Successive Refinement via Broadcast: Optimizing Expected Distortion of a Gaussian Source over a Gaussian Fading Channel

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1 Succeive Refinement via Broadcat: Optimizing Expected Ditortion of a Gauian Source over a Gauian Fading Channel Chao Tian, Member, IEEE, Avi Steiner, Student Member, IEEE, Shlomo Shamai (Shitz, Fellow, IEEE and Suha N. Diggavi, Member, IEEE Abtract We conider the problem of tranmitting a Gauian ource on a lowly fading Gauian channel, ubject to the mean quared error ditortion meaure. The channel tate information i known only at the receiver but not at the tranmitter. The ource i aumed to be encoded in a ucceive refinement manner, and then tranmitted over the channel uing the broadcat trategy. In order to minimize the expected ditortion at the receiver, optimal power allocation i eential. We propoe an efficient algorithm to compute the optimal olution in linear time O(M, when the total number of poible dicrete fading tate i M. Moreover, we provide a derivation of the optimal power allocation when the fading tate i a continuum, uing the claical variational method. The propoed algorithm a well a the continuou olution i baed on an alternative repreentation of the capacity region of the Gauian broadcat channel. Index Term Broadcat trategy, joint ource-channel coding, power allocation, ucceive refinement. I. INTRODUCTION Fading channel occur naturally a a model in wirele communication. For low fading, the receiver can uually recover the channel tate information (CSI accurately, however the tranmitter only know the probability ditribution of CSI, but not the realization. Such uncertainty can caue ignificant ytem performance degradation, and the broadcat trategy wa ued in [], [] a an approach to combat thi detrimental effect. In thi trategy, ome information can only be decoded when the fading i le evere, which i uperimpoed on the information that can till be decoded when the fading i more evere. Thu the receiver can decode the information adaptively according to the realization of the channel tate. The imilarity to the degraded broadcat channel [3] (ee alo [4] i clear in thi context, particularly for channel with a finite number of fading tate. Generalizing thi view, when The work by C. Tian and S. Diggavi wa upported in part by Swi National Competence Center on Mobile Information and Communication Sytem (NCCR-MICS. The reearch by A. Steiner and S. Shamai wa upported by the EU, 7th framework program via NEWCOM++. The material in thi paper wa preented in part at 7 IEEE Information Theory Workhop on Information Theory for Wirele Network, Jul. 7, Bergen, Norway. C. Tian wa with School of Computer and Communication Science, Ecole Polytechnique Federale de Lauanne, Lauanne, CH5, Switzerland. He i now with AT&T Lab Reearch, 8 Park Avenue, Florham Park, NJ 793. A. Steiner and S. Shamai are with the Department of Electrical Engineering, Technion Irael Intitute of Technology, Haifa 3, Irael. S. Diggavi i with the School of Computer and Communication Science, Ecole Polytechnique Federale de Lauanne, Lauanne, CH5, Switzerland. the fading gain can take continuou value, the receiver can be taken a a continuum of uer in a broadcat channel. The broadcat trategy naturally matche the ucceive refinement (SR ource coding framework [5] [7], a the information decodable under the mot evere fading i protected the mot, and hould be ued to convey the bae layer information in the SR ource coding. A more information can be decoded when the channel i ubject to le fading, more SR encoded layer can be decoded, and the recontruction quality improve. In thi work, we conider thi cheme for a quadratic Gauian ource on a ingle input ingle output (SISO channel. In order to minimize the expected ditortion at the receiver, it i eential to find the optimal power allocation in the broadcat trategy, and thi i indeed our focu. It i worth noting that though in [] the objective function to be maximized i the expected rate, the cro layer deign approach of combining SR ource coding with broadcat trategy wa in fact uggeted (though not treated in that work. Initial effort on thi problem wa made by Seia et al. in [8], where the broadcat trategy coupled with SR ource coding wa compared with everal other cheme. The optimization problem wa formulated by dicretizing the continuou fading tate, and an algorithm wa devied when the ource coding layer are aumed to have the ame rate. Thi algorithm, however, doe not directly yield the optimal power allocation when the fading tate are dicrete and pre-pecified, nor doe it give a cloed-form olution for the continuou cae. Etemadi and Jafarkhani alo conidered thi problem in [9], and provided an iterative algorithm by eparating the optimization problem into two ub-problem. In two intereting recent work [], [], Ng et al. provided a recurive algorithm to compute the optimal power allocation for the cae with M poible fading tate, with wort cae complexity of O( M ; moreover, by directly taking the limit of the optimal olution for the dicrete cae, a olution wa given for the continuou cae optimal power allocation, under the aumption that the optimal power allocation i concentrated in a ingle interval. Similar problem were conidered in [] [5] in the high SNR regime from the perpective of ditortion exponent. Our contribution in the preent work i two-fold: firtly, we propoe a new algorithm that can compute in linear time, i.e., of O(M complexity, the optimal power allocation for the cae with M fading tate; econdly, we provide a derivation of the continuou cae optimal power allocation olution by

