Approximate Capacity of Fast Fading Interference Channels with no CSIT

Transcription

1 Approximate Capacity of Fast Fading Interference Channels with no CSIT Joyson Sebastian, Can Karakus, Suhas Diggavi Abstract We develop a characterization of fading models, which assigns a number called logarithmic Jensen s gap to a given fading model. We show that as a consequence of a finite logarithmic Jensen s gap, approximate capacity region can be obtained for fast fading interference channels FF-IC for several scenarios. We illustrate three instances where a constant capacity gap can be obtained as a function of the logarithmic Jensen s gap. Firstly for an FF-IC with neither feedback nor channel state information at transmitter CSIT, if the fading distribution has finite logarithmic Jensen s gap, we show that a rate-splitting scheme based on average interference-to-noise ratio inr can achieve its approximate capacity. Secondly we show that a similar scheme can achieve the approximate capacity of FF-IC with feedback and delayed CSIT, if the fading distribution has finite logarithmic Jensen s gap. Thirdly, when this condition holds, we show that point-to-point codes can achieve approximate capacity for a class of FF-IC with feedback. We prove that the logarithmic Jensen s gap is finite for common fading models, including Rayleigh and Nakagami fading, thereby obtaining the approximate capacity region of FF-IC with these fading models. I. INTRODUCTION The -user Gaussian IC is a simple model that captures the effect of interference in wireless networks. Significant progress has been made in the last decade in understanding the capacity of the Gaussian IC [0], [6], [7], []. In practice the links in the channel could be time-varying rather than static. Characterizing the capacity of FF-IC without CSIT has been an open problem. In this paper we make progress in this direction by obtaining the capacity region of certain classes of FF-IC without CSIT within a constant bits/s/hz. Shorter versions of this work appeared in [0], [9] with outline of proofs. This version has complete proofs. This work was supported in part by NSF grants 5453 and

2 A. Related work Previous works have characterized the capacity region to within a constant gap for the IC where the channel is known at the transmitter and receiver. The capacity region of the -user IC without feedback was characterized to within bit/s/hz in [7]. In [], Suh and Tse characterized the capacity region of the IC with feedback to within bits/s/hz. These results were based on the Han-Kobayashi scheme [0], where the transmitters use superposition coding splitting their messages into common and private parts, and the receivers use joint decoding. Other variants of wireless networks based on the IC model have been studied in literature. The interference relay channel IRC, which is obtained by adding a relay to the -user interference channel IC setup, was introduced in [7] and was further studied in [3], [4], [3], [9]. In [8], Wang and Tse studied the IC with receiver cooperation. The IC with source cooperation was studied in [6], [9]. When the channels are time varying most of the existing techniques for IC cannot be used without CSIT. In [8], Farsani showed that if each transmitter of FF-IC has knowledge of the inr to the non-corresponding receiver, the capacity region can be achieved within bit/s/hz. The idea of interference alignment [5] has been extended to FF-IC to obtain the degrees of freedom DoF region for certain cases. The degrees of freedom region for the MIMO interference channel with delayed CSIT was studied in [6]. Their results show that when all users have single antenna, the DoF region is same for the cases of no CSIT, delayed CSIT and instantaneous CSIT. The results from [] show that DoF region for FF-IC with output feedback and delayed CSIT is contained in the DoF region for the case with instantaneous CSIT and no feedback. Kang and Choi [] considered interference alignment for the K-user FF-IC with delayed channel state feedback and showed a result of K K+ DoF. They also showed the same DoF can be achieved using a scaled output feedback, but without channel state feedback. Therefore, the above works have characterizations for DoF for several fading scenrios, and also show that for single antenna systems, feedback is not very effective in terms of DoF. However, as we show in this paper, the situation changes when we look for more than DoF, and for approximate optimality of the entire capacity region. In particular, we allow for arbitrary channel gains, and do not limit ourselves to For Tx the non-corresponding receiver is Rx and similarly for Tx the non-corresponding receiver is Rx

3 3 SNR-scaling results. In particular, we show that though the capacity region is same within a constant for the cases of no CSIT, delayed CSIT and instantaneous CSIT, there is a significant difference with output feedback. When there is output feedback and delayed CSIT the capacity region is larger than that for the case with with no feedback and instantaneous CSIT in contrast to the DoF result from []. This gives us a finer understanding of the role of CSIT as well as feedback in FF-IC with arbitary and potentially asymmetric link strengths, and is one of the main contributions of this paper. Some simplified fading models have been introduced to characterize the capacity region of the FF-IC in the absence of CSIT. In [7], Wang et al. considered the bursty IC, where the presence of interference is governed by a Bernoulli random state. The capacity of one-sided IC under ergodic layered erasure model, which considers the channel as a time-varying version of the binary expansion deterministic model [], was studied in [], [30]. The binary fading IC, where the channel gains, the transmit signals and the received signals are in the binary field was studied in [4], [5] by Vahid et al. In spite of these efforts, the capacity region of FF-IC without CSIT is still unknown, and this paper presents what we believe to be the first approximate characterization of the capacity region of FF-IC without CSIT, for a class of fading models satisfying the regularity condition, defined as the finite logarithmic Jensen s gap. B. Contribution and outline In this paper we first introduce the notion of logarithmic Jensen s gap for fading models. This is defined in Section III as a number calculated for a fading model depending on the probability distribution for the channel strengths. It is effectively the supremum of log E [link strength] E [log link strength] over all links and operating regimes of the system. We show that common fading models including Rayleigh and Nakagami fading have finite logarithmic Jensen s gap, but some fading models like bursty fading [7] have infinite logarithmic Jensen s gap. Subsequently we show the usefulness of logarithmic Jensen s gap in obtaining approximate capacity regions of FF-ICs without CSIT. We show that Han-Kobayashi type rate-splitting schemes [0], [6], [7], [] based on inr, when extended to rate-splitting schemes based on E [inr] for the FF- ICs, give the capacity gap as a function of logarithmic Jensen s gap, yielding the approximate However, we can also use our results to get the generalized DoF studied in [7] for the FF-IC. This shows that for generalized DoF, feedback indeed helps, as shown in our results.

4 4 capacity characterization for fading models that have finite logarithmic Jensen s gap. Since our rate-splitting is based on E [inr], it does not need CSIT. The constant gap capacity result is first obtained for FF-IC without feedback or CSIT. We also show that for the FF-IC without feedback, instantaneous CSIT cannot improve the capacity region over the case with no CSIT, except for a constant gap. We subsequently study FF-IC with feedback and delayed CSIT to obtain a constant gap capacity result. In this case, having instantaneous CSIT cannot improve the capacity region over the case with delayed CSIT. The usefulness of logarithmic Jensen s gap is further illustrated by analyzing a scheme based on point-to-point codes for a class of FF-IC with feedback, where we again obtain capacity gap as a function of logarithmic Jensen s gap. Our scheme is based on amplify-and-forward relaying, similar to the one proposed in []. It effectively induces a -tap inter-symbol-interference ISI channel for one of the users and a point-to-point feedback channel for the other user. The work in [] had similarly shown that an amplify-and-forward based feedback scheme can achieve the symmetric rate point, without using rate-splitting. Our scheme can be considered as an extension to this scheme, which enables us to approximately achieve the entire capacity region. Our analysis also yields a capacity bound for a -tap fading ISI channel, the tightness of the bound again determined by the logarithmic Jensen s gap. The paper is organized as follows. In section II we describe the system setup and the notations. In section III we develop the logarithmic Jensen s gap characterization for fading models. We illustrate a few applications of logarithmic Jensen s gap characterization in the later sections: in section IV, by obtaining approximate capacity region of FF-IC without feedback, in section V, by obtaining approximate capacity region of FF-IC with feedback and delayed CSIT, and in section VI, by developing point-to-point codes for a class of FF-IC with feedback. II. MODEL AND NOTATION We consider the two-user FF-IC Y l = g lx l + g lx l + Z l Y l = g lx l + g lx l + Z l where Y i l is the channel output of receiver i Rxi at time l, X i l is the input of transmitter i Txi at time l, Z i l CN 0, is complex AWGN noise process at Rxi, and g ij l is the time-variant random channel gain. The channel gain processes {g ij l} are independent

5 5 across links i, j as well as over time. The transmitters are assumed to have no knowledge of the channel gain realizations, but the receivers do have full knowledge of their corresponding channels. We assume that the phase of g ij l is uniformly distributed in [0, π], g ij l is distributed according to φ ij, i, j {, }, and Φ := {φ ij } i,j {,}. We assume average power constraint n n l= X il, i =, at the transmitters, and assume Txi has a message W i {,., NR i}, for a block length of N, intended for Rxi for i =,, and W, W are independent of each other. We denote SNR i := E [ g ii ] for i =,, and INR i := E [ g ij ] for i j. For the instantaneous interference channel gains we use inr i := g ij, i j. Note that we allow for arbitrary channel gains, and do not limit ourselves to SNR-scaling results, but get an approximate characterization of the FF-IC capacity region. Z N 0, Tx X g + Y Rx g g Z N 0, Tx X g + Y Rx Figure. The channel model without feedback. Under the feedback model Figure, after each reception, each receiver reliably feeds back the received symbol and the channel states to its corresponding transmitter. For example, at time l, Tx receives Y l, g l, g l from Rx. Thus X l is allowed to be a function of W, {Y k, g k, g k} k<l. We define symmetric FF-IC to be a FF-IC such that g g g d and g g g c, all of them still being independent. Here g d and g c are dummy random variables according to which the direct links and cross links are distributed. We denote SNR := E [ g d ], and INR := E [ g c ], for the symmetric case. We use the vector notation g = [g, g ], g = [g, g ] and g = [g, g, g, g ]. For schemes involving multiple blocks phases we use the notation X in k, where k is the user index, i is the block phase index and N is the number of symbols per block. The notation X i k j indicates the jth symbol in the i th block phase of k th user. We explain this in Figure 3.

6 6 Delay Y, g, g Z N 0, Tx X g + Y Rx g g Z N 0, Tx X g + Y Rx Delay Y, g, g Figure. The channel model with feedback. X i k j : Notation for one symbol. Block i Block i Block i + User k Blocklength N... j X in k : Notation for the N symbols of the block taken together. Figure 3. The notation for schemes involving multiple blocks phases. The natural logarithm is denoted by ln and the logarithm with base is denoted by log. Also we define log + := max log, 0. For obtaining approximate capacity region of ICs, we say that a rate region R achieves a capacity gap of δ if for any R, R C, R δ, R δ R, where C is the capacity region of the channel. III. A LOGARITHMIC JENSEN S GAP CHARACTERIZATION FOR FADING MODELS Definition. For a given fading model, let Φ = { φ : g ij φ, for some i, j {, } } be the set of all probability distributions, that the fading model induce on the channel link strengths g ij, across all operating regimes of the system. We define logarithmic Jensen s gap c JG of the fading model to be c JG = sup log a + E [W ] E [log a + W ]. 3 a R +,W φ Φ

7 7 In other words it is the smallest value of c such that log a + E [W ] E [log a + W ] c, 4 for any a 0, for any φ Φ, with W distributed according to φ. The following lemma converts requirement in Definition to a simpler form. Lemma. The requirement log a + E [W ] E [log a + W ] c for any a 0, is equivalent to log E [W ] E [log W ] = E [log W ] c, where W = W. E[W ] Proof: We first note that letting a = 0 in the requirement log a + E [W ] E [log a + W ] c shows that log E [W ] E [log W ] = E [log W ] c is necessary. To prove the converse, note that ξ a = log a + E [W ] E [log a + W ] 0 due to Jensen s inequality. Taking derivative with respect to a and again using Jensen s inequality we get [ ] ln ξ a = a + E [W ] E 0. 5 a + W Hence ξ a achieves the maximum value at a = 0 in the range [0,. Hence we have the equivalent condition log E [W ] E [log W ] c, 6 which is equivalent to E [log W ] c. 7 Hence it follows that for any distribution that has a point mass at 0 for example, bursty interference model [7], we do not have a finite logarithmic Jensen s gap, since it has E [log W ] =. Now we discuss a few distributions that can be easily shown to have a finite logarithmic Jensen s gap. Note that any finite c, which satisfies Equation 4, is an upper bound to the logarithmic Jensen s gap c JG. A. Gamma distribution Gamma distribution generalizes some of the commonly used fading models, including Rayleigh and Nakagami fading. The probability density function for Gamma distribution is given by f w = wk e w θ θ k Γk for w > 0, where k > 0 is the shape parameter, and θ > 0 is the scale parameter. 8

8 8 Proposition 3. If the elements of Φ are Gamma distributed with shape parameter k, they satisfy Equation 4 with constant c = loge α log + α for any 0 < α k. Proof: Using Lemma, it is sufficient to prove log E [W ] E [log W ] loge α log + α. It is known for the Gamma distribution that E [W ] = kθ and E [ln W ] = ψ k + ln θ, where ψ is the digamma function. Therefore log E [W ] E [log W ] = loge ln k ψ k 9 We first use the following property of digamma function ψ k = ψ k + k, 0 and then use the inequality from [4] ln k + < ψ k + < ln k + e γ. Hence log E [W ] E [log W ] < loge ln k ln k + + k = log e α log +. α The last step follows because the function involved is decreasing in k in the range 0, and since it is assumed 0 < α k. Corollary 4. If the elements of Φ are are exponentially distributed which corresponds to Rayleigh fading, they satisfy Equation 4 with constant c = Proof: In Rayleigh fading model the g ij is exponentially distributed. The exponential distribution itself is a special case of Gamma distribution with k =. Substituting α = in we get log E [W ] E [log W ] > Nakagami fading can be obtained as a special case of the Gamma distribution; in this case the logarithmic Jensen s gap will depend upon the parameters used in the model. B. Weibull distribution The probability density function for Weibull distribution is given by f w = k w k e w/λ k 3 λ λ

9 9 for x > 0 with k, λ > 0. Proposition 5. If the elements of Φ are Weibull distributed with parameter k, they satisfy Equation 4 with c = γ loge α + log Γ + α for any 0 < α k, where γ is Euler s constant. Proof: For Weibull distributed W, we have E [W ] = λγ + k and E [ln W ] = ln λ γ k, where Γ denotes the gamma function and γ is the Euler s constant. Hence for 0 < α k, it follows that log E [W ] E [log W ] γ log e α + log Γ +. 4 α Using Lemma concludes the proof. Note that exponential distribution can be specialized from Weibull distribution as well, by setting k =. Hence we get the tighter gap in the following corollary. Corollary 6. If the elements of Φ are exponentially distributed, they satisfy Equation 4 with constant c = In the following table we summarize the values we obtain as upper bound on logarithmic Jensen s gap, according to Definition and Equation 4 for different distributions. Table I UPPER BOUND OF LOGARITHMIC JENSEN S GAP FOR DIFFERENT DISTRIBUTIONS Fading Model c Rayleigh 0.83 Gamma k = 0.86 Gamma k = 0.40 Gamma k = Weibull k = 0.83 Weibull k = 0.4 Weibull k = 3 0. C. Other distributions Here we give a lemma that can be used together with Lemma to verify whether a given fading model has a finite logarithmic Jensen s gap.

10 0 Lemma 7. If the cumulative distribution function F w of W satisfies F w aw b w [0, ɛ] for some a 0, b > 0, and 0 < ɛ, then over E [ln W ] ln ɛ + aɛ b ln ɛ aɛb b. 5 Proof: The condition in this lemma ensures that the probability density function fw grows slow enough as w 0 so that fw ln w is integrable at 0. Also the behavior for large values of w is not relevant here, since we are looking for a lower bound on E [ln W ]. The detailed proof is in Appendix A. Hence if the cumulative distribution of the channel gain grows polynomially in a neighborhood of 0, the resulting logarithm becomes integrable, and thus it is possible to find a finite constant c for the Equation 4. In the following sections we make use of the logarithmic Jensen s gap characterization to derive approximate capacity results for FF-ICs. IV. APPROXIMATE CAPACITY REGION OF FF-IC WITHOUT FEEDBACK In this section we make use of the logarithmic Jensen s gap characterization to obtain the approximate capacity region of FF-IC with neither feedback nor CSIT. Theorem 8. For a non-feedback FF-IC with a finite logarithmic Jensen s gap c JG, the rate region R NF B described by 6a 6g is achievable with λ pk = min INR k, : R E [ log + g + λ p g ] R E [ log + g + λ p g ] R + R E [ log + g + g ] + E [ log + λ p g + λ p g ] R + R E [ log + g + g ] + E [ log + λ p g + λ p g ] R + R E [ log + λ p g + g ] + E [ log + λ p g + g ] R + R E [ log + g + g ] + E [ log + λ p g + g ] 6a 6b 6c 6d 6e + E [ log + λ p g + λ p g ] 3 6f R + R E [ log + g + g ] + E [ log + λ p g + g ] + E [ log + λ p g + λ p g ] 3 6g

11 and the region R NF B has a capacity gap of at most c JG + bits/s/hz. Proof: This region is obtained by a rate-splitting scheme that allocates the private message power proportional to. The analysis of the scheme and outer bounds are similar to that in E[inr] [7]. See subsection IV-B for details. Corollary 9. Instantaneous CSIT cannot improve the capacity region of the FF-IC except by a constant. Proof: Our outer bounds in subsection IV-B for the non-feedback IC allow for instantaneous CSIT at the receivers. These outer bounds are within constant gap of the rate region R NF B achieved without CSIT. Hence the proof. Corollary 0. Delayed CSIT cannot improve the capacity region of the FF-IC except by a constant. Proof: This follows from the previous corollary, since instantaneous CSIT is always better than delayed CSIT. Remark. The previous two corollaries are for FF-IC with users and single antennas. It does not contradict the results for MISO broadcast channel, X-channel, MIMO IC and multi-user IC where delayed CSIT or instantaneous CSIT can improve capacity region by more than a constant [], [3], [], [5], [6]. Corollary. Within a constant gap, the capacity region of the FF-IC without feedback is same as the capacity region of IC without fading with equivalent channel strengths SNR i := E [ g ii ] for i =,, and INR i := E [ g ij ] for i j. Proof: This is an application of the logarithmic Jensen s gap result. The rate region of non-feedback case in given in Equations 6a to 6g can be reduced to the rate region for a channel without fading. Let R NF B be the approximately optimal Han-Kobayashi rate region of IC [7] with equivalent channel strengths SNR i := E [ g ii ] for i =,, and INR i := E [ g ij ] for i j. Then for a constant c we have R NF B R NF B R NF B c. 7 This can be verified by proceeding through each inner bound equation.for example, consider the

12 first inner bound Equation 6a R E [ log + g + λ p g ]. The corresponding equation in R NF B is R log + SNR + λ p INR. Now log + SNR + λ p INR a E [ log + g + λ p g ] 8 b log + SNR + λ p INR c JG 9 where a is due to Jensen s inequality and b is by applying logarithmic Jensen s gap result twice. Due to 8, 9 it follows that the first inner bound equation for fading case is in constant gap with that of static case. Similarly by proceeding through each inner bound equation, it follows that for a constant c. R NF B R NF B R NF B c A. Discussion It is useful to view Theorem 8 in the context of the existing results for the ICs. It is known that for ICs, one can approximately achieve the capacity region by performing superposition coding and allocating a power to the private symbols that is inversely proportional to the strength of the interference caused at the unintended receiver. Consequently, the received interference power is at the noise level, and the private symbols can be safely treated as noise, incurring only a constant rate penalty. At first sight, such a strategy seems impossible for the fading IC, where the transmitters do not have instantaneous channel information. What Theorem 8 reveals with the details in subsection IV-B is that if the fading model has finite logarithmic Jensen s gap, it is sufficient to perform power allocation based on the inverse of average interference strength to approximately achieve the capacity region. We compare the symmetric rate point achievable for the non-feedback symmetric FF-IC in Figure 4. The fading model used is Rayleigh fading. The inner bound in numerical simulation is from Equations 0a to 0g in subsection IV-B according to the choice of distributions given in the same subsection. The outer bound is plotted by simulating Equations 4a to 4g in subsection IV-B. The SNR is varied after fixing loginr. The simulation yields a capacity gap logsnr of.48 bits/s/hz for α = 0.5 and a capacity gap of.5 bits/s/hz for α = 0.5. Our theoretical analysis for FF-IC gives a capacity gap of c JG +.83 bits/s/hz independent of α, using data from Table I in Section III.

13 3 5 0 R sym 5,=.5, Outer bound,=.5, Inner bound,=.5, Outer bound,=.5, Inner bound LogSNR Figure 4. Comparison of outer and inner bounds with given α = loginr for non-feedback symmetric FF-IC at the symmetric logsnr rate point. For high SNR, the capacity gap is approximately.48 bits/s/hz for α = 0.5 and.5 bits/s/hz for α = 0.5 from the numerics. Our theoretical analysis yields gap as.83 bits/s/hz independent of α. B. Proof of Theorem 8 From [6] we obtain that a Han-Kobayashi scheme for IC can achieve the following rate region for all p u p u p x u p x u. Note that we use Y i, g i instead of Yi in the actual result from [6] to account for the fading. R I X ; Y, g U 0a R I X ; Y, g U 0b R + R I X, U ; Y, g + I X ; Y, g U, U 0c R + R I X, U ; Y, g + I X ; Y, g U, U 0d R + R I X, U ; Y, g U + I X, U ; Y, g U 0e R + R I X, U ; Y, g + I X ; Y, g U, U + I X, U ; Y, g U 0f R + R I X, U ; Y, g + I X ; Y, g U, U + I X, U ; Y, g U. 0g Now similar to that in [7], choose the Gaussian input distribution U k CN 0, λ ck, X pk CN 0, λ pk, k {, }

14 4 X = U + X p X = U + X p 3 where λ ck + λ pk = and λ pk = min INR k,. Here we introduced the rate-splitting using the average inr. On evaluating the region described by 0a 0g with this choice of input distribution, we get the region described by 6a 6g; the calculations are deferred to Appendix B. Claim 3. An outer bound for the non-feedback case is given by 4a 4g R E [ log + g ] 4a R E [ log + g ] [ ] 4b R + R E [ log + g + g ] + E log + g + g [ ] 4c R + R E [ log + g + g ] + E log + g + g [ ] [ ] 4d R + R E log + g + g + g + E log + g + g + g 4e [ ] R + R E [ log + g + g ] + E log + g + g + g [ ] + E log + g + g 4f [ ] R + R E [ log + g + g ] + E log + g + g + g [ ] + E log + g + g. 4g Proof: The outer bounds 4a and 4b are easily derived by removing the interference from the other user by providing it as side-information. The outer bound in Equation 4e follows from [7, Theorem ]. Those in Equation 4f and Equation 4g follow from [7, Theorem 4]. We just need to modify the theorems from [7] for the fading case by treating Y i, g i as output, and using the i.i.d property of the channels. We illustrate the procedure for Equation 4g in Appendix C. Equation 4e and Equation 4f can be derived similarly.

15 5 The derivation of outer bounds 4c and 4d uses similar techniques as for Equation 4g. We derive Equation 4d in Appendix C. Equation 4d follows due to symmetry. Claim 4. The gap between the inner bound 6a 6g and the outer bound 4a 4g for the feedback case is at most c JG + bits/s/hz. Proof: The proof for the capacity gap uses the logarithmic Jensen s gap property of the fading model. Denote the gap between the first outer bound 4a and first inner bound 6a by δ, δ for the second pair and so on. By inspection δ and δ. Now [ ] δ 3 = E log + g + g E [ log + λ p g + λ p g ] + 5 [ ] a E log + g E [ log + λ p g ] + + c JG 6 + INR b + c JG 7 The step a follows from Jensen s inequality and logarithmic Jensen s gap property of g. The step b follows because λ p = min INR, INR. Similarly we can bound the other + δ s and gather the inequalities as: δ, δ 8 δ 3, δ 4 + c JG 9 δ 5 + c JG 30 δ 6, δ c JG 3 For δ 5, δ 6, and δ 7 we have to use the logarithmic Jensen s gap property twice and hence c JG appears. We note that δ is associated with bounding R, δ with R, δ 3, δ 4, δ 5 with R + R, δ 6 with R + R and δ 7 with R + R. Hence it follows that the capacity gap is at most max δ, δ, δ 3, δ 4, δ 5, δ 6 3, δ 7 3 cjg + bits/s/hz. V. APPROXIMATE CAPACITY REGION OF FF-IC WITH FEEDBACK In this section we make use of the logarithmic Jensen s gap characterization to obtain the approximate capacity region of FF-IC with output and channel state feedback, but transmitters having no prior knowledge of channel states. Under the feedback model, after each reception, each receiver reliably feeds back the received symbol and the channel states to its corresponding

16 6 transmitter. For example, at time l, Tx receives Y l, g l, g l from Rx. Thus X l is allowed to be a function of W, {Y k, g k, g k} k<l. The model is described in section II and is illustrated with Figure in the same section. Theorem 5. For a feedback FF-IC with a finite logarithmic Jensen s gap c JG, the rate region R F B described by 3a 3f is achievable for 0 ρ with λ pk = min INR k, ρ : R E [ log g + g + ρ Re g g + ] R E [ log + ρ g ] + E [ log + λ p g + λ p g ] R E [ log g + g + ρ Re gg + ] R E [ log + ρ g ] + E [ log + λ p g + λ p g ] R + R E [ log g + g + ρ Re gg + ] 3a 3b 3c 3d + E [ log + λ p g + λ p g ] 3e R + R E [ log g + g + ρ Re g g + ] + E [ log + λ p g + λ p g ] 3f and the region R F B has a capacity gap of at most c JG + bits/s/hz. Proof: The proof is in subsection V-A. Corollary 6. Instantaneous CSIT cannot improve the capacity region of the FF-IC with feedback and delayed CSIT except for a constant. Proof: Our outer bounds in subsection V-A allow for feedback and instantaneous CSIT at the receivers. These outer bounds are within constant gap of the rate region R F B achieved using only feedback and delayed CSIT. Hence the proof. Corollary 7. Within a constant gap, the capacity region of the FF-IC with feedback and delayed CSIT is same as the capacity region of a feedback IC without fading with equivalent channel strengths SNR i := E [ g ii ] for i =,, and INR i := E [ g ij ] for i j. Proof: This is again an application of the logarithmic Jensen s gap result. The proof is given in Appendix D.

17 7 A. Proof of Theorem 5 Note that since the receivers know their respective incoming channel states, we can view the effective channel output at Rxi as the pair Y i, g i. Then the block Markov scheme of [, Lemma ] implies that the rate pairs R, R satisfying R I U, U, X ; Y, g R I U ; Y, g U, X + I X ; Y, g U, U, U R I U, U, X ; Y, g R I U ; Y, g U, X + I X ; Y, g U, U, U R + R I X ; Y, g U, U, U + I U, U, X ; Y, g R + R I X ; Y, g U, U, U + I U, U, X ; Y, g 33a 33b 33c 33d 33e 33f for all p u p u u p u u p x u, u p x u, u are achievable. We choose the input distribution according to U CN 0, ρ, U k CN 0, λ ck, X pk CN 0, λ pk 34 X = U + U + X p 35 X = U + U + X p 36 with 0 ρ, λ ck + λ pk = ρ and λ pk = min INR k, ρ. With this choice of λ pk we perform the rate-splitting according to the average inr in place of rate-splitting based on the constant inr. On evaluating the terms in 33a 33f for this choice of input distribution, we get the inner bound described by 3a 3f; the calculations are deferred to Appendix E. An outer bound for the feedback case is given by 37a 37f with 0 ρ : R E [ log g + g + Re ρg g + ] R E [ log + [ ρ g ] ρ ] g + E log + + ρ g R E [ log g + g + Re ρgg + ] R E [ log + [ ρ g ] ρ ] g + E log + + ρ g 37a 37b 37c 37d

18 8 R + R E [ log g + g + Re ρgg + ] [ ρ ] g + E log + + ρ g 37e R + R E [ log g + g + Re ρg g + ] [ ρ ] g + E log + + ρ g, 37f The outer bounds can be easily derived following the proof techniques from [, Theorem 3] using E [X X] = ρ, treating Y i, g i as output, and using the i.i.d property of the channels. The calculations are deferred to deferred to Appendix F. Claim 8. The gap between the inner bound 3a 3f and the outer bound 37a 37f for the feedback case is at most c JG + bits/s/hz. Proof: Denote the gap between the first outer bound and inner bound by δ, for the second pair denote the gap by δ, and so on. We have δ = E [ log g + g + Re ρg g + ] E [ log g + g + ρ Re g g + ] + 38 a = E [ log g + g + g g ρ cos θ + ] E [ log g + g + g g ρ cos θ + ] + 39 b + g + g + + g + g ρ g g = E log + g + g + + g + g ρ g g + 40 c 4 where a is because phases of g, g are independently uniformly distributed in [0, π] yielding Re g g = g g cos θ with an independent θ Unif [0, π], b is using the fact that for p > q π p+ p log p + q cos θ dθ = log q and c is because the numerator in π 0 the fraction is less than the denominator. Now we consider the gap δ between the second inequality 37b of the outer bound and the second inequality 3b of the inner bound. [ ρ ] g δ = E log + + ρ g E [ log + λ p g + λ p g ] + 4

19 9 a E [ log + ρ INR + ρ g ] log + ρ INR + cjg E [ log + λ p g + λ p g ] + 43 [ ρ ] g E log + + ρ E [ log + λ p g ] + + c JG 44 INR b + c JG where a follows by using logarithmic Jensen s gap property on g and Jensen s inequality. The step b follows because ρ + ρ INR = 45 min, ρ = λ p INR ρ INR Similarly by inspection of the other bounding inequalities we can gather the inequalities on the δ s as: δ, δ 3 47 δ, δ 4, δ 5, δ 6 c JG + 48 We note that δ, δ is associated with bounding R, δ 3, δ 4 with R, δ 5, δ 6 with R + R. Hence it follows that the capacity gap is at most max δ, δ, δ 3, δ 4, δ 5, δ 6 cjg + bits/s/hz. VI. APPROXIMATE CAPACITY OF FEEDBACK FF-IC USING POINT-TO-POINT CODES As the third illustration for the usefulness of logarithmic Jensen s gap, we propose a strategy that does not make use of rate-splitting, superposition coding or joint decoding for the feedback case, which achieves the entire capacity region for -user symmetric FF-ICs to within a constant gap. This constant gap is dictated by the logarithmic Jensen s gap for the fading model. Our scheme only uses point-to-point codes, and a feedback scheme based on amplify-and-forward relaying, similar to the one proposed in []. The main idea behind the scheme is to have one of the transmitters initially send a very densely modulated block of data, and then refine this information using feedback and amplifyand-forward relaying for the following blocks, in a fashion similar to the Schalkwijk-Kailath scheme [8], while treating the interference as noise. Such refinement effectively induces a -tap point-to-point inter-symbol-interference ISI channel at the unintended receiver, and a

20 0 point-to-point feedback channel for the intended receiver. As a result, both receivers can decode their intended information using only point-to-point codes. Consider the symmetric fading interference channel, where the channel statistics are symmetric and independent, i.e., g ii l g d and g ij l g c, for i j. We consider n transmission phases, each phase having a block length of N. For Tx, generate nnr codewords X N,..., X nn i.i.d according to CN 0,. Tx encodes its message W { },..., nnr onto X N,..., X nn. For Tx, generate nnr codewords X N = X N i.i.d according to CN 0, and let it encode its message W { },..., nnr N onto X = X N. Note that for Tx the coding block length is N, whereas it is nn for Tx. Tx sends X in in phase i. Tx sends X N = X N in phase. At the beginning of phase i >, Tx receives Y i N = g i N X i N + g i N X i N + Z i N 49 from feedback. It can remove g i N X i N from Y i N to obtain g i N X i N + Z i N. Tx then transmits the resulting interference-plus-noise after power scaling as X in, i.e. Thus in phase i >, Rx receives Y in = g in = g in X in X in = gi N X i N + Z i N INR + g in X in + Z in 5 g i N X i N + Z i N + g in X in + Z in 5 + INR and feeds it back to Tx for phase i +. The transmission scheme is summarized in Table II. Note that for phase i = Tx receives Y N = g N X N + g N X N + Z N 53 and for phase i > Tx observes a block ISI channel since it receives Y in = g in X in + g in g i N X i N + Z i N + INR = g in X in + g in g i N + INR X i N + + Z in 54 Z in 55 where Z in = Z in + gin Z i N +INR.

21 Table II TRANSMITTED SYMBOLS IN n-phase SCHEME FOR SYMMETRIC FF-IC WITH FEEDBACK User Phase Phase.. Phase n X N X N.. X nn X N g N X N +Z N +INR.. g n N X n N +Z n N +INR At the end of n blocks, Rx collects Y N = Y N,..., Y nn and decodes W such that N X W, Y N is jointly typical where X N = X N,..., X nn treating X N = X N as noise. At Rx, channel outputs over n phases can be combined with an appropriate scaling so that the interference-plus-noise at phases {,..., n } are successively canceled, i.e., an effective point-to-point channel can be generated through Ỹ N = n i= n j=i+ g jn +INR Y in see the analysis in the subsection VI-A for details. Note that this can be viewed as a block version of the Schalkwijk-Kailath scheme [8] and the references therein. Given the effective channel Ỹ N, the receiver can simply use point-to-point typicality decoding to recover W, treating the interference in phase n as noise. Theorem 9. For a symmetric FF-IC with a finite logarithmic Jensen s gap c JG, the rate pair [ ]] R, R = log + SNR + INR 3c JG, E [log + g d + INR is achievable by the scheme. The scheme together with switching the roles of users and timesharing, achieves the capacity region of symmetric feedback IC within 3c JG + bits/s/hz. Proof: The proof follows from the analysis in the following subsection. A. Analysis of Point-to-Point Codes for Symmetric FF-ICs We now provide the analysis for the scheme, going through the decoding at the two receivers and then looking at the capacity gap for the achievable region. Decoding at Rx : At the end of n blocks Rx collects Y N = decodes W such that X N W, Y N is jointly typical, where X N =,... Y nn and X N,... X nn. Y N The joint typicality is considered according the product distribution p N X, Y, where p X, Y = p X,... X n, Y,... Y n 56

22 is a joint Gaussian distribution, dictated by the following equations that arise from our n-phase scheme: And for i =, 3,..., n: Y = g X + g X + Z 57 Y i = g i X i + g i g i X i + Z i + INR + Z i 58 with X i, X, Z i being i.i.d CN 0,. Essentially X, Z i are both Gaussian noise for Rx. Using standard techniques it follows that for the n-phase scheme as N user can achieve the rate [log E n ] KY where K n Y n denotes the determinant of covariance K Y X n matrix for the the n-phase scheme defined in the following pattern K Y = [ + g + g ] K Y = g + g g + g +INR + gg +INR g g g +INR g + g + K Y 3 = g 3 + g3 g + +INR + g g3g +INR 0 g g 3g +INR g + g g + +INR + 0 g g g g gg +INR +INR g + g + where g i g d i.i.d and g i], g i g c i.i.d. Letting n, Rx can achieve ] the rate R = lim [log E. We need to evaluate lim [log E. The n n n KY n K Y X n following lemma gives an lower bound on n E [log K Y n ]. Lemma 0. n n E [log K Y n ] n log ˆKY n 3c JG KY n K Y X n where ˆK Y n is obtained from K Y n by replacing g i s,g i s with INR and g i s with SNR. Proof: The proof involves expanding the matrix determinant and repeated application of the logarithmic Jensen s gap property. The details are given in Appendix G. Subsequently we use the following lemma in bounding lim log ˆKY n n. n

23 3 Lemma. If A = [ a ], A = a etc. with a > 4 b, then b lim inf n b a b 0, A 3 = b a a b 0 b a n log A n log a., A 4 = a b 0 0 b a b 0 0 b a b 0 0 b a Proof: The proof is given in Appendix H. For the n-phase scheme, the ˆK Y n matrix has the form A n, as defined in Lemma after identifying a = + INR + SNR and b = SNRINR +INR. Note that with this choice a > 4 b holds due to AM-GM Arithmetic Mean Geometric Mean inequality. Hence we have lim inf n n log + INR + SNR ˆKY n log using Lemma. Also, K Y X n is a diagonal matrix of the form 59 K Y X n = diag Hence using Jensen s inequality lim sup n g n + INR +, g n + INR +,..., g n E [ log K Y X n ] lim sup n = log + INR +, g + n INR n log + INR + + INR INR + INR Hence by combining Lemma 0, Equation 59 and Equation 63, we get is achievable. R log + INR + SNR 3c JG 64 Decoding at Rx : For user we can use a block variant of Schalkwijk-Kailath scheme ] [8] to achieve R = E [log + gd. The key idea is that the interference-plus-noise sent +INR in subsequent slots can indeed refine the symbols of the previous slot. The chain of refinement over n phases compensate for the fact that the information symbols are sent only in the first phase. We have Y N = g N X N + g N X N + Z N 65

24 4 and Y in = g in g i N X i N + Z i N + INR + g in X in + Z in 66 for i >. Now let Ỹ N = n n g jn i= j=i+ +INR Y in. We have Ỹ N = n n i= = g nn + + = g N j=i+ g n N g jn Y in 67 + INR X n N + Z n N + INR g nn g n N + INR g n N + g nn X nn + Z nn X n N g nn g n N g n N g n 3N + INR j= + Z n N + INR X n 3N + Z n 3N + INR + n g jn + g N X N + g N X N + Z N + INR n j= g jn X N + INR + g n N X n N + Z n N + g n N X n N + Z n N 68 + g nn X nn + Z nn. 69 due to cross-cancellation. Now Rx decodes for its message from Ỹ N. Hence Rx can achieve the rate R lim inf n [ n E n log + j= g j + INR g + g n where g,..., g n g d being i.i.d and g n g c. Hence it follows that ] R E [log + g d + INR is achievable. 3 Capacity gap: We can obtain the following outer bounds from Theorem 5 for the special case of symmetric fading statistics. R, R E [ log g d + g c + ] 7 [ ] R + R E log + g d + g c + E [ log g d + g c + g d g c + ] 73 ] 70 7

25 5 where Equation 7 is obtained from Equation 37b and Equation 37d by setting ρ = 0 note that ρ = 0 yields the loosest version of outer bounds in Equation 37b and Equation 37d. Similarly Equation 73 is a looser version of outer bound Equation 37e independent of ρ. The outer bounds reduce to a pentagonal region with two non-trivial corner points see Figure 5. Our n-phase scheme can achieve the two corner points within + 3c JG bits/s/hz for each user. The proof is using logarithmic Jensen s gap property and is deferred to Appendix I. R Achievable by n phase schemes 4.5 bits/s/hz Outer bound R Figure 5. Illustration of bounds for capacity region for symmetric FF-IC. The corner points of the outer bound can be approximately achieved by our n-phase schemes. The gap is approximately 4.5 bits/s/hz for the Rayleigh fading case. B. An auxiliary result: Approximate capacity of -tap fast Fading ISI channel Consider the -tap fast fading ISI channel described by Y l = g d l X l + g c l X l + Z l, 74 where g d CN 0, SNR and g d CN 0, INR are independent fading known only to the receiver and Z CN 0,. Also we assume a power constraint of E [ X ] on the transmit symbols. Our analysis for R can be easily modified to obtain a closed form approximate expression for this channel. This gives rise to the following corollary on the capacity of fading ISI channels. Corollary. The capacity C F ISI of the -tap fast fading ISI channel is bounded by log + SNR + INR + C F ISI log + SNR + INR 3c JG,

26 6 where the channel fading strengths is assumed to have a logarithmic Jensen s gap of c JG. Proof: The proof is given in Appendix J. VII. CONCLUSION We introduced the notion of logarithmic Jensen s gap and demonstrated that it can be used to obtain approximate capacity region for FF-ICs. We proved that the rate-splitting schemes for ICs [7], [6], [], when extended to the fast fading case give capacity gap as a function of the logarithmic Jensen s gap. Our analysis of logarithmic Jensen s gap for fading models like Rayleigh fading show that rate-splitting is approximately optimal for such cases. We then developed a scheme for symmetric FF-ICs, which can be implemented using point-to-point codes and can approximately achieve the capacity region. An important direction to study will be to see if similar schemes with point-to-point codes can be extended to general FF-ICs. Also our schemes are not approximately optimal for bursty IC since it does not have finite logarithmic Jensen s gap, it would be interesting to study if the schemes can be extended to bursty IC and then to any arbitrary fading distribution. APPENDIX A PROOF OF LEMMA 7 We have F w aw b for w [0, ɛ] where a 0, b > 0, ɛ > 0. Now using integration by parts we get E [ln W ] = ˆ 0 ˆ ɛ 0 f w ln w 75 f w ln w + ˆ ɛ f w ln w 76 ˆ ɛ = [F w ln w] ɛ 0 F w ˆ w + f w ln w 77 0 [ aw b ln w ] ˆ ɛ ɛ aw b + ln ɛ 78 0 w aɛ b ln ɛ aɛb b 0 ɛ + ln ɛ. 79 Note that ln w is negative in the range [0,, thus we get the desired inequalities in the last two steps.

27 7 APPENDIX B PROOF OF ACHIEVABILITY FOR NON-FEEDBACK CASE We evaluate the term in the first inner bound inequality 0a. The other terms can be similarly evaluated. I X ; Y, g U a = I X ; Y U, g = h Y U, g h Y X, U, g h Y U, g = h g X + g X + Z U, g = h g X + g X p + Z g variance g X + g X p + Z g = g + λ p g I [ X ; Y, g U = E log g + λ p g + ] + log πe 85 h Y X, U, g = h g X + g X + Z X, U, g 86 = h g X p + Z g 87 = E [ log + λ p g ] + log πe 88 [ b E log + ] g + log πe INR 89 c log + log πe 90 = + log πe, 9 I U, U, X ; Y, g E [ log g + λ p g + ] 9 where a uses independence, b is because λ pi INR i, and c follows from Jensen s inequality. APPENDIX C PROOF OF OUTER BOUNDS FOR NON-FEEDBACK CASE Note that we have the notation g = [g, g, g, g ], S = g X + Z, and S = g X + Z. Our outer bounding steps are valid while allowing X i to be a function of W, g n, thus letting transmitters have instantaneous and future CSIT. On choosing a uniform distribution of messages we get nr + R ɛ n I W ; Y n, g n + I W ; Y n, g n + I W ; Y n, g n 93

28 8 = I W ; Y n g n + I W ; Y n g n + I W ; Y n g n 94 I W ; Y n, S n g n + I W ; Y n g n + I W ; Y n, S n X n, g n 95 = I W ; S n g n + I W ; Y n S n, g n + I W ; Y n g n + I W ; S n X n, g n + I W ; Y n X n, S n, g n 96 = h S n g n h S n W, g n + h Y n S n, g n h Y n W, S n, g n + h Y n g n h Y n W, g n + h S n X n, g n h S n X n, W, g n + h Y n X n, S n, g n h Y n X n, W, S n, g n 97 = h S n g n h Z n + h Y n S n, g n h S n g n + h Y n g n h S n g n + h S n g n h Z n + h Y n X n, S n, g n h Z n 98 = h Y n S n, g n + h Y n g n + h Y n X n, S n, g n h Z n h Z n 99 a [ h Yi S i, g n h Z i ] + [ h Yi g n h Z i ] + [ h Yi X i, S i, g n h Z i ] 00 [ = E g n h Yi S i, g n h Z i ] [ + E g n h Yi g n h Z i ] [ + E g n h Yi X i, S i, g n h Z i ] 0 [ ] b ne log + g + g + g + ne [ log + g + g ] [ ] + ne log + g + g 0 where a is due to the fact that conditioning reduces entropy and b follows from Equations [7, 50], [7, 5] and [7, 5]. Note that in the calculation of step b we allow the symbols X i, X i to depend on g n, but since g n is available in conditioning the calculation proceeds similar to that in [7]. nr + R ɛ n 03 I W ; Y n, g n + I W ; Y n, g n 04 = I W ; Y n g n + I W ; Y n g n 05 I W ; Y n g n + I W ; Y n, S n X n, g n 06 = I W ; Y n g n + I W ; S n X n, g n + I W ; Y n X n, S n, g n 07

29 9 = h Y n g n h Y n W, g n + h S n X n, g n h S n X n, W, g n + h Y n X n, S n, g n h Y n X n, W, S n, g n 08 = h Y n g n h S n g n + h S n g n h Z n + h Y n X n, S n, g n h Z n 09 = h Y n g n + h Y n X n, S n, g n h Z n h Z n 0 a [ h Yi g n h Z i ] + [ h Yi X i, S i, g n h Z i ] [ = E g n h Yi g n h Z i ] [ + E g n h Yi X i, S i, g n h Z i ] b ne [ log [ ] + g + g ] + ne log + g + g 3 where a is due to the fact that conditioning reduces entropy and b again follows from Equations [7, 5] and [7, 5]. Let R NF B APPENDIX D PROOF OF COROLLARY 7 be the approximately optimal Han-Kobayashi rate region of feedback IC [] with equivalent channel strengths SNR i := E [ g ii ] for i =,, and INR i := E [ g ij ] for i j. Then for a constant c we have R F B R F B R F B c. 4 This can be verified by proceeding through each inner bound equation. For example, consider the first inner bound Equation 3a R E [ log g + g + ρ Re g g + ]. The corresponding equation in R NF B is R log + SNR + INR + ρ SNR INR +. Now E [ log g + g + ρ Re g g + ] a log + SNR + INR 5 log + SNR + INR + ρ SNR INR + 6 where a is due to Jensen s inequality and independence of g, g. Also E [ log g + g + ρ Re g g + ] a = E [ log g + g + g g ρ cos θ + ] 7

30 30 b E [ log g + g + ] 8 c log SNR + INR + c JG 9 d log SNR + INR + ρ SNR INR + c JG 0 where a is because phases of g, g are independently uniformly distributed in [0, π] yielding Re g g = g g cos θ with an independent θ Unif [0, π], b is using the fact that for p > q π p+ p log p + q cos θ dθ = log q log p, c is using the logarithmic π 0 Jensen s gap result twice and d is because SNR +INR ρ SNR INR. It follows from Equations 6 and 0, that the first inner bound for fading case is within constant gap with the first inner bound of the static case. Now consider the second inner bound Equation 3b R E [ log + ρ g ] + E [ log + λ p g + λ p g ] and the corresponding equation R log + ρ INR + log + λp SNR + λ p INR 3c JG from R F B. We have E [ log + ρ g ] + E [ log + λ p g + λ p g ] log + ρ INR + log + λp SNR + λ p INR 3 due to Jensen s inequality. And E [ log + ρ g ] + E [ log + λ p g + λ p g ] log + ρ INR + log + λp SNR + λ p INR 3c JG 4 using the logarithmic Jensen s gap result thrice. It follows from Equations 3 and 4, that the second inner bound for fading case is within constant gap with the second inner bound of the static case. Similarly by proceeding through each inner bound equation, it follows that for a constant c. R F B R F B R F B c

31 3 APPENDIX E PROOF OF ACHIEVABILITY FOR FEEDBACK CASE We evaluate the term in the first inner bound inequality 33a. The other terms can be similarly evaluated. I U, U, X ; Y, g a = I U, U, X ; Y g 5 = h Y g h Y g, U, U, X, 6 variance Y g = variance g X + g X + Z g, g 7 = g + g + g g E [X X ] + g g E [X X ] + 8 = g + g + ρ Re g g + 9 h Y g, U, U, X = h g X + g X + Z g, U, U, X = h g X p + Z g 30 3 = E [ log + λ p g ] + log πe 3 [ b E log + ] g + log πe INR 33 c log + log πe 34 = + log πe, 35 I U, U, X ; Y, g E [ log g + g + ρ Re g g + ] 36 where a uses independence, b is because λ pi INR i, and c follows from Jensen s inequality. APPENDIX F PROOF OF OUTER BOUNDS FOR FEEDBACK CASE Following the methods in [], we let E [X X ] = ρ. We have the notation g = [g, g ], g = [g, g ], g = [g, g, g, g ], S = g X + Z, and S = g X + Z. We let E [X X] = ρ = ρ e iθ. All of our outer bounding steps are valid while allowing X i to be a function of W, Y i, g n, thus letting transmitters have full CSIT along with feedback. On choosing a uniform distribution of messages we get nr ɛ n a I W ; Y n g n 37

32 3 b h Yi g i h Zi 38 = E gi [ h Yi g i = g i h Zi ] 39 c = E g [ h Yi g i = g h Zi ] 40 R E [ log g + g + ρ g g + ρg g + ] 4 where a follows from Fano s inequality, b follows from the fact that conditioning reduces entropy, and c follows from the fact that g i are i.i.d. Now we bound R in a second way as done in []: nr ɛ n I W ; Y n, g n I W ; Y n, g n, Y n, g n, W = I W ; g n, W + I W ; Y n, Y n g n, W = 0 + I W ; Y n, Y n g n, W = h Y n, Y n g n, W h Y n, Y n g n, W, W = [ h Yi, Y i g n, W, Y i, Y i = [ h Yi g n, W, Y i, Y i ] [h Zi + h Z i ] 47 ] + [ h Yi g n, W, Y i, Y i [h Z i + h Z i ] 48 a = [ ] [ ] h Yi g n, W, Y i, Y i, X i + h Yi g n, W, Y i, Y i, S i, X i [h Z i + h Z i ] 49 b [ h Yi g i, X i h Zi ] + [ h Yi g i, S i, X i h Zi ] 50 c [ = E g h Yi X i, g i = g h Z i ] [ + E g h Yi S i, X i, g i = g h Z i ] 5 d R E [ log + [ ρ g ] ρ ] g + E log + + ρ g 5 where a follows from the fact that X i is a function of W, Y i, g n and S i is a function of Y i, X i, g n, b follows from the fact that conditioning reduces entropy, c follows from the fact that g i are i.i.d., and d follows from [, 43]. The other outer bounds can be derived ]

33 33 similarly following [] and making suitable changes to account for fading as we illustrated in the previous two derivations. APPENDIX G FADING MATRIX The calculations are given in Equations 53,54. E [log K Y n ] [ = E log g n + g n g n + + K Y n + INR ] g n g n g n K Y n + INR 53 E [log + INR + SNR K Y n INR INR g n + INR K Y n ] 3c JG 54 The first step 53, is by expanding the determinant. We use the logarithmic Jensen s gap property thrice in the second step 54. This is justified because the coefficients of { g n, g n, g n } from Equation 53 are non-negative due to the fact that all the matrices involved are covariance matrices, and the coefficients themselves are independent of { g n, g n, g n }. Note that K Y n depend on g n but not on g n. This procedure can be carried out n times and it follows that: lim n n E [log K Y n ] lim n n log ˆKY n 3c JG 55 where ˆK Y n is obtained from K Y n by replacing g i s,g i s with INR and g i s with SNR. APPENDIX H MATRIX DETERMINANT: ASYMPTOTIC BEHAVIOR The following recursion easily follows: A n = a A n b A n 56