We develop and analyze low-complexity cooperative diversity protocols that combat fading induced by multipath propagation in wireless networks. The underlying techniques exploit space diversity available through cooperating terminals' relaying signals for one another. We outline several strategies employed by the cooperating radios, including fixed relaying schemes such as amplify-and-forward and decode-and-forward, selection relaying schemes that adapt based upon channel measurements between the cooperating terminals, and incremental relaying schemes that adapt based upon limited feedback from the destination terminal. We develop performance characterizations in terms of outage events and associated outage probabilities, which measure robustness of the transmissions to fading, focusing on the high signal-to-noise ratio (SNR) regime. Except for fixed decode-and-forward, all of our cooperative diversity protocols are efficient in the sense that they achieve full diversity (i.e., second-order diversity in the case of two terminals), and, moreover, are close to optimum (within 1.5 dB) in certain regimes. Thus, using distributed antennas, we can provide the powerful benefits of space diversity without need for physical arrays, though at a loss of spectral efficiency due to half-duplex operation and possibly at the cost of additional receive hardware. Applicable to any wireless setting, including cellular or ad hoc networks-wherever space constraints preclude the use of physical arrays-the performance characterizations reveal that large power or energy savings result from the use of these protocols.

Cooperative diversity in wireless networks: Efficient protocols and outage behavior

We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems.

http://www.inf.fu-berlin.de/lehre/WS11/Wireless/papers/CodAhlswede00.pdf

Network information flow

A relay channel consists of an input x_{l} , a relay output y_{1} , a channel output y , and a relay sender x_{2} (whose transmission is allowed to depend on the past symbols y_{1} . The dependence of the received symbols upon the inputs is given by p(y,y_{1}|x_{1},x_{2}) . The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)If y is a degraded form of y_{1} , then C \: = \: \max \!_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})} . 2)If y_{1} is a degraded form of y , then C \: = \: \max \!_{p(x_{1})} \max_{x_{2}} I(X_{1};Y|x_{2}) . 3)If p(y,y_{1}|x_{1},x_{2}) is an arbitrary relay channel with feedback from (y,y_{1}) to both x_{1} \and x_{2} , then C\: = \: \max_{p(x_{1},x_{2})} \min \,{I(X_{1},X_{2};Y),I \,(X_{1};Y,Y_{1}|X_{2})} . 4)For a general relay channel, C \: \leq \: \max_{p(x_{1},x_{2})} \min \,{I \,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}) . Superposition block Markov encoding is used to show achievability of C , and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.

/pdf/capacity-theorems-for-the-relay-channel-5432o2a5q4.pdf

Capacity theorems for the relay channel

Let \{(X_{k}, Y_{k}) \}^{ \infty}_{k=1} be a sequence of independent drawings of a pair of dependent random variables X, Y . Let us say that X takes values in the finite set \cal X . It is desired to encode the sequence \{X_{k}\} in blocks of length n into a binary stream of rate R , which can in turn be decoded as a sequence \{ \hat{X}_{k} \} , where \hat{X}_{k} \in \hat{ \cal X} , the reproduction alphabet. The average distortion level is (1/n) \sum^{n}_{k=1} E[D(X_{k},\hat{X}_{k})] , where D(x,\hat{x}) \geq 0, x \in {\cal X}, \hat{x} \in \hat{ \cal X} , is a preassigned distortion measure. The special assumption made here is that the decoder has access to the side information \{Y_{k}\} . In this paper we determine the quantity R \ast (d) , defined as the infimum ofrates R such that (with \varepsilon > 0 arbitrarily small and with suitably large n )communication is possible in the above setting at an average distortion level (as defined above) not exceeding d + \varepsilon . The main result is that R \ast (d) = \inf [I(X;Z) - I(Y;Z)] , where the infimum is with respect to all auxiliary random variables Z (which take values in a finite set \cal Z ) that satisfy: i) Y,Z conditionally independent given X ; ii) there exists a function f: {\cal Y} \times {\cal Z} \rightarrow \hat{ \cal X} , such that E[D(X,f(Y,Z))] \leq d . Let R_{X | Y}(d) be the rate-distortion function which results when the encoder as well as the decoder has access to the side information \{ Y_{k} \} . In nearly all cases it is shown that when d > 0 then R \ast(d) > R_{X|Y} (d) , so that knowledge of the side information at the encoder permits transmission of the \{X_{k}\} at a given distortion level using a smaller transmission rate. This is in contrast to the situation treated by Slepian and Wolf [5] where, for arbitrarily accurate reproduction of \{X_{k}\} , i.e., d = \varepsilon for any \varepsilon >0 , knowledge of the side information at the encoder does not allow a reduction of the transmission rate.

/pdf/the-rate-distortion-function-for-source-coding-with-side-15w82p0ta1.pdf

The rate-distortion function for source coding with side information at the decoder

The benefits of using modulation codes in two dimensions in multichannel recording include improved clocking and improved ratios of user bits to recorded bits. Recent work has described a class of d/sub x/, k/sub y/ codes in two dimensions which satisfy the d/sub x/ constraint in one dimension and the k/sub y/ constraint in the other. In tape systems this method is extremely vulnerable to dropouts. A new class of two-dimensional run-length codes that operate using the usual d/sub x/, k/sub y/ constraint along the track is proposed, with an additional k/sub y/ constraint across the tracks. In this approach the horizontal d/sub x/, k/sub x/ code is allowed a much larger k/sub x/ constraint since it is no longer the sole carrier of clocking information. Capacities are calculated for a range of codes and number of channels, and an example of the construction of one such code is given. An extension of the codes that preserves clocking during channel loss is described. >

A new class of two-dimensional RLL recording codes

Two novel combined error-correcting/run-length-limited (ECC/RLL) trellis codes are presented. One is a two-state

Combined ECC/RLL codes

We propose a new write channel model for bit-patterned media recording that reflects the data dependence of write synchronization errors. It is shown that this model accommodates both substitution-like errors and insertion-deletion errors whose statistics are determined by an underlying channel state process. We study information theoretic properties of the write channel model, including the capacity, symmetric information rate, Markov-1 rate and the zero-error capacity.

Write Channel Model for Bit-Patterned Media Recording

A Write Once Memory (WOM) is a storage device that consists of cells that can take on q possible linearly-ordered values, with the added constraint that rewrites can only increase a cell's value. In the binary case, each cell can change from the level zero to the level one only once. Examples of WOMs include punch cards, optical disks, and more recently flash memories. A length-n, t-write WOM-code is a coding scheme that allows t messages to be stored in n cells. If in the i-th write we write one of M i messages, then the rate of the i-th write is the ratio of the number of bits written to the WOM to the total number of cells used, i.e., log 2 (M i )/n. The rate of the WOM-code is the sum of all individual rates in all writes. In this paper, we review a recent construction of binary two-write WOM-codes. The construction is generalized for two-write WOM-codes with q levels per cell. Then, we show how to use such a code with ternary cells in order to construct three and four-write WOM-codes. This construction is used recursively in order to generate a family of t-write WOM-codes for all t. Another generalized construction is given which provides us with more ways to construct families of WOM-codes. Finally, we give a comparison between our codes and the best known WOM-codes in order to show that the WOM-codes constructed here outperform all previously known WOM-codes for 3 ≤ t ≤ 10.

Multiple-write WOM-codes

Almost all nonbinary codes were designed for the Hamming metric. The Lee metric was defined by Lee in 1958 . Golomb and Welch (1968) and Berlekamp, 1968a , Berlekamp, 1968b have designed codes for the Lee metric. In this paper, we derive all the discrete, memoryless, symmetric channels matched to the Lee metric, and investigate general properties of Lee metric block codes. Finally, a class of cyclic Lee metric codes is defined and the number of information symbols is discussed.

Jack K. Wolf

Papers

A new class of two-dimensional RLL recording codes

Combined ECC/RLL codes

Write Channel Model for Bit-Patterned Media Recording

Multiple-write WOM-codes

On channels and codes for the Lee metric