2 the claical variational method [6]. Our derivation for the continuou cae olution i more general than that in [] a it remove the retriction that the optimal power allocation i concentrated in a ingle interval. Both the algorithm and the derivation rely on an alternative repreentation of the Gauian broadcat channel capacity, which appeared in [7]. The dual problem of minimizing power conumption ubject to a given expected ditortion contraint i alo dicued. Several pecific example are given a illutration. The broadcat trategy coupled with SR ource coding can be conidered a a ource-channel eparation approach with optimized cro-layer reource allocation. More explicit joint ource-channel coding approache, for example thoe in [3], [8] [], may provide better performance in the cenario being conidered, however thi apect i beyond the cope of the preent work. We neverthele note that the algorithm propoed in the current work i extremely efficient, which make it poible to ue the broadcat trategy coupled with SR coding a a benchmark for future invetigation into thi joint ource-channel coding problem. Thi i indeed ueful ince an outer bound which i non-trivial yet imple to compute i o far lacking depite extenive reearch (ee [] and the comment therein. Moreover, a pointed out recently by Steinberg [], in certain application uch a medical imaging, it i important to enforce the requirement that the recontruction i a determinitic function of the ource obervation (the common knowledge requirement, and under thi requirement, the broadcat trategy coupled with SR ource coding i in fact optimal for the Gauian ource and channel etting conidered in thi work. The ret of the paper i organized a follow. In Section II we give the ytem model and ome preliminarie, and in Section III the new algorithm i provided and it optimality i proved; the dual problem of minimizing the power conumption ubject to an expected ditortion contraint i alo conidered. In Section IV we give the derivation for the continuou cae olution, and Section V ome pecial cae of fading ditribution are conidered. Finally Section VI conclude the paper. II. SYSTEM MODEL AND PRELIMINARIES We aume a memoryle ource {X i } i generated independently and identically according to a zero-mean unit variance real-valued Gauian ditribution. The channel i given by the model Y c = hx c + N c, ( where X c i the complex-valued channel input and Y c i the channel output, h i the (random multiplicative channel fading coefficient with h =, and N c CN(, i the zero-mean complex circularly-ymmetric independently and identically ditributed (i.i.d. Gauian additive noie. We conider a lowly fading channel model, where each channel codeword conit of a length-l c channel ymbol block, and the realization of the multiplicative fading coefficient i independent acro block. Source ymbol of block length-l are encoded into a ingle channel codeword, X l Fig.. l c X c p ( i = p i R M M M R i } M R Xˆ l D = exp( b i j= The broadcat approach for minimizing the expected ditortion. and there i a ource channel mimatch factor defined a b = l c /l ; thi mimatch factor can alo be interpreted a the bandwidth expanion/compreion factor. Thi ytem i illutrated in Fig.. Each channel block i aumed to be ufficiently long to approach channel capacity, a well a the rate-ditortion limit, however till much horter than the dynamic of the lowly fading proce. It i clear that without lo of generality each channel ue in the complex domain i equivalent to two channel ue in the real domain ubject to the ame power contraint a Y r = X r + N r, ( where X r i the real-value channel input, and R i the (random channel power gain, and N r N(, i the zeromean unit-variance real-valued Gauian additive noie in the channel. From here on, we hall adopt thi equivalent channel, and it i clear that the definition of ource-channel mimatch factor b already take thi into conideration. For the cae with a finite number of fading tate, the M poible power gain in an increaing order < <... < M are ditributed according to a probability ma function p i uch that M p i =. The tranmitter ha an average power contraint P, and if power P i i allocated to the i-th layer in the broadcat trategy, the i-th layer channel rate R i i given by R i = log( + i P i M + i j=i+ P j = log( + P i / i + M j=i+ P, (3 j where we ue natural logarithm. From the econd expreion, the equivalence to broadcat on a et of channel with different noie variance i clear. Let n i / i, which implie n > n >... > n M are the equivalent noie power on the channel. The layer correponding to maller value of i (and larger value of n i will be referred to a the lower layer, which i conitent with the intuition that they are ued to tranmit the better protected lower layer of the SR ource coding. Since the Gauian ource i ucceively refinable [6], the receiver with power gain i can thu recontruct the ource i

3 3 within ditortion D i = exp( b. (4 Combining (3 and (4, the problem we wih to olve i eentially the following minimization over the power allocation (P,P,...,P M, b M i min p i P j ( + / j + M k=j+ P (5 k ubject to: P i, i =,,...,M, M P i P. When the fading tate i continuou, the denity of the power gain ditribution i then given by f(, which i aumed to be continuou, and differentiable almot everywhere. In thi cae, the tak i to find a power allocation denity function P(, or it cumulative function, which minimize the expected ditortion. III. THE NEW ALGORITHM AND ITS OPTIMALITY In thi ection, we conider the optimization problem when the channel ha a finite number of fading tate. The difficulty i that the optimization problem a defined in (5 i not convenient for computation due to it complicated form. In fact it i not immediately clear that the function being optimized i a convex function of (P,P,...,P M, neverthele it i indeed clear that the rate-region of the degraded broadcat channel i alway convex. Thu our approach i to convert the problem given in (5 into a convex problem, by utilizing another characterization of R = (R,R,...,R M for the Gauian broadcat channel capacity. Such an alternative characterization wa given in [7], and we tart by tranlating it into our notation. A. Rederivation of the equivalent repreentation The rate vector R on the boundary of the Gauian broadcat capacity region correpond to a power allocation P,P,...,P M. Solving the value of P i in term of the rate vector R give an equivalent ytem of equation, and through further implification, we have i = i mp (n i n i+ exp n m, i m j=m m =,,3,...,M, (6 where we define n M+. The RHS of the above equation i monotonically decreaing in m, hence for any non-negative rate vector R, provided that (6 i atified for m =, i.e., P = i n i+ exp n, i (n The equation given in [7] appear to have a minor typographical error that the inner ummation wa given a i Rj. there mut be a power allocation uch that (6 i atified for all m, and it i on the boundary of the capacity region. Thu we can alternatively characterize the capacity region a C = (R,R,...,R M : R m, m =,,...,M, M (n i n i+ exp n P. (7 Moreover, the function on the left hand ide of thi inequality i convex in (R,R,...,R M, which i clear by oberving it um-exponential form. We can now reformulate the optimization problem a a tandard convex programming problem: min M p i exp( b ubject to: R i, i =,,...,M, M (n i n i+ exp ( i n P. If the optimal rate allocation R can be found, then the correponding optimal power allocation can be recovered from thi olution. B. The Lagrangian formulation and the algorithm Now conider the Lagrangian form L = M p i exp( b M ν i R i M + λ (n i n i+ exp n P. (8 The Karuh-Kuhn-Tucker (KKT condition require that L R m = for the optimal olution [], which ubequently yield L R m = + λ M bp i exp( b i=m M (n i n i+ exp i=m ν m =, m =,,...,M. (9 Taking the difference between L L R m = and R m+ = give m m bp m exp( b + λ(n m n m+ exp = ν m ν m+, m =,,...,M, ( where for convenience we define ν M+. λ and ν m, m =,,...,M are the Lagrangian multiplier. Furthermore, the complementary lackne require ν m R m =, m =,,...,M; the power contraint hould be atified with

4 4 equility, or λ =. Note that the et of equation in ( alo implie the et of equation (9, and thu they are equivalent. Since the optimization problem i a convex programming problem, the KKT condition i both neceary and ufficient for an optimal olution []. Clearly if the quantity bp m κ m n m n m+ i monotonically increaing, we can et ν i = for i =,,..., M and find an explicit olution, provided that the power contraint i not violated; however thi i not true in general. Neverthele, the κ factor play an extremely important role, which in fact reveal certain inherent tructure of the layer. We oberve the following imple fact to motivate the algorithm: whenever two conecutive layer yield two κ factor that are not increaing, ome combination of the layer mut occur uch that one layer i aigned zero rate. Thi i becaue otherwie the reulting rate vector would include negative rate, which i invalid. To be more precie, two conecutive layer are aid to be combined when the higher layer i not allocated any poitive rate (power, uch that the two layer can be treated a a ingle aggregated layer with the um-probability ma, and the power gain of the lower layer. The following algorithm can be ued to find the optimal rate allocation. In the equel, when a layer i aigned zero rate, it will be called ineffective; otherwie it will be called effective. For implicity, define κ M+ =. A layer i labeled active if it i a reult of combination of layer in the immediate previou loop. An intuitive explanation i given in the next ubection; the reader are encouraged to browe the algorithm below and read the intuitive explanation firt in their initial reading. Combination of layer to reach a monotonic κ equence. a Aign n m = n m n m+ and calculate κ m for m =,,...,M. Label all the layer effective and active. Let r =. b Denote the lower effective neighbor of layer i a i, and it upper effective neighbor layer a i +. Start from i k = i, for all the a r active layer i,i,...,i ar : i If i k > and κ i κ ik : label layer i k k ineffective and combine it with it current lower effective neighbor layer j. Update p j = p j + p ik, n j = n j + n ik, a well a κ j value accordingly. Label j a active. ii If κ ik κ i + : label layer i + k ineffective and k combine it with it current lower effective neighbor layer j. Update p j, n j and κ j value accordingly. Label j a active. iii If k < a r, increment k by and return to ((bi. c If after the above loop, any layer remain active: increment r by and return to (b. Denote the number of effective layer by K. For all the effective layer i k, k =,,...,K, let exp(r ik = κ /(b+ i k /κ /(b+ i k ; in other word for all the effective = κ /(b+. Aign the layer we have exp( i k i k ineffective layer rate zero. 3 Check power conumption. a Let i ko be the lowet effective layer, and define P n = P + n iko. b Let c If λ /(b+ = K k=k o (n ik n ik+ exp ( i k P n. bp iko, ( λ(n iko n iko+ then reduce R iko by (b+ log λ, and the algorithm terminate; otherwie, label i ko ineffective (R iko =, increment k o by, update P n = P + n iko, and return to (3b. C. The correctne of the algorithm and it complexity Intuitively, the layer are claified into two kind: thoe with κ value lower than or equal to their lower neighbor (the firt kind, and thoe with κ value higher than their lower neighbor (the econd kind; ee Fig.. The algorithm combine the firt kind layer to their lower neighbor in each tep, and then continue thi operation until no layer of the firt kind exit in the reulting equence. The reulting rate allocation i valid, if the κ equence i indeed monotonically increaing, the rate are non-negative and the power contraint i atified. A few more comment are in order: In Step, we eek to form a monotonic equence of κ i by combining conecutive layer, uch that Step can provide meaningful rate. To do thi, we combine (remove all the layer that are monotonically non-increaing. Only the neighbor of thoe layer whoe κ value were updated in the immediate previou loop need to be conidered, becaue thi i the only cae that a change of claification may occur. In Step 3, we need to aure that the total power i ued up by adjuting the value of λ. However, thi ha to be done uch that the lowet layer till ha poitive rate, which i the condition in (. If thi i not poible, the lowet effective layer i eliminated; thi condition i checked repeatedly for the reduced layer until it i atified. 3 In the loop of Step (b, we emphaize that the layer i combined with it current effective lower layer, becaue the layer i k (or i k may become ineffective in the previou tep. The complexity of the algorithm i O(M. Step i clearly of O(M complexity. In Step 3, λ i updated le than M time. A cloe inpection of the ummation in the numerator reveal that each time it can be done with O( complexity, and thu Step 3 i of O(M complexity. The complexity of Step i more ubtle. The value of κ i can be computed in O(M complexity. Denote the number of loop in Step a r ; denote the number of layer with κ value lower than or equal to it lower neighbor (the firt kind in the r-th loop

