On Constant Power Water-filling

On Constant Power Water-filling Wei Yu and John M. Cioffi Electrical Engineering Departent Stanford University, Stanford, CA94305, U.S.A. eails: {weiyu,cioffi}@stanford.edu Abstract This paper derives a rigorous perforance bound for the constant-power water-filling algorith for ISI channels with ulticarrier odulation and for i.i.d. fading channels with adaptive odulation. Based on the perforance bound, a very-low coplexity logarith-free power allocation algorith is proposed. Theoretical worstcase analysis and siulation show that the approxiate water-filling schee is close to optial. I. Introduction When a counication channel is corrupted by severe fading or by strong intersybol interference, the adaptation of transit signal to the channel condition can typically bring a large iproveent to the transission rate. Adaptation is possible when the channel state is available to the transitter, usually by a channel estiation schee and a reliable feedbac echanis. With perfect channel inforation, the proble of finding the optial adaptation strategy has been uch studied in the past. If the channel can be partitioned into parallel independent subchannels by assuing i.i.d. fading statistics for the fading channel, or by the discrete Fourier transfor for the intersybol interference channel, the optial transit power adaptation is the well-nown water-filling procedure. In water-filling, ore power is allocated to better subchannels with higher signal-to-noise ratio SNR, so as to axiize the su of data rates in all subchannels, where in each subchannel the data rate is related to the power allocation by Shannon s Gaussian capacity forula log + SNR. However, because the capacity is a logarithic function of power, the data rate is usually insensitive to the exact power allocation, except when the signal-to-noise ratio is low. This otivates the search for sipler power allocation schees that can perfor close to the optial. Approxiate water-filling schees often greatly siplify transitter and receiver design, and they have been the subject of considerable study. In the ulticarrier context, Chow [] epirically discovered that as long as the optial bandwidth is used, a constant-power allocation has a negligible perforance loss copared to true waterfilling. The sae phenoenon is observed in the adaptive This wor was supported by a Stanford Graduate Fellowship and by France Teleco. In this paper, log is used to denote logarith of base ; ln is used to denote logarith of base e. odulation setting []. There has been several perforance bounds on constant-power water-filling reported in the literature. Aslanis [3] copared the worst case difference between a true water-filling and a constant-power water-filling, and derived a bound based on the SNR cutoff value. Schein and Trott [4] derived a different bound also based on SNR. The current wor extends the existing results in several directions. First, a worst-case perforance bound is derived using a novel approach based on convex analysis, and the bound is valid for SNR. Secondly, it is shown that the new perforance bound can be used to design a very-low coplexity power allocation algorith with a bounded worst-case perforance. In particular, the algorith is shown to be at ost 0.66 bits/sec/hz away fro capacity on a Rayleigh channel, and it often perfors uch closer to capacity in practice. The rest of the paper is organized as follows. Section II forulates the water-filling proble, and derives the new bound. Section III proposes a new low-coplexity power adaptation algorith. Section IV applies the bound to the Rayleigh fading channel. Siulation results are presented in Section V, and conclusions are drawn in Section VI. II. Sub-optial Water-filling A. Proble Forulation We choose to forulate the proble in the adaptive odulation fraewor because it is slightly ore general than the ulticarrier setting. The counication channel is odeled as: Y i = νi Xi+Ni, where i is the discrete tie index, Xi and Y i are scalar input and output signals respectively, Ni is the additive white Gaussian noise, which is independent and identically distributed with a constant variance σ,and νi is the ultiplicative channel fading coefficient. For siplicity, νi, the squared agnitude of the fading coefficient, is assued to be independent and identically distributed with a probability distribution ρν. The capacity for this fading channel under an average transit power constraint when both the transitter and the receiver have perfect and instantaneous channel side inforation was characterized by Goldsith and Variaya [5]. They proposed a water-filling-in-tie solution and proved

