The Gaussian Channel with Noisy Feedback

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1 The Gaussia Chael with Noisy Feedback Youg-Ha Kim Departmet of ECE Uiversity of Califoria, Sa Diego La Jolla, CA 9093, USA Amos Lapidoth Sigal ad Iformatio Processig Lab. ETH Zurich 809 Zurich, Switzerlad Tsachy Weissma Iformatio Systems Lab. Staford Uiversity Staford, CA 94305, USA Abstract Upper ad lower bouds are derived o the reliability fuctio of the additive white Gaussia oise chael with output fed back to the trasmitter over a idepedet additive white Gaussia oise chael. Special attetio is paid to the regime of very low feedback oise variace ad it is show that the reliability fuctio is asymptotically iversely proportioal to the feedback oise variace. This result shows that the oise i the feedback lik, however small, reders the commicatio with oisy feedback fudametally differet from the perfect feedback case. For example, it is demostrated that with oisy feedback, liear codig schemes fail to achieve ay positive rate. I cotrast, a asymptotically optimal codig scheme is devised, based o a three-phase detectio/retrasmissio protocol, which achieves a error expoet iversely proportioal to the feedback oise variace for ay rate less tha capacity. I. INTRODUCTION This paper studies the discrete-time additive white Gaussia oise chael with a oisy feedback lik depicted i Fig.. We W P i= EX i P Z N(0, I) Z i X i(w, Ỹ i ) Fig.. Ỹ i Z i Z N(0, ɛ I) Gaussia chael with oisy feedback. Y i Ŵ (Y ) wish to commuicate a message idex W {,,..., e R } over the forward additive white Gaussia oise chael Y i = X i + Z i, () where X i, Z i, Y i, respectively, deote the chael iput, the additive Gaussia oise, ad the chael output. Let further Ỹ i deote a oisy versio of Y i over the feedback (backward) additive white Gaussia oise chael Ỹ i = Y i + Z i, () where Z i is the Gaussia oise i the backward lik. We assume that the forward oise process {Z i } i= ad the backward oise process { Z i } i= are idepedet of each other, ad respectively idepedet ad idetically distributed accordig to N(0, ) ad N(0, ɛ ). We defie a (e R, ) code with the ecodig fuctios f () i : (W, Ỹ i ) X i, i =,,...,, (3) satisfyig the expected average power costrait EXi (W ; Ỹ i ) P, (4) i= ad the decodig fuctio φ () : Y Ŵ. Thus, the ecoder has the causal access to the oisy feedback Ỹ. The probability of error P e () is defied by P () e = Pr(W Ŵ (Y )) = e R Pr(W Ŵ (Y ) W = w), e R w= where W is distributed uiformly over {,..., e R } ad is idepedet of (Z, Z ). A rate R (ats per trasmissio) is said to be achievable if there exists a sequece of (e R, ) codes such that the associated probability of error P e () teds to 0 as. The feedback capacity C FB is defied as the supremum of all achievable rates. Because the forward chael () is memoryless, the availability of feedback eve if oiseless (i.e., ɛ = 0) does ot icrease the capacity as show by Shao [5]. Thus, we focus o the reliability fuctio E FB (R; P, ɛ ) of the Gaussia chael with oisy feedback as a fuctio of trasmissio rate R, power costrait P, ad feedback oise variace ɛ. The reliability fuctio E FB (R; P, ɛ ) is defied as the rate of decay for the error probability of the optimal sequece of (e R, ) codes, i.e., E FB (R; P, ɛ ) = lim sup () log P e,opt(r), where P () e,opt(r) deotes the ifimum of the probability of error over all (e R, ) codes. We will use the otatio E(R; P ) to deote the reliability of the additive white Gaussia oise chael without feedback uder sigal-to-oise ratio P. While perfect feedback does ot icrease the capacity of a memoryless chael, it ca improve the reliability dramatically. I their celebrated work [4], Schalkwijk ad Kailath showed that a simple liear feedback codig scheme achieves the capacity. More surprisigly, the associated probability of error decays doubly expoetially. I terms of the reliability fuctio, we have the ifiite reliability, i.e., E FB (R; P, ɛ =0) =. This fasciatig result has bee later /07/$5.00 c 007 IEEE 46

2 exteded i may directios. Pisker [3], Kramer [9], ad Zigagirov [7] showed that the probability of error ca be made to decay as fast as a arbitrary level of ested expoetials. For owhite Gaussia chaels, Butma [3] studied a variat of the Schalkwijk Kailath codig scheme that achieves a rate higher tha the ofeedback capacity, which was later show to be optimal [8]. The Schalkwijk Kailath scheme has bee also exteded optimally to two-user Gaussia multiple access chaels with feedback by Ozarow [] ad to Costa s writig o dirty paper chael [4] with feedback []. Much less explored is how the feedback oise affects the reliability fuctio. There are a few papers i the literature o oisy feedback, icludig Kashyap [7], Laveberg [0], ad Draper ad Sahai [6], but these papers have a fudametally differet ature from the curret work. While Kashyap [7] focused o the reliability, his setup allowed the codig over the feedback chael with expoetially large power. Laveberg [0] cosidered the Gaussia chael with ifiite badwidth ad his focus was o orthogoal keyig both i the forward chael ad o the feedback chael. A recet work by Draper ad Sahai [6] deals with the variable-legth codig over discrete chaels with oisy feedback uder a differet error criterio. I compariso, this paper studies the block codig over the discrete-time Gaussia chael with ucoded feedback. The reaso is clear to exclude variable-legth codig schemes, i which the duratio of commuicatio is a radom variable that depeds o the chael behavior. Such schemes, which were studied by Burashev [] ad others i the oiseless feedback case, tur out to be extremely fragile with oisy feedback. Due to the oise i the feedback chael, the trasmitter ad receiver may ot be i agreemet as to whether the commuicatio has bee cocluded, ad this may ot oly cause a decodig error of the preset message, but also create havoc i all subsequet uses of the codig scheme. I order to uderstad the effect of the feedback oise o the reliability fuctio E FB (R; P, ɛ ), we focus especially o the regime 0 < ɛ, i.e., the feedback oise has a small, but ozero variace. We will ultimately establish the followig result. Theorem. For all rates 0 < R < C(P ) ad all ɛ > 0, there exist costats 0 < K (R, P ), K (R, P ) < such that the reliability fuctio E FB (R; P, ɛ ) is bouded by K (R, P ) ɛ < E FB (R; P, ɛ ) < K (R, P ) ɛ. (5) To prove Theorem, we preset upper ad lower bouds o the reliability fuctio. I Sectio II, three upper bouds o the reliability fuctio are give. The first method, which is derived o a geie-aided setup, yields a upper boud o the oisy feedback reliability fuctio E FB (R; P, ɛ ) i terms of the ofeedback reliability fuctio E(R; P ) at rate R ad sigal-to-oise ratio P : ( E FB (R; P, ɛ ) E R; P ( + ) ɛ ) ɛ. (6) This boud suffices to demostrate that very oisy feedback is like o feedback lim E FB(R; P, ɛ ) = E(R; P ) ɛ ad that the reliability fuctio E FB (R; P, ɛ ) is fiite for every ɛ > 0. (O the other had, recall that E FB (R; P, 0) =.) I fact, (6) implies that or, writte iformally, lim ɛ 0 ɛ E FB (R; P, ɛ ) <, E FB (R; P, ɛ ) = O(/ɛ ). (7) The boud (6) is, however, rather loose i that it is positive eve for R > C. To remedy this deficiecy we itroduce a secod upper boud E FB (R; P, ɛ ) f(r, P, ɛ ), (8) for all C(P/( + ɛ )) < R < C(P ), where C(P ) is defied as log( + P ) ad f(r, P, ɛ ) teds to zero as R C(P ). Thus, as a fuctio of the rate R, E FB (R; P, ɛ ) is cotiuous at R = C(P ) for each ɛ > 0. (O the other had, we have E FB (R; P, 0) = for all R < C.) The derivatio of (8) is based o a chage-of-measure argumet, a applicatio of Cramér s theorem from large deviatios theory, ad the strog coverse for chaels with perfect feedback at rates exceedig capacity. Our third upper boud is also based o a chage-of-measure argumet. By chagig the measure so that the feedback is useless, we ca show that E FB (R; P, ɛ ) E(R; P/σ ) + γ (9) for ay γ, σ satisfyig Λ ɛ,σ (γ) = E(R; P/σ ), where Λ ɛ,σ (γ) is the Fechel Legedre trasform of a certai radom variable ad ca be characterized aalytically. This third upper boud ca be easily geeralized to arbitrary o- Gaussia chaels. These upper bouds show that the feedback oise, however small it is, reders the feedback commuicatio fudametally differet from the perfect feedback case. This has importat implicatios o the desig of optimal codes. As a corollary of (6), i particular, E FB (R; P, ɛ ) < for all ɛ > 0, we will prove i Sectio III that liear feedback codig schemes such as the Schalkwijk Kailath codig scheme, which for f = 0 achieves ot oly the capacity but the double-expoetially decayig probability of error, fail to achieve ay positive rate. Motivated by (7), we preset i Sectio IV a codig scheme that achieves lim ɛ ( lim ) () log P e > 0, R < C(P ), (0) ɛ 0 or writte iformally, E FB (R; P, ɛ ) = Ω(/ɛ ). Combied with (7), this establishes Theorem. 47

3 The codig scheme that achieves (0) is based o a idea of simple detectio/retrasmissio protocol ad cosists of three phases ) ofeedback commuicatio, ) ackowledgmet, 3) retrasmissio. Each phase cotais, however, a uique idea that higes o the oise i the feedback lik. I the followig sectios, we provide some techical details o the upper ad lower bouds itroduced above. II. UPPER BOUNDS ON THE RELIABILITY FUNCTION I this sectio, we give three upper bouds o the reliability fuctio E FB (R; P, ɛ ) itroduced i Sectio I. Sice Ỹi = X i + Z i + Z i, we ca recast the ecodig fuctios (3) i the form: f () i : (W, S i ) X i, () where S i = Z i + Z i. Recall that (Z i, S i ), i =,,... are idepedet ad idetically distributed (i.i.d.) with Z i N(0, ), S i N(Z i, ɛ ). () Therefore, give a (e R, ) code, we ca write Pr(W Ŵ (Y ) W = w) = Pr((Z, S ) A w ), (3) for each w {,..., e R } with appropriately chose sets A,..., A e R that partitio R R. Uder this otatio, we start our discussio with the third upper boud (9). A. Upper Bouds via Chage of Measure Let (Z, S) be a geeric pair of radom variables distributed as the pair (Z i, S i ) i (), ad let (Z, S ) be a pair of idepedet Gaussias with Z N(0, σ ) ad S N(0, +ɛ ). Let further f ad f deote the respective desities of (Z, S) ad (Z, S ). Fially, let Λ ɛ,σ deote the Fechel Legedre trasform (see, e.g., [5]) of the radom variable log ( f (Z, S )/f(z, S ) ). This fuctio Λ ɛ,σ (γ) ca be characterized explicitly as follows. Let Λ ɛ,σ be the Fechel Legedre trasform of U σ +σ ɛ ɛ ɛ σ +ɛ ɛ UV + ɛ V with idepedet stadard Gaussias U, V, which ca be writte i closed form as Λ ɛ,σ (α) = ( α( + σ + ɛ (σ )) 4 + ) 4 + α (ɛ 4 (σ ) + (σ + ) + ɛ ( + σ 4 )) + log + 4+α (ɛ 4 ( σ ) +(+σ ) +ɛ (+σ 4 )) α ɛ. The we have Λ ɛ,σ (γ) = Λ ɛ,σ (γ + log[σ ɛ ( + ɛ )]). Propositio. For each σ > 0, E FB (R; P, ɛ ) γ + E where γ D(f f) is the uique solutio to Λ ɛ,σ (γ) = E(R; P/σ ) (R; Pσ ), (4) ad D(f f) is the relative etropy betwee the desities f ad f. Proof sketch.. Fix a (e R, ) code ({f () i }, φ () ) ad let A,..., A e R be the associated error evets defied as i (3). Let P deote the measure associated with the oises (Z i, S i ) i.i.d. (Z, S) ad let P deote the measure associated with (Z i, S i ) i.i.d. (Z, S ). Suppose we use the same code ({f () i }, φ () ) for the followig commuicatio sceario: Y i = X i (W, S,..., S i ) + Z i. (5) That is, the chael oises come from the measure P istead of P. The, because {Z i } ad {S i } are idepedet, the feedback {S i } is completely useless ad the error expoet achieved i (5) is o better tha the ofeedback reliability fuctio E(R; P/σ d ). (Because S i = S i, the power used uder the ew settig is idetical to that used i the origial settig.) We defie the set { B γ = (z, s ) : log f (z, s } ) f(z, s ) γ. The, we have P (A w ) P (A w B γ ) + P (Bγ) c = f (z, s )dz ds + P (Bγ) c A w B γ e γ A w B γ f(z, s )dz ds + P (B c γ) = e γ P (A w ) + P (B c γ). Whe averaged over w {,..., e R }, this implies P e e γ P e + P (B c γ), (6) where P e ad P e deote the error probabilities uder respective measures. We ow take a arbitrary sequece of (e R, ) codes ad choose γ such that Λ ɛ,σ(γ) = lim log P (Bγ) c > E(R; P/σ ). Sice E(R; P/σ ) lim sup log P e() by defiitio of the reliability fuctio, (6) imples that lim sup () log P e γ log( P e, () opt P (B γ ) ) γ + E(R; P/σ ). Fially the cotiuity ad strict mootoicity of Λ ɛ,σ (γ) i γ prove the desired result. Now we move o to the secod upper boud (8). The proof techique is agai based o a chage-of-measure argumet. This time, we defie the ew measure P as (Z, Z ) f δ = N(0, + δ) N(0, ɛ δ) (7) for δ (0, ɛ ), istead of the origial measure (Z, Z) f = N(0, ) N(0, ɛ ). (8) 48

4 Propositio. Let δ (0, ɛ ) be give. The, for all R (C(P/( + δ), C(P )), we have E FB (R; P, ɛ ) D(f f), (9) where D(f δ f) is the relative etropy betwee the desities f ad f defied i (7) ad (8). Proof sketch. Proceedig alog the similar lies as the proof of Propositio, we ca reach (6). (Agai observe that S d = S so that the power cosumptio stays the same.) Now whe R > C(P/( + δ)), the by the strog coverse of the codig theorem, P e() as. Thus we have E FB (R; P, ɛ ) γ, as log as B c γ is a large deviatios evet, amely, γ > D(f δ f). By solvig R for R = C(P/( + δ)), we ca rewrite the upper boud (9) as a fuctio of (R, P, ɛ ), leadig to (8). If R C with ɛ held fixed, or equivaletly, δ 0, the we have D(f δ f) 0. This proves that E FB(R; P, ɛ ) 0 as R C, so the reliability fuctio is cotiuous at R = C for every ɛ > 0. B. Upper Boud via Geie Cosider a geie-aided scheme i which the ecodig fuctios are allowed to deped o the S i sequece ocausally, that is, X i = f i (W, S ) istead of f i (W, S i ). Assume further that the decoder is also give access to S i additio to Y, i.e., Ŵ = φ(y, S ). By coditioig o S we the see that this ew chael is equivalet to the stadard ofeedback additive white Gaussia oise chael with oise variace equal to var(z i S i ) = var(z i Z i + Z i ) = ɛ /(ɛ + ). Obviously, the reliability fuctio of this ew problem domiates that of the origial problem, sice here the ecoder ad decoder are supplied with more iformatio. Therefore, we have proved the followig statemet. Propositio 3. E FB (R; P, ɛ ) E ( R; P ) (ɛ + ) ɛ. (0) Simple as the argumet leadig to it may be, the boud (6) is tighter tha (9) i may cases. Furthermore, from (6), we ca coclude that the oisy feedback ca be o more useful tha havig the power icrease by the factor of ( + ɛ )/ɛ i the absece of feedback. This observatio, combied with the sphere packig boud o the ofeedback reliability fuctio [6], [], implies that E FB (R; P, ɛ ) P ( + ɛ ) ɛ. () III. SENSITIVITY OF LINEAR FEEDBACK CODING SCHEMES It is relatively well-kow that the Schalkwijk Kailath codig scheme is sesitive to the oise i the feedback lik; see, for example, Schalkwijk [4, Part II, Sectio II-D] or Laveberg [0, p. 479]. I this sectio, we give a simple proof of this observatio ad exted it to a class of codig schemes based o liear ecodig of message ad feedback sigals. The basic idea is that ay successful liear feedback codig scheme that achieves a positive rate R > 0 uder ɛ > 0 must have the ifiite reliability for all rates R < R. This clearly cotradicts the upper boud E FB (R; P, ɛ ) = O(/ɛ ) obtaied i the previous sectio, ad therefore a liear feedback codig scheme caot achieve ay positive rate. We ow make this simple argumet more precise. First a sequece of (M, ) codes is called liear if the ecodig fuctios f () i ca be expressed i the followig form: X i = f () i (W, Ỹ i ) = L () i (θ(w ), Ỹ i ), i =,...,, where θ(w ) takes values i R k with dimesio k idepedet of, ad L () i is a liear fuctio of (θ, Ỹ i ). Propositio 4. For ay sequece of (M, ) liear feedback codes, if P () e 0, the M / 0. Proof sketch. Suppose there exists a sequece of liear feedback codes that achieves R > 0. For simplicity, we assume k =, i.e., θ R. Now from the positive achievable rate, there exists α > 0 such that I(θ; Y ) α for all. But from the liear structure of the ecodig fuctios ad the additive ature of the chael, we ca represet the chael as Y i = α i θ + ξ i, i =,,...,, for appropriately chose Gaussia radom variables (ξ,..., ξ ) idepedet of θ. Sice a Gaussia iput maximizes the mutual iformatio over the Gaussia chael, we thus have I(θ G; Y ) I(θ; Y ) α where θ G is a Gaussia radom variable with the same mea ad variace as θ. But from joit Gaussiaity of (θ G, Y ), it ca be easily verified that there exists a liear fuctio L(Y ) that ca be writte as L(Y ) = θ G + ξ () for some Gaussia radom variable ξ idepedet of θ G, so that I(θ G ; L(Y )) = I(θ G ; Y ) α. This further implies that Eξ (Eθ )/(e α ). This implies that for the chael (), we ca use the uiform message costellatio as i the origial Schalkwijk Kailath codig scheme, which achieves P e () exp( e (α R) ) for ay rate R < α. I particular, E FB (R; P, ɛ ) = for R < α, which clearly cotradicts the fact that E FB (R; P, ɛ ) < for all R ad all ɛ > 0. From a similar argumet, Propositio 4 ca be further exteded to cocateated codes with a liear feedback code as the ier code. IV. LOWER BOUNDS ON THE RELIABILITY FUNCTION I this sectio, we study lower bouds o the reliability fuctio E FB (R; P, ɛ ). I particular, we obtai a codig scheme that achieves the error expoet Ω(/ɛ ). We first cosider the case of of two-message commuicatio, which becomes useful later ad is iterestig o its ow. 49