5 5 i Original value D. The optimality of the algorithm i i i i-th layer The t -loop i-th layer The nd -loop i-th layer Since we are olving a convex optimization problem, and it obviouly atifie Slater condition, the KKT condition are ufficient for optimality. Thu the proof for optimality reduce to finding ν i, that atify ( and the complementary lackne condition ν i R i =, i =,,...,M, with the olution found by the algorithm; note that the power contraint i already atified with equality. Theorem : The algorithm given above find the optimal rate allocation. Proof: Since for the effective layer R ik > by definition, we may et ν ik = to atify the complementary lackne condition. There are everal cae that we need to conider next: The ineffective layer above the lowet effective layer. The (originally effective layer which are rendered ineffective by the power contraint, i.e., the layer that become ineffective in Step 3. 3 Other ineffective layer below the lowet effective layer, i.e., the layer that become ineffective in Step and are below the lowet effective layer. The 3 rd -loop i-th layer Fig.. An example of the algorithm: the line with dot on the top are the layer of the firt kind, and dahed line are the active layer after the previou loop; the active layer are not labeled before the firt tep. In the example, Step- of the algorithm terminate after three loop. At each loop, ome layer are combined with it lower neighbor layer (and removed. a b r. The complexity in the r-th loop i bounded by linear term of b r, however b r a r, becaue only the active layer and their lower effective neighbor layer can be of the firt kind. Moreover, notice that r r= a r M, ince it i upper bounded by the total number of layer made ineffective, further implied by the fact that a layer i active only when a layer i made ineffective and combined into it. Clearly we have a = M, and thu r a r M. The overall complexity i thu O(M, and then the converion into power allocation i of O(M complexity. We note that in order to achieve the O(M complexity of the given algorithm, a fairly involved data tructure i needed. More preciely, a doubly-linked lit to update effective layer, coupled together with a ingly-linked lit to update the active layer (which can be combined into one linked-lit appear mot appropriate. However, even a naive implementation without uch data tructure i of O(M complexity, ince Step terminate within at mot M iteration, and each iteration ha maximum complexity O(M. For the firt cae, uppoe ome of thee layer are between two effective layer I and J, I J; if there are ineffective layer above the highet effective layer, we take J = M +. From Step 3 of the algorithm we can eentially aign the value of ρ exp( I uch that b[p I + p I p J + p J ]ρ b + λ(n I n J ρ =. ( Since layer I and J are effective, we et ν I = ν J =. By expanding the condition in (, we have bp k ρ b + λ(n k n k+ ρ ν k + ν k+ =, k = I,I +,...,J. (3 Though the above equation (under the olution found by the algorithm uniquely pecify ν i, I < i < J, it i not clear yet whether thoe value are indeed non-negative. We need the following lemma to proceed, the proof of which i given after the proof of the theorem. Lemma : For the combined layer between layer-i and layer-j, given any j uch that I j J, we have κ j b j i=i p i j i=i n i b J i=j + p i J i=j + n κ + j. (4 i Now we are ready to prove the non-negativene of ν i for the firt kind of ineffective layer. The value of ν i, where I < i < J, i alo pecified by J J b p i ρ b + λ n i ρ ν i =, i=i i=i (5 b p i ρ b + λ n i ρ + ν i =. (6 i=i i=i

6 6 To ee ν i thu pecified i indeed non-negative, uppoe ν i < wa true, then J J b p i ρ b + λ n i ρ = ν i < (7 i=i i=i Lemma aert that for i we have κ i = b i i=i p i i i=i n i which implie that b p i ρ b + λ n i ρ i=i κ + i = b J i=i p i J i=i n. (8 i i=i J b p i ρ b + λ i=i However thi ubequently implie J b i=i J p i ρ b + λ i=i J i=i n i ρ < (9 n i ρ <, ( which would contradict (; alternatively we ee that (9 would contradict (6 if the uppoition wa true. Thi prove that for the firt kind of ineffective layer, the given ν i are non-negative. We next conider the econd kind of ineffective layer. Suppoe the originally effective layer i k, i k+,...,i k+h become ineffective due to the power contraint. Since they are all effective originally, we have by the monotonicity of the κ factor bp i k n ik n ik+ bp i k+ n ik+ n ik+... bp i k+h n ik+h n ik+h+, ( where we have ued p to denote the accumulated probability ma after the combining of layer in Step but before Step 3. By Step 3 of the algorithm we have λ > bp i k+h n ik+h n ik+h+. ( Thu we only need to how the following equation pecify a et of non-negative ν i for i = i k,i k+,...,i k+h : bp i k+h +λ(n ik+h n ik+h+ ν ik+h =, bp i k+h +λ(n ik+h n ik+h ν ik+h + ν ik+h =, bp i k+ +λ(n ik+ n ik+ ν ik+ + ν ik+ =, bp i k +λ(n ik n ik+ ν ik + ν ik+ =. (3 From the firt equation, we get ν ik+h = bp i k+h + λ(n ik+h n ik+h+, and it i non-negative becaue of (. From the econd equation, we have ν ik+h = bp i k+h + λ(n ik+h n ik+h + ν ik+h, (4 which i alo non-negative, becaue bp i k+h + λ(n ik+h n ik+h due to ( and (, and the lat term i non-negative from the proceeding argument. Continue thi line of argument, then it i clear ν ik ν ik+... ν ik+h. (5 By thi we conclude that ν i for the econd kind of ineffective layer are indeed non-negative. For the third kind of ineffective layer, we might have to plit any given layer in the i k,i k+,...,i k+h layer, in order to recover the original layer a well a the correponding ν value. However, from any one of equation in (3, it i een that bp i k+a + λ(n ik+a n ik+a+ = ν ik+a ν ik+a+. (6 Now following the ame line of argument a the firt kind ineffective layer, we can indeed find the deired non-negative ν value for all the plit layer in Step of the algorithm. Thi complete the proof for the optimality of the algorithm. It now remain to prove Lemma, before which we firt give the following ueful fact. Lemma : When all the quantitie are poitive If a b a b... a n b n e f e f... e n f n, then we have ai ei. bi fi If a a... a n c + c e e... e m, b b b n d + d f f f m c and c, d d then we have ai + c ei + c. bi + d fi + d Thee fact are traightforward by elementary calculation and thu the detailed proof i omitted. Proof of Lemma We ue an induction approach and how thi fact i true after any loop in Step of the algorithm. The tatement i clearly true after the firt loop, by the given monotonicity of the κ equence for any combined layer before the operation, and the firt fact in Lemma. Then conider r-th tep for Step (b. For a given index j, let κ j,r and κ+ j,r denote the appropriate quantitie after tep r ; i.e., the quantitie defined in Lemma in the interval I to J for which layer j i in. Denote the updated layer that the original layer j i in a i. By the procedure, uppoe the aggregated layer after the (r -th loop i < i <... < i n < i < j < j <... < j m are to be combined into a new aggregated layer in the r-th loop, which atify κ i κ i... κ in κ i κ j κ j... κ jm, (7 a well a κ j,r κ+ j,r. (8

