On the Reliability of Gaussian Channels with Noisy Feedback

Transcription

1 Forty-Fourth Annual Allerton Conference Allerton House, UIUC, Illinois, USA Sept 7-9, 006 WIIA.0 On the Reliability of Gaussian Channels with Noisy Feedback Young-Han Kim, Amos Lapidoth, Tsachy Weissman Abstract We derive bounds on the reliability of the additive white Gaussian noise channel Y = X +Z with output fed back to the transmitter over independent additive white Gaussian noise channel Ỹ = Y + Z. These bounds appear to be new even for transmission at zero rate. As a corollary, it is shown that linear feedback coding schemes of finite-dimensional constellation, such as the celebrated Schalkwijk Kailath coding scheme, not only fail to achieve the capacity, but in fact cannot achieve any positive rate. Our approach is applicable to the derivation of upper bounds on the error exponents in various other scenarios involving channels with feedback. I. INTRODUCTION That perfect feedback can dramatically improve the reliability of a memoryless channel is pointed out by Shannon in [8]. For the additive white Gaussian noise channel, this fact was reconfirmed by Schalkwijk Kailath [7] in the strong sense that perfect feedback allows for schemes under which the probability of error diminishes double-exponentially fast in block length at any rate below capacity. This double exponential decay has been further extended to the diminishing probability of any number of nested exponential levels by Pinsker [4], Kramer [8], Zigangirov [9]. Much less explored understood is how noise in the feedback link affects the achievable reliability cf. [6], [5], [] for some recent progress. In this paper we restrict attention, for concreteness, to the additive white Gaussian channel Y = X + Z with additive white Gaussian noise corrupted feedback Ỹ = Y + Z, derive bounds on the reliability function. The paper is organized as follows. After describing the problem setup in Section II, we review the case of perfect feedback in Section III. In subsequent two sections, we derive upper bounds on the reliability using two different methods. The first method, based on a change-of-measure argument, leads to a family of upper bounds on the reliability in Section IV. Section V presents another upper bound on the same exponent via a genie-aided argument. For lower bounds, we develop a simple binary messaging scheme in Section VI then show that true zero rate can be achieved by a concatenated code based on the Schalkwijk Kailath feedback coding scheme with error exponent blowing up as the feedback noise variance tends to zero. For positive rate, however, we show that any linear encoding scheme no only Young-Han Kim is with the Department of Electrical Computer Engineering, University of California, San Diego, La Jolla, CA 9093, USA yhk@ucsd.edu Tsachy Weissman is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USAtsachy@stanford.edu Amos Lapidoth is with the Department of Electrical Engineering, Swiss Federal Institute of Technology ETH, Zurich, CH-809, Switzerl lapidoth@isi.ee.ethz.ch fails to achieve capacity, but fails to reliably communicate at any positive rate. We conclude in Section VIII with a description of the way our bounds extend to the non- Gaussian to the discrete setting. Throughout log will denote the natural logarithm, capacity rate will be in nats per channel use. II. CHANNEL MODEL We wish to communicate a message index W {,...,e nr } over the additive white Gaussian noise channel Y i = X i + Z i, where X i, Z i, Y i, respectively, denote the channel input, the additive Gaussian noise, the channel output. Let further Ỹ i denote a noisy version of Y i, Ỹ i = Y i + Z i, where Z i is the Gaussian noise in the backward link. We define a e nr, n code with the encoding function of the form X i = X i W, Ỹ i 3 under the expected average power constraint EXi W; n Ỹ i P, i= the decoding function ŴY n {,...,e nr }. Thus, the encoder has the causal access to the noisy feedback. Equivalently, we can consider the encoding function to be of the form X i W, S i, 4 where S i = Z i + Z i, 5 since, given X i, there is a one-to-one transformation from Ỹ i to S i. The forward backward noise processes, {Z i } { Z i }, are independent of each other, independent identically distributed over time, with respective variances Z i N0, The probability of error P n e P n e = PrW ŴY n S i N0, ε. 6 is defined by = e nr e nr PrW ŴY n W = w w= with W Z n, S n independent of each other. 364

