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1 Discrete Adative Transmission for Fading Channels Lang Lin Λ, Roy D. Yates, Predrag Sasojevic WINLAB, Rutgers University 7 Brett Rd., NJ- fllin, ryates, sasojevg@winlab.rutgers.edu Abstract In this work we address otimal adative transmission olicies in slowvarying wireless environments. Continuous rate and ower assignments that maximize the average caacity for these channels have been derived reviously. Nevertheless, from a ractical oint of view, use of a finite number of ower and rate levels is imerative. Here, we address the maing from channel states of arbitrary distribution to a discrete set of ower level and code rate airs. Unlike earlier work, our design does not require that the transmitter knows the current channel state; one of L quantization levels is sufficient. For the roosed aroach, the limiting case of large L yields Goldsmith's well-known water-filling in time result. I. Introduction In the third generation cellular systems (see [] and []), adatation on the transmitter side is one of the core technologies leading to ower efficient design of high-seed wireless data communication systems. A tyical assumtion in the design of such systems is that the channel resonse varies slowly and the seminal work [] shows that the ergodic caacity of a slowly fading channel can be achieved using an adative coding scheme with multile codebooks each emloyed for a different channel state. Otimal adative transmission olicies [] require the knowledge of the current channel state at both the receiver and the transmitter. Furthermore, both ower and code rate assignments need to adat continuously to changes in the channel state. Both of these assumtions are widely adoted in further work on information theoretic asects of communication over fading channels [, ]. Unfortunately, in ractice it is difficult to deloy a system satisfying these two assumtions. In this aer, we roose a framework to exlore the information theoretical caacity, the highest achievable average rate, of finite-level discrete adatative transmission where the channel state sace is quantized to a finite number of intervals such that a transmitter will ick u a air from a finite set of code rate and ower level airs corresonding to the quantization level of the current channel state and transmit accordingly. Besides roosing a framework to exlore the caacity of finite-level designs, we also exlore the caacity-achieving olicies. It is demonstrated in this aer that finding an otimal olicy whichachieves the caacityisanintractable non-convex otimization roblem. One way of finding the Λ corresonding author otimal olicy is brute force maximization and its comlexity increases exonentially in the number of levels. In this aer, an convergent iterative algorithm is resented to numerically evaluate the lower bound of the caacity. It is shown that this algorithm roduces lower bounds close to the true caacitywhen the number of levels is small. Moreover, it enables us to see how effective the discrete adative transmission is, where brute force aroach may not feasible. II. System Model In this aer, we consider the multilicative flat fading channel model with comlex received signal Y = S + W; () where S is the channel fade state, is the comlex onedimensional transmitted signal, W is a circularly symmetric additive white Gaussian noise (AWGN) with variance N. The fading state S is a real random variable of unit mean with a robability density function f(s), distribution F (s), and domain S = fsjs g. Following [], it is assumed that the fading is slow and that the channel state is constant during the transmission of a single codeword so that the channel model () is sufficient for our uroses. The roosed adative transmission system quantizes anychannel state s to one of L levels v v v L, where v =. The L channel state quantization intervals are denoted by U l =[v l ;v l+ ) for l =; ;L ; where formally v L =. Note that the set fu ; ;U L g artitions the channel state sace S. When the channel is in state s U l, the transmitter generates codewords of code rate r l and transmits them at ower level l. As our channel is an AWGN channel for a fixed channel state s, the maximum mutual information is given by log( + l s=n ). Adoting the convenient notation R(x) = log( + x=n ), the maximum mutual information is R( l s). The set of quantization levels fv l g and the corresonding setofower and rate assignment airs f( l ;r l )g define the adative transmission olicy. More recisely, the trile of L element vectors (; r; v) =([ ; ; L ] > ; [v ; ;v L ] > ; [r ; ;r L ] > ) secifies the transmission olicy. As will be elaborated in the following sections, the transmission olicy characterizes

2 anumber of communication arameters including achievable average throughut, outage robability, and average ower. We note that given that s U l, the ower and rate assignment l and r l,andwe effectively have a non-adative olicy. Under the assumtions on our channel fading model, the fading is non-ergodic during the transmission of one codeword and here, we can use the established (see, e.g., []) outage robability definition for non-ergodic channels to define the conditional outage robability P out (r l ; l jl) =P [R( l s) r l js U l ]=P [R( l s) r l jl] In other words, given s U l, P out (r l ; l jl) is the robability that the channel state s U l is such that the maximum mutual information R( l s) is smaller than the desirable rate r l and hence, the assigned rate r l is not achievable using the ower level l. Given an L-level olicy (; v; r); the average outage robability (averaged over channel states s S)is P out (; v; r) = L P [s U l ] P out (r l ; l jl) When l = for all l, the average outage robability is equal to the achievable decoding error robability. Here, we adot the following convention when l = (and consequently r l = ) for some l, then there is no decoding error and no outage for s U l since no transmission is attemted, although the average throughut is affected. Based on the conditional outage robabilitywe define the average information rate. Our definition hinges on the assumtion that no information is successfully received when an outage occurs. In other words, if the channel state s is such that the assigned rate r l is not achievable and, therefore, communication is not reliable, Consequently, given an L-level olicy (; v; r), the average data rate (throughut) is defined as R L (; v; r) = L P [s U l ][ P out (r l ; l jl)] r l () Emloying the notation F (s ;s ) = F (s ) F (s ) = P [s» Ss ], the average ower for the olicy (; v; r) is ρ(; v; r) = L F (v l ;v l+ ) l () For an average ower constraint, the set of ossible L- level transmission olicies is ß L () =f(; v; r)j ρ(; v; r)» g () We can now define the caacity of the L-level adative system as max R L (; v; r) () (;v;r)ß L() Finding the otimal L-level olicy ( Λ ; v Λ ; r Λ ) that achieves caacity C L is, in the general case, a non-convex otimization roblem. s v q v q v (q ) v q v Fig.. Illustration of an arbitrary ower olicy assignment, rate assignment is not exlicitly shown. The shaded regions are the channel outage intervals. III. Proerties of Otimal Policies Couled with analytical methods, we resent sketches of ower allocation as a function of channel state serve to illustrate a number of interesting roerties of otimal olicies. One ossible olicy is deicted in Fig., where the heights of boxes indicate the ower levels assigned to their resective adatation intervals. Within each U l, the shaded region indicates states s U l for which theassigned rate r l does not ermit reliable communication. Note, that any outage interval is contiguous within its resective quantization interval U l and would include the left end oint v l of the interval U l. Thus, there can be at most L outage intervals. In the following, we develo an imlicit characterization of the assigned rates for an otimal olicy. Lemma For an otimal olicy ( Λ ; v Λ ; r Λ ) we have that r Λ l R( Λ l v Λ l );R( Λ l v Λ l+) () Since we are only interested in otimal olicies, we will, henceforth, assume that any olicy of interest (; v; r) also satisfies condition (). Consequently, for any olicy (; v; r) for which () holds and any quantization interval U l, continuity ofr( ) imlies we canfindachannel state q l U l such thatr l = R( l q l ). This defines a one to one maing between channel states fq ; ;q L g and the resective rate assignments fr ; ;r L g for a given ower olicy. Thus it will be more convenient to redefine transmission olicies of interest as vector triles (; v; q). Fig. is alotofsuch a (; v; q) olicy and there is no outage in an interval [v l ;v l+ ) or only a ortion of the interval [v l ;v l+ ) in outage. Now, we can rewrite the conditional outage robability as follows P out (r l ; l jl) =P [R( l s) R( l q l )js U l ] = P [s q l js U l ] (7) Using (7) and (), the average rate can be exressed in terms of the vector q as follows R L (; v; q) = L F (q l ;v l+ )R( l q l ) () Due to the length of this aer, we introduce all the lemmas and theorems without any roof excet the Theorem.

3 no outage, ower silled over th revious level s v q v (q ) v (q ) v (q ) v Fig.. The olicy achieves C L will have q = v and q = v. The otimal olicy (; v; q) achieves the average caacity max R L (; v; q); (9) (;v;q)ß L() where the definition of ß L () is given by (). From () and Fig., we again note that the outage region does not contribute to overall average rate. Thus, one could intuitively assume that the otimal olicy should minimize this region. Solving the non-convex otimization roblem (9) is hard. However, further characterization of the otimal solution is helful in simlifying the search for an otimal olicy. The following theorems describe roerties of an otimal solution. Lemma If R L (; q; v) =C L, L l= F (v l ;v l+ ) l = () In general, the average throughut R L is a function of both v and q, nevertheless, Theorem shows that there must be an otimal olicy for which v l = q l for all l>. Theorem If R L (; v; q) = C L, then there exists a olicy ( ; v ; q) such that q l = v l for all l > and R L ( ; v ; q) =C L. The relevance of Theorem is that it demonstrates that olicies of interest are fully determined by the vectors and q. For such olicies, the average throughut becomes R L (; q) = L while the average transmitted ower is F (q l ;q l+ )R( l q l ) () L ρ(; q) =F (;q ) + F (q l ;q l+ ) l () The otimal olicy air ( Λ ; q Λ ) is that which achieves max R L (; q); () (;q)ß L() where the set of L-level olicies of interest is now ß L () =f(; q)jρ(; q) =g () Although the new otimization roblem () is much simler, it remains non-convex and hard to solve in the general case. The next theorem shows that given any adatation artition, defined now by q, the otimal ower assignment is a water-filling olicy. Theorem Given q and >, the otimal ower allocation is l = F (q;q) F (;q ) N N q l + l> q + l = () where is the water-filling level that can be determined from () for any given and q. IV. Otimal Policy Comutation Finding an otimal olicy ( Λ ; q Λ )whichachieves the caacity in () is still an intractable non-convex otimization roblem. One way of finding the otimal olicy is brute force maximization of (). This aroach entails quantization of continuous olicy variables and q and can be taken for a small number of quantization levels L since its comlexity increases exonentially in L. Assuming the cdf F (s) is a strictly increasing function of s, instead of otimizing a olicy (; q), we roose an iterative algorithm which finds the equivalent olicy (; a); where a l = F (q l ) for all l based on the water-filling ower allocation (Theorem ) and a quantization otimization rocess, water-silling. The algorithm is as follows.. Choose initial owers l = μ and interval boundaries a ==L and a l = l=l for l =; ;L. Here, the average rate is zero over U as it is in outage.. (water-filling) Fix a and find based on the water-filling assignment of Theorem.. (reartitioning) If = and m is the first non-zero ower assignment, a Λ l = Λ l = a m =; l = a l+m ; l =; ;L m a L +( a L ) l m ow L m ; ; l = l+m ; l =; ;L m L ; ow Then, let new a l = a Λ l and l = Λ l for all l.. (water-silling) For l =; ; ;L Hold ; ; l ; l ; ; L and a ; ;a l ;a l+ ; ;a L constant andvary l and a l to maximize R L (; a) under the ower constraint ρ(; a) = μ. Hold and a excet a,max a (a a )R( F (a )) Λ.. Reeat stes,, and until convergence occurs where the rate increment after an iteration is less than a reset ffl. At the oint of convergence, the corresonding average rate is the lower bound of C L. This algorithm consists of three key oerations, waterfilling, reartitioning, and water-silling. The convergence follows from the fact that the average rate R L is nondecreasing during any ste in our algorithm and is uerbounded by C [].

4 7 Ergodic Caacity C.7. Pout L for C. a( a) a a a a a = Caacity (Bits/Hz/dim).. Outage Probability.. Average Received SNR (db) with N = a a a a a a a = Fig.. Caacity C L vs. outage P out with L = (no adatation) Fig.. Illustration of Ste of the reartition rocess for L =and m =. Ergodic Caacity C C C l l a l- a l a l+ Caacity (Bits/Hz/dim) Average Received SNR (db) with N = a l- a l Fig.. Illustration of Ste of the water-silling algorithm. The lower lot shows the increase in l as the boundary a l is increased. That is, as the boundary a l moves to the right, ower from interval U l =[a l ;a l+ ) sills over the boundary to fill the interval U l. The reartioning is illustrated in Fig.. At this ste, artitions with zero ower are merged and the U L is slit. This is to imrove the effectiveness of the following water-silling rocess. The water-silling rocess is illustrated in Fig.. By exressing the quantization intervals in terms of a, theav- erage ower assigned to a given interval is equal to the area of its resective rectangle. With the ower constraint μ l, in Fig., the sum of the areas of the shaded rectangles must remain the same to kee the same average transmitted ower. Consequently, when we shift a l to the right (or, equivalently increase the rate assignment r l ), ower sills from interval l to raise the ower