7 7 Then by the econd fact in Lemma, we have κ j,r κ+ j,r. Thi induction i apparently true for any loop in Step of the algorithm, thu the lemma i proved. E. The dual problem We can alo conider the dual problem of minimizing the power conumption for a given expected ditortion value. Thi can be done with eentially no change to the problem a M min (n i n i+ exp n P, ubject to: R i, i =,,...,M, M p i exp( b i D. The Lagrangian form i almot without change a L = λ M p i exp( b M ν i R i M + (n i n i+ exp n P. (9 The following condition imilar to ( can be derived, where again ν M+ m m λbp m exp( b + (n m n m+ exp = ν m ν m+ =, m =,,...,M. (3 From thi form we ee that the firt and econd tep of the algorithm can be ued without any change, and in the third tep of the algorithm, only very minor change are needed for thi dual problem. For implicity we combine the layer into the aggregated one and take the aggregated probability ma a p i. 3a Let i ko be the lowet effective layer, and define D = D p i ko. 3b Let 3c If λ /(b+ = D ( kk k=k o p i exp b. (3 i k λbp iko (n iko n iko+, then reduce R iko by (b+ log λ; otherwie, label i k o ineffective (R iko =, increment k o by, update D = D p i ko, and return to 3b. The proof of correctne and optimality remain virtually unchanged, and the complexity i alo unchanged. IV. VARIATIONAL DERIVATION OF THE CONTINUOUS CASE SOLUTION We next turn our attention to the cae of continuum of layer, which i in fact the cae conidered in []. To facilitate undertanding, we firt give a le technical derivation under the aumption that the optimal power allocation concentrate on a ingle interval of the power gain range, and how that thi i indeed true for ome probability denity function f(. Thi imple derivation provide important intuition for the general cae, baed on which a more general derivation i then given. For implicity, we firt aume f( ha upport on the entire non-negative real line [, ; later it i hown that thi aumption can be relaxed. A. A imple derivation for the ingle interval olution In thi ub-ection we give a imple derivation under the aumption that there i a unique interval [, ] which the power allocation concentrate on. The optimization problem can be reformulated a follow. Define I(i = exp(. (3 We take the number of layer to infinity, and the objective function the contraint become integral. The function I( we need to find i I( = exp( R(udu (33 where we convert back to the power gain intead of noie power n, and R( i the rate denity aociated with a fading gain. It i clear we can replace the inequality contraint by equality contraint without lo of optimality by writing the continuou counterpart of (6 a exp( R(udud = The term to be minimized i ( D(I = f(exp b I( d = P. (34 f( R(udu d = I( b d. (35 Note the additional condition that I( ha to be monotonically non-decreaing, and the boundary condition I( =. Ignoring the poitivity contraint I ( for now, take J(,I,I = f( I( b, G(,I,I = I(, the optimization problem can thu be written in the uual variational notation a minimize ubject to J(,I,I d, (36 G(,I,I d = P. (37

8 8 Next we aume there i a unique interval [, ] for which power allocation i non-zero. Under thi aumption, the objective function reduce to D(I = = J(,I,I d f( I( b d + F( + F( I( b, (38 where F( i the cumulative ditribution function of the fading gain random variable, i.e. F( = f(udu, and the contraint become P(I = = G(,I,I d I( d + I( = P. (39 We can write the Lagrangian form L(I = D(I+λ(P(I P. To find the extremal olution, we conider an increment q( on I(, and the increment of the Lagrangian functional i given by (q = L(I +q L(I. Take an arbitrary increment q( with q( = for / [, ] and q( =, and the principal linear part of the increment given in the following equation hould be zero (ee (7 p. 5, and pp. 4-5 in [6] h = δl(i = (J I + λg I dd [J I + λg I ] q(d h [ b( F( + (J I + λg I = q( + I( b+ + λ ] q( (4 where the lat term in the ummation i due to the term outide of the integration in (38 and (39. Since q( can be arbitrary, we have with J I + λg I d d [J I + λg I ] =, (4 J I = bf( I b+ (, G I =, J I = G I =, (4 which further implifie to ( bf( /(b+ I( =. (43 λ At thi point, it i clear that for I ( to be true, which i neceary for I( to be a valid olution, f( hould have non-negative derivative in any interval uch that (43 hold; in fact for any interval that poitive rate i allocated to, f( hould have trictly poitive derivative uch that I( i trictly increaing. If there i only one interval over the upport of f( where f( ha trictly poitive derivative, then the ingle interval olution aumption i indeed true. Now ince q( can be arbitrary, at thi variable end (pp. 5-9 in [6] a neceary condition for an extremum i b( F( I( b+ + λ =, (44 which give λ = b ( F( I( b+. (45 Becaue I( =, λ = bf(, the expreion of I( give one boundary condition F( = f(. (46 The lower bound i determined by the power contraint, from which we have I( ( /(b+ f( d = f( b/(b+ d + ( f( /(b+ f( = P +, (47 where in the econd equation we plit the integral into two part partitioned by =. We have thu found the unique extremal olution ( f( /(b+ I( = f( (48 in [, ] with the boundary condition pecified by (46 and (47. To find the correponding power allocation, define T( = T( = = P(rdr. We derive from (6 that ( f( f( I(r I(r dr /(b+ + ( f(rr f( /(b+ r dr. (49 For the continuou cae being conidered, the relation between T( and I( can alo be derived directly by olving a linear differential equation without uing (6, i.e., without taking the limit of the alternative channel capacity repreentation with a finite number of uer; ee the Appendix for detail. In fact, the derivation given in the Appendix can lead naturally to the contraint in (34 and (39. Through ome baic algebra, it can now be hown that (49 i in fact the ame olution a that in []. Thu the limit of the optimal olution of the dicrete cae in [] indeed converge to the extremal olution derived through the claical variational method. Moreover, the variational method derivation directly aert that f( ha a poitive derivative for any poitive power allocation interval; thi condition wa however lacking in the derivation in []. B. Derivation of the general cae olution Next we provide a more technical and complete derivation for the general cae when the power allocation doe not necearily concentrate in a ingle interval of the range of power gain. From the phyical meaning of I(, we aume that I( i a piecewie mooth continuou and differentiable function, and it i within thi function pace we eek the optimal olution. With the poitivity condition of I (,

