In this paper, we first consider a channel that is contaminated by two independent Gaussian noises @math and @math . The capacity of this channel is computed when independent noisy versions of @math are known to the transmitter and or receiver. It is shown that the channel capacity is greater then the capacity when @math is completely unknown, but is less then the capacity when @math is perfectly known at the transmitter or receiver. For example, if there is one noisy version of @math known at the transmitter only, the capacity is @math , where @math is the input power constraint and @math is the power of the noise corrupting @math . We then consider a Gaussian cognitive interference channel (IC) and propose a causal noisy dirty paper coding (DPC) strategy. We compute the achievable region using this noisy DPC strategy and quantify the regions when it achieves the upper bound on the rate.
One special case of ) is when a noisy version of @math is known only to the transmitter; our result in this case is a generalization of Costa's celebrated result @cite_6 . @cite_6 , it is shown that the achievable rate when the noise @math is perfectly known at the transmitter is equivalent to the rate when @math is known at the receiver, and this rate does not depend on the variance of @math . A new coding strategy to achieve this capacity was also introduced in @cite_6 and is popularly referred to as dirty paper coding (DPC). We generalize Costa's result to the case of noisy interference knowledge. We show that the capacity with knowledge of a noisy version of @math at the transmitter is equal to the capacity with knowledge of a statistically equivalent noisy version of @math at the receiver. However, unlike @cite_6 where the capacity does not depend on the variance of @math , in the general noisy side information case, the capacity decreases as the variance of @math increases.
@cite_6
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In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel (MAC) setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver. The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies (“lattice precoding”) can achieve positive rates, independent of the interference power. Furthermore, in some cases-which depend on the noise variance and power constraints-high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR-where the capacity region of the dirty MAC with strong interference approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1 2 log2(πe 6) ≈ 0.254 bit. Thus, the doubly dirty MAC is another instance of a network setting, like the Korner-Marton problem, where (linear) structured coding is potentially better than random binning. A generalization of the Gaussian dirty-paper problem to a multiple access setup is considered. There are two additive interference signals, one known to each transmitter but none to the receiver. The rates achievable using Costa's strategies (i.e. by a random binning scheme induced by Costa's auxiliary random variables) vanish in the limit when the interference signals are strong. In contrast, it is shown that lattice strategies ("lattice precoding") can achieve positive rates independent of the interferences, and in fact in some cases - which depend on the noise variance and power constraints - they are optimal. In particular, lattice strategies are optimal in the limit of high SNR. It is also shown that the gap between the achievable rate region and the capacity region is at most 0.167 bit. Thus, the dirty MAC is another instance of a network setup, like the Korner-Marton modulo-two sum problem, where linear coding is potentially better than random binning. Lattice transmission schemes and conditions for optimality for the asymmetric case, where there is only one interference which is known to one of the users (who serves as a "helper" to the other user), and for the "common interference" case are also derived. In the former case the gap between the helper achievable rate and its capacity is at most 0.085 bit. We study an interference network where equally-numbered transmitters and receivers lie on two parallel lines, each transmitter opposite its intended receiver. We consider two short-range interference models: the "asymmetric network," where the signal sent by each transmitter is interfered only by the signal sent by its left neighbor (if present), and a "symmetric network," where it is interfered by both its left and its right neighbors. Each transmitter is cognizant of its own message, the messages of the @math transmitters to its left, and the messages of the @math transmitters to its right. Each receiver decodes its message based on the signals received at its own antenna, at the @math receive antennas to its left, and the @math receive antennas to its right. For such networks we provide upper and lower bounds on the multiplexing gain, i.e., on the high-SNR asymptotic logarithmic growth of the sum-rate capacity. In some cases our bounds meet, e.g., for the asymmetric network. Our results exhibit an equivalence between the transmitter side-information parameters @math and the receiver side-information parameters @math in the sense that increasing decreasing @math or @math by a positive integer @math has the same effect on the multiplexing gain as increasing decreasing @math or @math by @math . Moreover---even in asymmetric networks---there is an equivalence between the left side-information parameters @math and the right side-information parameters @math . We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of @math non-colluding databases, each storing a coded version of @math messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is @math , where @math is the rate of the @math code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases. Lattice codes used under the Compute-and-Forward paradigm suggest an alternative strategy for the standard Gaussian multiple-access channel (MAC): The receiver successively decodes integer linear combinations of the messages until it can invert and recover all messages. In this paper, this strategy is developed and analyzed. For the two-user MAC, it is shown that without time-sharing, the entire capacity region can be attained with a single-user decoder as soon as the signal-to-noise ratios are above @math . A partial analysis is given for more than two users. Lastly the strategy is extended to the so-called dirty MAC where two interfering signals are known non-causally to the two transmitters in a distributed fashion. Our scheme extends the previously known results and gives new achievable rate regions. We study the degrees of freedom (DoF) of the @math -user interference channel with coordinated multipoint (CoMP) transmission and reception. Each message is jointly transmitted by @math successive transmitters, and is jointly received by @math successive receivers. We refer to this channel as the CoMP channel with a transmit cooperation order of @math and receive cooperation order of @math . Since the channel has a total of @math transmit antennas and @math receive antennas, the maximum possible DoF is equal to @math . We show that the CoMP channel has @math DoF if and only if @math . The key idea is that the zero forcing of the interference corresponding to the @math message at the decoder of the @math message, where @math , can be viewed as a shared responsibility between the @math transmitters carrying the @math message, and the @math receivers decoding the @math message. For the general case, we derive an outer bound that states that the DoF is bounded above by @math . For the special case with only CoMP transmission, i.e, @math , we propose a scheme that can achieve @math DoF for all @math K @math M_ t @math M_ t -1$ receivers, thereby allowing each of these receivers to enjoy 1 DoF, and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space. The achievability proofs are based on the notion of algebraic independence from algebraic geometry. We provide the first capacity approaching coding schemes that robustly simulate any interactive protocol over an adversarial channel that corrupts any @math fraction of the transmitted symbols. Our coding schemes achieve a communication rate of @math over any adversarial channel. This can be improved to @math for random, oblivious, and computationally bounded channels, or if parties have shared randomness unknown to the channel. Surprisingly, these rates exceed the @math interactive channel capacity bound which [Kol and Raz; STOC'13] recently proved for random errors. We conjecture @math and @math to be the optimal rates for their respective settings and therefore to capture the interactive channel capacity for random and adversarial errors. In addition to being very communication efficient, our randomized coding schemes have multiple other advantages. They are computationally efficient, extremely natural, and significantly simpler than prior (non-capacity approaching) schemes. In particular, our protocols do not employ any coding but allow the original protocol to be performed as-is, interspersed only by short exchanges of hash values. When hash values do not match, the parties backtrack. Our approach is, as we feel, by far the simplest and most natural explanation for why and how robust interactive communication in a noisy environment is possible. A Gaussian broadcast channel (GBC) with r single-antenna receivers and t antennas at the transmitter is considered. Both transmitter and receivers have perfect knowledge of the channel. Despite its apparent simplicity, this model is, in general, a nondegraded broadcast channel (BC), for which the capacity region is not fully known. For the two-user case, we find a special case of Marton's (1979) region that achieves optimal sum-rate (throughput). In brief, the transmitter decomposes the channel into two interference channels, where interference is caused by the other user signal. Users are successively encoded, such that encoding of the second user is based on the noncausal knowledge of the interference caused by the first user. The crosstalk parameters are optimized such that the overall throughput is maximum and, surprisingly, this is shown to be optimal over all possible strategies (not only with respect to Marton's achievable region). For the case of r>2 users, we find a somewhat simpler choice of Marton's region based on ordering and successively encoding the users. For each user i in the given ordering, the interference caused by users j>i is eliminated by zero forcing at the transmitter, while interference caused by users j<i is taken into account by coding for noncausally known interference. Under certain mild conditions, this scheme is found to be throughput-wise asymptotically optimal for both high and low signal-to-noise ratio (SNR). We conclude by providing some numerical results for the ergodic throughput of the simplified zero-forcing scheme in independent Rayleigh fading. This paper investigates the information-theoretic limits of energy-harvesting (EH) channels in the finite blocklength regime. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. We use the save-and-transmit strategy proposed by Ozel and Ulukus (2012) together with Shannon’s non-asymptotic achievability bound to obtain lower bounds on the achievable rates for both additive white Gaussian noise channels and discrete memoryless channels under EH constraints. The first-order terms of the lower bounds of the achievable rates are equal to @math and the second-order (backoff from capacity) terms are proportional to @math , where @math denotes the blocklength and @math denotes the capacity of the EH channel, which is the same as the capacity without the EH constraints. The constant of proportionality of the backoff term is found and qualitative interpretations are provided. We present a general framework for private information retrieval (PIR) from arbitrary coded databases that allows one to adjust the rate of the scheme to the suspected number of colluding servers. If the storage code is a generalized Reed--Solomon code of length @math and dimension @math , we design PIR schemes that achieve a PIR rate of @math while protecting against any @math colluding servers, for any @math . This interpolates between the previously studied cases of @math and @math and achieves PIR capacity in both of these cases asymptotically as the number of files in the database grows. We consider the generalized dirty-paper channel Y=X+S+N,E X sup 2 spl les P sub X , where N is not necessarily Gaussian, and the interference S is known causally or noncausally to the transmitter. We derive worst case capacity formulas and strategies for "strong" or arbitrarily varying interference. In the causal side information (SI) case, we develop a capacity formula based on minimum noise entropy strategies. We then show that strategies associated with entropy-constrained quantizers provide lower and upper bounds on the capacity. At high signal-to-noise ratio (SNR) conditions, i.e., if N is weak relative to the power constraint P sub X , these bounds coincide, the optimum strategies take the form of scalar lattice quantizers, and the capacity loss due to not having S at the receiver is shown to be exactly the "shaping gain" 1 2log(2 spl pi e 12) spl ap 0.254 bit. We extend the schemes to obtain achievable rates at any SNR and to noncausal SI, by incorporating minimum mean-squared error (MMSE) scaling, and by using k-dimensional lattices. For Gaussian N, the capacity loss of this scheme is upper-bounded by 1 2log2 spl pi eG( spl Lambda ), where G( spl Lambda ) is the normalized second moment of the lattice. With a proper choice of lattice, the loss goes to zero as the dimension k goes to infinity, in agreement with the results of Costa. These results provide an information-theoretic framework for the study of common communication problems such as precoding for intersymbol interference (ISI) channels and broadcast channels. We consider the problem of single-round private information retrieval (PIR) from @math replicated databases. We consider the case when @math databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of @math messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the @math -privacy constraint in this paper, where any @math databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of that can be retrieved privately (under the @math -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be @math , if @math . This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing @math databases from the system, with a penalty factor of @math , which signifies that even though the number of databases needed for PIR is effectively @math , the user still needs to access the entire @math databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the introduced by the Byzantine databases as opposed to in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes. Writing on dirty paper, a variation of the standard additive white Gaussian noise (AWGN) channel with the channel output is considered. The Gaussian noise noncausally known at the transmitter does not affect the capacity of the AWGN channel. This is known as WDP property. The WDP property for three Gaussian multiple user channels with known capacity region is established. The achievable rate regions of corresponding discrete channels is proved. Although those regions may be suboptimal in general, they turn out to be optimal for these Gaussian channels. Alternative proofs can be obtained by successive uses of Costa's WDP coding scheme, from which the WDP property can be established for more general noise and state distributions. This paper examines the problems of dirty-paper and dirty-tape coding with some additional side information (SI) at the decoder. In particular we focus on the situation in which the decoder SI is a noisy version of the SI at the encoder. We show that noisy SI at the decoder is advantageous for many channels in terms of capacity and, in general, the optimal coding strategy needs to take it into account. In particular, the capacities of the binary dirty-paper and dirty-tape channels are computed when the decoder SI is modeled as the output of a binary symmetric channel with the encoder SI as input. Moreover, the advantage offered by the decoder SI on the minimum achievable Eb NO in the dirty-tape scenario with Gaussian interference is demonstrated Let @math denote the first passage time to 1 of a standard Brownian motion. It is well known that as @math goes to infinity, @math goes to zero at rate @math , where @math equals @math . The goal of this note is to establish a quantitative, infinite dimensional version of this result. Namely, we will prove the existence of positive and finite constants @math and @math , such that for all @math , @math where @math ' denotes the natural logarithm, and @math is the Fukushima-Malliavin capacity on the space of continuous functions. We consider the problem of estimating an unknown signal @math from noisy linear observations @math . In many practical instances, @math has a certain structure that can be captured by a structure inducing convex function @math . For example, @math norm can be used to encourage a sparse solution. To estimate @math with the aid of @math , we consider the well-known LASSO method and provide sharp characterization of its performance. We assume the entries of the measurement matrix @math and the noise vector @math have zero-mean normal distributions with variances @math and @math respectively. For the LASSO estimator @math , we attempt to calculate the Normalized Square Error (NSE) defined as @math as a function of the noise level @math , the number of observations @math and the structure of the signal. We show that, the structure of the signal @math and choice of the function @math enter the error formulae through the summary parameters @math and @math , which are defined as the Gaussian squared-distances to the subdifferential cone and to the @math -scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of @math and is given by @math . We prove that its worst case NSE is achieved when @math and concentrates around @math . Secondly, we consider @math , for some @math . This time the NSE formula depends on the choice of @math and is given by @math . We then establish a mapping between this and the third estimator @math . Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE. Interference alignment has emerged as a powerful tool in the analysis of multiuser networks. Despite considerable recent progress, the capacity region of the Gaussian @math -user interference channel is still unknown in general, in part due to the challenges associated with alignment on the signal scale using lattice codes. This paper develops a new framework for lattice interference alignment, based on the compute-and-forward approach. Within this framework, each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer-valued coefficients and then solving these linear combinations for its desired codeword. For the special case of symmetric channel gains, this framework is used to derive the approximate sum capacity of the Gaussian interference channel, up to an explicitly defined outage set of the channel gains. The key contributions are the capacity lower bounds for the weak through strong interference regimes, where each receiver should jointly decode its own codeword along with part of the interfering codewords. As part of the analysis, it is shown that decoding @math linear combinations of the codewords can approach the sum capacity of the @math -user Gaussian multiple-access channel up to a gap of no more than @math bits. Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of @math messages from @math non-communicating replicated databases (each holds all @math messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. @math -private PIR is a generalization of PIR to include the requirement that even if any @math of the @math databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have @math databases, out of which any @math may fail to respond. For @math messages and @math databases out of which at least some @math must respond, we show that the capacity of @math -private and Robust PIR is @math . The result includes as special cases the capacity of PIR without robustness ( @math ) or @math -privacy constraints ( @math ). This paper develops a new communication strategy, ergodic interference alignment, for the K-user interference channel with time-varying fading. At any particular time, each receiver will see a superposition of the transmitted signals plus noise. The standard approach to such a scenario results in each transmitter-receiver pair achieving a rate proportional to 1 K its interference-free ergodic capacity. However, given two well-chosen time indices, the channel coefficients from interfering users can be made to exactly cancel. By adding up these two observations, each receiver can obtain its desired signal without any interference. If the channel gains have independent, uniform phases, this technique allows each user to achieve at least 1 2 its interference-free ergodic capacity at any signal-to-noise ratio. Prior interference alignment techniques were only able to attain this performance as the signal-to-noise ratio tended to infinity. Extensions are given for the case where each receiver wants a message from more than one transmitter as well as the “X channel” case (with two receivers) where each transmitter has an independent message for each receiver. Finally, it is shown how to generalize this strategy beyond Gaussian channel models. For a class of finite field interference channels, this approach yields the ergodic capacity region. Index Coding has received considerable attention recently motivated in part by real-world applications and in part by its connection to Network Coding. The basic setting of Index Coding encodes the problem input as an undirected graph and the fundamental parameter is the broadcast rate @math , the average communication cost per bit for sufficiently long messages (i.e. the non-linear vector capacity). Recent nontrivial bounds on @math were derived from the study of other Index Coding capacities (e.g. the scalar capacity @math ) by Bar- (2006), Lubetzky and Stav (2007) and (2008). However, these indirect bounds shed little light on the behavior of @math : there was no known polynomial-time algorithm for approximating @math in a general network to within a nontrivial (i.e. @math ) factor, and the exact value of @math remained unknown for any graph where Index Coding is nontrivial. Our main contribution is a direct information-theoretic analysis of the broadcast rate @math using linear programs, in contrast to previous approaches that compared @math with graph-theoretic parameters. This allows us to resolve the aforementioned two open questions. We provide a polynomial-time algorithm with a nontrivial approximation ratio for computing @math in a general network along with a polynomial-time decision procedure for recognizing instances with @math . In addition, we pinpoint @math precisely for various classes of graphs (e.g. for various Cayley graphs of cyclic groups) thereby simultaneously improving the previously known upper and lower bounds for these graphs. Via this approach we construct graphs where the difference between @math and its trivial lower bound is linear in the number of vertices and ones where @math is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse.