LQG Control Approach to Gaussian Broadcast Channels With Feedback

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST LQG Control Approach to Gaussian Broadcast Channels With Feedback Ehsan Ardestanizadeh, Member, IEEE, Paolo Minero, Member, IEEE, Massimo Franceschetti, Senior Member, IEEE Abstract A code for communication over the -receiver complex additive white Gaussian noise broadcast channel (BC) with feedback is presented analyzed using tools from the theory of linear quadratic Gaussian optimal control. It is shown that the performance of this code depends on the noise correlation at the receivers it is related to the solution of a discrete algebraic Riccati equation. For the case of independent noises, the sum rate achieved by the proposed code, satisfying average power constraint,is characterized as,thecoefficient quantifies the power gain due to the presence of feedback. This includes a previous result by Elia strictly improves upon the codes by Ozarow Leung by Kramer. When the noises are correlated, the prelog of the sum capacity of the BC with feedback can be strictly greater than 1. It is established that for all noise covariance matrices of rank the prelog of the sum capacity is at most, conversely, there exists a noise covariance matrix of rank for which the proposed code achieves this upper bound. This generalizes a previous result by Gastpar et al. for the two-receiver BC. Index Terms Broadcast channel (BC), capacity, feedback, linear quadratic control, network information theory. I. INTRODUCTION CONSIDER the communication problem over the -receiver complex additive white Gaussian noise (AWGN) broadcast channel (BC) with feedback depicted in Fig. 1. Here, a sender wishes to communicate independent messages to distinct receivers who observe the sequence of transmitted signals corrupted by, possibly correlated, AWGN sequences. The capacity region of this channel is an open problem. It is known that the presence of feedback from receivers to a sender improves communication performance over BCs. Specifically, Dueck [1] showed that feedback can enlarge the capacity Manuscript received February 15, 2011; revised November 10, 2011; accepted February 22, Date of publication May 16, 2012; date of current version July 10, This work was supported in part by the National Science Foundation under Awards CCF , CNS , CCF This paper was presented in part at the 48th Annual Allerton Conference on Communication, Control, Computation in part at the 2011 IEEE International Symposium on Information Theory. E. Ardestanizadeh was with the Department of Electrical Computer Engineering, University of California, San Diego, CA USA. He is now with ASSIA Inc., Redwood City, CA USA ( eardestani@assiainc.com). P. Minero is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN USA ( pminero@nd.edu). M. Franceschetti is with the Department of Electrical Computer Engineering, University of California, San Diego, CA USA ( massimo@ece.ucsd.edu). Communicated by S. Tatikonda, Associate Editor for Communications. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIT Fig. 1. -receiver AWGN BC with feedback. region of additive memoryless BCs with correlated noises at the receivers. Ozarow Leung [2] presented a code for the AWGN-BC with two receivers proved that feedback can be beneficial by providing a means of cooperation between the receivers the sender even when the noises at the receivers are independent, Kramer [3] extended the code in [2] to more than two receivers. Outer bounds for the AWGN-BC channel that are tighter than the cut-set bound [4] are presented in [2] [3]. These bounds are based on a physically degraded channel each receiver observes the channel outputs corresponding to receivers, use the fact that feedback cannot increase the capacity of the physically degraded BC [5], [6]. In this paper, we provide an inner bound on the capacity region of the AWGN-BC with feedback. We construct a code, which we refer to as LQG code, for communicating over the AWGN-BC with feedback we characterize its performance using tools from linear quadratic Gaussian (LQG) optimal control. The LQG code is derived based on a mapping from a feedback control problem to a linear code for the AWGN-BC with feedback. The set of achievable rates are determined by the eigenvalues of the open-loop matrix of a linear system the power constraint is related to the minimum power needed to stabilize the system using a feedback control signal. Specifically, the power needed by the LQG code for reliable transmission of messages encoded at a given set of rates depends on the correlation between the noises at the receivers is determined by the solution of a discrete algebraic Riccati equation (DARE) (see Theorem 1). We first evaluate the performance of the LQG code in the case of independent noises at the receivers, which is the most interesting in practice. We show that in this case the LQG code strictly outperforms the Ozarow--Leung (OL) code [2] for the two-receiver AWGN-BC with feedback the Kramer /$ IEEE

2 5268 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 code [3] for AWGN BCs with more than two receivers. By solving the corresponding DARE, it is shown that the LQG code achieves a sum rate of under average power constraint (see Theorem 2). Here, the real coefficient represents the power gain compared to the no-feedback sum capacity,forfixed it is an increasing function of, that is, more power allows more gain. In particular, as, the sum rate tends to, which is the same as the sum capacity of the single-input multiple-output (SIMO) channel obtained by allowing full cooperation among the receivers. Hence, the power gain due to feedback can be interpreted as the amount of cooperation among the receivers the sender established through feedback, which allows the transmitter to align the signals intended for different receivers use power more efficiently. At a high level, the OL Kramer codes are generalizations of the capacity achieving feedback scheme by Schalkwijk Kailath (SK)[7] for the AWGN channel, the sender iteratively transmits the error of the minimum mean square estimate (MMSE) of the transmitted message given the previous received channel outputs. Shayevitz Feder [8] recently showed that the SK scheme satisfies the posterior matching principle of optimality for point-to-point channels with feedback, according to which at every time step the sender should transmit a rom variable that is independent of the past channel outputs is relevant for the reconstruction of the transmitted message [8]. In the OL Kramer codes, at every time step the sender computes the MMSE error for each transmitted message given the most recent observation available only at the receiver interested in decoding the message, then transmits a linear combination of the corresponding MMSE errors. Each MMSE error is independent of the past channel output at the designated receiver, but is in general correlated with the channel outputs at all remaining receivers. It follows that the OL Kramer codes violate the posterior matching optimality principle for point-to-point channels. In the proposed LQG code, the sender also iteratively refines the estimates of the messages at the receivers by transmitting a linear combination of the estimation errors based on the most recent channel outputs. However, the receivers linear estimates are not the MMSE as in the SK scheme. Instead, they are chosen such that the steady-state power of the channel input is minimized (see Section V-B). Next, we consider the case of correlated noises at the receivers investigate how the sum capacity, the supremum of achievable sum rates, scales as the power at the transmitter increases. If this scales as as, then we refer to as the prelog 1 of the channel. We show that the prelog of the AWGN-BC with feedback depends on the rank of the correlation matrix of the noises at the receivers (see Theorem 3). Specifically, if the rank is, then the prelog can be at most. Conversely, for any, there exists a complex noise covariance matrix of rank for which the upper bound on the prelog is tight is achieved by the LQG code. In particular, the prelog is equal to for some 1 The prelog is also known as the number of degrees of freedom as it is equal to the number of orthogonal point-to-point channels with the same sum capacity. rank-one covariance matrix. This generalizes a previous result by Gastpar et al. [9], [10] for the two-receiver AWGN-BC to the case of receivers. Finally, we wish to mention some additional related works. Elia [11] followed a control-theoretic approach presented a linear code for the two-receiver AWGN-BC with independent noises which outperforms the OL code. When specialized to the case of two receivers independent noises, the LQG code has the same performance as Elia s code [11]. Wu et al. [12] applied the LQG theory to study Gaussian networks with feedback, the noises at the receivers are independent, but did not provide explicit solutions. It has been shown in [13] that the linear code proposed in [3] for the -sender multiple access channel (MAC) with feedback can be obtained by solving an LQG control problem. Along the same line, Anastasopoulos [14] formulated the communication problem over the MAC as a decentralized stochastic control problem. Coleman [15] gave a control-theoretic interpretation of the posterior matching scheme [8]. Finally, new inner bounds on the capacity region of memoryless BCs with feedback are derived in [16] [17]. The rest of this paper is organized as follows. Section II presents the problem definition. Section III discusses the pointto-point communication problem over the AWGN channel with feedback from a control-theoretic perspective. This viewpoint is then generalized in Section IV, which presents the LQG code for communicating over the -receiver AWGN-BC with feedback. The following two sections are devoted to studying the performance of the LQG code: in Section V we provide the analysis for the case of independent noises at the receivers, while in Section VI we characterize the prelog gain when the noises are correlated. Finally, Section VII concludes this paper. II. DEFINITIONS Consider the communication problem a sender wishes to communicate independent messages to distinct receivers by transmissions over the complex AWGN-BC channel with feedback depicted in Fig. 1. At each time, the channel outputs are given by is the complex transmitted symbol by the sender is the column vector of ones of length. The vector contains the complex channel outputs at time,thatis, is the channel output observed by receiver at time. The noise vector isassumedtobeindependent of the transmitted messages, independently identically distributed (i.i.d.) from a circular symmetric complex Gaussian distribution with zero mean covariance matrix. We refer to the channel (1) as the real AWGN-BC if. We assume that the output symbols are causally noiselessly fed back to the sender so that the transmitted symbol at time can depend on the message vector, the previous channel output vectors. (1)

3 ARDESTANIZADEH et al.: LQG CONTROL APPROACH TO GAUSSIAN BROADCAST CHANNELS WITH FEEDBACK 5269 Definition 1: A code for the AWGN-BC with feedback consists of : 1) discrete messages ; 2) an encoder that assigns a symbol to the message vector the previous channel output vectors for each ; 3) decoders, decoder assigns an estimate to each sequence. We assume that,,are independent uniformly distributed over the message sets. The probability of error for each receiver is defined as messages under (asymptotic block) power constraint existsasequenceof codes such that if there (2) holds. The next lemma establishes a useful connection between achievable MSE exponents, achievable rates, error exponents. Lemma 1: Let be an achievable MSE exponent vector with -dimensional messages under power constraint let be such that Definition 2: We say that is an achievable rate vector under (asymptotic block) power constraint if there exists a sequence of codes such that,, corre- is defined Definition 3: The error exponent vector sponding to an achievable rate vector as (2) Definition 4: The closure of the set of achievable rate vectors under power constraints is called the capacity region C. The sum capacity is defined as C the prelog is defined as 2 Definition 5: A code for the AWGN-BC with feedback consists of 1) message points in ; 2) an encoder that assigns a symbol to the message vector the previous channel output vectors for each ; 3) decoders, decoder assigns an estimate to each sequence. We assume that are independent uniformly distributed over the unit hypercube of dimension, i.e.,,. The mean square errors at time are defined as denotes the 2-norm of a complex vector denotes the conjugate transpose of. Definition 6: We say that is an achievable mean square error (MSE) exponent vector with -dimensional 2 In the special case of a real AWGN-BC, the term at denominator is replaced by. Then, the rate vector is achievable under power constraint the corresponding error exponent vector satisfies Proof: See Appendix A. Remark 1: The proof of Lemma 1 utilizes a technique developed in [8] for characterizing the set of achievable rates in a point-to-point channel the transmitted message is drawn from a continuous distribution. Lemma 1 states that achievability of a given MSE exponent vector is a sufficient condition for achievability of a set of rate vectors determined by the MSE exponents the message dimension. The corresponding error exponent is lower bounded by a function of the MSE exponent, the achievable rate, the message dimension. As a result, Lemma 1 provides a connection between the MSE a natural quantity for the continuous channels such as the AWGN-BC the notion of achievable rate which is based on discrete sets the probability of error. A similar connection between achievable rates error exponents has also been established in [8] for the posterior matching scheme. III. LQG APPROACH: THE REAL AWGN CHANNEL Before presenting the LQG code for the complex AWGN-BC with feedback, we first revisit the communication problem over the point-to-point real AWGN channel with feedback in Fig. 2(a), demonstrate how the theory of the LQG control can be used to design codes for communicating over such a channel. It is well known [11] that a capacity-achieving code for the real AWGN channel can be derived from the solution of an optimal control problem. However, here we provide a derivation of this result that naturally generalizes to the case of multiple receivers. We also explicitly show that the LQG code is asymptotically equivalent to the code by SK [7], [18], a special case of the posterior matching scheme [8], the sender recursively transmits the MMSE error of the message independent of the previous channel outputs. Let be the initial state of an unstable linear system with open-loop dynamics

4 5270 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 Lemma 2: If the control stabilizes the system with openloop mode,thenthe code described by (6) (7) achieves MSE exponent with 1-D message under power constraint equal to the asymptotic average control power (8) Proof: Combining(6)(7),wehave (9) Fig. 2. Coding scheme for the feedback communication over the real AWGN channel depicted in (a) can be derived base on the closed-loop dynamics of a feedback control system the feedback control signal is corrupted by AWGN, as depicted in (b). (a) Real AWGN channel with feedback. (b) Stabilization over the real AWGN channel. for some coefficient that is stabilized by a controller having full state information the control signal is corrupted by AWGN [see Fig. 2(b)]. The closed-loop dynamics of this system are given by. The mappings are referred to as the control, the linear dynamical system in (3) with unstable mode is simply referred to as the system. We say that a control is stabilizing if i.e., if the system is mean square stable. Given a control (5) for the system (3), we construct a sequence of codes for the point-to-point AWGN channel (4) with feedback (cf., Definition 5 in the special case of ) as follows. 1) Encoder: Given a continuous message,the encoder recursively forms transmits for each. 2) Decoder: The decoder sets as an estimate of the message, is recursively formed as (3) (4) (5) (6) (7) the last equality follows from the fact that. From (9), the MSE of the estimate at time is Since the control is stabilizing, we know hence Since, we conclude that MSE exponent is achievable under power constraint equal to (8). Lemma 2 implies that for a fixed any stabilizing control yields a code achieving MSE exponent under a power constraint determined by the asymptotic control power (8). The LQG code is the code corresponding to the control with minimum stationary power, which can be computed using the LQG control theory. It is known from the theory of LQG control [19] that the linear stationary control (10) attains the minimum asymptotic power (8), which is given by. Hence, from Lemma 2, the linear code corresponding to this optimal LQG control, which we refer to as the LQG code, achieves the MSE exponent under power constraint. By Lemma 1, when specialized to, we conclude that the LQG code achieves any rate under power constraint, hence is capacity achieving. A natural question to ask is what is the connection between the LQG code the SK code? By combining (6) (10), observe that in the LQG code the channel input is updated recursively as (11) with. The recursion converges, irrespective of the initial value, since. On the other h, the channel input update in the SK can be represented by the following time-varying recursion [20]: (12) with. Comparing (11) (12), we can see that the LQG code is asymptotically equivalent to the SK code if such that, in steady state,

5 ARDESTANIZADEH et al.: LQG CONTROL APPROACH TO GAUSSIAN BROADCAST CHANNELS WITH FEEDBACK 5271, being distinct points out-. The complex vector represents the state of the system with side the unit circle, i.e., at time denotes the vector of complex channel outputs of the AWGN-BC in (1), i.e., (15) Fig. 3. Control over the AWGN BC.. Moreover, we assume that is a vector of complex rom variables such that are drawn i.i.d. from a uniform distribution over. At every discrete time, the control input can depend on the state of the system up to at time,so tends to the minimum mean squared error estimate of given. By plugging (10) into (6), we obtain that the closed-loop dynamics of can be represented as.as the second moment of the state converges to since, Therefore, (16) for some function.werefertothesequence as the control. Since, the eigenvalues of are outside the unit circle the open-loop system (13) is unstable. We say that the control stabilizes the closed-loop system if as, the last equality follows from the fact that the optimal control is given by. Hence, the LQG code the SK code are asymptotically equivalent. IV. LQG CODE: THE AWGN BC WITH FEEDBACK In this section, we extend the control-theoretic approach describedinsectioniiitothecaseofthe -receiver complex AWGN-BC with feedback in (1) with message points 3 in, i.e., in Definition 5. We do so by considering the control problem depicted in Fig. 3, in which a -dimensional unstable dynamical system is stabilized by a controller having full state observation the scalar control signal is perturbed by, possibly correlated, AWGN noises, each affecting a different component of the state vector. We show that any controller stabilizing the system in the mean square sense yields a code for the -receiver AWGN-BC with feedback. In particular, the LQG code is obtained from the minimum average power control which can be computed using the LQG control theory. A. Code Design Based on a Control-Theoretic Approach Consider the linear dynamical system with open-loop matrix shown in Fig. 3 Given the system (13) the control (16), consider the following sequence of codes for the -receiver complex AWGN-BC with feedback (1). 1) Encoder: At each time the encoder recursively forms as in (13) transmits (17) 2) Decoders: At each time decoder forms an estimate for the desired message, is recursively formed as (18) The following lemma characterizes the set of MSE exponent vectors that can be achieved by the sequence of codes so generated. Lemma 3: Let be a stabilizing control for (13). Then, the MSE exponent vector is achievable with 2-D messages under power constraint equal to the asymptotic average control power (13) (14) Proof: For each,let, is as in (18).Then, 3 We can think of the message points as being complex numbers the two real components correspond to real imaginary parts, respectively.

6 5272 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 is as in (14). It follows that the recursion in (13) can then be rewritten as Hence,. Since the control is stabilizing,, we conclude that which proves the claim. B. LQG Code for the AWGN-BC With Feedback According to Lemma 3, for a fixed open-loop matrix, any stabilizing control yields a sequence of codes for the complex AWGN-BC with feedback that achieves MSE exponent vector under a power constraint determined by the asymptotic average power of the control. The LQG code corresponds to the control with minimum stationary power, which can be computed using the LQG control theory. Lemma 4: Let be given as in (14) (15), respectively, let be the covariance matrix of the noise vector in the AWGN-BC (1). Consider the problem of minimizing the asymptotic average control power Remark 2: The proof of the aforementioned lemma is omitted as it follows closely the proof for the stard LQG optimal control problem [19, Chapter 4] in the special case the cost function does not depend on the state. The main difference with respect to the classical proof is that in this setting the optimal control is required to stabilize the system (see Lemma 3). However, using stard techniques (cf. [21, Lemma 2.4]) it is possible to show that that there exists a unique solution to (22) such that the control in (21) is stabilizing, that is, is stable. Theorem 1: Let be such that in the case of the complex AWGN-BC in the case of the real AWGN-BC. Then, the rate vector is achievable under power constraint (24). Proof: CombiningLemmas1,3,4itfollowsimmediately that if, then rate vector is achievable under power constraint (24). Here, the rate is measured in bits per complex channel use. Since one channel use over the complex AWGN-BC corresponds to two channel uses over the real AWGN-BC, for the real part imaginary parts of the complex signal, respectively, it follows that the rate vector is achievable for the real AWGN-BC with feedback under power constraint (24). within the class of causal stabilizing controls.then, 1) The optimal control is linear stationary, i.e., (19) (20) C. Example: The Two-Receiver Real AWGN-BC With Correlated Noises Consider the special case of a two-receiver real AWGN-BC, let is given by (21) is the unique positive-definite solution to the DARE (22) such that the matrix is stable, that is, every eigenvalue of lies inside the unit circle. 2) Under the optimal control law (21), the system covariance matrix converges as to an asymptotic covariance matrix, which is the unique solution to the discrete algebraic Lyapunov equation (DALE) (23) within the class of positive-semidefinite symmetric matrices. 3) The minimum asymptotic average control power (19) is given by (24) for some a real correlation coefficient. By solving (22) plugging the solution into (24), we obtain that the symmetric rate is achievable under power constraint (25) Equating the right-h side of (25) to the target power constraint solving the resulting polynomial equation in yields the set of achievable symmetric rates as a function of the noise correlation coefficient. Specifically, the symmetric rate is achievable under power constraint, is the unique real positive root of the equation (26) Cardano s formula for the roots of a cubic equation with real coefficients provides the explicit analytic expression for.in

7 ARDESTANIZADEH et al.: LQG CONTROL APPROACH TO GAUSSIAN BROADCAST CHANNELS WITH FEEDBACK 5273 V. INDEPENDENT NOISES: POWER GAIN In this section, we consider the special case of a real AWGN-BC with independent noises, which is the case for most practical scenarios, the same noise variance, i.e., assuming that. We characterize an inner bound on the achievable symmetric rate as a function of the power constraint the number of receivers. To do so, we analyze the performance of the LQG code in the symmetric case the diagonal elements of in (14) are chosen (possibly suboptimally) as (27) Fig. 4. Achievable symmetric rates as a function of the channel noise correlation coefficient for the OL code the LQG code ( ). the special case, (26) reduces to ; hence, the LQG code achieves the symmetric rate region is real. For this choice of, in fact, it is possible to characterize the solution to the DARE (22). Lemma 5: [22, Lemma 12] Suppose that the diagonal elements of the open-loop matrix are as in (27). Then, the unique positive-definite solution to the DARE (22) is circulant with real eigenvalues satisfying i.e., it achieves the sum capacity region of the channel without feedback. Notice that in this case the received signals at the two receivers are identical, thus, the AWGN-BC channel degenerates into an AWGN point-to-point channel. When, instead, (26) reduces to the second-order polynomial equation whose roots are.itfollows that in this case the LQG code achieves the symmetric rate region The largest eigenvalue satisfies (28) (29) Theorem 2: If, the LQG code achieves the symmetric rate, under power constraint for the real AWGN-BC, Notice that is the unique solution in the interval to (30) thus, in this case the prelog of the sum capacity region is 2, as it was first established in [9] [10]. Fig. 4 compares the LQG code the OL code to outer bound derived in [2] based on the construction of a physically degraded channel receiver 2 has access to the channel output of both receivers. This bound states that if is an achievable rate for the AWGN-BC with feedback, then is the unique solution in to the following secondorder equation in : Remark 3: The quantity represents the power gain compared to the sum capacity of the same channel without feedback. This power gain can be interpreted as the amount of cooperation among the receivers established through feedback, which allows the transmitter to align signals intended for different receivers use power more efficiently. Proof: If the diagonal elements of the open-loop matrix areasin(27),thenbylemma5theuniquepositive-definite solution to the DARE (22) is circulant, therefore, has equal elements on the diagonal. Let satisfy, such that. By combining (28) (29) it follows that the largest real eigenvalue of satisfies the equation (31) Notice that the performance of the LQG code is strictly better thantheoneoftheolcodeforall. When the noises at the receivers are independent ( ), the code proposed by Elia in [11] has the same performance as the LQG code. Then, Theorem 1 for the real AWGN-BC (28) imply that the sum rate

8 5274 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 is achievable under under power constraint, satisfies (31), thus, the claim follows after applying the change of variable. Next, we show that Theorem 2 implies that the OL code the Kramer code are strictly suboptimal. A. Comparison With the OL Code. Hence, as In this section, we compare the OL code with the LQG code in the special case of a two-receiver real AWGN-BC with. The OL code can be represented as follows [20]. The channel input is set as the asymptotic channel input power is given by (35) are defined through the recursions (32) (33) for two independent continuous messages uniformly distributed over. Decoder forms an estimate for the desired message, is recursively formed as By Cesáro s lemma, Lemma 1, Lemma 3, it follows that the OL code achieves any symmetric rate under power constraint. We claim that is it possible to achieve the same set of rates under a power constraint which is strictly less than.tosee this, suppose that (32) (33) are modified as follows: (36) (37) is a time-invariant parameter chosen such that (36) (37) are stable the asymptotic channel input power is minimized. By letting, the aforesaid recursions can be rewritten as, is the closed-loop system matrix given by (34) for. It is shown in [2] that the recursions (32) (33) are mean square stable that converges as to a steady-state value.toevaluate,notice that the covariance matrix of the state variables can be computed recursively as It is easy to verify that the eigenvalues of are ; hence, we require (38) to ensure that (36) (37) are stable. The covariance matrix of the state vector satisfies the recursive equation. If (38) is satisfied, then it can be shown that It can be shown that (39) as, as,

9 ARDESTANIZADEH et al.: LQG CONTROL APPROACH TO GAUSSIAN BROADCAST CHANNELS WITH FEEDBACK 5275 to the the power needed by the LQG code to achieve the symmetric rate region. B. Comparison With the Kramer Code In this section, we compare the LQG code with the Kramer code [3], an extension of the OL code to the case of more than two receivers. We consider the case the asymptotic regime, for which explicit performance is provided in [3]. To characterize the performance of the LQG code, note that condition (30) can be written as Fig. 5. Power needed by a linear code to achieve the symmetric rate (i.e., ) in a two-receiver AWGN-BC with covariance as a function of the coefficient in the linear recursions (36) (37). It follows that As solution (41), the unique solution to (41) converges to the unique of the nonlinear equation (42) as. The parameter is chosen such that the asymptotic power is minimized: Differentiating with respect to,weobtain (40) so has two stationary points at 0, respectively. The latter point satisfies (38) corresponds to a minimum for (e.g., Fig. 5 is a plot of as a function of ); thus, we conclude that For, by solving (42) numerically we obtain that hence by Theorem 2 the symmetric rate is achievable, For the same power, the Kramer code achieves at most nats/channel use [3]. Hence, the LQG code outperforms the Kramer code. C. Comparison With the AWGN-MAC The LQG approach can be also applied to the AWGN-MAC with feedback. It is known [13] that the LQG code for AWGN-MAC has the same performance as the Kramer code [3], which achieves the linear sum capacity [22], the supremum sum rate achievable by linear codes. Let denote the symmetric sum rate achievable by the LQG code for the -sender AWGN MAC with feedback each sender has power constraint. Then, we have [13, Theorem 4] The corresponding minimum asymptotic power is is the unique solution to Direct comparison shows that Comparing with Theorem 2, it is not hard to see that therefore the asymptotic value used in the linear update of the OL code is strictly smaller than. Accordingly, it is easy to verify that for every i.e., the power needed by the OL code to achieve the symmetric rate region is strictly greater than (see also Fig. 5). This proves that the OL is suboptimal. On the other h, by setting into (25) we can see that is equal This shows that under the same sum power constraint,the sum rate achievable by the LQG code over MAC BC is equal. This connection between the MAC the BC is also considered in [12]. VI. CORRELATED NOISES: PRELOG GAIN In this section, we show that the capacity region of the complex AWGN-BC with feedback can be significantly larger than the capacity of the same channel without feedback when the

10 5276 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 noises at the receivers are correlated. We consider the highpower regime study the prelog as a function of covariance matrix, which represents the number of orthogonal point-to-point channels with the same sum capacity. Theorem 3: For all of rank Conversely, for any, there exists a covariance matrix such that for, is the matrix with eigenvalues on its diagonal. Let with (43) Observe that the noise covariance matrix so defined is circulant has diagonal entries equal to 1. Then, we have by Lemma 5 Proof: First, we prove the upper bound. By assumption contains linearly independent rows. Without loss of generality, suppose that these are the last rows. Assume that receivers share their received signals form a single receiver equipped with receive antennas let denote the vector of received signals by this multiple antenna receiver. The corresponding AWGN vector BC with feedback is specified by Therefore, for the symmetric choice of in (27), the LQG code achieves sum rate over the complex AWGN-BC under power constraint.itfollowsthat,, by assumption is full rank invertible. Now suppose that the sender wishes to send message to receiver,, under power constraint.since we made the optimistic assumption that a subset of receiver can cooperate, the sum capacity of this channel is an outer bound on the sum capacity of the original AWGN-BC. Note that for every,therate is upper bounded by the capacity of the point-to-point AWGN channel. The rate for the th receiver with multiple antenna is upper bounded by the capacity of a SIMO by assumption is invertible. Thus, the sum capacity of this channel is upper bounded by, therefore, the prelog can be at most. Next, we consider the proof of achievability. First, we consider the case show that there exists a rank-one covariance matrix for which. Suppose that the open-loop matrix is as in (27). By Theorem 1 for the complex AWGN-BC, the symmetric rate vector is achievable under the power constraint, is the circulant matrix in Lemma 5. Note that any circulant matrix can be written as, is the point discrete Fourier transform matrix with entries since for any rank-one covariance matrix,we conclude that for the covariance matrix in (43). Next, we show that for every aprelogequal to is achievable for some such that. Consider the following choice: denotes the zero matrix of dimension, is the identity matrix of dimension, is the circulant matrix having first row equal to the last column of the discrete Fourier transform matrix of dimension. Clearly,. On the other h, suppose that the transmitter communicates only to users, while the transmission rate for the remaining users is set to zero. We can use a similar argument as above show that the LQG code for the corresponding -receiver AWGN-BC with feedback noise covariance matrix achieves a prelog equal to. Remark 4: To achieve, we used the LQG code. However, the same prelog can be achieved even with codes which are less power efficient since we are considering only the prelog of the sum rate in the high-power regime. For instance, for the special case of, Gastpar et al. [9], [10] showed that the OL code, which is suboptimal, achieves prelog 2 for anticorrelated noises. Remark 5: The matrix that is shown to achieve the upper bound in the proof of achievability of Theorem 3 is a complex circulant matrix having the first row equal to the last column of a discrete Fourier transform of suitable dimensions. In general, the outer bound in Theorem 3 need not be attained in the case of a real AWGN-BC with feedback, the covariance matrix has real entries.

11 ARDESTANIZADEH et al.: LQG CONTROL APPROACH TO GAUSSIAN BROADCAST CHANNELS WITH FEEDBACK 5277 VII. CONCLUSION Using tools from control theory, we have presented a code for the -receiver AWGN-BC with feedback, called the LQG code, whichwehavethenusedtoinvestigatesomepropertiesofthe capacity region of this channel. When the noises at the receivers are independent, the prelog of the sum capacity is at most 1, so feedback can yield at most a power gain over the case without feedback. We have quantified the power gain achieved by the LQG code shown that in the case,thelqg code recovers a previous result of Elia which strictly improves upon the OL code. For the LQG code strictly improves upon the Kramer code. In the case the noises at the receivers are correlated, instead, the prelog of the sum capacity can be strictly greater than 1. We established that for all noise covariance matrixes of rank the prelog is at most, conversely, there exists a covariance matrix for which this upper bound is achieved by the LQG code. In particular, a prelog equal to is achievable for some circulant noise covariance matrix of rank 1. This generalizes previous results obtained by Gastpar et al. for the case. Although the LQG code is shown to achieve the best known inner bound on the capacity region for the AWGN-BC with feedback, the question of whether it is capacity achieving or not is open. It is well known [3] that the performance of the OL Kramer codes can possibly improve if the entire vector of observations available at each receiver ( not just the most recent one) is used to estimate the transmitted messages, but the analysis of the corresponding code seems analytically intractable [3]. One interesting question left for future research is to consider the generalization of the LQG code when the state at the encoder is allowed to depend on the last received channel outputs. Another question that remains open for further investigation is understing to what extent the posterior matching optimality principle for point-to-point channels generalize to BCs with feedback. Finally, the LQG approach exploited here could be in principle useful for other multiuser communication channels with feedback. as. Inequalities (46) (47) follow from the Chebyshev inequality (44), respectively. It follows from (48) that if,then (49) Given the sequence of codes satisfying (44),, we construct a sequence of codes such that. Let the discrete message be mapped to a message point, is a set message points in the unit hypercube of dimension.tosend,weusethegiven code the corresponding message point. The decoder first forms the estimate of the message point according to the given code, then chooses such that is the closest message point to. Next, we show that there exists a set of message points of size in the unit hypercube of dimension such that the distance between any two message points is greater than or equal to (50) for which is equivalent to, for the code described previously. The following argument, which is similar to one in [8, Lemma 3], completes proof of the first part. Define the event Then we have.let.then hence APPENDIX PROOF OF LEMMA 1 By assumption, we know that there exists a sequence of codes for such that the power constraint (2) holds. Consider (44) (51) considering (49) condition (50) holds. Since the volume of a hypersphere of dimension with radius is proportional to,thesizeof is upper bounded as for some constant, or considering (49),. Therefore, rate vector is achievable. To prove the second part, we combine (48) (51), conclude (45) (46) (47) (48) for.

12 5278 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 8, AUGUST 2012 REFERENCES [1] G. Dueck, Partial feedback for two-way broadcast channels, Inf. Control, vol. 46, pp. 1 15, [2] L. H. Ozarow C. S. K. Leung, An achievable region outer bound for the Gaussian broadcast channel with feedback, IEEE Trans. Inf. Theory, vol. 30, no. 4, pp , Jul [3] G. Kramer, Feedback strategies for white Gaussian interference networks, IEEE Trans. Inf. Theory, vol. 48, no. 6, pp , Jun [4] J. A. Thomas, Feedback can at most double Gaussain multiple access channel capacity, IEEE Trans. Inf. Theory, vol. 33, no. 5, pp , Sep [5] A. El Gamal, The feedback capacity of degraded broadcast channels, IEEE Trans. Inf. Theory, vol. IT-24, no. 3, pp , May [6] A. El Gamal, The capacity of the physically degraded Gaussian broadcast channel with feedback, IEEE Trans. Inf. Theory, vol. IT-27, no. 4, pp , Jul [7] J. P. M. Schalkwijk T. Kailath, A coding scheme for additive noise channels with feedback: I. No blimited constraint, IEEE Trans. Inf. Theory, vol. IT-12, no. 2, pp , Apr [8]O.ShayevitzM.Feder, Optimal feedback communication via posterior matching, IEEE Trans. Inf. Theory, vol. 57, no. 3, pp , Mar [9] M.Gastpar,A.Lapidoth,M.Wigger, When feedback doubles the prelog in AWGN networks, IEEE Trans. Inf. Theory, submitted for publication. [10] M. A. Wigger M. Gastpar, The pre-log of Gaussian broadcast with feedback can be two, presented at the IEEE Int. Symp. Inf. Theory, [11] N. Elia, When Bode meets Shannon: Control oriented feedback communication schemes, IEEE Trans. Automat. Control, vol. 49, no. 9, pp , Sep [12] W. Wu, S. Vishwanath, A. Arapostathis, Gaussian interference networks with feedback: Duality, sum capacity dynamic team problem, presented at the 44th Allerton Conf. Commun., Control, Comput., [13] E. Ardestanizadeh M. Franceschetti, LQG control approach to Gaussian broadcast channels with feedback, IEEE Trans. Autom. Control, to be published. [14] A. Anastasopoulos, A sequential transmission scheme for the multiple access channel with noiseless feedback, in Proc. 47th Annu. Allerton Conf. Commun., Control, Comput., 2009, pp [15] T. P. Coleman, A stochastic control viewpoint on posterior matchingstyle communication schemes, in Proc. IEEE Int. Symp. Inf. Theory, 2009, pp [16] O. Shayevitz M. A. Wigger, On the capacity of the discrete memoryless broadcast channel with feedback, IEEE Trans. Inf. Theory, to be published. [17] R. Venkataramanan S. S. Pradhan, An achievable rate region for the broadcast channel with feedback, IEEE Trans. Inf. Theory, 2011, to be published. [18] J. P. M. Schalkwijk, A coding scheme for additive noise channels with feedback: II. B-limited signals, IEEE Trans. Inf. Theory, vol. IT-12, no. 2, pp , Apr [19] D. P. Bertsekas, Dynamic Programming Optimal Control, 3rd ed. Nashua, NH: Athena Scientific, 2007, vol. 2. [20] A. El Gamal Y.-H. Kim, Network Information Theory. Cambridge, U.K.: Cambridge Univ. Press, [21] Y.-H. Kim, Feedback capacity of stationary Gaussian channels, IEEE Trans. Inf. Theory, vol. 56, no. 1, pp , Jan [22] E. Ardestanizadeh, M. A. Wigger, T. Javidi, Y.-H. Kim, Linear sum capacity for Gaussian multiple access channel with feedback, IEEE Trans. Inf. Theory, vol. 58, no. 1, pp , Jan [23] D. Tse P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, Ehsan Ardestanizadeh (S 08 M 11) received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2004, the M.S. Ph.D. degrees in electrical engineering from the University of California, San Diego, in , respectively. Since February 2011, he has been with ASSIA Inc., Redwood City, CA. Dr. Ardestanizadeh is the recipient of the 2010 Shannon Fellowship awarded by the Center for Magnetic Recording at the University of California. His research interests are in information theory, feedback communication, networked systems. Paolo Minero (M 11) received the Laurea degree (with highest honors) in electrical engineering from the Politecnico di Torino, Torino, Italy, in 2003, the M.S. degree in electrical engineering from the University of California at Berkeley in 2006, the Ph.D. degree in electrical engineering from the University of California at San Diego in He is an Assistant Professor in the Department of Electrical Engineering at the University of Notre Dame. Before joining the University of Notre Dame, he was a postdoctoral scholar at the University of California at San Diego for six months. His research interests are in communication systems theory include information theory, wireless communication, control over networks. Dr. Minero received the U.S. Vodafone Fellowship in , the Shannon Memorial Fellowship in Massimo Franceschetti (M 98 SM 11) received the Laurea degree (magna cum laude) in computer engineering from the University of Naples, Naples, Italy, in 1997, the M.S. Ph.D. degrees in electrical engineering from the California Institute of Technology, Pasadena, CA, in 1999, 2003, respectively. He is an Associate Professor in the Department of Electrical Computer Engineering, University of California at San Diego (UCSD). Before joining UCSD, he was a postdoctoral scholar at the University of California at Berkeley for two years. He has held visiting positions at the Vrije Universiteit Amsterdam, the Ecole Polytechnique Federale de Lausanne, the University of Trento. His research interests are in communication systems theory include rom networks, wave propagation in rom media, wireless communication, control over networks. Dr. Franceschetti is an Associate Editor for Communication Networks of the IEEE TRANSACTIONS ON INFORMATION THEORY ( ) has served as a Guest Editor for two issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATION. He was awarded the C. H. Wilts Prize in 2003 for best doctoral thesis in electrical engineering at Caltech; the S.A. Schelkunoff Award in 2005 for best paper in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION; a National Science Foundation (NSF) CAREER award in 2006, an ONR Young Investigator Award in 2007; the IEEE Communications Society Best Tutorial Paper Award in 2010.