9 9 the Lagrangian functional ubject to optimization can be written a ( f( L(I = I( b + λi( I (v( d (5 where v( i an arbitrary non-negative function; ee page 49 in [3]. The general extremum and ubidiary condition are then imilar a in the lat ub-ection, and we tate here for completene b f( I( b+ + λ + v ( = (5 and the complementary lackne condition are λ I( d P =, (5 I (v( = (53 We thu have the general olution for I( in ome interval where I (, ( bf( /b+ I( =, (54 λ which ha the ame form a given in the previou ub-ection. We firt oberve from the complementary lackne in (53 that for any I (, it mut be true that v( =. Thu for any interval [a,b] for which I (, it i een that bf( I b+ ( + λ =. (55 Thi implie that λ, becaue otherwie, I( = in thi interval, which clearly violate the power contraint. Then the firt complementary lackne condition require I( d P =. (56 Becaue for any interval [a,b] for which I ( >, we have bf( I b+ ( = λ, (57 it i clear that f( ha to have poitive derivative in any interval which are allocated with poitive rate (and power; if on the other hand thi condition i not atified, then I ( = for thi interval, and accordingly v( can be trictly poitive. Given the above dicuion, it i clear that dijoint interval with poitive rate (power allocation are eparated by interval where I ( =. Moreover, it will be hown in the equel that only a ingle continuou allocation interval may occur within a ingle interval where the derivative of f( i trictly poitive; we hall aume that the number of uch interval i finite. For implicity, aume that the firt uch interval ha the form [,a], i.e., the lower boundary i zero; thi condition can be relaxed a hown in the next ub-ection. Denote there are a total of K poitive power allocation interval, and let the i-th poitive power allocation interval be pecified by [ i,l, i,u ], and label the poitive f( derivative interval it belong to a the i-th uch interval, pecified by [l i,u i ]; i.e., [ i,l, i,u ] [l i,u i ]. For a piecewie mooth continuou extremal olution, the Weiertra-Erdmann (corner condition mut be atified (page 63 in [6]. Thee are given for every corner point c { i,l, i,u } K. The two condition are L I = c = L I = + c (L I L I = c = (L I. (58 L I = + c By ubtituting L(I from (5 into the corner condition (58, we get v( c = v( + c (59 and ( f( I( b + λi( + I (v( I (v( = c ( f( = I( b + λi( + I (v( I (v( = + c which further implifie into f( c I( c + b λi( c = f( c c I( + c + b λi(+ c c (6 (6 where (59 and (6 impoe certain continuity condition on v( and on I(, repectively. Thu v( = for [ i,l, i,u ] for all i =,,...,K, and i non-negative otherwie; from (59, we have v( + i,u = and v( i,l = for all i =,,...,K. The continuity of v( at the corner point doe not rely on the preumption that in the extremal olution, the power allocation in a ingle interval [l n,u n ] with poitive f( derivative doe not have within it more than one dijoint poitive power allocation ub-interval. We now prove that the preumption i indeed true. Suppoe otherwie, which implie the range (c,d [l n,u n ] i aigned with I ( =, and there exit c [l n,c and d (d,u n ] uch that I ( > in [c,c] and [d,d ]. We have that v(c = v(d =, (6 v ( = bf( I b+ (c λ for (c,d (63 bf(c I b+ (c λ =, c (64 where (6 i due to the preceding dicuion, (63 i due to (5, and (64 i due to (57, the continuity of I( and the continuou aumption on f(. Since f( ha trictly poitive derivative in (c,d, we have that in thi interval v ( = bf( I b+ (c λ >. (65 But thi contradict (6. Thu the uppoition can not be true, and there can be only one effective allocation interval in each uch firt kind interval [l n,u n ]. In order to determine all corner point c { i,l, i,u } K, we rewrite the original functional (5, a a piecewie opti-

10 δl(i = = K i,u i,l K i,u i,l (D I dd D I q(d + D I =i,u q( i,u D I =i,l q( i,l + K [ b F( ( i+,l F( i,u I( i,u b+ + λ ] q( i,u i,u i+,l ( bf( I b+ ( + λ K q(d + [ b F( ( i+,l F( i,u I( i,u b+ + λ ] q( i,u (67 i,u i+,l mization problem, and derive the variation. L(I = F(,l λ,l I (v(d,l K i,u ( f( + I( b + λi( I (v( d + i,l K F( i+,l F( i,u ( I( i,u b + λi( i,u i+,l i,u i,u i+,l I (v(dh (66 where we define K+,l, and thu F( K+,l =. The variation of L(I in (66 w.r.t. an arbitrary function q(, which atifie q ( = for / [ i,l, i,u ], i given by (67 at the top of thi page, where D(I f( + λ I( I( b I (v(. Note that for I( to remain a continuou function, it mut be true that q( i,u = q( i+,l. Hence the general olution extremal expreion i the ame a that afore-pecified, with an additional condition, ariing from the variable end-point problem imilar a in the previou ub-ection. Thi condition i given by b F( i+,l F( i,u I( i,u b+ + λ ( i,u i+,l =, (68 By uing the general olution for I( within [ i,l, i,u ] a given in (54, the condition (68 become ( F( i+,l F( i,u f( i,u =. (69 i,u i,u i+,l From the continuity condition on the corner point, i.e. I( i,u = I( i+,l the next condition i obtained f( i+,l i+,l = f( i,u i,u. (7 For i = K, the condition (69 implifie into which i (46 when K =. F( i,u = f( i,u i,u, (7 To ummarize, the extremal olution i given by ( /(b+ f( I( = f(,l, (7,l in the interval [ i,l, i,u ], i =,,...,K, and the boundary value are determined by F( i+,l F( i,u / i,u / i+,l = f( i,u i,u = f( i+,l i+,l, a well a the power contraint i =,,...,K, (73 I( d = P (74 Thee are the neceary condition to determine an extremum. C. Comment on relaxing the upport condition Until thi point, we have aumed that the pdf f( ha upport on the entire non-negative real line [, ], uch that the condition (69 and (7 (or (46 and (47 are ufficient to determine boundary value. Thi aumption can be relaxed to the cae that f( ha a compact upport, however certain complication will be introduced. To illutrate thi, we next treat one pecial cae where the upport of f( i on a ingle finite interval [ a, b ] and f( ha a unique poitive derivative interval [l,u ], uch that a l and b u ; the ame approach can be extended to more general upport. Without lo of generality, we can aume a = l and b > u. We hall follow the le general derivation a given in Section IV-A for implicity. The main difference now i that I( = i not necearily true if = a, i.e., it i poible for I( to be dicontinuou at a. In order to reolve thi difficult, firt aume > a, then indeed we have I( =, from which (46 and (47 follow. If the olution given by them indeed atifie l < u, then an extremal olution i pecified by thee boundarie. If on the other hand thi i not true, then = a and the condition I( = i not necearily true. Denote the value a I( a I. In thi cae, i.e. = a, the olution in [, ] can be written a I( = I ( f( f( a a /(b+. (75