l so that the shaded area stays constant. Thus we call the above algorithm water-silling. a l+ Fig.. C L with L =; ;. It can be roved that if the fading cdf F (s) satisfies F (s) + F (s) ; s S () [F (s)] the objective function in the water-silling ste will be convex. V. Numerical results In this section, we resent numerical results of C L and lower bounds which obtained by brute force and the roosed iterative algorithm, resectively. We comare the C L and lower bounds with the ergodic caacity C for Rayleigh fading. The most imortant oint is that C L of small L are very close to C. For a Rayleigh fading channel, the cdf is F (s) = e s s S (7) In this case, () holds and the convexity condition holds. We numerically evaluate C L and P out with L». In Fig., it is shown for a non-adative system with L =

5 Caacity (Bits/Sec/Hz/dim) 7 C C C C C C Average SNR (db) Fig. 7. C L with L». that there is a to 7 db ga between the ergodic caacity curve and the otimal non-ergodic throughut. The outage robability corresonding to this otimal solution ranges from 7% to %. It confirms there is a significant enalty on the caacity and a high outage robability when using a non-adative communication system in a fading environment. In articular, when the received ower is low, the outage robability is extremely high which imlies that the otimal q is quite large. Hence, most of time, the channel is oor and the channel state sq. Fig. shows that by alying an L = level adative transmission olicy, the required SNR for the same caacity can be reduced by aroximately db in comarison with using the non-adative olity. In other words, a - level adative system can eliminate about half of the SNR ga between the caacity curves of the ergodic case and the non-adative case. Furthermore, increasing L from to yields another db reduction in the SNR requirement. Note, C and C are obtained from exhaustive search. As exhaustive search becomes rohibitive for increasing L, we use the heuristic iterative algorithm to lower bound C L for large L. In Fig. 7, we resentlower bounds of C L for L». There is no ercetible difference between C L and the lower bounds roduced by the heuristic algorithm for L. Moreover, with these lower bounds, we note that L =levels, the caacity of our discrete design could be within db from the ergodic caacity when SNR is less than db. VI. Conclusion In this work, we otimize the design of an adative transmission system with a discrete set of ower levels and code rates for wireless fading environments. We show that our design rocedure yields results close to the well-known water-filling result []. Aendix Proof Theorem Suose v l+ q l+ for some» ll. We will construct a new olicy ( ; v ; q) ß L () such that v l+ = q l+ and R L ( ; v ; q) C L. If there is more than one l such thatv l+ q l+,we can reeat the same construction for each value of l. Let ^R = i=l F (q i ;v i+ )R(q i i ) () denote the contributions to R L from channel states s U l. Since R() =, we can write R L (; v; q) = ^R + F (ql ;v l+ )R(q l l )+F (v l+ ;q l+ )R() = ^R + F (ql ;q l+ )[ffr(q l l )+( ff)r()] where ff = F (q l ;v l+ )=F (q l ;q l+ ) satisfies» ff». Since R( ) isaconcave function, we observe that R L (; v; q)» ^R + F (ql ;q l+ )R(q l ff l ) (9) The right side of (9) can be achieved by a olicy in which weuseower l for channel states s [v l ;q l )andower ff l when s [q l ;q l+ ). By defining fi = F (v l ;q l )=F (v l ;q l+ ), the average ower over the states s [v l ;q l+ )is ^ = fi l +( fi)ff l ff l Since ^ ff l,we observe that (9) and the fact that R( ) is an increasing function imly R L = ^R + F (ql ;q l+ )R(q l ^) R L (; v; q) Furthermore, the transmission olicy ( ; v ; q) with ρ vi ql+ i = l + = v i i = l + ρ ^ i = l i = i = l i achieves the average rate R L ( ; v ; q) =RL while meeting the average ower constraint ρ( ; v )». References [] F. Adachi, M. Sawahashi, and H. Suda, Wideband DS-CDMA for next-generation mobile communications systems," IEEE Commun. Mag., vol., no. 9,. 9, Se. 99. [] K. Balachandran, R. Ejzak, S. Nanda, S. Vitebskiy, and S. Seth, GPRS- high-rate acket data service for North American TDMA digital cellular systems," IEEE Personal Commun., vol., no.,. 7, Jun [] A. Goldsmith and P. Varaiya, Caacity of fading channels with channel side information," IEEE Trans. Info. Theory, vol., no.,. 9 99, Nov [] D. Tse and S. Hanly, Multi-access fading channels Part I Polymatroid structure, otimal resource allocation and throughut caacities," IEEE Trans. Info. Theory, vol., no. 7,. 79, Nov. 99. [] G. Caire, G. Taricco, and E. Biglieri, Otimum ower control over fading channels," IEEE Trans. Info. Theory, vol., no.,. 9, Jul [] E. Telatar, Caacity ofmulti-antenna Gaussian channels," Euroean Trans. Telecomm., vol., no.,. 9, Nov.-Dec. 999.

4. Score normalization technical details We now discuss the technical details of the score normalization method.

4. Score normalization technical details We now discuss the technical details of the score normalization method. SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules