Energy-Distortion Tradeoff for the Gaussian Broadcast Channel with Feedback

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1 016 IEEE International Sympoium on Information Theory Energy-itortion Tradeoff for the Gauian Broadcat Channel with Feedback Yonathan Murin 1, Yonatan Kapi, Ron abora 3, and eniz Gündüz 4 1 Stanford Univerity, USA, Univerity of California, San iego, USA, 3 Ben-Gurion Univerity, Irael, 4 Imperial College London, UK Abtract Thi work focue on the imum tranmiion energy required for communicating a pair of correlated Gauian ource over a two-uer Gauian broadcat channel with noiele and caual channel output feedback (GBCF). We tudy the fundamental limit on the required tranmiion energy for broadcating a pair of ource ample, uch that each ource can be recontructed at it repective receiver to within a target ditortion, when the ource-channel bandwidth ratio i not retricted. We derive a lower bound and three ditinct upper bound on the imum required energy. For the upper bound we analyze three tranmiion cheme: Two cheme are baed on arate ource-channel coding, and apply coding over multiple ample of ource pair. The third cheme i baed on joint ource-channel coding obtained by extending the Ozarow-Leung (OL) tranmiion cheme, which applie uncoded linear tranmiion. Numerical imulation how that depite it implicity, the energy-ditortion tradeoff of the OL-baed cheme i cloe to that of the better aration-baed cheme, which indicate that the OL cheme i attractive for energy-efficient ource tranmiion over GBCF. I. INTROUCTION We tudy the energy-ditortion tradeoff (ET) for the tranmiion of a correlated pair of Gauian ource over a twouer Gauian broadcat channel (GBC) with noiele caual feedback, referred to a the GBCF. ET, recently propoed for the multiple-acce channel (MAC) and the point-to-point channel in [1], characterize the imum energy-per-ource ample required to achieve a target ditortion pair at the receiver, without contraining the ource-channel bandwidth ratio. In many practical cenario, e.g., atellite broadcating [], enor network meauring phyical procee [3], and in particular wirele body-area enor network [4], an extremely high energy efficiency i required for broadcating correlated obervation, while the available power i limited. It follow that there i a trong motivation for tudying the ET for broadcating correlated ource, and in the preent work we focu on ET for feedback-aited cenario, repreented by the GBCF. It i well known that, for loy ource tranmiion over memoryle Gauian point-to-point channel, with or without feedback, when the bandwidth ratio i fixed and the average power per channel ue i limited, arate ource and channel coding (SSCC) achieve the imum poible average ditortion [5, Thm. 3]. In [1, Cor. 1] it i further hown that SSCC i optimal in the ET ene a well: For any target ditortion level, the imal tranmiion energy i achieved by optimal loy compreion followed by the mot energy efficient channel coding. Thi work wa upported by Irael Science Foundation under grant 396/11. In contrat to the point-to-point etting, in multi-uer cenario joint deign of the ource and channel code can improve the performance [6]. However, we have a relatively limited undertanding of the impact of feedback on joint ourcechannel coding (JSCC) over multi-uer channel. The work [7] preented everal achievability cheme and a et of neceary condition for lolely tranmitting a pair of dicrete and memoryle correlated ource over a MAC with feedback. Loy tranmiion of correlated Gauian ource over a twouer Gauian MAC with feedback wa tudied in [8], in which ufficient condition a well a neceary condition for the achievability of a mean quare error (MSE) ditortion pair were derived. While [8] conidered only ource-channel coding with a unit bandwidth ratio, [1] tudied the ET for the tranmiion of correlated