11 The boundary condition (46 can till be derived. The other boundary condition derived from the power contraint i now given by ( f( /(b+ I f( a a d ( f( /(b+ ( + I f( a P =. a b (76 Thu for thi cae that = a, the olution i given by (75 with the value of and boundary function value I determined by (46 and (76. V. NUMERICAL EXAMPLES In thi ection, everal example are olved uing the olution developed in the previou ection. We tart with one example motivated by multiple acce with uer interference, then dicu the Rayleigh fading cae, finally a imple example i given for the general olution when the optimal power allocation doe not concentrate in a ingle interval. A. Interference multiple acce channel Conider here an additive white Gauian noie (AWGN channel with interfering uer. The goal by a ingle uer i to find the optimal power allocation of tranmitting in the preence of unknown interference by the other uer. The level of interference depend on the number of uer uing the multiple acce channel at a given tranmiion block. Such channel model were conidered, for example, in [4] [6]. More preciely, conider the following model y = x + N z i i + w (77 where y i the received ignal, and x i the multi-layer tranmitted real-valued ignal the uer can deign ubject to an average power contraint P. The additive interference z i are aumed to be Gauian ditributed, which are the ignal ent by other uer in thi time lot. Finally, w N(, i the additive noie in the channel. Every interference element z i i aociated with an average power level σi, and a random binary variable i to determine whether uer i i tranmitting at thi time lot. i i aumed to be i.i.d. Bernoulli random variable with Pr( i = = p on. The exact realization of the i are not known to uer j i, however it i aumed that the value of p on i known. Since the receiver ha no knowledge of z i and cannot attempt joint decoding, it ha to treat z i a AWGN when decoding x. Thu an equivalent channel model repreentation may be ueful, y = x + w, (78 + N σi i where w N(,. The equivalent fading gain j i then ( N j = + σi i, j =,.., N. (79 It i clear that there are N poible fading gain tate, which correpond to all poible tate of i. The optimal power allocation for multi-layer coding can be found by the dicrete layering algorithm preented in Section III-B. We demontrate by example of the optimal power allocation for two cae of interference ditribution. Figure 3.(a - 3.(d how the fading gain dicrete denity p i, the optimal dicrete power allocation P i and I i exp( i, repectively, where there are N = 4 interfering uer; the tranmit power i P = db; the bandwidth expanion i b = ; σ i, and p on are σ = (.95,.45,.55,.8, p on =.3 (8 The econd example i given in Figure 4.(a - 4.(d, where there are N = 6 interfering uer, with σ i, and p on pecified by σ = (.49,.9,.5,.87,.8,.46, p on =.5 (8 where the tranmit power and bandwidth expanion are the ame in both example. The reult may jut a well be generalized to the cae that p on i different for every uer, depending on the cheduling method in the ytem. B. The Rayleigh fading cae We conider here the SISO Rayleigh fading channel. Starting with the outage approach minimal ditortion, which i a ingle level coding ditortion bound. Thi i later compared to the minimal average ditortion with continuou broadcating. Outage approach: In cae there i only a ingle ource code and channel code, the tranmiion cheme i known a the outage approach. When channel condition allow, the data can be completely recovered, otherwie an outage event occur. Let the tranmiion rate be R R = log( + P, (8 where can be conidered a the fading gain threhold. For any, a rate R may be achieved, otherwie a failure occur uch that D =. Thi i in contrat with the continuou broadcating, where there i an outage region with different decodable rate depending on the fading gain realization. The obtained ditortion when decoding R i D = ( + P b. Under thi coding trategy, one can alo optimize the expected ditortion D,opt = min F( + ( + P b ( F(. (83 [, For a flat Rayleigh fading channel, the fading power ditribution i F( = e, the average ditortion then correpond to D,opt = min D( [, = min [, e + ( + P b e. (84

12 .4.3 p i, 4 interfering uer p i (a i 4 3 κ i κ i (b i.8.6 P i dicrete power allocation P i (c i I( i I( i (d i Fig. 3. An example of AWGN interference channel with N = 4 interfering uer (p on =.3, b =, P = db. (a pdf of the fading gain i, denoted by p i. (b The aociated κ i. (c Optimal dicrete power allocation P i. (d Optimal cumulative rate exponent I( i. The condition for extremum, i.e. following polynomial, d d D( =, yield the x b+ x + Pb = (85 where x + P. An explicit olution for,opt cannot be derived analytically in general for every b. However, for particular cae it can be analytically olved. For example, for b =, olving (85 give,,opt (b = = P ( + + 4P /P. (86 Continuou cae broadcat: We conider the average achievable ditortion for a SISO Rayleigh fading channel. Conider the following fading gain ditribution, F( = exp (, (87 where i the expected fading gain power. For thi ditribution the optimal power allocation i ingle interval continuou, and zero outide the interval [, ]. That can be immediately oberved from f( by taking it firt derivative d d f( = ( ( exp, (88 where d d f( on a ingle interval [,]. Then the upper bound [,] i determined by (46, which reduce to ( exp = ( exp (89 yielding =. Solving (47 give the other boundary value, denoted a,opt ; the condition (47 doe not lead to an analytical expreion, but can be olved numerically. Then the general expreion for I(, for the Rayleigh fading channel i given by,opt ( ( /(b+ I( = exp,opt,opt,opt < ( ( /(b+ >,opt exp,opt In Fig. 5, the average ditortion bound for Rayleigh fading channel are demontrated for three different value of bandwidth expanion value (b =.5,,. For every bandwidth expanion, the minimal average ditortion of the outage approach and broadcat approach are compared. It can be noticed that the larger b i, the larger i the broadcat.

13 3..5 p i, 6 interfering uer p i (a i 4 3 κ i κ i (b i.8.6 P i dicrete power allocation P i (c i.4.3 I( i I( i (d i Fig. 4. An example of AWGN interference channel with N = 6 interfering uer (p on =.3, b =, P = db. (a pdf of the fading gain i, denoted by p i. (b The aociated κ i. (c Optimal dicrete power allocation P i. (d Optimal cumulative rate exponent I( i.. parameter b. Average Ditortion 3 Continuou Broadcating, b=.5 Continuou Broadcating, b= Continuou Broadcating, b= Outage Approach, b=.5 Outage Approach, b= Outage Approach, b= SNR [db] Fig. 5. Minimal average ditortion, a comparion of outage approach and broadcat approach, for b =.5,,. C. An example with more than one power allocation interval Conider the following piece-wie mooth pdf f( (7c f (. + / [.,. ( 7c f( = f (. + 8/ [.,. (39c f (. + / (9 [.,3. ( 8c f ( / [3.,4.] where c f i a normalizing contant uch that f(d =, which can be computed to be c f Fig. 6 how pdf f(, f( and the optimal continuou power allocation denity P( with P = db. It i worth noting that at the upper bound for the firt effective power allocation interval and the lower bound of the econd interval, the value of f( i exactly the ame, a required by (73. gain, which can be defined a the SNR gain of the broadcat approach over the outage approach for the ame average ditortion value. Thu the benefit of the broadcat approach compared to the outage directly depend on the ytem deign VI. CONCLUSION We conidered the optimal power/rate allocation in the broadcat trategy, in order to minimize the expected ditortion of a quadratic Gauian ource tranmitted over a fading channel. A linear complexity algorithm i propoed, and it

14 f( f( P( Fig. 6. Optimal power allocation P( for f( given in (9 with the total power P = db. correctne and optimality are proved. Moreover, a derivation for the optimal allocation with a continuum of layer i given, uing the claical variational method. APPENDIX In thi appendix, we derive directly the relation between the power allocation and the rate allocation when there exit a continuum of uer, without relying on the alternative repreentation of the Gauian broadcat channel capacity with a finite number of uer given in [7]. Thi alo provide naturally the power contraint for the continuou cae given in (34 and (39. Denote the complementary power ditribution function by T(, i.e., T( = P(rdr a defined above (49; note that = without lo of generality we may aume T( = P. Then the incremental broadcating rate i R(d = T (d + T(. (9 By uing the definition I( = exp( we have the following relation R(udu a in (33, R(udu = log(i( (9 R(d = I (d I(. (93 In order to expre the power contraint a a function of I(, we need to olve the equation obtainable from (9 and (93, I (d I( = T (d + T(. (94 The above implifie by defining Q( I ( I( into T ( + Q(T( = Q(/. (95 The olution for the above tandard firt order differential equation i given by T( = c exp{ + exp{ duq(u} duq(u} c dr Q(r r exp{ r duq(u} (96 where c,c are contant. By ubtituting Q(u = I ( I(, the expreion for T( i implified a, T( = c I( + I( c dr I (r. (97 r Following the boundary condition T( = it i required to et c =, and by definition T( for every, therefore c =, hence T( = I( dr I (r. (98 r The above expreion i identical to (49, which i verified by another tep of of integration in part, that i T( = I( [I(r r ]r= r= + I( = I( dr I(r r (99 dr I(r r. ( It i worth noting that thi formula hold generally regardle the number of poitive power allocation interval. In the ingle interval cae, where T( i a decreaing function over [, ], and T( = P for and T( = for, (99 become the ubidiary condition a given in (39. In the general cae, (99 i equivalent to (56. REFERENCES [] S. Shamai, A broadcat trategy for the Gauian lowly fading channel, in Proc. IEEE International Sympoium on Information Theory, Ulm, Germany, 9 June-4 July 997, p. 5. [] S. Shamai and A. Steiner, A broadcat approach for ingle-uer lowly fading MIMO channel, IEEE Tran. Information Theory, vol. 49, no., pp , Oct. 3. [3] T. Cover, Broadcat channel, IEEE Tranaction on Information Theory, vol. 8, no., pp. 4, Jan. 97. [4] T. M. Cover and J. A. Thoma, Element of information theory. New York: Wiley, 99. [5] V. N. Kohelev, Hierarchical coding of dicrete ource, Probl. Pered. Inform., vol. 6, no. 3, pp. 3 49, 98. [6] W. H. R. Equitz and T. M. Cover, Succeive refinement of information, IEEE Tran. Information Theory, vol. 37, no., pp , Mar. 99. [7] B. Rimoldi, Succeive refinement of information: characterization of achievable rate, IEEE Tran. Information Theory, vol. 4, no., pp , Jan [8] S. Seia, G. Caire, and G. Vivier, Loy tranmiion over low-fading awgn channel: a comparion of progreive, uperpoition and hybrid approache, in Proc. IEEE International Sympoium on Information Theory, Adelaide, Autralia, Sep. 5, pp [9] F. Etemadi and H. Jafarkhani, Optimal layered tranmiion over quaitatic fading channel, in Proc. IEEE Sympoium Information Theory, Seattle, WA, Jul 6, pp