2 WIIA.0 Finally, let E FB R = E FB R; P, ε denote the reliability function associated with this communication problem at rate R, power constraint P, the feedback noise variance ε. As usual, the reliability function is defined as the rate of decay for error probability of the optimal sequence of e nr, n codes, i.e., E FB R = lim sup n log P n n e,optr. We will use the notation ER; P to denote the reliability of the additive white Gaussian noise channel without feedback under signal-to-noise ratio P. III. PERFECT FEEDBACK Here we give a brief review of the case when the feedback noise Z is constant, or equivalently, ε = 0. Our analysis is purely information theoretic thus very simple, compared to the original analysis of the Schalkwijk [6] the later one by Butman [], [3]. Let θ be one of e nr equally spaced real numbers on some interval, say, [, ], with distance between nearest neighbors. Initially, the transmitter sends X 0 = θ. With noiseless feedback, the channel input X i can be an arbitrary function of the previous output values Y0 i = Y 0,...,Y i. Hence, the transmitter can subsequently send X = α Y 0 X 0 = α Z 0 i X i = α i Z 0 Ẑ0Y, i =, 3,..., i where Ẑ0Y is the minimum mean square error MMSE estimate of Z 0 given Y,...,Y i, {α i } i= are chosen to satisfy the power constraint EXi = P for each i. From the orthogonality property of the MMSE estimate the joint Gaussianity, it is easy to see that X i, Z i, Y i are independent of Y i. Furthermore, from our choice of the scaling factor α i, {Y i } i= is i.i.d. N0, P +. Now we look at the mutual information IZ 0 ; Y n. On one h, we have IZ 0 ; Y n = hy n hy n Z 0 = hy i Y i hy i Z 0, Y i = = = i= i= i= hy i hx i + Z i Z 0, Y i hy i hz i Z 0, Y i hy i hz i i= = n log + P = nc. 7 On the other h, we have IZ 0 ; Y n = hz 0 hz 0 Y n = log varz 0 Y n, which combined with 7 implies that varz 0 Y n = e nc. Finally, upon receiving Y 0,..., Y n, the receiver forms the maximum likelihood estimate ˆθ n of θ as ˆθ n = Y 0 Ẑ0Y n = θ + Z 0 Ẑ0Y n Nθ, e nc performs the nearest neighbor decoding. It is easy to see that the decoding error happens only if ˆθ n is closer to the nearest neighbors of the true θ, that is, if N0, e nc >. Since = c 0 e nr with c 0 being a fixed constant depending only on the message constellation, the probability of error is given by P e n = Q c 0 e nc R where Qx = x exp t dt π is the stard Gaussian cumulative density function. Therefore, if R < C, we have P e n exp c 0 e nc R, that is, the probability of error decays doubly exponentially fast in block size n. In particular, this implies for all R < C. E FB R; P, ε = 0 = IV. UPPER BOUNDS VIA CHANGE OF MEASURE In this section, we derive upper bounds on the reliability E FB R; P, ε via a change-of-measure method. The idea is to change the joint law of the noises in the forward the backward links into one under which the noisy feedback is useless. An upper bound on the error exponent of interest is then given by the error exponent under the latter law which is a classical channel coding error exponent plus an additional penalty term stemming from the change of measure. Let Z, S be a generic pair of rom variables distributed as the pair Z i, S i of 5 6, we let Z, S be a pair of independent Gaussians with Z N0, σ S N0, +ε. Let further f f denote the respective densities of Z, S Z, S. Finally, let Λ ε,σ denote the Fenchel-Legendre transform see, e.g., [5] of the rom variable log f Z, S fz, S. 365

3 WIIA.0 We now present the main result of this section. The proof involves changing the law of the noise components in the d d original channel from Z i, S i = S, Z to S i, Z i = Z, S. The feedback in the latter case is useless, so the associated error exponent is ER; P/σ. Theorem. For every σ > 0, E FB P, ε, R γ + E NoFB P/σ, R 8 where γ uniquely solves in the region Λ ε,σ γ = E NoFBP/σ, R γ D f f = [ σ ε + σ ε ] + ε log[σ ε + ε ]. Note that this theorem gives a family of bounds, indexed by σ, which should be viewed as a parameter to be optimized over for a given value of R. The Fenchel-Legendre transform Λ ε,σ can be explicitly obtained. Towards this end note that fz, s = ε π exp f z, s = so that log f Z, S fz, S { + ε z zs + s } ε 9 { π σ + ε exp [ ]} z σ + s + ε 0 = log[σ ε + ε ] [ Z + σ ε + σ ε σ ε Z σ S + ε + S + ε σ + ε ε ] ε. Since both Z σ S +ε are stard Gaussian rom variables, it is evident from that Λ ε,σ γ = Λ ε,σ γ + log[σ ε + ε ], where Λ ε,σ is the Fenchel-Legendre transform of U σ + σ ε ε ε σ + ε ε UV + V ε, 3 with independent stard Gaussian rom variables U V. Fortunately, Λ ε,σ can be derived in closed form Λ ε,σ α = α + σ + ε σ α ε 4 σ + σ + + ε + σ 4 + log α ε 4 σ + + σ + ε + σ 4 α ε 4 which, taken with, gives the explicit form of Λ ε,σ γ. While we have obtained Λ ε,σ γ explicitly, solving for the γ that satisfies Λ ε,σ γ = E NoFB P/σ, R, namely the inverse function of Λ ε,σ, is elusive. It is therefore useful to express the bound of Theorem in parametric form. Towards this end, let Rs, e denote the rate-reliability function of the AWGN channel without feedback under signal-to-noise ratio P error exponent E, i.e., RE; P = { R such that ER; P = E, for 0 E < E0; P, 0, for E E0; P. 5 Note, in particular, that R0; P = log + P. 6 Theorem can be stated equivalently as follows. Theorem. For every σ > 0, E FB R; P, ε is upper bounded by the following curve, which is given in parametric form by [ R Λ ε,σ γ;p/σ, Λ ε,σ γ + γ] 7 where γ varies in the interval [ [ σ ε + σ ε ] ] + ε log[σ ε + ε ], γ max,σ, 8 γ max,σ is the value of γ for which Λ ε,σ γ = E0; P/σ. Although the functions E ; P R ; P are not explicitly known, bounds on these functions can be combined with the theorems above to obtain concrete bounds, as illustrated in the following corollaries. Corollary. For every σ > 0, the following curve, in parametric form, is an upper bound to the curve of E FB ; P, ε [ ] Λ ε,σ γ P/σ log + P/σ, Λ ε,σ γ + γ 9 where γ varies in the range given in 8 γ max,σ is the value of γ for which Λ ε,σ γ = P σ. Proof. The corollary follows by further bounding the bound in Theorem using the sphere packing bound []: ER; P E SP R; P P [ ] R, 0 log + P which also implies RE; P log + P [ E P ]. 366

4 WIIA Fig.. Upper bounds on E FB,, R. The purple, red, black, yellow, blue curves are the parametric bounds of Corollary for the respective values of σ:,.5,.,, 0.8. The green curve is the bound of Corollary 3. The green line higher of the two is the straight line bound on the sphere packing bound inequality 0, applied to the bound of Proposition. The lower green line, for comparison, is the straight line bound on the sphere packing bound of the same channel with no feedback. This line intersects the x-axis at the capacity of this channel, log nats. Thus, the region where the bounds can be useful is 0 R log. Figure shows the curves of Corollary, for the case of ε =, different values of σ. It is seen that there is no one value which dominates for all values of R, although there are values that are dominated for all values of R at the relevant range [0, C] blue curve, corresponding to σ = 0.8. Another immediate consequence of Theorem 0 is the following result. Corollary. E0; P, ε for any σ > 0 γ satisfying Λ ε,σ γ P + γ, σ P σ 3 γ σ + σ ε ε + ε log[σ ε + ε ]. For another working point, note that by choosing σ = P/e R smallest value of σ for which ER; P/σ = 0 in Theorem, we obtain: Corollary 3. For any R > 0, ER; P, ε P e R + ε ε + ε log Pε + ε e R. 4 The bound of Corollary 3, for the case ε =, is plotted in Figure green curve. Note that, as it should, this curve passes through the endpoints of the curves of Corollary. We conclude this section with the proof of Theorem. Proof of Theorem. Fix a particular coding scheme, for the setting of 6, operating at average power upper bounded by P. Let P e m denote its probability of error when transmitting the message m. In other words, for block length n, P e W=w = PrZ n, S n A w, 5 A w denoting the error set A w = {z n, s n : Ŵy n w}, 6 where Ŵyn in 6 denotes the decoder estimate under the fixed coding scheme the realized values z n, s n when encoding for the message m. Consider now the following scenario of communications with useless feedback: Y i = X i + Z i, X i = X i w, S i, 7 where {Z i } {S i } are independent white noises with Z i N0, σ, S i N0, + ε. 8 Let P e w denote the probability of error of the coding scheme associated with P e w, when operating in the useless feedback setting. Thus, denoting { B γ = z n, s n : n log f z n, s n } fz n, s n γ, 9 we have P e w = PrZ n, S n A w = f z n, s n dz n ds n A w = f z n, s n dz n ds n A w B γ + f z n, s n dz n ds n A w B c γ e nγ fz n, s n dz n ds n A w B γ + f z n, s n dz n ds n A w Bγ c e nγ fz n, s n dz n ds n A w + f z n, s n dz n ds n Bγ c = e nγ PrZ n, S n A w + Pr Z n, S n Bγ c = e nγ P e w + Pr Z n, S n Bγ c. 30 Averaging the two sides of 30 over w gives P e enγ P e + Pr Z n, S n Bγ c or, equivalently, 3 e nγ P e e nγ Pr Z n, S n B c γ Pe. 3 Assuming γ was chosen such that Pr Z n, S n Bγ c < P e, 3 is, in turn, equivalent to n log P e γ n log[p e Pr Z n, S n B c γ ]

5 WIIA.0 implying n log P e γ n log[p min n R; P, σ Pr Z n, S n Bγ c ], 34 where P n min R; P, σ denotes the minimum probability of error achievable with block-length l in the useless feedback setting of 7 8. Since, by definition of ER; P, lim n log P n min n R; P, σ = ER; P/σ, 35 it follows that E FB R; P, ε γ + ER; P/σ 36 for γ sufficiently large that lim inf n n log Pr Z n, S n Bγ c > ER; P/σ. 37 The lim inf in 8, however, is in fact a limit we can explicitly characterize. Indeed, by the definition of B γ in 9, Pr Z n, Z n Bγ c 38 = Pr n log f Z n, S n fz n, S n > γ 39 = Pr log f Z i, S i n fz i= i, S i > γ 40 [ ] thus, for γ E log f Z,S fz,s = D f f, Cramér s theorem cf. [5, Th...3] implies lim n n log Pr Z n, S n Bγ c = Λ γ, 4 where Λ is the Fenchel-Legendre transform of the rom variable log f Z,S fz,s. It follows by substitution of 4 in 37 that 8 holds for all γ D f f satisfying Λ γ > ER; P/σ, consequently, by the continuity strict monotonicity of Λ γ, for the value of γ satisfying Λ γ = ER; P/σ. V. UPPER BOUND VIA GENIE Consider a genie-aided scheme where encoding is allowed to depend on the S i sequence non-causally, i.e., to be of the form X i = X i W, S n instead of X i W, S i. Assume further that the decoder is also given access to S n in addition to Y n, i.e., Ŵ = ŴY n, S n. By conditioning on S n we then see that the capacity error exponent for this setting is exactly that for the stard additive white Gaussian noise channel with no feedback noise variance equals to VarZ i S i = VarZ i Z i + Z i = ε ε +. Of course, the capacity error exponent for this problem upper bound those of our problem, since here encoder decoder are supplied with more information. Therefore we have the following: Note that we have used here the fact that S n d = S n, which implies that the power used by the scheme in the useless feedback setting is identical to that used in the original setting. Proposition. E FB R; P, ε E R; Pε + ε. 4 Simple as the argument leading to it may be, the bound of Proposition is, in many cases, tighter than those of the previous section see Figure for a comparison in the case ε =. Furthermore, the bound allows us to conclude that the noisy feedback at least insofar as the fundamental limits go can be no more useful than having the power increase by P/ε in the absence of feedback. Furthermore, when combined with the sphere packing bound on E NoFB, Proposition gives E FB R; P, ε