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We present capacity achieving multilevel run-length-limited (ML-RLL) codes that can be decoded by a sliding window of size $2$ .

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Channel information at the transmitter can simplify the coding scheme and increase the achievable data rate over a multiple-input multiple-output (MIMO) fading channel. Feed- back from the receiver can be used to specify a precoding matrix, which selectively activates the strongest channel modes. We eval- uate the sum data rate per receive antenna when the precoding matrix is quantized with a random vector quantization (RVQ) scheme, assuming a matched filter, or linear Minimum Mean Squared Error (MMSE) receiver. Our results are asymptotic as the number of transmit and receive antennas increases with fixed ratio, for a fixed number of feedback bits per dimension. Numerical results show that given a target spectral efficiency, the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback. We also compare these results with a simpler reduced-rank scheme for quantizing the precoding matrix.

Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature. I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to
avoid introducing new errors in the Second Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink
Rotterdam, November 2004

We study the capacity of a single-user channel with multiple antennas and limited feedback. The receiver has perfect channel knowledge, and can relay B bits, which specify a beamforming vector, to the transmitter. We show that a random vector quantization scheme is asymptotically optimal, and give a simple expression for the associated capacity.

We consider a single-user, point-to-point communication system with M transmit and N receive antennas with independent flat Rayleigh fading between antenna pairs. The mutual information of the multi-input/multi-output (MlMO) channel is maximized when the transmitted symbol vector is a Gaussian random vector with covariance matrix Q. The optimal Q depends on how much channel state information is available at the transmitter. Namely, in the absence of any channel state information, the optimal Q is full-rank and isotropic, whereas with perfect channel knowledge, the optimal Q has columns which are the eigenvectors of the channel, and has rank at most min {M, N}. We assume that the receiver can feed back B bits to the transmitter (per codeword). The feedback bits are used to choose the columns of Q from a random set of i.i.d. vectors. We compute the mutual information as a function of both B and the rank of Q. Our results are asymptotic in the number of antennas, and show how much feedback is needed to achieve a rate, which is close to the capacity with perfect channel knowledge at the transmitter.

We study a multiple-antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t-1)2<sup>-B</sup>t-1/, where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space-time codes under certain conditions. Finally, we show that good beamformers are good packings of two-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.

We study the capacity of some channels whose conditional output
probability distribution depends on a state process independent of the
channel input and where channel state information (CSI) signals are
available both at the transmitter (CSIT) and at the receiver (CSIR).
When the channel state and the CSI signals are jointly independent and
identically distributed (i.i.d.), the channel reduces to a case studied
by Shannon (1958). In this case, we show that when the CSIT is a
deterministic function of the CSIR, optimal coding is particularly
simple. When the state process has memory, we provide a general capacity
formula and we give some more restrictive conditions under which the
capacity has still a simple single-letter characterization, allowing
simple optimal coding. Finally, we turn to the additive white Gaussian
noise (AWGN) channel with fading and we provide a generalization of some
results about capacity with CSI for this channel. In particular, we show
that variable-rate coding (or multiplexing of several codebooks) is not
needed to achieve capacity and, even when the CSIT is not perfect, the
capacity achieving power allocation is of the waferfilling type

Bell System Technical Journal, also pp. 623-656 (October)

Half-title pageSeries pageTitle pageCopyright pageDedicationPrefaceAcknowledgementsContentsList of figuresHalf-title pageIndex

This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bit-rates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multi-element array (MEA) technology, that is processing the spatial dimension (not just the time dimension) to improve wireless capacities in certain applications. Specifically, we present some basic information theory results that promise great advantages of using MEAs in wireless LANs and building to building wireless communication links. We explore the important case when the channel characteristic is not available at the transmitter but the receiver knows (tracks) the characteristic which is subject to Rayleigh fading. Fixing the overall transmitted power, we express the capacity offered by MEA technology and we see how the capacity scales with increasing SNR for a large but practical number, n, of antenna elements at both transmitter and receiver. We investigate the case of independent Rayleigh faded paths between antenna elements and find that with high probability extraordinary capacity is available. Compared to the baseline n = 1 case, which by Shannon’s classical formula scales as one more bit/cycle for every 3 dB of signal-to-noise ratio (SNR) increase, remarkably with MEAs, the scaling is almost like n more bits/cycle for each 3 dB increase in SNR. To illustrate how great this capacity is, even for small n, take the cases n = 2, 4 and 16 at an average received SNR of 21 dB. For over 99%

Information theory answers two fundamental questions in communication theory: what is the ultimate data compression (answer: the entropy H), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We will argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam's Razor: “The simplest explanation is best”) and to probability and statistics (error rates for optimal hypothesis testing and estimation). The relationship of information theory to other fields is discussed. Information theory intersects physics (statistical mechanics), mathematics (probability theory), electrical engineering (communication theory) and computer science (algorithmic complexity). We describe these areas of intersection in detail.

In certain communication systems where information is to be transmitted from one point to another, additional side information is available at the transmitting point. This side information relates to the state of the transmission channel and can be used to aid in the coding and transmission of information. In this paper a type of channel with side information is studied and its capacity determined.

We present two results on the Shannon capacity of M-ary (d,k) codes. First we show that 100-percent efficient fixed-rate codes are impossible for all values of (M,d,k), 0⩽d

We deal with the design and performance analysis of transmit-beamformers for multi-input multi-output (MIMO) systems, based on bandwidth-limited information that is fed back from the receiver to the transmitter. By casting the design of transmit-beamforming based on limited-rate feedback as an equivalent sphere vector quantization (SVQ) problem, we first consider multi-antenna beamformed transmissions through independent and identically distributed (i.i.d.) Rayleigh fading channels. We upper-bound the rate distortion function of the vector source, and also lower-bound the operational rate distortion performance achieved by the generalized Lloyd's algorithm. A simple and valuable relationship emerges between the theoretical distortion limit and the achievable performance, and the average signal to noise ratio (SNR) performance is accurately quantified. Finally, we study beamformer codebook designs for correlated Rayleigh fading channels. and derive a low-complexity codebook design that achieves near optimal performance.

In multiple antenna wireless systems, beamforming is a simple technique for guarding against the negative effects of fading. Unfortunately, beamforming requires the transmitter to have knowledge of the forward-link channel which is often not available a priori. One way of overcoming this problem is to design the beamforming vector using a limited number of feedback bits sent from the receiver to the transmitter. In limited feedback beamforming, the beamforming vector is restricted to lie in a codebook that is known to both the transmitter and receiver. Random vector quantization (RVQ) is a simple approach to codebook design that generates the vectors independently from a uniform distribution on the complex unit sphere. This correspondence presents performance analysis results for RVQ limited feedback beamforming

Investigates the problem of information storage in a defective medium where real-time noisy information is available on the defects at both the encoder and the decoder. The problem is modelled as communication over a finite state channel with noisy state information available at both sides, It is shown that this problem can be transformed into Shannon's channel with side information problem. An optimal coding strategy is described and the capacity is derived. In general the capacity is given in terms of strategies. It is further shown that in some cases of interest the capacity can be described without employing Shannon strategies.< >

We provide an overview of the extensive results on the Shannon capacity of single-user and multiuser multiple-input multiple-output (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about the underlying time-varying channel model and how well it can be tracked at the receiver, as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. For time-varying MIMO channels there are multiple Shannon theoretic capacity definitions and, for each definition, different correlation models and channel information assumptions that we consider. We first provide a comprehensive summary of ergodic and capacity versus outage results for single-user MIMO channels. These results indicate that the capacity gain obtained from multiple antennas heavily depends on the available channel information at either the receiver or transmitter, the channel signal-to-noise ratio, and the correlation between the channel gains on each antenna element. We then focus attention on the capacity region of the multiple-access channels (MACs) and the largest known achievable rate region for the broadcast channel. In contrast to single-user MIMO channels, capacity results for these multiuser MIMO channels are quite difficult to obtain, even for constant channels. We summarize results for the MIMO broadcast and MAC for channels that are either constant or fading with perfect instantaneous knowledge of the antenna gains at both transmitter(s) and receiver(s). We show that the capacity region of the MIMO multiple access and the largest known achievable rate region (called the dirty-paper region) for the MIMO broadcast channel are intimately related via a duality transformation. This transformation facilitates finding the transmission strategies that achieve a point on the boundary of the MIMO MAC capacity region in terms of the transmission strategies of the MIMO broadcast dirty-paper region and vice-versa. Finally, we discuss capacity results for multicell MIMO channels with base station cooperation. The base stations then act as a spatially diverse antenna array and transmission strategies that exploit this structure exhibit signifi-
cant capacity gains. This section also provides a brief discussion of system level issues associated with MIMO cellular. Open problems in this field abound and are discussed throughout the paper.

We investigate the capacity of a system with multiple transmit and receive antennas, assuming that the transmitter and receiver both have access to (possibly defective) channel-state information. Two different special cases of a general system are studied in detail. Our main results are capacity expressions for these cases and a general conclusion that the encoder can be split into separate "space-time coding" and "direction weighting" or "beamforming," without capacity loss. We also present numerical results illustrating the dependence of capacity on the parameters of a quantization scheme providing channel-state information to the transmitter from the receiver. These results have high practical value since the assumptions behind them are closely related to the ones of the closed-loop mode in the UMTS/WCDMA standard.

In this paper, we propose a combined adaptive power control and beamforming framework for optimizing multiple-input/multiple-output (MIMO) link capacity in the presence of feedback-link capacity constraint. The feedback channel is used to carry channel state information only. It is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block. We show that the hybrid design could achieve the optimal MIMO link capacity, and we derive a computationally efficient algorithm to search for the optimal design under a specific average power constraint. Finally, we shall illustrate that a minimum mean-square error spatial processor with a successive interference canceller at the receiver could be used to realize the optimal capacity. We found that feedback effectively enhances the forward channel capacity for all signal-to-noise ratio (SNR) values when the number of transmit antennas (n<sub>T</sub>) is larger than the number of receive antennas (n<sub>R</sub>). The SNR gain with feedback is contributed by focusing transmission power on active eigenchannel and temporal power waterfilling . The former factor contributed, at most, 10log<sub>10</sub>(n<sub>T</sub>/n<sub>R</sub>) dB SNR gain when n<sub>T</sub>>n<sub>R</sub>, while the latter factor's SNR gain is significant only for low SNR values.

We investigate the effects of fading correlations in multielement
antenna (MEA) communication systems. Pioneering studies showed that if
the fades connecting pairs of transmit and receive antenna elements are
independently, identically distributed, MEAs offer a large increase in
capacity compared to single-antenna systems. An MEA system can be
described in terms of spatial eigenmodes, which are single-input
single-output subchannels. The channel capacity of an MEA is the sum of
capacities of these subchannels. We show that the fading correlation
affects the MEA capacity by modifying the distributions of the gains of
these subchannels. The fading correlation depends on the physical
parameters of MEA and the scatterer characteristics. In this paper, to
characterize the fading correlation, we employ an abstract model, which
is appropriate for modeling narrow-band Rayleigh fading in fixed
wireless systems

Employing multiple antennas at the transmitter is a well-established technique for providing diversity advantage in wireless systems. Transmit beamforming relies on the assumption of current channel knowledge at the transmitter, which is unrealistic when the forward and reverse links are separated in frequency. One solution to this problem is for the receiver to send a small number of feedback bits that convey channel information to the transmitter. Feedback design techniques have been proposed over the past few years, but they were derived using the assumption of spatially uncorrelated Rayleigh fading. This correspondence addresses the design of limited feedback beamformers when the channel is correlated.

This paper is concerned with M-ary runlength-limited (RLL) codes
for nonbinary recording channels. The codes have fixed-rate finite state
encoders, sliding block decoders, and large coding density. Five codes
are given achieving coding densities of 24 bit/minimum-recorded-mark as
compared with 1-1.5 for binary recording channels. The codes are
93-98.5% efficient and either achieve or come close to achieving the
fewest number of encoder states possible. One of these codes has been
implemented in a prototype system that supports M=6 discrete recording
levels

Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space-time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to ill-conditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter-which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini-Study distances. Simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback.

We compare the capacity of dirty-paper coding (DPC) to that of time-division multiple access (TDMA) for a multiple-antenna (multiple-input multiple-output (MIMO)) Gaussian broadcast channel (BC). We find that the sum-rate capacity (achievable using DPC) of the multiple-antenna BC is at most min(M,K) times the largest single-user capacity (i.e., the TDMA sum-rate) in the system, where M is the number of transmit antennas and K is the number of receivers. This result is independent of the number of receive antennas and the channel gain matrix, and is valid at all signal-to-noise ratios (SNRs). We investigate the tightness of this bound in a time-varying channel (assuming perfect channel knowledge at receivers and transmitters) where the channel experiences uncorrelated Rayleigh fading and in some situations we find that the dirty paper gain is upper-bounded by the ratio of transmit-to-receive antennas. We also show that min(M,K) upper-bounds the sum-rate gain of successive decoding over TDMA for the uplink channel, where M is the number of receive antennas at the base station and K is the number of transmitters.

Consider a codebook containing N unit-norm complex vectors in a K-dimensional space. In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I<sub>max</sub>) is often desirable. Relying on tools from combinatorial number theory, we construct analytically optimal codebooks meeting, in certain cases, the Welch lower bound. When analytical constructions are not available, we develop an efficient numerical search method based on a generalized Lloyd algorithm, which leads to considerable improvement on the achieved I<sub>max</sub> over existing alternatives. We also derive a composite lower bound on the minimum achievable I<sub>max</sub> that is effective for any codebook size N.

A coding theorem is proved for memoryless channels when the channel state feedback of finite cardinality can be designed. Channel state information is estimated at the receiver and a function of the estimated channel state is causally fed back to the transmitter. The feedback link is assumed to be noiseless with a finite feedback alphabet, or equivalently, finite feedback rate. It is shown that the capacity can be achieved with a memoryless deterministic feedback and with a memoryless device which select transmitted symbols from a codeword of expanded alphabet according to current feedback. To characterize the capacity, we investigate the optimization of transmission and channel state feedback strategies. The optimization is performed for both channel capacity and error exponents. We show that the design of the optimal feedback scheme is identical to the design of scalar quantizer with modified distortion measures. We illustrate the optimization using Rayleigh block-fading channels. It is shown that the optimal transmission strategy has a general form of temporal water-filling in important cases. Furthermore, while feedback enhances the forward channel capacity more effectively in low-signal-to noise ratio (SNR) region compared with that of high-SNR region, the enhancement in error exponent is significant in both high- and low-SNR regions. This indicates that significant gain due to finite-rate channel state feedback is expected in practical systems in both SNR regions.

We define a duality between Gaussian multiple-access channels (MACs) and Gaussian broadcast channels (BCs). The dual channels we consider have the same channel gains and the same noise power at all receivers. We show that the capacity region of the BC (both constant and fading) can be written in terms of the capacity region of the dual MAC, and vice versa. We can use this result to find the capacity region of the MAC if the capacity region of only the BC is known, and vice versa. For fading channels we show duality under ergodic capacity, but duality also holds for different capacity definitions for fading channels such as outage capacity and minimum-rate capacity. Using duality, many results known for only one of the two channels can be extended to the dual channel as well.

We consider a multiuser multiple-input multiple- output (MIMO) Gaussian broadcast channel (BC), where the transmitter and receivers have multiple antennas. Since the MIMO BC is in general a nondegraded BC, its capacity region remains an unsolved problem. We establish a duality between what is termed the "dirty paper" achievable region (the Caire-Shamai (see Proc. IEEE Int. Symp. Information Theory, Washington, DC, June 2001, p.322) achievable region) for the MIMO BC and the capacity region of the MIMO multiple-access channel (MAC), which is easy to compute. Using this duality, we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the MIMO BC. We also show that the dirty paper achievable region achieves the sum-rate capacity of the MIMO BC by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the MIMO BC.

Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple-input multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector; both are hard to realize. In this article, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent and identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.

The existence of 100% efficient (i.e., capacity-achieving)
fixed-rate codes for input-constrained, noiseless channels is guaranteed
provided the channel has rational capacity. A class of M-ary
runlength-limited (M,d,∞) constraints was shown in previous work
to have rational capacity. In this correspondence we present a code
construction procedure for obtaining 100% efficient codes with the
fewest number of encoder states for all (M,d,∞) constraints with
rational capacity. The decoders are sliding-block decoders with sliding
window size d+1

Presents two results on the Shannon capacity of M-ary (d,k) codes.
First the authors show that 100% efficient fixed-rate codes are
impossible for all values of (M,d,k), 0⩽d<k<∞,
M<∞, thereby extending a result of Ashley and Siegel (1987) to
M-ary channels. Second, they show that for k=∞, there exist an
infinite number of 100% efficient M-ary (d,k) codes, and they construct
three such capacity-achieving codes

The capacity region of frame-synchronous and frame-asynchronous,
discrete, two-user multiple-access channels with finite memory is
obtained. Frame synchronism refers to the ability of the transmitters to
send their code words in unison. The absence of frame synchronism in
memoryless multiple-access channels is known to result in the removal of
the convex hull operation from the expression of the capacity region. It
is shown that when the channel has memory, frame asynchronism rules out
nonstationary inputs to achieve any point in the capacity region,
thereby allowing only coding strategies that involve cooperation in the
frequency domain but not in the time domain. This restriction
drastically reduces the capacity region of some multiple-access channels
with memory, and in particular the total capacity of the channel, which
is invariant to frame asynchronism for memoryless channels

The scientific responsibility rests with its authors

- Leuven

Leuven, project OT/05/40 (Large rank structured matrix computations), CoE EF/05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research–Flanders (Belgium), G.0423.05 (RAM: Rational modeling: optimal conditioning and stable algorithms), and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The scientific responsibility rests with its authors.

Germany (e-mail: immink@exp-math.uni-esen.de)

- Essen

Essen, Germany (e-mail: immink@exp-math.uni-esen.de).