5 A. Feedback Codig for Two Messages Suppose we wish to commuicate oe of two messages, amely, W {, }, over the oisy feedback chael (), (). From Propositio 3, the optimal error expoet is upper bouded by E FB (biary; P, ɛ ) := lim () log P e P ( + ɛ ) ɛ. O the other had, we have the followig lower boud, which meets the upper boud up to the same costat. Propositio 5. E FB (biary; P, ɛ ) P ɛ. Proof sketch. Suppose W = is set. At time, we sed X = a, ad subsequetly sed X i = a Z i Z i for some positive costat a, the value of which will be specified later. I words, each trasmissio cacels the effect of oises i both forward ad backward chaels. If W = is to be commuicated, the we use a istead of a. After trasmissios, the decoder declares Ŵ = if i= Y i > 0 ad declares Ŵ = otherwise. Coditioed o W =, it is easy to verify that i Y i N(a, +( )ɛ ). By symmetry, this implies that P e (). = exp( a /(ɛ )), while the expected power satisfies E [ i= (X i) ] a + + ɛ =: Q. I other words, we ca achieve lim () log P e = Q ɛ ɛ. Now by operatig oly fractio γ of the time, we ca achieve the error expoet γ(q ɛ )/(ɛ ) uder the power costrait γq. Fially, by takig γ 0 with P = γq held fixed, we have the desired result. B. Three-Phase Protocol Here we sketch a codig scheme that achieves the error expoet lim sup () log P e = Ω(/ɛ ). This codig scheme has three phases. I the first phase, the chael is used as a stadard Gaussia chael without feedback ad at the ed of the first phase, the ecoder guesses from the oisy feedback Ỹ whether the receiver Y has decoded the message correctly. The, depedig o this decisio, the ecoder commuicates i the secod phase a biary message SUCCESS or FAILURE, i a maer that will be described shortly. Fially i the third phase, if a FAIRLURE has bee declared, the message is retrasmitted, this time with very high power for a very short period of time; otherwise, othig is trasmitted. This codig scheme turs out to achieve the desired error expoet uder the give power costrait. Because the differece Z = Ỹ Y betwee the true output ad the oisy observatio has a small variace ɛ, we ca devise a decodig scheme for the first phase (ofeedback codig) so that Y ad Ỹ ca agree upo the correct message with high probability. There are two type of evets to be dealt with. The first type, called Type I (false positive), is the evet that the oisier receiver Ỹ decodes the message correctly, but the true receiver Y does ot. The secod type, called Type II (false egative), is the evet that Ỹ decodes the message icorrectly, so the FAILURE is declared. The basic igrediet for the first-phase code desig is the feedback decoder (o the trasmitter side) that achieves a reasoably small Type II error probability (with positive error expoet), but a very small Type I error probability (with the error expoet /ɛ ). As we saw before, the secod phase also has the expoet /ɛ. Fially, for the third phase, we use a large amout of power (expoetial i block size) ad retrasmit the message via the uiform message costellatio codig as i the Schalkwijk Kailath codig. The error expoet for the third phase is ifiite. But the average power spet ca be made to stay the same, sice the probability of retrasmissio, which is less tha the probability of Type II error, is also expoetially small. REFERENCES [] A. E. Ashikhmi, A. Barg, ad S. N. Litsy, A ew upper boud o the reliability fuctio of the Gaussia chael, IEEE Tras. If. Theory, vol. IT-46, o. 6, pp , 000. [] M. V. Burashev, Data trasmissio over a discrete chael with feedback: Radom trasmissio time, Problems of Iformatio Trasmissio, vol., o. 4, pp. 0 30, 976. [3] S. Butma, A geeral formulatio of liear feedback commuicatio systems with solutios, IEEE Tras. If. Theory, vol. IT-5, o. 3, pp , May 969. [4] M. H. M. Costa, Writig o dirty paper, IEEE Tras. If. Theory, vol. IT-9, o. 3, pp , 983. [5] A. Dembo ad O. 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