] e ρ(s s )+Z + Z s S. () Then, the instantaneous I is defined [8] by I (γ, p(γ; )) (p i (γ; ) ). () i= For a fixed target transmission rate, the outage probability is defined [8] by P out (p(γ; ),P,) PI (γ, p(γ; )) <} () where the probability is with respective to the random γ. III. POWE ALLOCATION SCHEES A. Optimal Power Allocation: ercury/water-filling The outage probability minimization problem is minimize P out (p(γ; ),P,) subject to p(γ; ) () p(γ; ). We refer to the solution of the above problem as the optimal power allocation, denoted by p opt (γ; ). Since the outage probability is minimized when the instantaneous I () is maximized for each realization of γ, we have[8] p opt (γ; ) = arg max p(γ;) p(γ;) I (γ, p(γ; )). () Let SE (ρ) denote the SE incurred in the estimation of an equiprobable -QA symbol over the AWGN channel with SN ρ. y the Lagrangian duality, the Karush Kuhn Tucker (KKT) conditions [], and the relationship d dρ IAW (ρ) = ln SE (ρ) between I and SE [], the solution of () is given [] by p opt i (γ; ) = ( λ ), γi λ SE, <λ where the SN threshold λ is chosen so that the average power constraint is satisfied with equality. The optimal solution () is often referred to as mercury/water-filling (WF) []. Substituting () into () yields the outage probability with the optimal power allocation P i: λ ( SE () ( )) } λ <. (8). Uniform Power Allocation with Thresholding In a UPAT scheme, nonzero constant power is assigned to a set of selected blocks while zeropower is assigned to the other blocks. Let γ o γ o γ o be the ordered γ sequence. Then, a UPAT scheme is given by p UPAT i (γ; ) = N UPAT, γ o N UPAT+, <γ o N UPAT+ where N UPAT <is the number of zero-power blocks. Therefore, a UPAT scheme is completely defined by how to determine the value of N UPAT. The optimal value of N UPAT that maximizes the instantaneous I is N UPAT = arg max I AW n< i=n+ ( n γo i (9) ). () The UPAT with NUPAT is referred to as the optimal UPAT, denoted by p UPAT (γ; ). Substituting (9) and () into () yields the outage probability with the optimal UPAT ( ) } P NUPAT γi o <. () i=n UPAT + C. Examples We now discuss a set of examples that show power allocations of WF and the optimal UPAT, corresponding instantaneous Is, and the power loss ΔP that indicates the additional P required for the optimal UPAT to achieve the same instantaneous I as WF. The examples are with respect to two specific realizations of γ but provide some insight into the outage probability results that come from a random γ, discussed in the next section. We consider a specific channel gain vector h i } i= with =, where its elements were i.i.d. drawn according to N C (, ), normalized and ordered such that i= h i = and h h (see Fig. and Fig. ). Two different values of P, P = d and P =d are examined. Similar discussions are presented in [], focusing on WF and waterfilling. In this paper, however, we are mainly interested in the comparison between WF and the optimal UPAT. The ordering is merely intended to clearly show the dependency of power allocation on the channel gains

3 Channel gain or Norm. Tx Power ercury/water filling, QPSK ercury/water filling, QA ercury/water filling, QA waterfilling The optimal UPAT, QPSK The optimal UPAT, QA The optimal UPAT, QA h i Channel gain or Norm. Tx Power ercury/water filling, QPSK ercury/water filling, QA ercury/water filling, QA waterfilling The optimal UPAT, QPSK The optimal UPAT, QA The optimal UPAT, QA h i 8 9 lock Index 8 9 lock Index Fig.. Power allocation results of mercury/water-filling (optimal), waterfilling, and the optimal UPAT, when γ = γ ( i= = d). Fig.. Power allocation results of mercury/water-filling (optimal), waterfilling, and the optimal UPAT, when γ = γ ( i= = d). TALE I INSTANTANEOUS I AND ADDITIONAL P EQUIED Y p UPAT (γ,) I (p opt ) I (p UPAT ) ΔI ΔP (γ, )..88. %. d (γ, ).8..8 %. d (γ, ) %. d (γ, ) %. d (γ, ) %. d (γ, ).9.8. %. d Three constellation sizes, =,, are considered. Let γ = P h i } i= and γ = P h i } i=. Then, γ = d and γ = d. Since WF and the optimal UPAT depend on (γ,), there are different cases, i.e., =,, for each γ. The power allocations of WF and the optimal UPAT are shown in Fig. and Fig. in case γ = γ and γ = γ, respectively. For each γ, we also present the power allocations of waterfilling that does not depend on. We note the following observations. First, all the power allocation schemes activate (or assign nonzero power to) a small portion of the blocks when γ = γ, while activating most of the blocks when γ = γ. When γ = γ, the optimal UPAT and waterfilling are similar to the UPA, regardless of. Second, WF is similar to waterfilling except the case of (γ = γ, =), especially in the cases of (γ = γ, = ), (γ = γ, = ), and (γ = γ, = ). Third, in the case of (γ = γ, =), WF assigns power almost inversely proportionally to, which is completely different from waterfilling and the optimal UPAT. For each case, Table I shows the instantaneous I of WF, I (p opt ), the instantaneous I of the optimal UPAT, I (p UPAT ), the loss ΔI of the optimal UPAT in instantaneous I compared to WF, and the power loss ΔP.We note the following observations. First, in all the cases, ΔI is insignificant, i.e., only.% even in the case of (γ = γ, = ) where the power allocations are significantly different between WF and the optimal UPAT. Second, ΔP is significant in the case of (γ = γ, =), while it is insignificant in the other cases. The results in Fig., Fig., and Table I can be explained by the following properties of (ρ). Note that a sum of (p i ) is the objective function in the relevant optimization problems () and (). (i) Given, ifρ is so low that (ρ) log, (ρ) is not much different from log ( + ρ). In particular, (ρ) log ( + ρ) in the low ρ regime []. (ii) Given, ifρ is so high that (ρ) log, (ρ) slowly approaches its maximum log as ρ increases. Therefore, in this regime, (ρ) does not vary much as ρ changes. Due to property (i), if (γ,) is such that () log for most of the blocks, e.g., (γ = γ, = ), (γ = γ, = ), and (γ = γ, = ), WF is not much different from waterfilling, i.e., cutting off weakest blocks and then assigning more power to stronger blocks. This is because the solution to the problem () is waterfilling if (p i ) is replaced by log ( + p i ). In this case, as shown in [], [] for Gaussian inputs, ΔI and ΔP are not significant. Due to property (ii), on the contrary, if (γ,) is such that () log for most of the blocks, e.g., (γ = γ, = ), WF tends to assign power inversely proportionally to []. Intuitively, this is because with relatively small power, stronger blocks can still provide per-block I (p i ) close to log. y assigning more power to weaker blocks, WF increases per-block I of weaker blocks, while minimally decreasing per-block I of stronger blocks. This is how WF maximizes the average of per-block I, i.e., the instantaneous I. In contrast, the optimal UPAT just activates most of the blocks due to the uniform power constraint and therefore the optimal UPAT tends to be the UPA. Interestingly, although the power allocations of WF and the optimal UPAT are quite different from each other, ΔI is not significant since the

4 Outage Probability =. =. =. ercury/water filling The optimal UPAT =.8 Additional Power equired by the Optimal UPAT [d] 9 8 QPSK QA QA = = = QA P [d] 8 Fig.. Outage performance of the optimal power allocation (mercury/waterfilling) and the optimal UPAT for different target transmission rates ( =.,.,.,.8) when the number of blocks is ( =) and =(QPSK). Fig.. UPAT suboptimality for various (,, ): each curve indicates additional P required for the optimal UPAT at outage rate to achieve the same outage probability as mercury/water-filling. instantaneous I of the optimal UPAT is close to log and there is not much room to improve. However, even though ΔI is small, ΔP can be significant. This is because the optimal UPAT is similar to the UPA and therefore if P increases, a large portion of the increased power is assigned to the blocks that could provide per-block I close to log without the increased power, which makes increasing P very inefficient in terms of increasing the instantaneous I. Therefore, a large ΔP is needed to increase the instantaneous I by a small amount. IV. NEA-OPTIALITY OF THE OPTIAL UPAT In this section, we discuss the outage performance loss from the optimal UPAT compared to WF. We first show that the optimal UPAT performs close to WF as long as the constellation size is sufficiently large. We then show that the constellation size should be sufficiently large for both WF and the optimal UPAT in order not to suffer from a huge performance degradation. A. The Optimal UPAT is Near-Optimal If log Fig. shows the outage probability of WF and the optimal UPAT when =(QPSK), =.,.,.,.8, and =. We observe that when =., the optimal UPAT performs close to WF. ut, as increases, the performance loss from the optimal UPAT increases. In particular, the loss is significant when log, e.g., at outage rate, the loss is about danddfor =. and =.8, respectively. The performance loss of the optimal UPAT for various values of,, and is summarized in Fig.. Each curve in the figure indicates the additional P required for the optimal UPAT to achieve the same outage probability as WF. The results clearly show that the loss due to the optimal UPAT is marginal when log, increases with, and becomes significant when log. Intuitively, the above results can be explained as follows. y the definition of the outage probability (), when log, the outage events occur when () log for most of the blocks. As discussed in the previous section, for γ realizations such that (γ i) log for most of the blocks, WF is not much different from waterfilling and the additional power required for the optimal UPAT to achieve the same instantaneous I as WF is not significant. When log, in contrast, the outage events occur unless (γ i) log for most of the blocks. For γ realizations such that (γ i) log for most of the blocks, the optimal power allocation tends to be inversely proportional to, while the optimal UPAT tends to be the UPA. In this case, the additional power required for the optimal UPAT is significant, as discussed in the previous section. An asymptotic behavior of these observations is proved by the following proposition. Proposition : lim P out (pupat (γ; ),P,) P out =. (popt (γ; ),P,). A System Should Operate in the egime Where log The following property of the outage probability is important for subsequent discussion. Lemma : P out (p(γ; ),P,) is a decreasing function of the constellation size. Proof: Since (ρ) is an increasing function of [], [], I (γ, p(γ; )) is an increasing function of. The proof follows by (). Lemma implies that the outage probability with - QA inputs not only provides an analytical insight into the asymptotic behavior for large but also serves as a lower bound for the outage probability of any finite. The proof is omitted due to space limitations.

5 QPSK QA ound (Gain of QA) Outage Probability =. =. =.8 Gain in Average Transmit Power [d] Gain of QA over QPSK The optimal UPAT Gain of QA over QA ercury/water filling Gain of QA over QA =. P [d] Fig.. Outage performance gain of -QA over QPSK with the optimal power allocation (mercury/water-filling) when =. Now, we show through numerical results that the condition log is a necessary prerequisite for a system to perform well. Fig. shows the performance gain of -QA over QPSK with WF when =. We observe that when =., QPSK and -QA exhibit a similar performance. ut, as increases, the gain of -QA increases. In particular, when is close to the maximum achievable rate of QPSK (i.e., log =), the gain is significant, e.g., d gain at outage probability when =.8. Fig. summarizes the gain from using a larger for various values of and when =. The dashed line indicates the gain in P from using one-step larger at outage probability. The solid line indicates the gain of - QA, which bounds the gain from using any larger (finite). We note the following observations. If log, the gain from using a larger is marginal for both WF and the optimal UPAT. In contrast, if log, the performance of them can be significantly improved by even one-step larger. In other words, both WF and the optimal UPAT perform poorly in this regime, due to having too small. Intuitively, these results can be explained as follows. If log, the outage events occur when () log for most of the blocks. For γ realizations such that (γ i) log for most of the blocks, increasing does not significantly increase the instantaneous I, since (γ i) (γ i), <, for most of the blocks. Therefore, increasing does not significantly decrease the outage probability. In contrast, if log, the outage events occur unless (γ i) log for most of the blocks. For γ realizations such that (γ i) log for most of the blocks, increasing can significantly increase the instantaneous I, since (γ i) (γ i), <, for most of the blocks. Therefore, increasing can substantially decrease the outage probability. C. Summary If log the optimal UPAT performs close to WF; otherwise, the suboptimality of the optimal UPAT can be Fig.. Gain from a larger constellation size in average power P at outage probability when =. significant. However, if log, both WF and the optimal UPAT perform poorly and therefore a system should avoid operating in this regime. In conclusion, for a given target transmission rate, the optimal UPAT is near-optimal as long as the constellation size is chosen appropriately not to limit the performance. EFEENCES [] T.. Cover and J. A. Thomas, Elements of Information Theory. Wiley,. [] A. Lozano, A.. Tulino, and S. Verdu, Optimum power allocation for parallel gaussian channels with arbitrary input distributions, IEEE Trans. Inf. Theory, vol., no., July. [] D. Dardari, Ordered subcarrier selection algorithm for OFD- ased high-speed WLANs, IEEE Trans. Wireless Communi., vol., no., Sep.. [] H. Kwon and. D. ao, Uniform bit and power allocation with subcarrier selection for soded OFD systems, in Proc. IEEE VTC Spring,. []. Schein and. Trott, Sub-optimal power spectra for colored Gaussian channels, in Proc. IEEE Int. Symp. Inf. Theory, 99. [] W. Yu and J.. Cioffi, Constant-power waterfilling: Performance bound and low-complexity implementation, IEEE Trans. Communi., vol., no., Jan.. [] K. D. Nguyen, A. Guillén i Fàbregas, and L. K. asmussen, Power allocation for block-fading channels with arbitrary input constellations, IEEE Trans. Wireless Communi., 9. [8] G. Caire, G. Taricco, and E. iglieri, Optimum power control over fading channels, IEEE Trans. Inf. Theory, 999. [9] J. G. Proakis, Digital Communications. cgraw-hill,. [] T. F. Wong, Numerical calculation of symmetric capacity of ayleigh fading channel with PSK/QPSK, IEEE Communi. Letters, vol., no. 8,. [] S. oyd and L. vandenberghe, Convex Optimization. Cambridge University Press,. [] D. Guo, S. Shamai, and S. Verdu, utual information and minimum mean-square error in Gaussian channels, IEEE Trans. Inf. Theory, vol., no., Apr.. [] G. D. Forney, Jr. and G. Ungerboeck, odulation and coding for linear Gaussian channels, IEEE Trans. Inf. Theory, vol., no., 998. [] G. Caire K.. Kumar, Information theoretic foundations of adaptive coded modulation, Proceedings of the IEEE, vol. 9, Issue., Dec..