We introduce a novel mechanism, called timid/bold coding, by which feedback can be used to improve coding performance. For a certain class of DMCs, called compound-dispersion channels, we show that timid/bold coding allows for an improved second-order coding rate compared with coding without feedback. For DMCs that are not compound dispersion, we show that feedback does not improve the second-order coding rate. Thus we completely determine the class of DMCs for which feedback improves the second-order coding rate. An upper bound on the second-order coding rate is provided for compound-dispersion DMCs. We also show that feedback does not improve the second-order coding rate for very noisy DMCs. The main results are obtained by relating feedback codes to certain controlled diffusions.
Linear coding schemes have been the main choice of coding for the additive white Gaussian noise broadcast channel (AWGN-BC) with noiseless feedback in the literature. The achievable rate regions of these schemes go well beyond the capacity region of the AWGN-BC without feedback. In this paper, a concatenating coding design for the $K$-user AWGN-BC with noisy feedback is proposed that relies on linear feedback schemes to achieve rate tuples outside the no-feedback capacity region. Specifically, a linear feedback code for the AWGN-BC with noisy feedback is used as an inner code that creates an e…
In this paper, we investigate the additive Gaussian noise channel with noisy feedback. We consider the setup of linear coding of the feedback information and Gaussian signaling of the message (i.e. Cover-Pombra Scheme). Then, we derive the upper and lower bounds on the largest achievable rate for this setup. We show that these two bounds can be obtained by solving two convex optimization problems. Finally, we present some simulations and discussion.
Optimal coding over the additive white Gaussian noise channel under the peak energy constraint is studied when there is noisy feedback over an orthogonal additive white Gaussian noise channel. As shown by Pinsker, under the peak energy constraint, the best error exponent for communicating an M-ary message, M >= 3, with noise-free feedback is strictly larger than the one without feedback. This paper extends Pinsker's result and shows that if the noise power in the feedback link is sufficiently small, the best error exponent for conmmunicating an M-ary message can be strictly larger than the one…
Network coding has the potential to improve the overall throughput of a network by combining different streams of data and forwarding them. In wireless networks, the wireless channel provide an excellent medium for physical layer network coding as signals from different transmitters are combined automatically by the wireless channel. In such scenarios, it would be interesting to investigate protocols and algorithms which can optimally relay information. In this paper, we look at a four-node two-way or bidirectional relay network, and propose a relay protocol which can relay information efficie…
We consider the problem of zero error source coding with limited feedback when side information is present at the receiver. First, we derive an achievable rate region for arbitrary joint distributions on the source and the side information. When all source pairs of source and side information symbols are observable with non-zero probability, we show that this characterization gives the entire rate region. Next, we demonstrate a class of sources for which asymptotically zero feedback suffices to achieve zero-error coding at the rate promised by the Slepian-Wolf bound for asymptotically lossless…
The model of wiretap channel (WTC) is important as it constitutes the essence of physical layer security (PLS). Wiretap channel with noiseless feedback (WTC-NF) is especially interesting as it shows what can be done when a private feedback is available. The already existing secret key based feedback coding scheme focuses on generating key from the feedback and using this key to protect part of the transmitted message. It has been shown that this secret key based feedback coding scheme is only optimal for the degraded WTC-NF, and finding an optimal feedback scheme for the general WTC-NF motivat…
We show that for the multi-antenna Gaussian multi-access channel (MAC) and broadcast channel (BC) with perfect feedback, the rate regions achieved by linear-feedback coding schemes (called linear-feedback capacity regions) coincide when the same total input-power constraint is imposed on both channels and when the MAC channel matrices are the transposes of the BC channel matrices. Such a pair of MAC and BC is called dual. We also identify sub-classes of linear-feedback coding schemes that achieve the linear-feedback capacity regions of these two channels and present multi-letter expressions fo…
Coding schemes for discrete memoryless multicast networks (DM-MN) with rate-limited feedback from the receivers and relays to the transmitter are proposed. The schemes improve over the noisy network coding proposed by Lim et al.. For the single relay channel with relay-transmitter feedback, our coding schemes recover Gabbai and Bross's results, and strictly improve on noisy network coding, distributed decode-forward coding proposed by Lim et al., and all known lower bounds on the achievable rates proposed for the setup without feedback. The coding schemes are based on block-Markov coding, supe…
This paper considers the memoryless input-constrained binary erasure channel (BEC). The channel input constraint is the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s in the input sequence be separated by at least $d$ $0$s. We consider a scenario where there is causal, noiseless feedback from the decoder. We demonstrate a simple, labelling-based, zero-error feedback coding scheme, which we prove to be feedback capacity-achieving, and, as a by-product, obtain an explicit characterization of the feedback capacity. Our proof is based on showing th…
We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a Hidden Markov Model (HMM). We propose a Decision Feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian Wolf limits.
We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a Hidden Markov Model (HMM). We propose a Decision Feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian Wolf limits.
A coding scheme is proposed for the memoryless Gaussian broadcast channel with correlated noises and feedback. For all noise correlations other than -1, the gap between the sum-rate the scheme achieves and the full-cooperation bound vanishes as the signal-to-noise ratio tends to infinity. When the correlation coefficient is -1, the gains afforded by feedback are unbounded and the prelog is doubled. When the correlation coefficient is +1 we demonstrate a dichotomy: If the noise variances are equal, then feedback is useless, and otherwise, feedback affords unbounded rate gains and doubles the pr…
The Ising channel, which was introduced in 1990, is a channel with memory that models Inter-Symbol interference. In this paper we consider the Ising channel with feedback and find the capacity of the channel together with a capacity-achieving coding scheme. To calculate the channel capacity, an equivalent dynamic programming (DP) problem is formulated and solved. Using the DP solution, we establish that the feedback capacity is the expression $C=(\frac{2H_b(a)}{3+a})\approx 0.575522$ where $a$ is a particular root of a fourth-degree polynomial and $H_b(x)$ denotes the binary entropy function. …
The error exponent of Markov channels with feedback is studied in the variable-length block-coding setting. Burnashev's classic result is extended and a single letter characterization for the reliability function of finite-state Markov channels is presented, under the assumption that the channel state is causally observed both at the transmitter and at the receiver side. Tools from stochastic control theory are used in order to treat channels with intersymbol interference. In particular the convex analytical approach to Markov decision processes is adopted to handle problems with stopping time…
We consider both channel coding and source coding, with perfect past feedback/feedforward, in the presence of side information. It is first observed that feedback does not increase the capacity of the Gel'fand-Pinsker channel, nor does feedforward improve the achievable rate-distortion performance in the Wyner-Ziv problem. We then focus on the Gaussian case showing that, as in the absence of side information, feedback/feedforward allows to efficiently attain the respective performance limits. In particular, we derive schemes via variations on that of Schalkwijk and Kailath. These variants, whi…
Existing fixed-length feedback communication schemes are either specialized to particular channels (Schalkwijk--Kailath, Horstein), or apply to general channels but either have high coding complexity (block feedback schemes) or are difficult to analyze (posterior matching). This paper introduces a new fixed-length feedback coding scheme which achieves the capacity for all discrete memoryless channels, has an error exponent that approaches the sphere packing bound as the rate approaches the capacity, and has $O(n\log n)$ coding complexity. These benefits are achieved by judiciously combining fe…
Recently, the physical layer security (PLS) of the communication systems has been shown to be enhanced by using legal receiver's feedback. The present secret key based feedback scheme mainly focuses on producing key from the feedback and using this key to protect part of the transmitted message. However, this feedback scheme has been proved only optimal for several degraded cases. The broadcast channel with mutual secrecy requirement (BC-MSR) is important as it constitutes the essence of physical layer security (PLS) in the down-link of the wireless communication systems. In this paper, we inv…
In this paper, we propose an approach to numerically compute the feedback capacity of stationary finite dimensional Gaussian channels and construct (arbitrarily close to) capacity-achieving feedback codes. In particular, we first extend the interpretation of feedback communication over stationary finite dimensional Gaussian channels as feedback control systems by showing that, the problem of finding stabilizing feedback controllers with maximal reliable transmission rate over Youla parameters coincides with the problem of finding strictly causal filters to achieve feedback capacity derived in …
We consider the problem of communication over a two-user Additive White Gaussian Noise Broadcast Channel (AWGN-BC) with an AWGN Multiple Access (MAC) active feedback. We describe a constructive reduction from this setup to the well-studied setup of linear-feedback coding over the AWGN-BC with noiseless feedback (and different parameters). This reduction facilitates the design of linear-feedback coding schemes in the (passive) noiseless feedback regime, which can then be easily and constructively transformed into coding schemes in the MAC feedback regime that attain the exact same rates. Our co…
We provide a practical implementation of the rubber method of Ahlswede et al. for binary alphabets. The idea is to create the "skeleton" sequence therein via an arithmetic decoder designed for a particular $k$-th order Markov chain. For the stochastic binary symmetric channel, we show that the scheme is nearly optimal in a strong sense for certain parameters.
We consider transmission of discrete memoryless sources (DMSes) across discrete memoryless channels (DMCs) using variable-length lossy source-channel codes with feedback. The reliability function (optimum error exponent) is shown to be equal to $\max\{0, B(1-R(D)/C)\},$ where $R(D)$ is the rate-distortion function of the source, $B$ is the maximum relative entropy between output distributions of the DMC, and $C$ is the Shannon capacity of the channel. We show that, in this setting and in this asymptotic regime, separate source-channel coding is, in fact, optimal.
We propose a new coding scheme for the discrete memoryless two-user multi-access channel (MAC) with rate-limited feedback. Our scheme combines ideas from the Venkataramanan-Pradhan scheme for perfect feedback with ideas from the Shaviv-Steinberg scheme for rate-limited feedback. Our achievable region includes the Shaviv-Steinberg achievable region and this inclusion can be strict. For general MACs and for sufficiently large feedback rates, our scheme outperforms the Shaviv-Steinberg scheme as it achieves the same rate region as the Venkataramanan-Pradhan scheme for perfect feedback (which cann…
A coding theorem is proved for a class of stationary channels with feedback in which the output Y_n = f(X_{n-m}^n, Z_{n-m}^n) is the function of the current and past m symbols from the channel input X_n and the stationary ergodic channel noise Z_n. In particular, it is shown that the feedback capacity is equal to $$ \limp_{n\to\infty} \sup_{p(x^n||y^{n-1})} \frac{1}{n} I(X^n \to Y^n), $$ where I(X^n \to Y^n) = \sum_{i=1}^n I(X^i; Y_i|Y^{i-1}) denotes the Massey directed information from the channel input to the output, and the supremum is taken over all causally conditioned distributions p(x^n…
In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $τ\ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $τ<1$ and $n\to\infty$. Additionally, we …
In this paper, we propose a new coding scheme for symmetric Gaussian interference channels with feedback based on the ideas of time-varying coding schemes. The proposed scheme improves the Suh-Tse and Kramer inner bounds of the channel capacity for the cases of weak and not very strong interference. This improvement is more significant when the signal-to-noise ratio (SNR) is not very high. It is shown theoretically and numerically that our coding scheme can outperform the Kramer code. In addition, the generalized degrees-of-freedom of our proposed coding scheme is equal to the Suh-Tse scheme i…
Limited feedback is a paradigm for the feedback of channel state information in wireless systems. In multiple antenna wireless systems, limited feedback usually entails quantizing a source that lives on the Grassmann manifold. Most work on limited feedback beamforming considered single-shot quantization. In wireless systems, however, the channel is temporally correlated, which can be used to reduce feedback requirements. Unfortunately, conventional predictive quantization does not incorporate the non-Euclidean structure of the Grassmann manifold. In this paper, we propose a Grassmannian predic…
This paper presents a variable-length decision-feedback scheme that uses tail-biting convolutional codes and the tail-biting Reliability-Output Viterbi Algoritm (ROVA). Comparing with recent results in finite-blocklength information theory, simulation results for both the BSC and the AWGN channel show that the decision-feedback scheme using ROVA can surpass the random-coding lower bound on throughput for feedback codes at average blocklengths less than 100 symbols. This paper explores ROVA-based decision feedback both with decoding after every symbol and with decoding limited to a small number…
This paper quantifies the fundamental limits of variable-length transmission of a general (possibly analog) source over a memoryless channel with noiseless feedback, under a distortion constraint. We consider excess distortion, average distortion and guaranteed distortion ($d$-semifaithful codes). In contrast to the asymptotic fundamental limit, a general conclusion is that allowing variable-length codes and feedback leads to a sizable improvement in the fundamental delay-distortion tradeoff. In addition, we investigate the minimum energy required to reproduce $k$ source samples with a given f…
An adaptive randomized distributed space-time coding (DSTC) scheme is proposed for two-hop cooperative MIMO networks. Linear minimum mean square error (MMSE) receiver filters and randomized matrices subject to a power constraint are considered with an amplify-and-forward (AF) cooperation strategy. In the proposed DSTC scheme, a randomized matrix obtained by a feedback channel is employed to transform the space-time coded matrix at the relay node. The effect of the limited feedback and feedback errors are considered. Linear MMSE expressions are devised to compute the parameters of the adaptive …
A general time-varying feedback coding scheme is proposed for $M$-user fully connected symmetric Gaussian interference channels. Based on the analysis of the general coding scheme, we prove a theorem which gives a criterion for designing good time-varying feedback codes for Gaussian interference channels. The proposed scheme improves the Suh-Tse and Kramer inner bounds of the channel capacity for the cases of weak and not very strong interference when $M=2$. This capacity improvement is more significant when the signal-to-noise ratio (SNR) is not very high. In addition, our coding scheme can b…
In this paper, we propose a novel adaptive modulation and coding (AMC) algorithm dedicated to reduce the feedback frequency of the channel state information (CSI). There have been already plenty of works on AMC so as to exploit the bandwidth more efficiently with the CSI feedback to the transmitter. However, in some occasions, frequent CSI feedback is not favorable in these systems. This work considers finite-state Markov chain (FSMC) based channel prediction to alleviate the feedback while maximizing the overall throughput. We derive the close-form of the frame error rate (FER) based on chann…
Random linear network coding (RLNC) is asymptotically throughput optimal in the wireless broadcast of a block of packets from a sender to a set of receivers, but suffers from heavy computational load and packet decoding delay. To mitigate this problem while maintaining good throughput, we partition the packet block into disjoint generations after broadcasting the packets uncoded once and collecting one round of feedback about receivers' packet reception state. We prove the NP-hardness of the optimal partitioning problem by using a hypergraph coloring approach, and develop an efficient heuristi…
Recently, a new class of error-control codes, the polar codes, have attracted much attention. The polar codes are the first known class of capacity-achieving codes for many important communication channels. In addition, polar codes have low-complexity encoding algorithms. Therefore, these codes are favorable choices for low-complexity devices, for example, in ubiquitous computing and sensor networks. However, the polar codes fall short in terms of finite-length error probabilities, compared with the state-of-the-art codes, such as the low-density parity-check codes. In this paper, in order to …
In this paper, a general binary-input binary-output (BIBO) channel is investigated in the presence of feedback and input constraints. The feedback capacity and the optimal input distribution of this setting are calculated for the case of an $(1,\infty)$-RLL input constraint, that is, the input sequence contains no consecutive ones. These results are obtained via explicit solution of a corresponding dynamic programming optimization problem. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The po…
Gaussian channels with memory and with noiseless feedback have been widely studied in the information theory literature. However, a coding scheme to achieve the feedback capacity is not available. In this paper, a coding scheme is proposed to achieve the feedback capacity for Gaussian channels. The coding scheme essentially implements the celebrated Kalman filter algorithm, and is equivalent to an estimation system over the same channel without feedback. It reveals that the achievable information rate of the feedback communication system can be alternatively given by the decay rate of the Cram…
The goal of combining beamforming and space-time coding in this work is to obtain full-diversity order and to provide additional received power (array gain) compared to conventional space-time codes. In our system, we consider a quasi-static fading environment and we incorporate both high-rate and low-rate feedback channels with possible feedback errors. To utilize feedback information, a class of code constellations is proposed, inspired from orthogonal designs and precoded space-time block codes, which is called generalized partly orthogonal designs or generalized PODs. Furthermore, to model…
We investigate the maximum coding rate for a given average blocklength and error probability over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blockleng…
The use of open-loop coding can be easily extended to a closed-loop concatenated code if the channel has access to feedback. This can be done by introducing a feedback transmission scheme as an inner code. In this paper, this process is investigated for the case when a linear feedback scheme is implemented as an inner code and, in particular, over an additive white Gaussian noise (AWGN) channel with noisy feedback. To begin, we look to derive an optimal linear feedback scheme by optimizing over the received signal-to-noise ratio. From this optimization, an asymptotically optimal linear feedbac…
In this paper, we first study a two-user interference channel with generalized feedback. We establish an inner bound on its capacity region. The coding scheme that we employ for the inner bound is based on an appropriate combination of Han-Kobayash rate splitting and compress-and-forward at the senders. Each sender compresses the channel output that is observes using a compression scheme that is à-la Lim et al. noisy network coding and Avestimeher et al. quantize-map-and-forward. Next, we study an injective deterministic model in which the senders obtain output feedback only intermittently. Sp…
The use of open-loop coding can be easily extended to a closed-loop concatenated code if the channel has access to feedback. This can be done by introducing a feedback transmission scheme as an inner code. In this paper, this process is investigated for the case when a linear feedback scheme is implemented as an inner code and, in particular, over an additive white Gaussian noise (AWGN) channel with noisy feedback. To begin, we look to derive the optimal linear feedback scheme by optimizing over the received signal-to-noise ratio. From this optimization, an asymptotically optimal linear feedba…
A single-letter achievable rate region is proposed for the two-receiver discrete memoryless broadcast channel with generalized feedback. The coding strategy involves block-Markov superposition coding, using Marton's coding scheme for the broadcast channel without feedback as the starting point. If the message rates in the Marton scheme are too high to be decoded at the end of a block, each receiver is left with a list of messages compatible with its output. Resolution information is sent in the following block to enable each receiver to resolve its list. The key observation is that the resolut…
In this paper, the posterior matching scheme proposed by Shayevits and Feder is extended to the Gaussian broadcast channel with feedback, and the error probabilities and achievable rate region are derived for this coding strategy by using the iterated random function theory. A variant of the Ozarow-Leung code for the general two-user broadcast channel with feedback can be realized as a special case of our coding scheme. Furthermore, for the symmetric Gaussian broadcast channel with feedback, our coding scheme achieves the linear-feedback sum-capacity like the LQG code and outperforms the Krame…
In this paper, we consider the problem of minimizing the multicast decoding delay of generalized instantly decodable network coding (G-IDNC) over persistent forward and feedback erasure channels with feedback intermittence. In such an environment, the sender does not always receive acknowledgement from the receivers after each transmission. Moreover, both the forward and feedback channels are subject to persistent erasures, which can be modelled by a two state (good and bad states) Markov chain known as Gilbert-Elliott channel (GEC). Due to such feedback imperfections, the sender is unable to …
Space-time codes leverage the availability of multiple antennas to enhance the reliability of communication over wireless channels. While space-time codes have initially been designed with a focus on open-loop systems, recent technological advances have enabled the possibility of low-rate feedback from the receiver to the transmitter. The focus of this paper is on the implications of this feedback in a single-user multi-antenna system with a general model for spatial correlation. We assume a limited feedback model, that is, a coherent receiver and statistics along with B bits of quantized chan…
Multiple input multiple output (MIMO) precoding is an efficient scheme that may significantly enhance the communication link. However, this enhancement comes with a cost. Many precoding schemes require channel knowledge at the transmitter that is obtained through feedback from the receiver. Focusing on the natural common fusion of orthogonal frequency division multiplexing (OFDM) and MIMO, we exploit the channel correlation in the frequency and spatial domain to reduce the required feedback rate in a frequency division duplex (FDD) system. The proposed feedback method is based on Huffman codin…
We consider the problem of communication over a multi-path network in the presence of a causal adversary. The limited-view causal adversary is able to eavesdrop on a subset of links and also jam on a potentially overlapping subset of links based on the current and past information. To ensure that the communication takes place reliably and secretly, resilient network codes with necessary redundancy are needed. We study two adversarial models - additive and overwrite jamming and we optionally assume passive feedback from decoder to encoder, i.e., the encoder sees everything that the decoder sees…
Current approaches to the practical implementation of network coding are batch-based, and often do not use feedback, except possibly to signal completion of a file download. In this paper, the various benefits of using feedback in a network coded system are studied. It is shown that network coding can be performed in a completely online manner, without the need for batches or generations, and that such online operation does not affect the throughput. Although these ideas are presented in a single-hop packet erasure broadcast setting, they naturally extend to more general lossy networks which e…
We investigate the maximum coding rate achievable over a two-user broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. Specifically, upon decoding the common message, each decoder sends a stop signal to the encoder, which transmits continuously until it receives both stop signals. For the point-to-point case, Polyanskiy, Poor, and Verdú (2011) recently demonstrated that variable-length coding combined with stop feedback significantly increases the speed at which the maximum coding rate converges to capacity. This speed-up manifests…
Theoretical analysis has long indicated that feedback improves the error exponent but not the capacity of single-user memoryless channels. Recently Polyanskiy et al. studied the benefit of variable-length feedback with termination (VLFT) codes in the non-asymptotic regime. In that work, achievability is based on an infinite length random code and decoding is attempted at every symbol. The coding rate backoff from capacity due to channel dispersion is greatly reduced with feedback, allowing capacity to be approached with surprisingly small expected latency. This paper is mainly concerned with V…
The problem of three-user multiple-access channel (MAC) with noiseless feedback is investigated. A new coding strategy is presented. The coding scheme builds upon the natural extension of the Cover-Leung (CL) scheme; and uses quasi-linear codes. A new single-letter achievable rate region is derived. The new achievable region strictly contains the CL region. This is shown through an example. In this example, the coding scheme achieves optimality in terms of transmission rates. It is shown that any optimality achieving scheme for this example must have a specific algebraic structure. Particularl…
An $l$-link binary CEO problem is considered in this paper. We present a practical encoding and decoding scheme for this problem employing the graph-based codes. A successive coding scheme is proposed for converting an $l$-link binary CEO problem to the $(2l-1)$ single binary Wyner-Ziv (WZ) problems. By using the compound LDGM-LDPC codes, the theoretical bound of each binary WZ is asymptotically achievable. Our proposed decoder successively decodes the received data by employing the well-known Sum-Product (SP) algorithm and leverages them to reconstruct the source. The sum-rate distortion perf…