Gauian ource over a twouer Gauian MAC with and without feedback, when the bandwidth ratio i not retricted. Previou work on GBCF mainly tudied the channel coding apect which applie to independent and uniformly ditributed meage [9] [11]. JSCC of correlated Gauian ource over GBCF wa alo tudied in [1], in which the imal number of channel ue required to achieve a target MSE ditortion pair wa characterized for three linear encoding cheme, uing uncoded tranmiion: The firt cheme i a JSCC cheme baed on the coding cheme of Ozarow and Leung [9], to which we hall refer a the OL cheme; the econd cheme i a JSCC cheme baed on the linear quadratic Gauian (LQG) coding cheme of [11], to which we hall refer a the LQG cheme; and the third cheme i a JSCC cheme whoe parameter are obtained uing dynamic programg (P). 1 We note that the advantage of linear and uncoded tranmiion, a implemented in the OL and LQG cheme, include a low computational complexity, low coding delay, and low torage requirement. In the preent work we analyze loy ource coding over GBCF uing SSCC and JSCC broadcating cheme baed on a different performance metric the ET. Main Contribution: Thi i the firt work toward characterizing the ET in GBCF. We derive lower and upper bound on the imum energy required to achieve a target MSE ditortion pair when tranmitting a pair of Gauian ource over a two-uer GBCF, without contraining the number of channel ue per ource ample. The propoed lower bound i baed on cut-et argument, and the upper bound are obtained 1 In the preent work we dicu only the former two cheme, ince the cheme baed on P become analytically and computationally untractable a the number of channel ue goe to infinity /16/$ IEEE 189

2 016 IEEE International Sympoium on Information Theory uing three tranmiion cheme: Two SSCC cheme and an uncoded JSCC cheme. The firt SSCC cheme jointly compree the two ource equence into a ingle bit tream, and tranmit thi tream uing a channel code, deigned for the tranmiion of a common meage. The econd SSCC cheme arately encode each ource equence into two ditinct bit tream, and then tranmit the reulting two bit tream via the LQG channel code of [11]. It i hown that, in term of the imum energy per bit, the LQG code provide no gain compared to orthogonal tranmiion, from which we conclude that the firt SSCC cheme, which jointly compree the equence into a ingle tream, i more energy efficient. A both SSCC cheme apply coding over multiple ample of the ource pair, they require a high computational complexity, long delay, and large torage. Thu, motivated by the low computational complexity, hort coding delay, and mall torage requirement of the uncoded OL cheme preented in [1], we ue thi cheme to obtain a third upper bound on the ET. Numerical reult indicate that the SSCC cheme baed on joint compreion achieve a better ET compared to the JSCC OL cheme; yet, in many cenario the gap i quite mall. Moreover, in many application there i a contraint on the maximal allowed latency. In uch cenario, coding over large block of independent and identically ditributed (i.i.d.) pair of ource ample introduce unacceptable delay, and intantaneou tranmiion of each oberved pair of ource ample via the propoed JSCC-OL cheme may be preferable in order to atify the latency requirement, while maintaining high energy efficiency. The ret of thi paper i organized a follow: The problem formulation i detailed in Section II. Bound on the imum energy are derived in Section III. Numerical reult are decribed in Section IV, and concluding remark are provided in Section V. Notation: We ue capital letter to denote random variable, e.g., X, and boldface letter to denote column random vector, e.g., X, where the k th element of a vector X i denoted by X k, k 1. We ue an-erif font to denote matrice, e.g., Q. Finally, we ue E { }, ( ) T, log( ), and R to denote tochatic expectation, tranpoe, natural bai logarithm, and the et of real number, repectively. II. PROBLEM EFINITION Fig. 1 depict the two-uer GBCF, in which all the ignal are real. The encoder oberve m i.i.d. realization of a correlated and jointly Gauian pair of ource (S 1,j, S,j ) N (0, Q ), j = 1,..., m, where Q σ [ 1 ρ ] ρ 1, ρ < 1. Both ource generate their realization at a fixed rate. The tak of the encoder i to end the obervation of the i th ource Si,1 m, i = 1,, to the i th decoder (receiver) denoted by Rx i. The received ignal at time k at Rx i i given by: Y i,k = X k + Z i,k, i = 1,, (1) for k = 1,..., n, where the noie equence {Z 1,k, Z,k } n k=1, are i.i.d. over k = 1,,..., n, with (Z 1,k, Z,k ) N (0, Q z ), where Q z σz [ 1 ρ z ] ρ z 1, ρz < 1. Let Y k (Y 1,k, Y,k ). Rx i, Fig. 1: Gauian broadcat channel with correlated ource and feedback link. Ŝ m 1,1 and Ŝm,1 are the recontruction of Sm 1,1 and Sm,1, repectively. i=1,, ue it channel output equence Yi,1 n to etimate Sm i,1 via Ŝm i,1 = g i(yi,1 n ), g i : R n R m. The encoder map the oberved pair of ource equence and the noiele caual channel output obtained through the feedback link into a channel input via: X k = f k (S1,1, m S,1, m Y 1, Y,..., Y k 1 ), f k : R (m+k 1) R. We tudy the ymmetric GBCF with parameter (σ, ρ, σz, ρ z ), and define a (, E, m, n) code to be a collection of n encoding function {f k } n k=1 and two decoding function g 1, g, that atify the MSE ditortion contraint: m E{(S i,j Ŝi,j) } m, 0< σ, i = 1,, () j=1 and energy contraint: n E { Xk} me. (3) k=1 Contraint (3) reflect the energy per ource ample rather than per channel ue. Note that by defining P m n E, contraint (3) can be equivalently tated a 1 n n k=1 E { Xk} P, which i the well known average power contraint. Our objective i to characterize the imal E, for a given target MSE at each uer, uch that for all ɛ > 0 there exit m, n and a ( + ɛ, E + ɛ, m, n) code. We call thi imal value the ET, and denote it by E(). In the next ection we preent lower and upper bound on E(). III. BOUNS ON E() A. Lower Bound on E() Our firt reult i a lower bound on E(). We begin by defining the following rate-ditortion function, ee [14, Sec. III.B]: R S1 () 1 ( ) σ log (4a) ( 1 R S1,S () log σ (1+ ρ ) σ ), (1 ρ ) >σ(1 ρ ) ( ) 1 log σ 4 (1 ρ ), σ(1 ρ ). (4b) Note that R S1 () i the rate-ditortion function for the ource variable S 1, while R S1,S () i the rate ditortion function for jointly compreing the pair of ource {S 1, S } into a ingle tream of rate R R S1,S (). Note that [14, Sec. III.B] ue the function R S1,S ( 1, ) a it conider a different ditortion contraint for each ource. For the preent cae, in which the ame ditortion contraint i applied to both ource, R S1,S () can be obtained by etting 1 = = in [14, 1830

3 016 IEEE International Sympoium on Information Theory Eq. (10)], and thu we ue the implified notation R S1,S (). The lower bound on E() i tated in the following theorem: Theorem 1. The ET E() atifie E() E lb (), where: E lb ()=σ z log e max { R S1 (), (1+ρ z )R S1,S () }. (5) Proof Outline: A we conider a ymmetric etting, we can focu on the ditortion at Rx 1. To obtain (5), we derive two different lower bound. The firt bound i derived by ignoring Rx and requiring that the ditortion at Rx 1 will be. The econd bound i obtained by conidering the tranmiion of both ource over a point-to-point channel with two output Y 1 and Y. In [13, Sec. III] we explicitly compute the reulting rate bound, and how that, if a (+ɛ, E+ɛ, m, n), ɛ>0, code exit, then the firt bound yield R S1 () (E+ɛ), and the σz log e (E+ɛ) σz (1+ρz) log. e econd bound yield R S1,S () In the next ubection we analyze the ET for three achievability cheme thereby obtaining three upper bound on E(). While the conidered cheme have imple code contruction, analyzing their ET performance analyi i challenging. B. Upper Bound on E() via SSCC SSCC in multi-uer cenario carrie the advantage of modularity and eae of integration with the layered network architecture. In thi ubection we analyze the ET of two SSCC cheme: The firt cheme take advantage of the correlation between the ource and ignore the correlation between the noie component. The econd cheme ignore the correlation between the ource and aim at utilizing the correlation between the noie component. 1) The SSCC-ρ Scheme (Utilizing ρ ): Thi cheme utilize the correlation between the ource by firt jointly encoding the two ource equence into a ingle bit tream via the ource coding cheme propoed in [14, Thm. III.1]. Thi tep give rie to the rate-ditortion function tated in (4b). The reulting bit tream i then encoded via an optimal channel code for ending a common meage over the GBC (without feedback), and i tranmitted to both receiver. Note that the optimal code for tranmitting a common meage over GBCF with ρ z 0 i not known. When ρ z = 0, the capacity for ending a common meage over the GBCF i achievable uing an optimal point-to-point channel code which ignore the feedback. Thu, SSCC-ρ ue the correlation between the ource, but ignore the correlation among the noie component. The following theorem tate the ET achieved by thi cheme. Theorem. The SSCC-ρ cheme achieve the following ET: ( ) σ σ E (ρ) z log (1+ ρ ) e σ ()= (1 ρ ), >σ(1 ρ ) ( σ z σ 4 log (1 ρ ) e, σ(1 ρ ). (6) ) Proof: The optimal rate for jointly encoding the ource equence into a ingle bit tream i R S1,S (), given in (4b) [14, Sec. III.B]. Note that from thi tream both ource equence can be recovered to within a ditortion. The encoded bit tream i then tranmitted to the receiver via a capacity-achieving point-to-point channel code [15, Thm ] (note that thi code doe not need the caual feedback [15, Thm ]). Let Eb common denote the imum energyper-bit required for reliable tranmiion over the Gauian point-to-point channel [16]. From [16, pg. 105] we have Eb common = σz log e. A the conidered cheme i baed on ource-channel aration, the achievable ET i given by E() = Eb common R S1,S (), where R S1,S () i tated in (4b). Thi reult in the ET in (6). ) The SSCC-ρ z Scheme (Utilizing ρ z ): Thi cheme utilize the correlation among the noie component, which i available through the feedback link for channel encoding, but doe not ue the correlation between the ource in the compreion. Firt, each of the ource equence i encoded uing the optimal rate-ditortion ource code for calar Gauian ource [15, Thm ]. Then, the reulting two bit tream are ent over the GBCF uing the LQG channel coding cheme of [11]. The following theorem characterize the imum energy per ource ample required by thi cheme. Theorem 3. The ( SSCC-ρ ) z cheme achieve the ET: E (ρz) ()=σz σ log e. Proof: The encoder arately compree each ource equence at rate R S1 (), where R S1 () i given in (4a). Thu, from each encoded tream the correponding ource equence can be recovered to within a ditortion. Then, the two encoded bit tream are tranmitted to the receiver uing the LQG cheme of [11]. Let E LQG b denote the imum required energy per pair of encoded bit, required by the LQG cheme. In [13, Appendix. B] we how that for the ymmetric cenario E LQG b = σ z log e. Since two bit tream are tranmitted, the achievable ET i given by ( E() ) = E LQG b R S 1 (), yielding E (ρz) () = σz σ log e. Remark 1. Since E (ρz) () i independent of ρ z, the LQG cheme cannot take advantage of the correlation among the noie component to improve the imum energy per ource ample needed in the ymmetric etting. Indeed, an ET of () can alo be achieved by tranmitting the two bit tream via time haring over the GBCF without uing the feedback. E (ρz) Remark. We oberve that E (ρ) () E (ρz) (). For σ(1 ρ ) thi relationhip directly follow from the expreion of E (ρ) () and E (ρz) (). For >σ(1 ρ ) the above relationhip hold if the polynomial q() = (1+ ρ ) σ+σ (1 ρ 4 ) i poitive. Thi i atified a the the dicriant of q() i negative. We thu conclude that it i preferable to ue the correlation between the ource than the correlation between the noie component. C. An Upper Bound on E() via JSCC Next, we derive an upper bound on E() uing the uncoded JSCC OL cheme. Thi cheme equentially tranmit the ource pair (S 1,j, S,j ), j = 1,,..., m, without applying a ource code. We note that the OL cheme i deigned 1831

4 016 IEEE International Sympoium on Information Theory Fig. : Upper and lower bound on E() for σ = σ z = 1, and ρ z = 0.5. Solid line correpond to ρ = 0.9, while dahed line correpond to ρ =0.. for a fixed P = E n, and from condition (3) we obtain that P = E n 1 n n k=1 E { Xk}. An upper bound on E() i obtained by calculating the imal number of channel ue required by the OL cheme to achieve the target ditortion, denoted by K OL (P, ), and then obtaining the required energy via K OL(P,) k=1 E { Xk}. In the OL cheme, each receiver recurively etimate it intended ource. Uing the feedback, the tranmitter track the etimation error at the receiver, and end a linear combination of thee error, where the cheme i terated after K OL (P, ) channel ue. For a detailed decription of the propoed OL-baed JSCC cheme we refer the reader to [13, Sec. V.A]. Let E OL- () denote the imal energy per ource-pair ample, required to achieve an MSE of uing the JSCC OL cheme. The following theorem preent an upper bound on E OL- (), and therefore on E(). Theorem 4. Let th σ ( ρz ρ ) ρ z. Then, E OL- () E OL (), where: ( σ z σ 3 ρ z log (1+ ρ ) +( ρ z)( σ ), )+σ ρ th, ( ) E OL ()= σz log( ( ρz ρ )σ ( ρ z) (7) ( )) ρ z log ( ρz)(1+ ρ ) ρ z ρ, < th. Proof Outline: In [13, Sec. V.B] we derive an upper bound on K OL (P, ), denoted by KOL ub (P, ). We further how that K ub OL (P,) K OL(P,) 1 and that Kub OL (P, ) a P 0. A both ource generate their realization at a fixed rate, the above limit ratio implie that the bandwidth ued by the OL cheme increae to infinity a P 0. Next, recalling that E OL ()= P K OL (P, ), we evaluate P KOL ub (P, ) a P 0, and obtain an upper bound on E OL- (). The detailed proof i provided in [13, Sec. V.C]. Remark 3. It can be oberved that when the ource are independent, i.e., ρ = 0, then E OL () =E (ρ) () =E (ρz) (), for all 0 σ. When ( ρ) 1 and ρ z 1 then E OL () E lb () σz σ log e, in thi cae we alo have () E lb () and E (ρz) () E OL (). E (ρ) Remark 4. In thi work we did not analyze the ET of JSCC uing the LQG cheme, E LQG (). The reaon i two-fold: Fig. 3: Upper and lower bound on E() for σ = σ z = 1, ρ =0.8. Solid line correpond to ρ z =0.9, while dahed line correpond to ρ z = 0.9. analytic tractability and practical relevance. We note that in [1, Sec. 4] we adapted the LQG cheme for [11] to the tranmiion of correlated Gauian ource over GBCF. We have oberved in [1] that obtaining a cloed-form expreion for E LQG () eem intractable. Yet, uing the reult and analyi of [1] one can find good approximation for E LQG (). We alo howed in [1] that, in the context of JSCC, and in contrat to the reult of [17] for the channel coding problem, when the duration of tranmiion i finite and the tranmiion power i very low, the OL cheme outperform the LQG cheme. Thi concluion i expected to hold for the ET a well. Indeed, numerical imulation indicate that the LQG cheme of [1, Sec. 4] achieve roughly the ame imum energy a the SSCC-ρ z cheme, while in Section IV we how that the OL cheme outperform the SSCC-ρ z cheme. IV. NUMERICAL RESULTS Next, we numerically compare E lb (), E (ρ) and E OL (). We et σ = σz = 1 and conider everal value (), E (ρz) () of ρ z and ρ. Fig. depict E lb (), E (ρ) (), E (ρz) () and E OL () for ρ z = 0.5, and for two value of ρ : ρ = 0. and ρ = 0.9. A E (ρz) () i not a function of ρ, it i plotted only once. It can be oberved that when ρ = 0., E (ρ) (), E (ρz) () and E OL () are almot the ame. Thi follow becaue when the correlation between the ource i low, the gain from accounting for thi correlation are alo low. Furthermore, when ρ = 0., the gap between the lower bound and the upper bound i evident. On the other hand, when ρ = 0.9, both SSCC-ρ and OL ignificantly improve upon SSCC-ρ z. Thi follow a SSCC-ρ z doe not take advantage of the correlation among the ource. It can further be oberved that when the ditortion i low, there i a mall gap between OL and SSCC-ρ, while when the ditortion i high, OL and SSCC-ρ require roughly the ame amount of energy. Thi i alo upported by Fig 4. We conclude that a the SSCCρ cheme encode over long equence of ource ample, it better exploit the correlation among the ource compared to the OL cheme. Fig. 3 depict E lb (), E (ρ) (), E (ρz) () and E OL () v. () and, for ρ = 0.8, and for ρ z { 0.9, 0.9}. A E (ρ) 183

5 016 IEEE International Sympoium on Information Theory Fig. 4: Normalized exce energy requirement of the OL cheme over the SSCC-ρ cheme, ρz = 0.5. Fig. 5: Normalized exce energy requirement of the SSCC-ρz cheme over the OL cheme, ρz = 0.5. Ez () are not function of ρz, we plot them only once. It can be oberved that when ρz = 0.9, Elb (), E () and EOL () are very cloe to each other, a wa analytically concluded in Remark 3. On the other hand, for ρz = 0.9 the gap between the bound i large. Note that analytically comparing E (), Ez () and EOL () for any i difficult. Our numerical imulation ugget that E () EOL () Ez (), for all value of, ρ, ρz. For example, Fig. 4 depict the difference EOL () E () for ρz = 0.5, and for all value of and ρ. It can be oberved that for low ρ, or for high, E () EOL (). On the other hand, when ρ i large and i low, then the SSCC-ρ cheme improve upon the OL cheme. Fig. 5 depict the difference Ez () EOL () for ρz = 0.5. It can be oberved that larger ρ reult in a larger gap. V. C ONCLUSIONS Thi work i a firt tep toward characterizing the optimal ET for ending correlated Gauian ource over GBCF, where no retriction i placed on the ignaling cheme or the ource-channel bandwidth ratio. In particular, we firt lower bounded the imum required energy per ource-pair ample uing cut-et argument, and then upper bounded it by analyzing three tranmiion cheme: Two different SSCC cheme, and the uncoded JSCC OL cheme. We concluded that while SSCC-ρ uccefully exploit the correlation between the ource by jointly encoding the ource equence into a ingle bit tream, SSCC-ρz doe not exploit the correlation between the noie component ince the LQG channel coding cheme of [11] achieve the ame imum energy per pair of bit a orthogonal tranmiion. Thi lead to the concluion that SSCC-ρ outperform SSCC-ρz. Numerical reult indicate that SSCC-ρ outperform the OL cheme a well. On the other hand, the gap between the energy requirement of the two cheme i rather mall. We note that, in the SSCC-ρ cheme coding take place over multiple ample of ource pair which introduce high computational complexity, large delay, and require large amount of torage. On the other hand, the OL cheme applie linear and uncoded tranmiion to each ource ample pair arately, which require low computational complexity, hort delay, and limited torage. Therefore, the OL cheme provide an attractive alternative for energy efficient tranmiion over GBCF. R EFERENCES [1] A. Jain,. G und uz, S. R.Kulkarni, H. V. Poor, and S. Verd u, Energyditortion tradeoff in Gauian joint ource-channel coding problem, IEEE Tran. Inf. Theory, vol. 58, no. 5, pp , May 01. [] F. Alag oz, and G. G ur, Energy efficiency and atellite networking: A holitic overview, Proc. of IEEE, vol. 99, no. 11, pp , Nov [3] T. Rault, A. Bouabdallah, and Y. 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