On the K-user Cognitive Interference Channel with Cumulative Message Sharing Sum-Capacity

Transcription

1 03 EEE nternational Symposium on nformation Theory On the K-user Cognitive nterference Channel with Cumulative Message Sharing Sum-Capacity Diana Maamari, Daniela Tuninetti and Natasha Devroye Department of Electrical and Computer Engineering University of llinois at Chicago, Chicago L 60607, USA dmaama, danielat, uic.edu Abstract This paper considers the K-user cognitive interference channel with one primary and K secondary/cognitive transmitters with a cumulative message sharing structure, i.e., cognitive transmitter i, i : K], non-causally knows all messages of the users with index less than i. We first propose a computable outer bound valid for any memoryless channel and show the sum-rate to be achievable for the symmetric K- user Linear Deterministic Channel. nterestingly, for the K- user channel having only the K-th transmitter know all other messages is sufficient to achieve the sum-capacity, i.e., cognition at transmitters to K is not needed. Next, the sum-capacity of the symmetric Gaussian noise channel is characterized to within a constant additive and multiplicative gap, which depend on K. As opposed to other interference channel models, a single scheme suffices for both the weak and strong interference regimes. Moreover it is only required for transmitters to K to have, in addition to their own message, non-causal message knowledge of the transmitter s message.. NTRODUCTON The cognitive radio channel, first introduced in ], consists of two source-destination pairs in which one of the transmitters called the secondary transmitter has non-causal knowledge of the message of the other transmitter known as the primary transmitter. This non-causal message knowledge idealizes a cognitive radio s ability to overhear other transmissions and exploit them to either cancel them out at their own receiver or aid in their transmission. For the state-of-the-art on the -user cognitive channel we refer the reader to ], 3]. n this paper we extend the -user cognitive interference channel model to K-users. The K-user cognitive interference channel CFC consists of one primary and K secondary, or cognitive, users. We assume a cumulative message sharing CMS cognition structure introduced in 4] for the 3-user channel, and extended here to K-users, whereby user is the primary user, and cognitive users i, i : K], know the messages of user through i. The cumulative message cognition model is inspired by the concept of overlaying, or layering multiple co-existing cognitive networks. The first layer consists of the primary users. Each additional cognitive layer transmits simultaneously with the previous layers overlay given the lower layers codebooks. We idealize higher layers being able to obtain lower layers messages through non-causal cognition. Past Work. The literature on multi-user cognitive interference channels is limited, in part since the -user counterpart is not yet fully understood ], 3]. The only other work on a K-CFC with K > 3 is that of 5], 6] to the best of our knowledge. n 5] the channel model consists of one primary user and K parallel cognitive users; each cognitive user only knows the primary message in addition to their own message thus not a cumulative message structure; for this channel model the capacity in the very strong interference regime is obtained 5]. n 6] the sum capacity of the K-user linear deterministic cognitive interference channel with cumulative message sharing is obtained. n 4], 6] 0] different 3-user cognitive channels are considered which differ from the one considered here either in the number of transmitter/receivers, or in the message sharing/cognition structure in all but 4], 6], 7]. n 7], several types of 3- user cognitive interference channels are proposed: that with cumulative message sharing CMS as considered here, that with primary message sharing where the message of the single primary user is known at both cognitive transmitters, and finally cognitive only message sharing CoMS where there are two primary users who do not know each others message and a single cognitive user which knows both primary messages. n 6] the sum-capacity for the 3-user channel with CMS for the linear deterministic channel LDC is obtained our work generalizes this to K-users and to the Gaussian noise channel, where a constant gap to capacity is obtained. Contributions and Outline. We consider the K-user CFC with CMS K-CFC-CMS, with the following contributions: n Section we obtain a novel outer bound region that reduces to the outer bound of ] for the -user case partially presented in 6] for the LDC. The bound is valid for any memoryless channel and any number of users, and does not contain auxiliary random variables and is thus computable for many channels including the Gaussian noise channel. n Section V we derive the sum-capacity for the symmetric K-user LDC exactly; the outer bound was obtained in 6] and here we show achievability. nterestingly, the scheme only requires one user to be fully cognitive of the messages of all other users, which considerably relaxes the CMS requirement. 3 n Section V we derive the sum-capacity for the symmetric K-user Gaussian noise channel to within a constant additive and multiplicative gap. The additive gap is a function of the number of users and grows as K log K ; the proposed achievable scheme can be thought of as a MMObroadcast scheme where only one encoding order is possible due to the CMS mechanism; interestingly, a single scheme suffices for both the weak and strong interference regimes; /3/$ EEE 034

2 03 EEE nternational Symposium on nformation Theory W W W W3 W W X X X3 Fig.. h h h h 3 h 3 h 3 h h 3 h 33 Z Z Z3 The Gaussian 3-CFC-CMS moreover the achievable scheme only requires the first K cognitive users to know the message of the primary user but not those of other cognitive users, again considerably relaxing the CMS requirement. The multiplicative gap is K and is achieved by having all users beam-form to the primary user.. CHANNELMODEL The K-CFC-CMS channel consists of: channel inputs X i X i, i : K], channel outputs Y i Y i, i : K], a memoryless channel PY,..., Y K X,..., X K, and independent messages W i known to users,,..., i, i : K]. A code with non-negative rate vector R,..., R K and block length N is defined by: messages W i, i : K], uniformly distributed over : NRi ], encoding functions Xi N W,..., W i, i : K], decoding functions Ŵi Y N i, i : K], and probability of error P e N : max i :K] PŴi W i ]. The capacity of the K- CFC-CMS channel consists of all rates R,..., R K such that P e N 0 as N. A. The Gaussian Noise Channel The single-antenna complex-valued K-CFC-CMS with Additive White Gaussian Noise AWGN, shown in Fig. for the case K 3, has input-output relationship Y l h li X i Z l, l : K], i :K] where, without loss of generality, the inputs are subject to the power constraint E X i ], i : K], and the noises are marginally the joint distribution does not matter as the receivers cannot cooperate proper-complex Gaussian random variables with parameters Z l N 0,, l : K]. The channel gains h ij, i, j : K], are constant. Without loss of generality we may assume the direct links h ii, i : K] to be real-valued and non-negative since a receiver can compensate for the phase of one of its channel gains. B. The Linear Deterministic Approximation The Linear Deterministic approximation of the Gaussian Noise Channel at high SNR LDC was first introduced in ] and for the K-CFC-CMS has input-output relationship: Y l S m n li X i, l : K], i :K] where m : max i,j :K] {n ij }, S is the binary shift matrix of dimension m, X i and Y i are binary vectors of length m, Y Y Y3 W W W3 and the channel gains n li, l, i : K], are non-negative integers. The channel in can be thought of as the high SNR approximation of the channel in with their parameters related as n ij log h ij, i, j : K].. OUTERBOUND FOR THE K-USER CFC WTH CMS We obtain a general and computable outer bound next. The sum-capacity bound is obtained by giving S i as side information to receiver i, i : K], where S i : S i, W i, Yi N ] starting with S :. With this nested side information, the mutual information terms can be expressed in terms of entropies which may be recombined in ways that can be easily single-letterized. This form of the side information allows us to extend the result from the 3-user case 6] to any number of users. Other partial sum-rate bounds are obtained with similar side information structure. Theorem. The capacity region of a general memoryless K- CFC-CMS is contained in the region defined by i : K] R i Y i ; X i:k] X :i ], 3a R j Y j ; X j:k] X :j ], Y :j ], i : K] ji ji 3b for some joint input distribution P X,...,X K, where X S : {X i : i S} for some index set S : K]. Moreover, each rate bound in 3b may be tightened with respect to the channel joint conditional distribution as long as the channel conditional marginal distributions are preserved. Proof: The the individual rate bounds in 3a are trivial cut-set bounds, we therefore present the proof of 3b: N R j ɛ N Yj N ; W j Fano s inequality ji ji ji ji ki ji Yj N, W :i ], W i:j ], Yi:j ] N ; W j Y N i:j] ; W j W :j ] j ki jk ki ki t Yk N ; W j W :j ], Yi:k ] N Yk N ; W j W :j ], Yi:k ] N independence of messages chain rule Yk N ; W k:k] W :i ], W i:k ], Yi:k ] N N Y k,t ; X k:k],t X :k ],t, Y i:k ],t. Finally, by introducing a time-sharing random variable and by conditioning reduces entropy, we arrive at 3b. 035

3 03 EEE nternational Symposium on nformation Theory V. SUM-CAPACTY OF THE LDC K-CFC-CMS For the LDC, we show a sum-capacity achieving scheme that only requires cognition of all messages at one transmitter. For simplicity we consider the symmetric case only. n a symmetric LDC all direct links have the same strength n ii n d 0, i : K], and all the cross links have the same strength n li n c α n d 0, l, i : K], l i. Theorem. The sum-capacity bound in 3b for i is achievable for the symmetric LDC K-CFC-CMS. Proof: For the symmetric LDC K-CFC-CMS the sumcapacity in was shown to be upper bounded by 6], ]: K k R { k K max{, α} α for α for α. 4 n d The discontinuity at α in 4 follows from the fact that when n d n c the channel reduces to a K-user MAC with sum-capacity given by max HY n d. To show the achievability of 4, let U j, j : K], be a vector composed of i.i.d. Bernoulli/ bits. Let the transmit signals be X j U j, j : K ], ] K X K nc 0 nc n d n c] 0 nd n c] n c 0 nd n c] n d n c] j ] 0 nc n c 0 nc n d n c] U 0 nd n c] n c K, nd n c] where 0 n m indicates the all zero matrix of dimension n m, n the identity matrix of dimension n, and x] : max{x, 0}. Recall that operations are on GF. With these choices, it may be shown that Y l S m n d S m nc X l. Then, since the matrix S m n d S m nc is full rank for n d n c, receiver l, l : K], decodes U l from S m n d S m nc Y l X l. Hence receiver l, l : K ], can decode m max{n d, n c } n d max{, α} bits since X l U l, while receiver K can decode the lower n d n c ] n d max{, α} α bits of U K from X K. Remark: nterestingly, with the proposed scheme, receivers from to K are interference free, while receiver K decodes n c bits of the interference function K j U j. Hence, cognition is only needed at one transmitter in all interference regimes, i.e., our sum-capacity result holds for all symmetric LDC cognitive channels where user i is cognizant of any subset including the empty set of the messages of users with index less than i. V. SUM-CAPACTY OF THE GAUSSAN K-CFC-CMS n this section we derive the sum-capacity for the symmetric Gaussian channel to within a constant gap. We denote the direct link gains as h d, which can be taken to be realvalued and non-negative without loss of generality, and the interference link gains as h i. For this channel, we may show constant additive and multiplicative gaps which depend on K but not the channel parameters. To do so, we first evaluate our U j outer bound, then obtain a generic achievability scheme, and finally show the additive and multiplicative gaps. A. Upper Bound For the symmetric Gaussian K-CFC-CMS with h d h i the bound in 3b with i is further upper bounded as R u u u X X u,, X K ; Y u, Y,, X u, Y u X K,, X K ; h d X h i X i Z K u i K X u,, X K ; h d X u h i X i Z u X l, h i K iu iu X i Z l, l : u ] Xl X K ; h d X K Z K, h i X K Z l, l : K ] h h K d X h i X i Z hz K u h i h h d h i ]X u Z u Z u hz u h d X K Z K hi X K K K Z l hz K. Finally, by the Gaussian maximizes entropy principle: k R k log h d K h i 5a h d h i K log K log 5b h d log K h i. 5c For h i h d all received signals are statistically equivalent, therefore the K-CFC-CMS is equivalent to a K-user MAC with correlated inputs, whose sum-capacity is bounded by R k log K h d. k B. Achievable Scheme n this section we describe a scheme which will be used to show a constant gap to the symmetric upper bound derived in Section V-A. nspired by the capacity achieving strategy for the Gaussian MMO-BC, we introduce a scheme that uses Dirty Paper Coding DPC with encoding order 3 K. We denote by Σ l the covariance matrix corresponding to the message intended for decoder l, 036

4 03 EEE nternational Symposium on nformation Theory l : K], as transmitted across the K antennas/transmitters. The input covariance matrix is K ] CovX,..., X K ] Σ l : Σ l, k : K], k,k 6a where the constraints on the diagonal elements correspond to the input power constraints. Moreover, since message l can only be broadcasted by transmitters with index larger than l, we further impose ] Σl 0 for all k < l K. 6b k,k The achievable rate region is then the set of non-negative rates R,..., R K that satisfy, for h l : h l,h l,... h l,k ] h R l log l Σ lh l h K, 7 l kl Σ k h l for l : K] and all possible CovX,..., X K ] complying with 6, with the convention that K kk Σ k 0. n particular we consider the transmit signals X α U, X j γ j U j β j U ZF j α j U, j : K ], K X K γ K U K β K U ZF j α K U, j where U l, U ZF l are i.i.d. N 0,, l : K], and the coefficients {α j, β j, γ j } j :K] are such that α, γ j β j α j, j : K ], γ K β K K α K, in order to satisfy the power constraints. Notice the negative sign for β K, which we shall use to implement zero-forcing of the aggregate interference K j U ZF j. Moreover, all transmitters cooperate in beam forming U to receiver. These two facts can be easily seen by observing that for β... β K : β X l γ l U l, γ : α l. With these choices the message covariance matrices are Σ aa, a : α,..., α K ] T, Σ j γ j e j e j β e j e K e j e K, j : K], where e j indicates a length-k vector of all zeros except for a one in position j, j : K], indicates the Hermitian transpose, and where β β... β K. We next express the channel vectors h l for the symmetric Gaussian channel as K h l h d h i e l h i e k, l : K]. k By noticing that h l e j δl j] h d h i h i, l : K], where δk] is the Kronecker s delta function, the following rates see ] are achievable h d h i K j α j R log h i K k γ, 8a k h d h i β h d γ j R j log h i K kj γ, 8b k R K log h d γ K, 8c for j : K ] and with α expj h i notice the phase of α which allows coherent combining at receiver of the different signals carrying U, i.e., all users beamform to the primary receiver. C. Constant Additive Gap Theorem 3. The sum-capacity bound in 3b is achievable for the symmetric Gaussian K-CFC-CMS to within 6 bits per channel use for K 3 and to within K log K 3.88 bits per channel use for K 4. Proof: We now choose the parameters in 8 so as to match the upper bound in 5. We recall that user K is the most cognitive user and can therefore pre-code the whole interference seen at its receiver by using DPC; by doing so, receiver K would not have anything to treat as noise besides the Gaussian noise itself. We therefore interpret the term K h i in 5c as the fraction of power transmitter K dedicates to its own signal. This is as setting γ K K h i in 8c. This choice guarantees that 8c exactly matches 5c. Next we match the upper bound term in 5b to the achievable rates in 8b by setting γ j 0, j, K ], β h i γ K. However, from the power constraint for user K, β γ K K, which imposes the following condition K 4 K h i γ K 0. K The above condition cannot be satisfied for K 4; for K 3 it requires that γ 3 h i h i, which can be satisfied by h i. Therefore, in the following we shall assume h i and set γ j 0, j, K ] and γ K β K K K h i K 4 h i γ 3 3 hi h i K 3, 037

5 03 EEE nternational Symposium on nformation Theory which implies { 0 K 4 α K β γ K hi h i K 3. Finally, for j : K ] { K 3 α j β j K K K h i K 4 h i h i K 3. The rates may then be bounded for K 4 using h i : h d R K log K h i hd h i K R j log, j : K ], K K hd h i K 3K R log, and for K 3 R 3 log h d h i hd h i R log, R log. hd h i By taking the difference between the upper bound in 5 and the derived achievable-rates, we find that the gap is upper bounded by, for K 4: GAP K log K log exp, where we used K log e /K, and for K 3 GAP log log h d h i h d h i log 6 log. For h i <, set β j α j 0, γ j for j : K] so that h d R l log K l h i. The gap to the upper bound is at most K GAP K log logk log l K l which is smaller than the gap for h i. Details on the calculation of the gaps are found in ]. D. Constant Multiplicative Gap We next consider the sum-capacity to within a multiplicative gap, more relevant at low SNR than additive gaps. Theorem 4. The symmetric sum-capacity is achievable to within a factor K by beamforming to the primary user. C j. Proof: The rate of user j in 3a is upper bounded by : log h d K j h i C, j : K]; this implies that the sum-rate is upper bounded by K C. Consider an achievability scheme in which all users beamform to user : this achieves the sum-rate R R K C. This is to within a factor K of the upper bound, proving Th. 4. nterestingly, the achievable scheme only requires user K to be cognitive of the messages of all other users, while the rest of users only need to know the message of user in addition to their own message; this considerably relaxes the CMS requirement. V. CONCLUSON We presented a general outer bound region for the K- user cognitive interference channel with cumulative message sharing. This computable outer bound was used to show that the symmetric sum-capacity is exactly achievable for the linear deterministic channel and to within constant gap for the Gaussian noise channel. While the focus of this paper was on cumulative message sharing, we note that, interestingly, the same sum-capacity outer bound may be achieved with a different message sharing structure with less message knowledge at the cognitive transmitters. ACKNOWLEDGMENT The work of the authors was partially funded by NSF under award The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the NSF. REFERENCES ] N. Devroye, P. Mitran, and V. Tarokh, Achievable rates in cognitive radio channels, EEE Trans. nf. Theory, vol. 5, no. 5, pp , May 006. ] S. Rini, D. Tuninetti, and N. Devroye, New inner and outer bounds for the discrete memoryless cognitive interference channel and some new capacity results, EEE Trans. nf. Theory, vol. 57, no. 7, pp , Jul. 0. 3], On the capacity of the gaussian cognitive interference channel: new inner and outer bounds and capacity to within bit, EEE Trans. nf. Theory, 0. 4] K. Nagananda and C. Murthy, nformation theoretic results for threeuser cognitive channels, in Proc. EEE Global Telecommun. Conf., ] A. Jafarian and S. Vishwanath, On the capacity of multi-user cognitive radio networks, in Proc. EEE nt. Symp. nf. Theory. EEE, 009, pp ] D. Maamari, N. Devroye, and D. Tuninetti, The sum-capacity of the linear deterministic three-user cognitive interference channel, in Proc. EEE nt. Symp. nf. Theory, Cambridge, MA, July 0. 7] K. G. Nagananda, P. Mohapatra, C. R. Murthy, and S. Kishore, Multiuser cognitive radio networks: An information theoretic perspective, 8] K. Nagananda, C. Murthy, and S. Kishore, Achievable rates in threeuser interference channels with one cognitive transmitter, in nt. Conf. on Signal Proc. and Comm. SPCOM, 00. 9] M. Mirmohseni, B. Akhbari, and M. Aref, Capacity bounds for the three-user cognitive z-interference channel, in th Canadian Workshop on nformation Theory CWT, 0, pp ] M. G. Kang and W. Choi, The capacity of a three user interference channel with a cognitive transmitter in strong interference, in Proc. EEE nt. Symp. nf. Theory, 0. ] A. Avestimehr, S. Diggavi, and D. Tse, A deterministic approach to wireless relay networks, Proc. Allerton Conf. Commun., Control and Comp., Sep ] D. Maamari, D. Tuninetti, and N. Devroye, Approximate sum-capacity of k-user cognitive interference channels with cumulative message sharing, 038