We consider a line network of nodes connected by additive white Gaussian noise channels and equipped with local feedback. We study the velocity at which information spreads over this network. For the transmission of a data packet, we derive an explicit positive lower bound on the velocity for any packet size. Furthermore, we consider streaming, that is, transmission of data packets that is generated at a given average arrival rate. We show that a positive velocity exists as long as the arrival rate is below the individual Gaussian channel capacity and provide an explicit lower bound. Our analysis involves applying pulse-amplitude modulation to the data (successively in the streaming case) and using linear mean-squared error estimation at the network nodes. Due to the analog-linear nature of the scheme, the results extend to any additive noise. For general noise, we derive exponential error-probability bounds. Moreover, for (sub-)Gaussian noise, we show doubly-exponential behavior, which reduces to the celebrated Schalkwijk-Kailath scheme when considering a single node. By viewing the constellation as an “analog source”, we also provide bounds on the exponential decay of the mean-squared error of source transmission over the network.

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate$R_{\mathrm{h}}$to the transmitter and/or the receiver. We derive upper and lower bounds on the optimal achievable$\alpha-\mathbf{th}$moment of the estimation error and show that they coincide for small values of$\alpha$and for high values of$R_{\mathrm{h}}$. The upper bound relies on a recently proposed channel-coding scheme that effectively conveys$R_{\mathrm{h}}$bits essentially error-free and the rest of the rate—over the same AWGN channel without help, with the error-free bits being allocated to the most significant bits of the quantized parameter.

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate$R_{\mathrm {h}}$to the transmitter and/or the receiver. We derive upper and lower bounds on the optimal achievable$\alpha $-th moment of the estimation error and show that they coincide for small values of$\alpha $and for high values of$R_{\mathrm {h}}$. The upper bound relies on a recently proposed channel-coding scheme that effectively conveys$R_{\mathrm {h}}$bits essentially error-free and the rest of the rate—over the same AWGN channel without help, with the error-free bits being allocated to the most significant bits of the quantized parameter. We then concentrate on the setting with a total transmit energy constraint, for which we derive achievability results for both channel coding and parameter modulation for several scenarios: when the helper assists only the transmitter or only the receiver and knows the noise, and when the helper assists the transmitter and/or the receiver and knows both the noise and the message. In particular, for the message-informed helper that assists both the receiver and the transmitter, it is shown that the error probability in the channel-coding task decays doubly exponentially. Finally, we translate these results to those for continuous-time power-limited AWGN channels with unconstrained bandwidth. As a byproduct, we show that the capacity with a message-informed helper that is available only at the transmitter can exceed the sum of the capacity without help and the help rate$R_{\mathrm {h}}$.

We consider the problem of in-order packet transmission over a cascade of packet-erasure links with acknowledgment (ACK) signals, interconnected by relays. We treat first the case of transmitting a single packet, in which ACKs are unnecessary, over links with independent identically distributed erasures. For this case, we derive tight upper and lower bounds on the probability of arrive failure within an allowed end-to-end communication delay over a given number of links. When the number of links is commensurate with the allowed delay, we determine the maximal ratio between the two—coined information velocity—for which the arrive-failure probability decays to zero; we further derive bounds on the arrive-failure probability when the ratio is below the information velocity, determine the exponential arrive-failure decay rate, and extend the treatment to links with different erasure probabilities. We then elevate all these results for a stream of packets with independent geometrically distributed interarrival times, and prove that the information velocity and the exponential decay rate remain the same for any stationary ergodic arrival process and for deterministic interarrival times. We demonstrate the significance of the derived fundamental limits—the information velocity and the arrive-failure exponential decay rate—by comparing them to simulation results.

It is common to assess the "memory strength" of a stationary process by looking at how fast the normalized log– determinant of its covariance submatrices (i.e., entropy rate) decreases. In this work, we propose an alternative characterization in terms of the normalized trace–inverse of the covariance submatrices. We show that this sequence is monotonically non-decreasing and is constant if and only if the process is white. Furthermore, while the entropy rate is associated with one-sided prediction errors (present from past), the new measure is associated with two-sided prediction errors (present from past and future). Minimizing this measure is then used as an alternative to Burg’s maximum-entropy principle for spectral estimation.

We consider the problem of transmitting a source over an infinite-bandwidth additive white Gaussian noise channel with unknown noise level under an input energy constraint. We construct a universal scheme that uses modulo-lattice modulation with multiple layers; for each layer, we employ either analog linear modulation or analog pulse position modulation (PPM). We show that the designed scheme with linear layers requires less energy than existing solutions to achieve the same quadratically increasing distortion profile with the noise level; replacing the linear layers with PPM layers offers an additional improvement.

We study the problem of transmitting a source sample with minimum distortion over an infinite-bandwidth additive white Gaussian noise channel under an energy constraint. To that end, we construct a joint source–channel coding scheme using analog pulse position modulation (PPM) and bound its quadratic distortion. We show that this scheme outperforms existing techniques, since its quadratic distortion attains both the exponential and polynomial decay orders of Burnashev’s outer bound. We supplement our theoretical results with numerical simulations and comparison to existing schemes.

It is common to assess the “memory strength” of a stationary process by looking at how fast the normalized log–determinant of its covariance submatrices (i.e., entropy rate) decreases. In this work, we propose an alternative characterization in terms of the normalized trace–inverse of the covariance submatrices. We show that this sequence is monotonically non-decreasing and is constant if and only if the process is white. Furthermore, while the entropy rate is associated with one-sided prediction errors (present from past), the new measure is associated with two-sided prediction errors (present from past and future). Minimizing this measure is then used as an alternative to Burg’s maximum-entropy principle for spectral estimation. We also propose a counterpart for non-stationary processes, by looking at the average trace–inverse of subsets.

We study the problem of transmitting a source sample with minimum distortion over an infinite-bandwidth additive white Gaussian noise channel under an energy constraint. To that end, we construct a joint source–channel coding scheme using analog pulse position modulation (PPM) and bound its quadratic distortion. We show that this scheme outperforms existing techniques since its quadratic distortion attains both the exponential and polynomial decay orders of Burnashev’s outer bound.

We address the recently suggested problem of causal lossless coding of randomly arriving source samples. We construct variable-to-fixed coding schemes and show that they outperform the previously considered fixed-to-variable schemes when traffic is high, in terms of both delay and Age of Information by appealing to tools from queueing theory. We supplement our theoretical bounds with numerical simulations.

We consider a discrete-time linear quadratic Gaussian networked control setting where the (full information) observer and controller are separated by a fixed-rate noiseless channel. We study the event-triggered control setup in which the encoder may choose to either transmit a packet or remain silent. We recast this problem into that of fixed-rate quantization with an extra symbol that corresponds to the silence event. This way, controlling the average transmission rate is possible by constraining the minimal probability of the silence symbol. We supplement our theoretical framework with numerical simulations.

We consider the problem of communications over the binary symmetric channel with feedback, where the information sequence is made available in a causal, possibly random, fashion. We develop a real-time variant of the renowned Horstein scheme and provide analytical guarantees for its error-probability exponential decay rate. We further use the scheme to stabilize an unstable control plant over a binary symmetric channel and compare the analytical guarantees with its empirical performance as well as with those of anytime-reliable codes.

The problem of sending a secret message over the Gaussian multiple-input multiple-output (MIMO) wiretap channel is studied. While the capacity of this channel is known, it is not clear how to construct optimal coding schemes that achieve this capacity. In this paper, we use linear operations along with successive interference cancellation to attain effective parallel single-antenna wiretap channels. By using independent scalar Gaussian wiretap codebooks over the resulting parallel channels, the capacity of the MIMO wiretap channel is achieved. The derivation of the schemes is based upon joint triangularization of the channel matrices. We find that the same technique can be used to rederive capacity expressions for the MIMO wiretap channel in a way that is simple and closely connected to a transmission scheme. This technique allows to extend the previously proven strong security for scalar Gaussian channels to the MIMO case. We further consider the problem of transmitting confidential messages over a two-user broadcast MIMO channel. For that problem, we find that derivation of both the capacity and a transmission scheme is a direct corollary of the proposed analysis for the MIMO wiretap channel.

We consider the problem of sequential transmission of Gauss-Markov sources. We show that in the limit of large spatial block lengths, greedy compression with respect to the squared error distortion is optimal; that is, there is no tension between optimizing the distortion of the source in the current time instant and that of future times. We then extend this result to the case where at time t a random compression rate rtis allocated independently of the rate at other time instants. This, in turn, allows us to derive the optimal performance of sequential coding over packet-erasure channels with instantaneous feedback. For the case of packet erasures with delayed feedback, we connect the problem to that of compression with side information that is known at the encoder and may be known at the decoder - where the most recent packets serve as side information that may have been erased, and demonstrate that the loss due to a delay by one time unit is rather small.

In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an independent identically distributed ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented that was shown to be optimal in the limit of high signal-to-noise ratios, where the sum capacity is dictated by the minimal (“bottleneck”) channel gain. In this paper, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions that transform the channel into effective parallel scalar dirty multiple-access channels. This approach, however, suffers from a “bottleneck” effect for each effective scalar channel and, therefore, the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high signal-to-noise ratios, under an equal rank assumption. This approach is then extended to any number of transmitters. Finally, an application to physical-layer network coding for the MIMO two-way relay channel is presented.

We consider the problem of stabilizing an unstable plant driven by bounded noise over a digital noisy communication link, a scenario at the heart of networked control. To stabilize such a plant, one needs real-time encoding and decoding with an error probability profile that decays exponentially with the decoding delay. The works of Schulman and Sahai over the past two decades have developed the notions of tree codes and anytime capacity, and provided the theoretical framework for studying such problems. Nonetheless, there has been little practical progress in this area due to the absence of explicit constructions of tree codes with efficient encoding and decoding algorithms. Recently, linear time-invariant tree codes were proposed to achieve the desired result under maximum-likelihood decoding. In this work, we take one more step towards practicality, by showing that these codes can be efficiently decoded using sequential decoding algorithms, up to some loss in performance (and with some practical complexity caveats). We supplement our theoretical results with numerical simulations that demonstrate the effectiveness of the decoder in a control system setting.

In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an i.i.d. ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented, which was shown to be optimal in the limit of high signal-to-noise ratios (SNRs), where the sum-capacity is dictated by the minimal (“bottleneck”) channel gain. In the present work, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions, which transform the channel into effective parallel scalar dirty multiple-access channels. This approach however suffers from a “bottleneck” effect for each effective scalar channel and therefore the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high SNRs, under an equal rank assumption.

A framework is developed for decode-and-forward-based relaying using standard coding and decoding that are good for the single-input single-output (SISO) additive white Gaussian noise channel. The framework is applicable to various scenarios and is demonstrated for several important cases. Each of these scenarios is transformed into an equivalent Gaussian multiple-input multiple-output (MIMO) common-message broadcast problem, which proves useful even when all links are SISO ones. Over the effective MIMO broadcast channel, a recently developed Gaussian MIMO common-message broadcast scheme is applied. This scheme transforms the MIMO links into a set of parallel SISO channels with no loss of mutual information, using linear pre- and post-processing combined with successive decoding. Over these resulting SISO channels, off-the-shelf scalar codes may be used.

We consider the problem of transmitting confidential messages over a two-user broadcast multiple-input multiple-output (MIMO) channel. Surprisingly, the capacity region of this setting under a covariance matrix constraint was shown by Liu et al. to be rectangular. That is, there is no tension, and both users can attain their respective MIMO wiretap capacities, simultaneously. In this work, we provide a new derivation of this result by proposing an alternative achievability scheme for the corner point of the capacity region. This derivation, in addition to being considerably shorter and simpler than the original, also provides a practical transmission scheme, in the sense that the codes used are scalar (single-antenna) ones. We use two main ingredients. The first is the explicit optimal input covariance matrix of Bustin et al. for the MIMO wiretap channel under a covariance matrix constraint, which we also re-derive in a simple manner. The second is a dirty-paper variant of a recently proposed optimal scheme for the MIMO wiretap channel, which uses scalar codes. The proposed treatment demonstrates the connection between the confidential broadcast problem and the MIMO wiretap one: the former almost reduces to the latter, except for the use of dirty-paper coding which is not mandatory in MIMO wiretap; the work sheds light on the reason for this difference.

A long-standing question in coding theory is whether code ensembles having a low-density parity check (LDPC) matrix can attain capacity under belief propagation (BP) decoding. An affirmative answer to this problem was recently given by the special class of spatially-coupled LDPC code ensemble. In this work, we provide a simple derivation of a different LDPC code ensemble that approaches capacity under BP decoding, following the classical approach of serial concatenation. This LDPC code ensemble is constructed by concatenating a high-rate outer LDPC code with an inner random convolutional one. The analysis of the concatenated-coding framework takes a particularly simple - “black box” - form. Specifically, the joint effect of the particular inner code and the binary-input memoryless output-symmetric (BMS) channel is encapsulated in a single parameter - the Bhattacharyya parameter, which is maximal for the binary symmetric channel (BSC). This implies that an inner convolutional code designed for the BSC achieves good performance over all BMS channels with a given capacity. Moreover, the performance guarantee of the outer LDPC code under BP decoding is dictated solely by this parameter. This, in turn, implies that the overall concatenated code approaches capacity under BP decoding for all BMS channels with a given capacity, simultaneously.

The problem of transmitting a common message to multiple users over the Gaussian multiple-input multiple-output broadcast channel is considered, where each user is equipped with an arbitrary number of antennas. A closed-loop scenario is assumed, for which a practical capacity-approaching scheme is developed. By applying judiciously chosen unitary operations at the transmit and receive nodes, the channel matrices are triangularized so that the resulting matrices have equal diagonals, up to a possible multiplicative scalar factor. This, along with the utilization of successive interference cancellation, reduces the coding and decoding tasks to those of coding and decoding over the single-antenna additive white Gaussian noise channel. Over the resulting effective channel, any off-the-shelf code may be used. For the two-user case, it was recently shown that such joint unitary triangularization is always possible. In this paper, it is shown that for more than two users, it is necessary to carry out the unitary linear processing jointly over multiple channel uses, i.e., space-time processing is employed. It is further shown that exact triangularization, where all resulting diagonals are equal, is still not always possible, and appropriate conditions for the existence of such are established for certain cases. When exact triangularization is not possible, an asymptotic construction is proposed, that achieves the desired property of equal diagonals up to edge effects that can be made arbitrarily small, at the price of processing a sufficiently large number of channel uses together.

The problem of sending a secret message over the multiple-input multiple-output (MIMO) wiretap Gaussian channel is studied. While the capacity of this channel is known, it is not clear how to construct optimal coding schemes that achieve this capacity. In this work we show how to use linear operations along with successive interference cancellation in order to reduce the problem to that of designing optimal codes for the single-antenna additive-noise Gaussian wiretap channel. Much like popular communication techniques in the absence of an eavesdropper, the data is carried over parallel streams. The design approach is flexible enough to allow for using the same scalar wiretap code over all streams, or alternatively to use different scalar wiretap codes over parallel sub-channels without successive interference cancellation. This approach is applicable to more involved secrecy settings, by adjusting the linear operations performed by the encoder, and by jointly processing several channel uses.

The problem of sending a secret message over the Gaussian multiple-input multiple-output wiretap channel is studied. In a recent work, we have proposed a layered coding scheme where a scalar wiretap code is used in each layer, and successive interference cancellation (SIC) is carried at the legitimate receiver. By a proper rate allocation across the layers, we showed that this scheme satisfies the secrecy constraint at the eavesdropper and achieves the secrecy capacity. However, the existence of the scalar codes was based upon a random coding argument. In this work we take a further step and show how the scheme can be based upon any codes that are good for the ordinary (non-secrecy) additive white Gaussian noise channel. As any stage of the SIC process is equivalent to achieving a corner point of a Gaussian multiple-access channel (MAC) capacity region, the class of codes used needs to be good for the MAC under SIC. Since in the secrecy analysis of our layered scheme, it suffices at each stage to consider a genie-aided eavesdropper that performs SIC, the coding task reduces to guaranteeing secrecy for corner points of induced MACs to the eavesdropper. Structured generation of such codes from ordinary ones is discussed.

The Gaussian parallel relay network, introduced by Schein and Gallager, consists of a concatenation of a Gaussian additive broadcast channel from a single encoder to a layer of relays followed by a Gaussian multiple-access channel from the relays to the final destination (decoder), where all noises are independent. This setup exhibits an inherent conflict between digital and analog relaying; while analog relaying [known as amplify-and-forward (A&F)] suffers from noise accumulation, digital relaying (known as decode-and-forward) looses the potential coherence gain in combining the relay noises at the decoder. For a large number of relays, the coherence gain is large, and thus analog relaying has better performance; however, it is limited to white channels of equal bandwidth. In this paper, we present a generalization of the analog approach to the case of bandwidth mismatch. Our strategy, coined rematch and forward (R&F), is based upon applying joint source-channel coding techniques that belong to a certain class of maximally analog schemes. Using such techniques, R&F converts the bandwidth of the broadcast section to that of the multiple-access section, creating an equivalent matched-bandwidth network over which A&F is applied. It is shown that this strategy exploits the full bandwidth of the individual channels, without sacrificing the coherence gain offered by A&F. Specifically, for given individual-link capacities, R&F remains within a constant gap from the network capacity for any number of relays and any bandwidth ratio between the sections. Finally, the approach is extended to the case of colored channels.

We address the problem of communication over arbitrarily permuted parallel Gaussian channels, where the permutation is known only to the receiver. We present a practical transmission scheme, that allows to transmit over this channel using off-the-shelf codes, in conjunction with linear processing and successive interference cancellation. The scheme is based on the approach of joint matrix triangularization. Explicit precoding matrices are derived for up to six parallel channels.

Multicasting is the general method of conveying the same information to multiple users over a broadcast channel. In this work, the Gaussian MIMO broadcast channel is considered, with multiple users and any number of antennas at each node. A “closed loop” scenario is assumed, for which a practical capacity-achieving multicast scheme is constructed. In the proposed scheme, linear modulation is carried over time and space together, which allows to transform the problem into that of transmission over parallel scalar sub-channels, the gains of which are equal, except for a fraction of sub-channels that vanishes with the number of time slots used. Over these sub-channels, off-the-shelf fixed-rate AWGN codes can be used to approach capacity.

This work considers practical implementation of the decode-and-forward relaying protocol for the full-duplex Gaussian relay channel. Unlike previous works which developed coding techniques tailored to this protocol, it is shown that standard codes which are good for the Gaussian scalar channel of fixed signal-to-noise ratio suffice to approach the theoretical performance promised by this protocol. The proposed technique employs only linear operations and successive interference cancelation in conjunction with fixed signal-to-noise ratio base codes, and the achievable rate is solely dictated by the performance of these base codes. The same approach and results carry over to the multiple-antenna case as well.

This work considers the joint source-channel problem of transmitting a Gaussian source over a two-user multiple-input multiple-output (MIMO) broadcast channel. We show the existence of non-trivial channels, where the optimal distortion pair (which for high signal-to-noise ratios equals the point-to-point distortions of the individual users) may be achieved. A condition for existence of a joint triangularization of the MIMO channels which shapes the ratio of the diagonals to a desired form is derived. Whenever possible, all diagonal elements but one are made equal. We then employ a hybrid digital-analog scheme to the source, where the digital part is sent over the equal subchannels and the analog refinement is sent over the remaining one.

In a recent work, a capacity-achieving scheme for the common-message two-user MIMO broadcast channel, based on single-stream coding and decoding, was described. This was obtained via a novel joint unitary triangularization which is applied to the corresponding channel matrices. In this work, the triangularization is generalized, to any (finite) number of matrices, allowing multi-user applications. To that end, multiple channel uses are jointly treated, in a manner reminiscent of space-time coding. As opposed to the two-user case, in the general case there does not always exist a perfect (capacity-achieving) solution. However, a nearly optimal scheme (with vanishing loss in the limit of large blocks) always exists. Common-message broadcasting is but one example of communication networks with MIMO links which can be solved using an approach coined “Network Modulation”; the extension beyond two links carries over to these problems.

The physical-layer network coding (PNC) approach provides improved performance in many scenarios over “traditional” relaying techniques or network coding. This work addresses the generalization of PNC to wireless scenarios where network nodes have multiple antennas. We use a recent matrix decomposition, which allows, by linear pre- and post-processing, to simultaneously transform both channel matrices to triangular forms, where the diagonal entries, corresponding to both channels, are equal. This decomposition, in conjunction with precoding, allows to convert any two-input multiple-access channel (MAC) into parallel MACs, over which single-antenna PNC may be used. The technique is demonstrated using the two-way relay channel with multiple antennas. For this case it is shown that, in the high signal-to-noise regime, the scheme approaches the cut-set bound, thus establishing the asymptotic network capacity.

State-dependent channels have received much attention over the years, due to their relevance in many different network and multi-user communication scenarios. Nonetheless, previous treatments of this problem assumed that all of the state is available in the same manner: causally, non-causally or non-causally with a finite look-ahead. Yet, in many realistic situations, different parts of the state are known in a different manner. We consider the case where the state is composed of several parts, where each part is known with a different look-ahead. Specifically, we derive the capacity for the case where part of the state is known non-causally to the transmitter, whereas the other part is known only causally, and demonstrate that there are cases in which this capacity can be strictly larger that the capacity of the case where the state is known in a causal fashion, and strictly smaller than the capacity of the same channel, where the state is available non-causally. We note that the treatment in this work provides a unified framework for treating the causal state-information case, the non-causal state-information case, as well as a mixture of the two.

The problem of multicasting common data to several users over multiple-input multiple-output (MIMO) Gaussian channels is studied. A closed-loop setup is considered where the channel matrices are known to the transmitter and respective receivers. An incremental-redundancy (rateless) scenario is considered, where the effective rate is measured by the time that each user needs to stay online until it is able to decode the message. A practical transmission scheme for the two-user case is proposed which, by linear pre - and post-processing combined with successive decoding and interference cancellation, transforms the two MIMO channels into a set of parallel channels with no loss of mutual information, where each user needs to tune in for a duration of time proportional to its individual capacity. This scheme is used for designing a practical transmission scheme for the Gaussian MIMO half-duplex relay channel. We then turn to the related scenario of transmission to a single user over a MIMO channel with unknown but constant signal-to-noise ratio (SNR), for which we develop an optimal low-complexity hybrid ARQ coding scheme, which is optimal for two SNRs and propose a scheme for more SNRs, the loss of which vanishes when the SNRs are high. Finally, we show that even when applied to single-input single-output (“scalar”) channels, the scheme provides a practical solution for cases not covered by previous work.

A dirty-paper channel is considered, where the transmitter knows the interference sequence up to a constant multiplicative factor, known only to the receiver. Lower bounds on the achievable rate of communication are derived by proposing a coding scheme that partially compensates for the imprecise channel knowledge.We focus on a communication scenario where the signal-to-noise ratio is high. Our approach is based on analyzing the performance achievable using lattice-based coding schemes. When the power of the interference is finite, we show that the achievable rate of this lattice-based coding scheme may be improved by a judicious choice of the scaling parameter at the receiver. We further show that the communication rate may be improved, for finite as well as infinite interference power, by allowing randomized scaling at the transmitter.

The Gaussian parallel relay network problem consists of transmitting a message from a single source node to a single destination node, through a layer of parallel relay nodes. The source is connected to the relays by a Gaussian broadcast channel, while the relays are connected to the destination by a Gaussian multiple access channel. When the channels are all white with the same bandwidth, and the relays cannot decode the message, the best known strategy is "amplify and forward", which achieves the coherence gain of multiple relays. We propose a strategy which achieves this gain even when the noises are colored or the channels have different bandwidths. To that end we use analog modulo-lattice modulation of the codewords in the BC, and then forward the estimated codeword by each of the relays to the MAC. This modulation allows the relays to re-match the signal to the optimal spectrum of the MAC, thus demonstrating how a channel problem can gain from a joint source/channel approach. We show that this strategy is asymptotically optimal in some limiting cases, and that it outperforms the known alternatives in most other cases, where the optimum is unknown. We also demonstrate how to improve the achievable rate in the original white problem, for some signal to noise ratio values.

A dirty paper channel is considered, where the transmitter knows the interference sequence up to a constant multiplicative factor, known only to the receiver. We derive lower bounds on the achievable rate of communication by proposing a coding scheme that partially compensates for the imprecise channel knowledge.We focus on a communication scenario where the Gaussian noise is small while the interference is strong. Our approach is based on analyzing the performance achievable using extended Tomlinson-Harashima like coding schemes. When the power of the interference is finite, we show that this may be achieved by a judicious choice of the scaling parameter at the receiver. We further show that the communication rate may be improved, for finite as well as infinite interference power, by allowing randomized scaling at the transmitter.