a coding theore based on finite partitions of channel fading statistics, i.e. ν is restricted to tae finite values ν,ν, ν, with probabilities p,p, p.inthis case, the axiization proble becoes: ax p log + S S σ s.t. p S S, 3 S 0, 4 where S is the average transit power constraint, and the axiization is over all power allocation policies S based on the instant channel fading state. Letting p = reduces the proble to the ulticarrier setting. The solution to this optiization proble is the wellnown water-filling procedure. Our interest is in finding approxiate solutions with provable worst-case perforance. B. Duality Gap The optiization proble belongs to the class of convex prograing probles, where a convex objective function is to be iniized subject to a convex constraint set. A general for of a convex proble is the following: in x f 0 x 5 s.t. f i x 0, 6 where f i x, i =0,, are convex functions. f 0 x is called the prial objective. The Lagrangian of the optiization proble is defined as: Lx, λ =f 0 x+λ f x+ λ f x, 7 where λ i are positive constants. The dual objective is defined to be gλ =inf x Lx, λ. It is easy to see that gλ is a lower bound on the optial f 0 x: f 0 x f 0 x+ λ i f i x 8 i inf f 0 z+ λ i f i z 9 z i gλ. 0 So, gλ in f 0 x. x This is the lower bound that we will use to investigate the optiality of approxiate water-filling algoriths. The difference between the prial objective f 0 x and the dual objective gλ is called the duality-gap. A central result in convex analysis [6] is that when the prial proble is convex, the duality gap reduces to zero at the optiu. C. Lower Bound The above general result is now applied to the waterfilling proble. First, axiizing the data rate is equivalent to iniizing its negative. The capacity is a concave function of power, so its negative is convex. The constraints are linear, so they are convex as well. Associate dual variable λ to the power constraint, and µ to each of the positivity constraints on S, the Lagrangian is then: LS,λ,µ = p log + S σ [ ] +λ p S S + µ S. The dual objective gλ, µ is the infiu of the Lagrangian over prial variables S. At the infiu, the partial derivative of the Lagrangian with respect to S ust be zero: L /σ =0= p S +S /σ ln + λ p µ, 3 fro which the classical water-filling condition S + σ = λ µ /p ln. 4 is obtained. This condition, together with the constraints of the original prial proble, the positivity constraints on the dual variables, and the so-called copleentary slacness constraints, for the Karush-Kuhn-Tucer KKT condition, which is sufficient and necessary in this case. Substituting the water-filling condition 4 into gives the dual objective: ν /σ p log gλ, µ = λ µ /p λp µ σ ln λ S + ln 5 The dual objective is always convex, and it is a lower bound to the prial objective for all positive λ and µ. In particular, substituting the dual variables as in 4 gives the following duality gap Γ, which is defined as the difference between the prial and the dual objectives: σ / Γ= p S + σ + λ / ln S ln. 6 gλ, µ is a lower bound to the iniization proble. So gλ, µ is an upper bound to the rate axiization proble. To avoid notational inconvenience, the rest of the paper will be speaing of only the duality gap.

To express the gap exclusively in prial variables S, a suitable λ needs to be found. A sall λ is desirable because it aes the duality gap sall. Since S + σ / ln = λ µ, 7 p and recall that λ and µ need to be non-negative, the sallest non-negative λ is then λ =ax S + σ / ln = in{s + σ / } ln. 8 Assue that the approxiate water-filling algorith satisfies the power constraint p S S with equality 3, the above gives the following: Γ= ln p S in{s j + σ /ν j } j S S + σ / 9 The preceeding developent is suarized in the following theore: Theore : For the optiization proble, if S 0 is a power allocation strategy that satisfies the power constraint with equality, then the achievable data rate using S is at ost Γ bits/sec per Hz away fro the optial water-filling solution, where Γ is expressed in 9. This result is a general bound to all approxiate water-filling algoriths. For exaple, it can be used to bound the perforance of power allocation strategies with integer-bit constraint 4. It is clear that if exact waterfilling is used, i.e. when S +σ / is a constant whenever S > 0, the gap reduces to zero. Therefore, the cost of not doing water-filling is the decrease in the denoinator in the second ter. The siplicity of the above expression aes it quite useful in deriving new results, as it shall soon be seen. D. Constant Power Adaptation We now turn our attention to the particular class of constant-power adaptation algoriths. As entioned before, log + SNR is ore sensitive to SNR when SNR is low. So, it aes sense that the critical tas in waterfilling should be to ensure that low SNR subchannels are allocated the correct aount of power. In particular, those subchannels that would be allocated zero power in exact water-filling should not receive a positive power in an approxiate water-filling algorith, for otherwise, the power is alost wasted. This intuition allowed Chow 3 When the power constraint is not satisfied with equality, S ust be used in the second su in 9 instead of p S. 4 Equation 9 can be used to show that integer-bit restriction costsatost/ ln bits/sec per Hz by noticing an integer bit allocation algorith essentially doubles S + σ / in allocating each additional bit. Unfortunately, this bound is rather loose. [] to observe that a constant-power allocation strategy, where the transitter allocates zero power to subchannels that would receiver zero power in exact water-filling, but allocates constant power in subchannels that would receive positive power in exact water-filling is often close to the optial. In this section, this intuition will be ade precise using the gap bound derived before. Consider the following class of constant-power allocation strategies where beyond a cut-off point, ν 0, all subchannels are allocated the sae power: { S0 if ν S = ν 0. 0 0 if <ν 0 Here, the subchannels are assued to be ordered so that ν l whenever l. If the sae cut-off point ν 0 is used as in exact water-filling, we have, σ S 0 + in σ in ν 0 ν 0 <ν 0 σ. The first inequality is true because in the transission band i.e. when ν 0 the constant-power allocation is an suboptial strategy, therefore the inial su of power and noralized noise is less than the water level σ /ν 0. The second inequality holds because the subchannels are ordered. Equation allows us to replace the in {S + σ / } ter in the gap forula by S 0 +in {σ / }. In this case, 9 becoes: S 0 ln Γ = p S 0 +in j {σ /ν j } S 0 S 0 + σ / S 0 σ / in j {σ /ν j } = p S 0 + σ / S 0 +in j {σ /ν j } σ / p S 0 + σ, / where denotes the nuber of channel states with positive power allocation. Notice that an iediate constant σ / bound can be obtained by replacing S 0 + σ with. / In this case, Γ / ln =.44 bits/sec/hz is an upper bound to the axiu capacity loss for constant-power allocation algoriths. But this is usually too loose to be of practical interest. Instead, we can siplify the notation using the fact that the nuber of bits allocated in each subchannel is given by log +S 0 /σ. In this case, Γ can be written in a particularly siple for: Γ p b, 3 ln where b is the nuber of bits allocated in each subchannel. Note that b are not restricted to integer values in the above bound. Also note that the crucial assuption for the bound to hold is in {S + σ / } =

P A in σ v Fig.. S 0 B * σ v Constant-Power Allocation 0 σ v S 0 +in {σ / }. Having the sae cut-off point as in exact water-filling is a sufficient but not necessary. Thus, we have the following theore. Theore : For a constant-power allocation strategy of the for 0 that satisfies the power constraint with equality, if in {S + σ / } = S 0 +in {σ / },then it is at ost p b / ln bits/sec/hz away fro the water-filling optial, where the su is over all subchannels that are allocated S 0 aount of power, and b is the nuber of bits allocated in subchannel, i.e. b =log+s 0 /σ. Fig. illustrates the theore graphically. As long as level A is lower than level B, the achievable rate is bounded by 3. Note that subchannels with low SNR and hence low bit allocation are precisely those contributing ost to the bound, thus confiring the intuition that low SNR subchannels are the ost sensitive to power is-allocation. III. Low Coplexity Adaptation The crucial condition in Theore is in {S + σ / } = S 0 +in {σ / }. This condition says that the bound is valid only if not too few subchannels are used. The condition is trivially satisfied, for exaple, by putting equal power in all subchannels. In that case, b will be nearly for any subchannels, and the duality gap becoes large although still bounded by the constant.44 bits/sec/hz. Therefore, it is of interest to use as few subchannels as possible without violating the condition so as to siultaneously ae the nuber of ters in the suation sall, and ae each individual ter sall since fewer subchannels iplies larger S 0,whichin ter iplies saller b. This suggests that a siple power allocation strategy which sets the cut-off point to be the largest that satisfies S 0 + σ /ν σ /ν + is close to the optial. Graphically, an algorith that tries to find the sallest so that level A is less than level B has the sallest duality gap. This fact is used to devise the following algorith: Algorith : Assue that the channel gain s are ordered so that ν ν ν.letν 0 be the cut-off point so that a constant power S 0 is allocated for all ν 0. Let be the largest such that ν 0. The following steps find the with the sallest duality gap:. Set =.. Copute S 0 = S/ p. 3. If σ /ν + S 0 + σ /ν,set =, repeat step. Otherwise, set = + and go to the next step. 4. Copute b =log+s 0 /σ for =,,. Then, R = p b is at ost p b / ln bits/sec/hz away fro capacity. Two properties of this algorith ae it attractive. First, unlie ost previous low coplexity bit-loading ethods e.g. [], where the boundary point is found by finding the cut-off point that gives the highest data rate, this algorith finds the optial cut-off point without actually coputing the data rate achieved in each step, and is therefore free of logarithic operations. The ost expensive operation in this algorith is the single division in each step, thus aing its coplexity very low. Secondly, this algorith has a provable worst-case perforance bound as given by Theore. Finally, we note that a binary search of the cut-off point can be used to further iprove the algorith s efficiency. IV. Rayleigh Channel The bound developed previously can be explicitly coputed if channel fading statistics are nown. In particular, for a Rayleigh fading channel, it can be shown that the constant-power adaptation strategy is only a sall fraction of one bit away fro capacity. In a wireless channel where a large nuber of scatterers contribute to the signal at the receiver, application of the central liit theore leads to a zero-ean coplex Gaussian odel for the channel response. The envelope of the channel response at any tie instant has a Rayleigh distribution, whose square agnitude is exponentially distributed, p ν ν = Ω e ν/ω,whereω,theaverage channel gain, paraeterizes the set of all Rayleigh distributions. Fixing Ω, the constant-power control strategy is deterined by the average power constraint, or alternatively by the cut-off value ν 0. The low coplexity power allocation algorith says that the constant power allocated in each state S 0 should be such that { } σ S 0 +in = σ. 4 ν ν 0

The Rayleigh distribution has a non-zero probability for arbitrarily large aplitudes of ν, so the above reduces to S 0 = σ /ν 0. Curiously, note that the constant-power allocation algorith allocates a constant power S 0 to all subchannels that can support at least one bit/second/hz with S 0. Now, using the gap bound, the spectral efficiency for an optial constant-power allocation with cut-off ν 0 is bounded within the following constant fro capacity: Γν 0 = ln ν 0 σ /ν σ /ν 0 + σ /ν Ω e ν/ω dν 5 By a change of variable t = ν/ω andalsot 0 = ν 0 /Ω, define t 0 e t ft 0 = dt, 6 t 0 t + t 0 the duality gap can be expressed as Γν 0 = ln f v0. 7 Ω The authors are not aware of a closed-for expression for the integral. Nuerical evaluation reveals that it has a single axiu occurring at about t 0 =0.39, and the value of the axiu is about 0.840. The duality gap is largest when the power constraint is such that the cutoff point ν 0 = 0.39Ω. In this worst case, the average data rate is.363 bits/sec/hz, and the duality gap is 0.840/ ln 0.66 bits/sec/hz away fro capacity. The following theore suarizes the result. Theore 3: For a flat i.i.d. Rayleigh fading channel with perfect side inforation at the transitter and the receiver, assuing infinite granularity on the channel state partition, a constant-power adaptation ethod should allocate S 0 to all subchannels that could support at least one bit with S 0,whereS 0 is deterined fro the power constraint. In this case, the resulting spectral efficiency is at ost 0.66 bits/sec/hz away fro capacity. V. Siulation Siulation results on the Rayleigh channel are now presented. The average channel gain Ω is chosen to be -0dB. In Fig., the average spectral efficiencies of the exact water-filling and the low-coplexity constantpower allocation are plotted against the average power constraint together with the duality-gap bound. The average power constraint is noralized by setting noise power σ =0dB. The two curves are indistinguishable. For Rayleigh channels, the constant-power allocation ethod perfors even better than the bound suggests, and it has a truly negligible loss copared to the exact water-filling. Note that the constant-power allocation ethod is designed using the bound. So, while the bound could be loose, the algorith designed using the bound wors very well. Average Spectral Efficiency bps/hz 6 5 4 3 Exact Waterfilling Constant Power Upper Bound 0 0 5 0 5 0 5 30 Average Power Constraint db Fig.. Spectral efficiency of exact water-filling and constant-power allocation on Rayleigh channel with Ω=-0dB. VI. Conclusion Approxiate power adaptation algoriths are investigated in this paper. A rigorous perforance low bound for sub-optial power allocation is derived. A verylow coplexity constant-power adaptation ethod is proposed using the bound derived. The low-coplexity algorith has the desirable properties of having a provable worst-case perforance and being logarith-free. The perforance bound is applied to Rayleigh fading channels, and it is shown that constant-power adaptive odulation is at ost 0.66 bits/sec/hz away fro capacity. Siulation results suggest that the actual gap is even saller. References [] P. S. Chow, Bandwidth optiized digital transission techniques for spectrally shaped channels with ipulse noise, Ph.D. thesis, Stanford University, 993. [] A. J. Goldsith and S. Chua, Variable-rate variable-power MQAM for fading channels, IEEE Transactions on Counications, pp. 986 99, Nov. 997. [3] J. T. Aslanis, Coding for Counication Channels with Meory, Ph.D. thesis, Stanford University, 989. [4] B. Schein and M. Trott, Sub-optial power spectra for colored gaussian channels, in International Syposiu on Inforation Theory ISIT, 997. [5] A. Goldsith and P. Varayia, Capacity of fading channel with channel side inforation, IEEE Transactions on Inforation Theory, vol. 43, no. 6, pp. 986 99, Nov 997. [6] S. Boyd and L. Vandenberghe, Convex optiization with engineering applications, 998, Course Notes, EE364, Stanford University.