P ε + ε 43 implying that E FB R; P, ε increases with small ε essentially no faster than P ε. The following section shows that this bound is very tight at R = 0. Finally, we note that the bound of Proposition can potentially be tightened by denying the decoder of the genieaided scheme access to S n. The non-feedback reliability on the right-h side of 4 can thus be replaced by the error exponent of the corresponding dirty paper problem [4], [0]. Unfortunately, it is as yet unknown whether the latter is strictly smaller than the stard nonfeedback reliability. VI. ZERO RATE AND SMALL FEEDBACK NOISE In this section we consider the asymptotic regime of zero rate i.e., R 0 feedback noise of very small variance i.e., ε 0. Specializing inequality 43 to this regime yields lim sup ε E FB R; P, ε P 44 ε 0 for all R. We shall first show that this upper bound is achievable in the two-message setting. Denoting the two-message reliability by E FB,binary P, ε, we shall thus show that lim inf ε 0 ε E FB, binary P, ε P. 45 Towards this end, we consider the following scheme for communicating one bit, 0 or. Suppose we wish to communicate. Then we take the channel input as X i = a S i = a Z i Z i 46 for some positive a whose value is to be determined. The channel output will then be Y i = X i + Z i = a Z i + Z i Z i. 47 If 0 is to be communicated, then we use a instead of a. The decoder declares that was sent if n i= Y i > 0; otherwise, it decides on 0. Thus, conditioned on being sent, n Y i = na + Z n Z i i= i= Nna, + n ε,

6 WIIA.0 which implies P n e = P n i= Y i na + n ε na + n ε na = Q + n ε. = exp n a ε, 49 while the average power is E[X i ] = a + + ε. 50 Thus we have the following lower bound on the reliability. Proposition. For all a > 0, ε > 0 E FB,binary a + + ε, ε a ε 5 by operating only fraction α of the time for any 0 < α <, E FB,binary αa + + ε, ε α a ε. 5 We can now use this proposition to prove 45. Proposition implies that for P >, lim inf ε 0 ε E FB,binary P, ε P, consequently, for any 0 α Q >, 53 lim inf ε 0 ε E FB,binary αq, ε α Q. 54 It follows that for any P > 0 0 < α < min{, P }, by taking Q = P/α in 54, lim inf ε 0 ε E FB,binary P, ε P α, 55 implying 45 when taking α 0. Now we consider the true zero rate i.e., any subexponential number of messages. The basic idea is to use k iterations of Schalkwijk Kailath coding as inner code to transform the noisy feedback channel into a stard additive white Gaussian noise channel without feedback under new signal-to-noise ratio P k. Then, by employing the optimal code for this new channel, we can achieve the error exponent EkR; P k. k In other words, we are using the concatenated coding with the k feedback iterations as the innner code the optimal Gaussian nonfeedback code as the outer code. The details follow. Let + P + ε α = + ε β = P + P + ε. If the transmitter uses the Schalkwijk Kailath described in Section III with the noisy output Ỹ in place of the real output Y, we have X i = α i X ˆX Ỹ i = αx i βỹi where ˆXi Ỹ i = EX Ỹ i. From our analysis of the Schalkwijk Kailath coding scheme, it is straightward to check that + ε varx ˆX Ỹ n n = P + P + ε. It is also easy to check that i ˆX Ỹ i = αβ α j Ỹ j. j= Now let ˆX Y i denote the true receiver s estimate of X. Then, var ˆX Y i ˆX Ỹ i i var αβ α j Y j Ỹj Therefore, j= i = var αβ α j Zj j= i = αβ α j ε varx ˆX Y n j= Pε + P + ε. varx ˆX Y n + ˆX Y n ˆX Y n varx ˆX Y n + var ˆX Y n ˆX Y n + varx ˆX Y n / var ˆX Y n ˆX Y n / + ε n Pε P + P + ε + + ε + P + ε n ε + P + P + ε + ε + P =: V n P, ε. Now take [ n = n ε log ε ] = log + P + ε. Then, it is easy to check that V n P, ε 4Pε + P + ε 369

7 WIIA.0 so that the n iterations of the Schalkwijk Kailath coding scheme gives a nonfeedback Gaussian channel with effective signal-to-noise ratio P n P, ε = P V n P, ε + ε + P 4Pε. By concatenating the optimal code for stard Gaussian nonfeedback channels as the outer code, we thus achieve E FB R; P, ε En R; P n n. Finally, taking R 0, we have E FB 0; P, ε E0; P n n = Ω ε log/ε. VII. FRAGILITY OF LINEAR FEEDBACK CODING In this section, we show that applying the stard Schalkwijk Kailath coding scheme or its variants directly does not achieve any positive rate, not to mention the capacity of the channel. More precisely, we prove the following result: Proposition 3. Consider any sequence of M n, n codes, with average power bounded by P, the following structure: the message index W {,...,e nr } is first mapped to a finite dimensional constellation point as θw R k for some finite k independent of n, then encoded linearly for each time index i as X i W, Ỹ i = L i θw, Ỹ i for some affine map L i. Assume further that the Then, P n e 0 imples that logm n 0. n This result is rather surprising, especially because the linear feedback coding scheme such as the Kailath Schalkwijk coding scheme its variants have been very successful in many communication scenarios under perfect feedback, such as the Gaussian nonwhite feedback channel [7], Gaussian multiple access channel [3], [9], writing-on-dirty paper with feedback [], to name a few. Proof. Proof by contradiction. Suppose there exists a sequence of linear feedback coding schemes that achieve R > 0. For simplicity, we assume that k =. Then according to the structure of the linear coding scheme, we have θ = θw R, which, combined with the positive achievable rate, implies that there exists α > 0 such that n Iθ; Y n α for all n. Now from the restriction of linear feedback coding the additive nature of the channel, we can represent the channel as Y i = α i θ + ξ i, i =,,...,n for a Gaussian rom vector ξ,...,ξ n. Since a Gaussian input maximizes the mutual information over Gaussian channels, we have n I θ; Y n n Iθ; Y n α where θ NEθW, varθw. Now from joint Gaussianity of θ, Y n, it is easy to check that there exists a linear function LY n such that I θ; LY n = I θ; Y n > α 56 LY n = θ + ξ where ξ N0, Eξ is independent of θ. Furthermore, from 56, we can easily lower bound Eξ as Eξ Eθ e nα. But suppose we use the uniform message constellation for the channel θ LY n, as in the original Schalkwijk Kailath coding scheme. Then, we can achieve P n e exp enα R for any rate R < α. In particular, E FB R; P, ε = for R < α, which contradicts our upper bounds stating E FB R; P, ε < for all R. VIII. CONCLUDING REMARKS We have obtained upper bounds on the reliability of Gaussian channels with noisy feedback, which show that the reliability E FB R; P, ε under power constraint P noisy feedback with noise variance ε is finite for every positive ε in fact E FB R; P, ε = ER; O/ε in low ε asymptotics. As a matching lower bound, we presented a feedback coding scheme that achieves E FB R = 0; P, ε ER = 0; O/ε δ for all δ > 0. However, the problem is still open which feedback coding scheme can achieve the error exponent that blows up as ε 0 at strictly positive rate, if this is indeed possible. Linear feedback coding schemes, which are wellknown to be very effective under perfect feedback in many other scenarios, fail to achieve this goal in the strong sense that they cannot achieve a positive rate of reliable communication. Although we have restricted our attention to the Gaussian channel with noisy feedback, our approach techniques 370

8 WIIA.0 are applicable more generally. One straightforward generalization is to the case of non-gaussian channels. It is readily verified that the proof of Theorem carries over to the case where the noise components are generally distributed. More specifically, if instead of 6 we have Z fz Z fs that are independent, as before, S = Z + Z, we can take Z, S to be a pair of independent variables where Z can be arbitrarily distributed f z S d = S. Then, we have E FB R; P, f γ + ER; P, f 57 for any γ D f f for which Λ ε,σ γ ER; P, f, where Λ ε,σ is the Fenchel Legendre transform of the rom variable log f Z,S fz,s. Note that the restriction S d = S guarantees that the input constraint P used by the scheme in the useless feedback setting is the same as in the original