We consider the problem of distributed joint source-channel coding of correlated Gaussian sources over a Gaussian Multiple Access Channel (GMAC). There may be side information at the decoder and/or at the encoders. First we specialize a general result (for transmission of correlated sources over a MAC with side information) to obtain sufficient conditions for reliable transmission over a Gaussian MAC. This system does not satisfy the source-channel separation. We study and compare three joint source-channel coding schemes available in literature. We show that each of these schemes is optimal under different scenarios. One of the schemes, Amplify and Forward (AF) which simplifies the design of encoders and the decoder, is optimal at low SNR but not at high SNR. Another scheme is asymptotically optimal at high SNR. The third coding scheme is optimal for orthogonal Gaussian channels. We also show that AF is close to the optimal scheme for orthogonal channels even at high SNR.
In this paper, we give a distributed joint source channel coding scheme for arbitrary correlated sources for arbitrary point in the Slepian-Wolf rate region, and arbitrary link capacities using LDPC codes. We consider the Slepian-Wolf setting of two sources and one destination, with one of the sources derived from the other source by some correlation model known at the decoder. Distributed encoding and separate decoding is used for the two sources. We also give a distributed source coding scheme when the source correlation has memory to achieve any point in the Slepian-Wolf rate achievable reg…
A joint source-channel coding (JSCC) scheme based on hybrid digital/analog coding is proposed for the transmission of correlated sources over discrete-memoryless two-way channels (DM-TWCs). The scheme utilizes the correlation between the sources in generating channel inputs, thus enabling the users to coordinate their transmission to combat channel noise. The hybrid scheme also subsumes prior coding methods such as rate-one separate source-channel coding and uncoded schemes for two-way lossy transmission, as well as the correlation-preserving coding scheme for (almost) lossless transmission. M…
We give a complete characterization of the correlated random coding secrecy capacity of arbitrarily varying wiretap channels (AVWCs). We apply two alternative strong secrecy criteria, which both lead to the same multi-letter formula. The difference of these criteria lies in the treatment of correlated randomness, they coincide in the case of uncorrelated codes. On the basis of the derived formula, we show that the correlated random coding secrecy capacity is continuous as a function of the AVWC, in contrast to the discontinuous uncorrelated coding secrecy capacity. In the proof of the secrecy …
This paper studies the random-coding exponent of joint source-channel coding for the multiple-access channel with correlated sources. For each user, by defining a threshold, the messages of each source are partitioned into two classes. The achievable exponent for correlated sources with two message-dependent input distributions for each user is determined and shown to be larger than that achieved using only one input distribution for each user. A system of equations is presented to determine the optimal thresholds maximizing the achievable exponent. The obtained exponent is compared with the o…
This paper deals with a universal coding problem for a certain kind of multiterminal source coding system that we call the complementary delivery coding system. In this system, messages from two correlated sources are jointly encoded, and each decoder has access to one of the two messages to enable it to reproduce the other message. Both fixed-to-fixed length and fixed-to-variable length lossless coding schemes are considered. Explicit constructions of universal codes and bounds of the error probabilities are clarified via type-theoretical and graph-theoretical analyses. [[Keywords]] multiterm…
Source and channel coding over multiuser channels in which receivers have access to correlated source side information is considered. For several multiuser channel models necessary and sufficient conditions for optimal separation of the source and channel codes are obtained. In particular, the multiple access channel, the compound multiple access channel, the interference channel and the two-way channel with correlated sources and correlated receiver side information are considered, and the optimality of separation is shown to hold for certain source and side information structures. Interestin…
Non-adaptive joint source network coding of correlated sources is discussed in this paper. By studying the information flow in the network, we propose quantized network coding as an alternative for packet forwarding. This technique has both network coding and distributed source coding advantages, simultaneously. Quantized network coding is a combination of random linear network coding in the (infinite) field of real numbers and quantization to cope with the limited capacity of links. With the aid of the results in the literature of compressed sensing, we discuss theoretical and practical feasi…
A new approach to joint source-channel coding is presented in the context of communicating correlated sources over multiple access channels. Similar to the separation architecture, the joint source-channel coding system architecture in this approach is modular, whereby the source encoding and channel decoding operations are decoupled. However, unlike the separation architecture, the same codeword is used for both source coding and channel coding, which allows the resulting hybrid coding scheme to achieve the performance of the best known joint source-channel coding schemes. Applications of the…
Noisy network coding, which elegantly combines the conventional compress-and-forward relaying strategy and ideas from network coding, has recently drawn much attention for its simplicity and optimality in achieving to within constant gap of the capacity of the multisource multicast Gaussian network. The constant-gap result, however, applies only to Gaussian relay networks with independent noises. This paper investigates the application of noisy network coding to networks with correlated noises. By focusing on a four-node Gaussian relay chain network with a particular noise correlation structur…
The problem of broadcasting a pair of correlated Gaussian sources using optimal separate source and channel codes is studied. Considerable performance gains over previously known separate source-channel schemes are observed. Although source-channel separation yields suboptimal performance in general, it is shown that the proposed scheme is very competitive for any bandwidth compression/expansion scenarios. In particular, for a high channel SNR scenario, it can be shown to achieve optimal power-distortion tradeoff.
In this paper, we study a half-duplex two-way relay channel (TWRC) with correlated sources exchanging bidirectional information. In the case, when both sources have the knowledge of correlation statistics, a source compression with physical-layer network coding (SCPNC) scheme is proposed to perform the distributed compression at each source node. When only the relay has the knowledge of correlation statistics, we propose a relay compression with physical-layer network coding (RCPNC) scheme to compress the bidirectional messages at the relay. The closed-form block error rate (BLER) expressions …
We study the transmission of correlated sources over discrete memoryless (DM) multiple-access-relay channels (MARCs), in which both the relay and the destination have access to side information arbitrarily correlated with the sources. As the optimal transmission scheme is an open problem, in this work we propose a new joint source-channel coding scheme based on a novel combination of the correlation preserving mapping (CPM) technique with Slepian-Wolf (SW) source coding, and obtain the corresponding sufficient conditions. The proposed coding scheme is based on the decode-and-forward strategy, …
In this paper, distributed (or multiterminal) source coding with one distortion criterion and correlated messages is considered. This problem can be also called ``Berger-Yeung problem with correlated messages''. It corresponds to the source coding part of the graph-based framework for transmission of a pair of correlated sources over the multiple-access channel (MAC) where one is lossless and the other is lossy. As a result, the achievable rate-distortion region for this problem is provided. It is an information-theoretic characterization of the rate of exponential growth (as a function of the…
We consider the distributed source coding system of $L$ correlated Gaussian sources $Y_i,i=1,2,...,L$ which are noisy observations of correlated Gaussian remote sources $X_k, k=1,2,...,K$. We assume that $Y^{L}={}^{\rm t}(Y_1,Y_2,$ $..., Y_L)$ is an observation of the source vector $X^K={}^{\rm t}(X_1,X_2,..., X_K)$, having the form $Y^L=AX^K+N^L$, where $A$ is a $L\times K$ matrix and $N^L={}^{\rm t}(N_1,N_2,...,N_L)$ is a vector of $L$ independent Gaussian random variables also independent of $X^K$. In this system $L$ correlated Gaussian observations are separately compressed by $L$ encoders…
The problem of reliable transmission of correlated sources over the broadcast channel, originally studied by Han and Costa, is revisited. An alternative characterization of their sufficient condition for reliable transmission is given, which includes results of Marton for channel coding over broadcast channels and of Gray and Wyner for distributed source coding. A ``minimalistic'' coding scheme is presented, which is based on joint typicality encoding and decoding, without requiring the use of Cover's superposition coding, random hashing, and common part between two sources. The analysis of th…
A general lossless joint source-channel coding scheme based on linear codes is proposed and then analyzed in this paper. It is shown that a linear code with good joint spectrum can be used to establish limit-approaching joint source-channel coding schemes for arbitrary sources and channels, where the joint spectrum of the code is a generalization of the input-output weight distribution.
In this paper we provide sufficient conditions for lossy transmission of functions of correlated data over a multiple access channel (MAC). The conditions obtained can be shown as generalized version of Yamamoto's result. We also obtain efficient joint source-channel coding schemes for transmission of discrete and continuous alphabet sources to recover the function values. Keywords: Joint source-channel coding, Graph coloring, Lipschitz functions, Correlated sources.
Lossy coding of correlated sources over a multiple access channel (MAC) is studied. First, a joint source-channel coding scheme is presented when the decoder has correlated side information. Next, the optimality of separate source and channel coding, that emerges from the availability of a common observation at the encoders, or side information at the encoders and the decoder, is investigated. It is shown that separation is optimal when the encoders have access to a common observation whose lossless recovery is required at the decoder, and the two sources are independent conditioned on this co…
In this paper, we investigate the joint design of channel and network coding in bi-directional relaying systems and propose a combined low complexity physical network coding and LDPC decoding scheme. For the same LDPC codes employed at both source nodes, we show that the relay can decodes the network coded codewords from the superimposed signal received from the BPSK-modulated multiple-access channel. Simulation results shown that this novel joint physical network coding and LDPC decoding method outperforms the existing MMSE network coding and LDPC decoding method over AWGN and complex MAC cha…
In this paper, we use linear codes to study zero-error Slepian-Wolf coding of a set of sources with deviation symmetry, where the sources are generalization of the Hamming sources over an arbitrary field. We extend our previous codes, Generalized Hamming Codes for Multiple Sources, to Matrix Partition Codes and use the latter to efficiently compress the target sources. We further show that every perfect or linear-optimal code is a Matrix Partition Code. We also present some conditions when Matrix Partition Codes are perfect and/or linear-optimal. Detail discussions of Matrix Partition Codes on…
We consider the distributed source coding system for $L$ correlated Gaussian observations $Y_i, i=1,2, ..., L$. Let $X_i,i=1,2, ..., L$ be $L$ correlated Gaussian random variables and $N_i,$ $i=1,2,... L$ be independent additive Gaussian noises also independent of $X_i, i=1,2,..., L$. We consider the case where for each $i=1,2,..., L$, $Y_i$ is a noisy observation of $X_i$, that is, $Y_i=X_i+N_i$. On this coding system the determination problem of the rate distortion region remains open. In this paper, we derive explicit outer and inner bounds of the rate distortion region. We further find an …
In this paper we consider the communication problem that involves transmission of correlated sources over broadcast channels. We consider a graph-based framework for this information transmission problem. The system involves a source coding module and a channel coding module. In the source coding module, the sources are efficiently mapped into a nearly semi-regular bipartite graph, and in the channel coding module, the edges of this graph are reliably transmitted over a broadcast channel. We consider nearly semi-regular bipartite graphs as discrete interface between source coding and channel c…
Existing polar coding schemes for the two-user interference channel follow the original idea of Han and Kobayashi, in which component messages are encoded independently and then mapped by some deterministic functions (i.e., homogeneous superposition coding). In this paper, we propose a new polar coding scheme for the interference channel based on the heterogeneous superposition coding approach of Chong, Motani and Garg. We prove that fully-joint decoding (the receivers simultaneously decode both senders' common messages and the intended sender's private message) in the Han-Kobayashi strategy c…
Achievability and converse results for the lossy transmission of correlated sources over Shannon's two-way channels (TWCs) are presented. A joint source-channel coding theorem for independent sources and TWCs for which adaptation cannot enlarge the capacity region is also established. We further investigate the optimality of scalar coding for TWCs with discrete modulo additive noise as well as additive white Gaussian noise. Comparing the distortion of scalar coding with the derived bounds, we observe that scalar coding achieves the minimum distortion over both families of TWCs for independent …
We consider a distributed source coding problem of $L$ correlated Gaussian observations $Y_i, i=1,2,...,L$. We assume that the random vector $Y^{L}={}^{\rm t} (Y_1,Y_2,$ $...,Y_L)$ is an observation of the Gaussian random vector $X^K={}^{\rm t}(X_1,X_2,...,X_K)$, having the form $Y^L=AX^K+N^L ,$ where $A$ is a $L\times K$ matrix and $N^L={}^{\rm t}(N_1,N_2,...,N_L)$ is a vector of $L$ independent Gaussian random variables also independent of $X^K$. The estimation error on $X^K$ is measured by the distortion covariance matrix. The rate distortion region is defined by a set of all rate vectors f…
A coding problem for correlated information sources is investigated. Messages emitted from two correlated sources are jointly encoded, and delivered to two decoders. Each decoder has access to one of the two messages to enable it to reproduce the other message. The rate-distortion function for the coding problem and its interesting properties are clarified.
We are interested in how to best communicate a real valued source to a number of destinations (sinks) over a network with capacity constraints in a collective fidelity metric over all the sinks, a problem which we call joint network-source coding. It is demonstrated that multiple description codes along with proper diversity routing provide a powerful solution to joint network-source coding. A systematic optimization approach is proposed. It consists of optimizing the network routing given a multiple description code and designing optimal multiple description code for the corresponding optimiz…
In this paper, we propose {\em distributed network compression via memory}. We consider two spatially separated sources with correlated unknown source parameters. We wish to study the universal compression of a sequence of length $n$ from one of the sources provided that the decoder has access to (i.e., memorized) a sequence of length $m$ from the other source. In this setup, the correlation does not arise from symbol-by-symbol dependency of two outputs from the two sources (as in Slepian-Wolf setup). Instead, the two sequences are correlated because they are originated from the two sources wi…
In this paper, we consider the application of Reed- Solomon (RS) channel coding for joint error correction and cooperative network coding on non-binary phase shift keying (PSK) modulated signals. The relay first decodes the RS channel coded messages received each in a time slot from all sources before applying network coding (NC) by the use of bit-level exclusive OR (XOR) operation. The network coded resulting message is then channel encoded before its transmission to the next relay or to the destination according to the network configuration. This scenario shows superior performance in compar…
In this paper, we study the upper and the lower bounds on the joint source-channel coding error exponent with decoder side-information. The results in the paper are non-trivial extensions of the Csiszar's classical paper [5]. Unlike the joint source-channel coding result in [5], it is not obvious whether the lower bound and the upper bound are equivalent even if the channel coding error exponent is known. For a class of channels, including the symmetric channels, we apply a game-theoretic result to establish the existence of a saddle point and hence prove that the lower and upper bounds are th…
We consider a cache-aided communications system in which a transmitter communicates with many receivers over an erasure broadcast channel. The system serves as a basic model for communicating on-demand content during periods of high network congestion, where some content can be pre-placed in local caches near the receivers. We formulate the cache-aided communications problem as a joint cache-channel coding problem, and characterise some information-theoretic tradeoffs between reliable communications rates and cache sizes. We show that if the receivers experience different channel qualities, th…
The second-order achievable rate region in Slepian-Wolf source coding systems is investigated. The concept of second-order achievable rates, which enables us to make a finer evaluation of achievable rates, has already been introduced and analyzed for general sources in the single-user source coding problem. Analogously, in this paper, we first define the second-order achievable rate region for the Slepian-Wolf coding system to establish the source coding theorem in the second- order sense. The Slepian-Wolf coding problem for correlated sources is one of typical problems in the multi-terminal i…
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…
In this work we investigate the behavior of the distortion threshold that can be guaranteed in joint source-channel coding, to within a prescribed excess-distortion probability. We show that the gap between this threshold and the optimal average distortion is governed by a constant that we call the joint source-channel dispersion. This constant can be easily computed, since it is the sum of the source and channel dispersions, previously derived. The resulting performance is shown to be better than that of any separation-based scheme. For the proof, we use unequal error protection channel codin…
We investigate joint network and channel coding schemes for networks when relay nodes are not capable of performing channel coding operations. Rather, channel encoding is performed at the source node while channel decoding is done only at the destination nodes. We examine three different decoding strategies: independent network-then-channel decoding, serial network and channel decoding, and joint network and channel decoding. Furthermore, we describe how to implement such joint network and channel decoding using iteratively decodable error correction codes. Using simple networks as a model, we…
We propose a joint source-channel-network coding scheme, based on compressive sensing principles, for wireless networks with AWGN channels (that may include multiple access and broadcast), with sources exhibiting temporal and spatial dependencies. Our goal is to provide a reconstruction of sources within an allowed distortion level at each receiver. We perform joint source-channel coding at each source by randomly projecting source values to a lower dimensional space. We consider sources that satisfy the restricted eigenvalue (RE) condition as well as more general sources for which the randomn…
In this letter, we propose two power allocation schemes based on the statistical channel state information (CSI) and instantaneous s->r CSI at transmitters respectively for a 2-N-2 cooperative multicast system with non-regenerative network coding.Then the isolated precoder and the distributed precoder are respectively applied to the schemes to further improve the system performance by achieving the full diversity gain. Finally, we demonstrate that joint instantaneous s->r CSI based power allocation and distributed precoder design achieve the best performance.
In this paper, we consider joint network and LDPC coding for practically implementing the denosie-and-forward protocol over bi-directional relaying. the closed-form expressions for computing the log-likelihood ratios of the network-coded codewords have been derived for both real and complex multiple-access channels. It is revealed that the equivalent channel observed at the relay is an asymmetrical channel, where the channel input is the XOR form of the two source nodes.
This paper considers a degraded Gaussian broadcast channel over which Gaussian sources are to be communicated. When the sources are independent, this paper shows that hybrid coding achieves the optimal distortion region, the same as that of separate source and channel coding. It also shows that uncoded transmission is not optimal for this setting. For correlated sources, the paper shows that a hybrid coding strategy has a better distortion region than separate source-channel coding below a certain signal to noise ratio threshold. Thus, hybrid coding is a good choice for Gaussian broadcast chan…
In this paper, we consider the problem of robust joint source-channel coding over an additive white Gaussian noise channel. We propose a new scheme which achieves the optimal slope of the signal-to-distortion (SDR) curve (unlike the previously known coding schemes). Also, we propose a family of robust codes which together maintain a bounded gap with the optimum SDR curve (in terms of dB). To show the importance of this result, we drive some theoretical bounds on the asymptotic performance of delay-limited hybrid digital-analog (HDA) coding schemes. We show that, unlike the delay-unlimited case…
The secrecy performance of a source-channel model is studied in the context of lossy source compression over a noisy broadcast channel. The source is causally revealed to the eavesdropper during decoding. The fidelity of the transmission to the legitimate receiver and the secrecy performance at the eavesdropper are both measured by a distortion metric. Two achievability schemes using the technique of hybrid coding are analyzed and compared with an operationally separate source-channel coding scheme. A numerical example is provided and the comparison results show that the hybrid coding schemes …
In this paper the multicasting of independent parallel Gaussian sources over a binary erasure broadcasted channel is considered. Multiresolution embedded quantizer and layered joint source-channel coding schemes are used in order to serve simultaneously several users at different channel capacities. The convex nature of the rate-distortion function, computed by means of reverse water-filling, allows us to solve relevant convex optimization problems corresponding to different performance criteria. Then, layered joint source-channel codes are constructed based on the concatenation of embedded sc…
A joint communication and channel state estimation problem is investigated, in which reliable information transmission over a noisy channel, and high-fidelity estimation of the channel state, are simultaneously sought. The tradeoff between the achievable information rate and the estimation distortion is quantified by formulating the problem as a constrained channel coding problem, and the resulting capacity-distortion function characterizes the fundamental limit of the joint communication and channel estimation problem. The analytical results are illustrated through case studies, and further i…
We present and analyze a joint source-channel coding strategy for the transmission of a Gaussian source across a Gaussian channel in n channel uses per source symbol. Among all such strategies, our scheme has the following properties: i) the resulting mean-squared error scales optimally with the signal-to-noise ratio, and ii) the scheme is easy to implement and the incurred delay is minimal, in the sense that a single source symbol is encoded at a time.
In this paper, we investigate communication strategies for the multiple access channel with feedback and correlated sources (MACFCS). The MACFCS models a wireless sensor network scenario in which sensors distributed throughout an arbitrary random field collect correlated measurements and transmit them to a common sink. We derive achievable rate regions for the three-node MACFCS. First, we study the strategy when source coding and channel coding are combined, which we term full decoding at sources. Second, we look at several strategies when source coding and channel coding are separated, which …
Low-density parity-check (LDPC) codes with the parity-based approach for distributed joint source channel coding (DJSCC) with decoder side information is described in this paper. The parity-based approach is theoretical limit achievable. Different edge degree distributions are used for source variable nodes and parity variable nodes. Particularly, the codeword-averaged density evolution (CADE) is presented for asymmetrically correlated nonuniform sources over the asymmetric memoryless transmission channel. Extensive simulations show that the splitting of variable nodes can improve the coding e…
We study the transmission of a set of correlated sources $(U_1,\cdots,U_K)$ over a Gaussian multiple access relay channel with time asynchronism between the encoders. We assume that the maximum possible offset ${\mathsf{d_{max}}}(n)$ between the transmitters grows without bound as the block length $n \rightarrow \infty$ while the relative ratio ${\mathsf{d_{max}}(n) / n}$ of the maximum possible offset to the block length asymptotically vanishes. For such a joint source-channel coding problem, and under specific gain conditions, we derive necessary and sufficient conditions for reliable commun…
Motivated by video coding applications, the problem of sequential coding of correlated sources with encoding and/or decoding frame-delays is studied. The fundamental tradeoffs between individual frame rates, individual frame distortions, and encoding/decoding frame-delays are derived in terms of a single-letter information-theoretic characterization of the rate-distortion region for general inter-frame source correlations and certain types of potentially frame specific and coupled single-letter fidelity criteria. The sum-rate-distortion region is characterized in terms of generalized directed …
A general lossless joint source-channel coding (JSCC) scheme based on linear codes and random interleavers for multiple-access channels (MACs) is presented and then analyzed in this paper. By the information-spectrum approach and the code-spectrum approach, it is shown that a linear code with a good joint spectrum can be used to establish limit-approaching lossless JSCC schemes for correlated general sources and general MACs, where the joint spectrum is a generalization of the input-output weight distribution. Some properties of linear codes with good joint spectra are investigated. A formula …
The groupcast index coding problem is the most general version of the classical index coding problem, where any receiver can demand messages that are also demanded by other receivers. Any groupcast index coding problem is described by its \emph{fitting matrix} which contains unknown entries along with $1$'s and $0$'s. The problem of finding an optimal scalar linear code is equivalent to completing this matrix with known entries such that the rank of the resulting matrix is minimized. Any row basis of such a completion gives an optimal \emph{scalar linear} code. An index coding problem is said …
The problem of characterizing the capacity region for Gaussian broadcast channels with receiver message side information appears difficult and remains open for N >= 3 receivers. This paper proposes a joint network and Gelfand-Pinsker coding method for 3-receiver cases. Using the method, we establish a unified inner bound on the capacity region of 3-receiver Gaussian broadcast channels under general message side information configuration. The achievability proof of the inner bound uses an idea of joint interference cancelation, where interference is canceled by using both dirty-paper coding at …
We study the joint source-channel coding problem of transmitting a discrete-time analog source over an additive white Gaussian noise (AWGN) channel with interference known at transmitter.We consider the case when the source and the interference are correlated. We first derive an outer bound on the achievable distortion and then, we propose two joint source-channel coding schemes. The first scheme is the superposition of the uncoded signal and a digital part which is the concatenation of a Wyner-Ziv encoder and a dirty paper encoder. In the second scheme, the digital part is replaced by the hyb…
A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The mo…
We construct a joint coordination-channel polar coding scheme for strong coordination of actions between two agents $\mathsf X$ and $\mathsf Y$, which communicate over a discrete memoryless channel (DMC) such that the joint distribution of actions follows a prescribed probability distribution. We show that polar codes are able to achieve our previously established inner bound to the strong noisy coordination capacity region and thus provide a constructive alternative to a random coding proof. Our polar coding scheme also offers a constructive solution to a channel simulation problem where a DM…
We consider the problem of transmission of several distributed sources over a multiple access channel (MAC) with side information at the sources and the decoder. Source-channel separation does not hold for this channel. Sufficient conditions are provided for transmission of sources with a given distortion. The source and/or the channel could have continuous alphabets (thus Gaussian sources and Gaussian MACs are special cases). Various previous results are obtained as special cases. We also provide several good joint source-channel coding schemes for a discrete/continuous source and discrete/co…