setting. A similar bound can be obtained for finite-alphabet channels with modulo-additive noise. For this case, in the absence of power or cost constraints, the restriction S d = S is no longer required. The approach behind the bound of Section V can also be extended to the non-gaussian discrete cases. As another direction of extensions, we can allow encoding of the output signal Y over the backward channel Ỹ = X + Z under the feedback function X i = X i Y i with power constraint P. Thus, we use the Gaussian two-way channel for one-way information flow. Moreover, instead of block coding, we may use the variable-length coding. With these two additional power, we can show that the reliability is lower-bounded as E FB R; P, P, ε ER; O/ε. Unfortunately, our upper bounding techniques are not applicable to the encoded feedback. Also for the variable-length coding with noisy feedback, the problem formulation involves some subtlety, because the transmitter the receiver cannot agree upon the stopping time of communication, due to the uncertainty stemming from noise in the feedback link. These other extensions will be detailed elsewhere. ACKNOWLEDGMENT Discussions with Amir Dembo are gratefully acknowledged. REFERENCES [] A. E. Ashikhmin, A. Barg, S. N. Litsyn, New Upper Bound on the Reliability Function of the Gaussian Channel, IEEE Trans. Inform. Theory, vol. IT-46, Number 6, pp , September 000. [] S. Butman, A general formulation of linear feedback communication systems with solutions, IEEE Trans. Inf. Theory, vol. IT-5, no. 3, pp , May 969. [3], Linear feedback rate bounds for regressive channels, IEEE Trans. Inf. Theory, vol. IT-, no. 3, pp , May 976. [4] M. H. M. Costa, Writing on Dirty Paper, IEEE Trans. Inform. Theory, vol. IT-9 pp , May 983. [5] A. Dembo O. Zeitouni, Large Deviations Techniques Applications, nd ed. Springer, New York, 997. [6] S. C. Draper A. Sahai, Noisy feedback improves communication reliability, In Int. Symp. Inf. Th., Seattle, Washington, July 006 [7] Y.-H. Kim, Feedback capacity of stationary Gaussian channels, submitted to IEEE Trans. Info. Theory, Feb available at arxiv.org/cs.it/0609 [8] A. J. Kramer, Improving communication reliability by use of an intermittent feedback channel, IEEE Trans. Inf. Theory, vol. IT-5, pp. 5 60, Jan [9] G. Kramer M. Gastpar, Dependence balance the Gaussian multiaccess channel with feedback, in IEEE Information Theory Workshop, Punta del Este, Uruguay, March 006, pp [0] T. Liu, P. Moulin, R. Koetter, On error exponents of modulo lattice additive noise channels, IEEE Trans. Inf. Theory, vol. IT-5, no., pp , Feb 006. [] N.C. Martins T. Weissman, Coding Schemes for Additive White Noise Channels with Feedback Corrupted by Quantization or Bounded Noise, available at the arxiv. [] N. Merhav T. Weissman, Coding for the feedback Gel f- Pinsker channel the feedforward Wyner-Ziv source, Int. Symp. Inf. Th., Adelaide, Australia, September 005. [3] L. H. Ozarow, The capacity of the white Gaussian multiple access channel with feedback, IEEE Trans. Inf. Theory, vol. IT-30, no. 4, pp , 984. [4] M. S. Pinsker, The probability of error in block transmission in a memoryless Gaussian channel with feedback, Problemy Peredači Informacii, vol. 4, no. 4, pp. 3 9, 968. [5] A. Sahai T. Şimşek, On the variable-delay reliability function of discrete memoryless channels with access to noisy feedback, In IEEE Inf. Th. Workshop, San Antonia, Texas, 004. [6] J. P. M. Schalkwijk, A Coding Scheme for Additive Noise Channels with Feedback-II: B-limited Signals, IEEE Trans. Inform. Theory, vol. pp , Apr [7] J. P. M. Schalkwijk T. Kailath, A Coding Scheme for Additive Noise Channels with Feedback-I: No Bwidth Constraint, IEEE Trans. Inform. Theory, vol. pp , Apr [8] C. E. Shannon, The Zero-Error Capacity of a Noisy Channel, IRE Trans. Inform. Theory, vol. IT-, pp. 8 9, Sept [9] K. Sh. Zigangirov, Upper Bounds for the Error Probability for Channels with Feedback, Prob. Inf. Trans., Volume 6, Number, April- June, 970, pp